The needs of the 21st century, now evident as we are well into it, have led us to rethink the social function of education: the educational system should empower and equip young people with skills to face the unexpected. This does not imply dismantling the entire traditional educational system, but rather rethinking many aspects and building upon them to improve. These demands for change are fueled by contributions from fields such as neuroeducation and cognitive sciences in recent years. We know, for example, that emotions influence learning (Kort et al., 2001; Pekrun et al., 2002), that we retain information better when we give it meaning (Tyng et al., 2017), and that learning is an active process (Michael, 2006). Fortunately, many stakeholders, from teachers to university researchers, including the official curricula of countries leading in education, are beginning to incorporate this vision.

The educational model of the last century was mainly oriented towards the transmission of academic content of a preparatory nature (used to prepare students for further studies) with a fundamentally selective function: to distinguish students “suitable” for university from those who would directly enter the labor market after completing compulsory education.

Today, schools have more comprehensive objectives, focused on developing competencies (Burns and Klingstedt, 1973). When we talk about competencies, we go beyond content: they are knowledge, attitudes, values, skills, and behaviors (Gervais, 2016) that precisely serve to deepen and give meaning to the content.

In this paradigm shift, the role of the teacher has also undergone profound changes. As mentioned, most teachers previously followed rather deductive methodologies: starting with theories or general concepts to later show their application to specific cases or situations. Thus, the teacher assumed a rather transmissive role; they were the main source of information that the student had to memorize and often superficially assimilate. Today, in the internet era, the role of the teacher has changed to become a guide who accompanies the student in their process to achieve more “meaningful and deep” participatory learning (Mayer, 2002). The teacher, therefore, follows inductive methods: learning begins with specific cases or situations and then works to discover the theories or general concepts that unite them and give them meaning. Moreover, an approach that aims to be more strategic is adopted (teaching is adapted to the individual needs and learning styles of students) rather than simply applying the curriculum.

Mathematics education, which is part of this grand transformation, has also experienced an evolution over time. To understand this evolution and its current focus on competency-based learning through processes, several questions must be considered: What is “mathematical competence”? Which teaching strategies are most effective for competency-based and meaningful learning of mathematics? How can we foster the development of socio-emotional skills in our students and establish connections with other areas?

These are some of the questions we address in this document, where we delve into key concepts such as mathematical richness, learning trajectories, the use of manipulative materials, the importance of differentiating instruction, and assessment as a tool to improve the educational experience in the mathematics classroom.

Human beings are social creatures: we share experiences, learnings, and emotions with other members of our species. Life in society has allowed us to build cultures, languages, norms, and shared values that have been key to our survival and evolution as a species. Therefore, since we are social beings, we can think that the origins of mathematics education (initially understood as the transmission of information) go hand in hand with the origins of mathematics itself.

The history of mathematics as we understand it today dates back to the Mesolithic era when societies began to use (and represent) numbers and operations to measure time, the quantity of animals or grain, the dimensions of land, etc. An example of these early mathematical practices is the Ishango bone, an archaeological artifact found in what is now the Democratic Republic of the Congo, dating back to approximately 20,000 BC. The bone has a series of markings with mathematical significance (Huylebrouck, 2019), and some experts have interpreted it as a numbering system or a tool for counting or performing basic arithmetic operations. Others suggest that the markings could be a lunar calendar. It is not until several millennia later, around 3000 BC, that we find the first evidence of a numbering system that we can interpret today. These are found in Mesopotamian civilizations such as the Sumerians, who are said to have invented writing (Friberg, 1984). A Sumerian counts the cows he has before grazing them and records it by making marks on a clay tablet. This is how the first written numbers appear. Another Sumerian asks what these marks are, and the first explains it to him. Almost at the same time as the first mathematical ideas, we can say that mathematics education appears. Evidently, in the early stages, we cannot speak of formal education, let alone mathematics didactics, but somehow the foundations were established.

Without intending to be exhaustive, several later civilizations, such as the Egyptian, Chinese, Roman, Hindu, and especially the Greek, formalized this transmission of knowledge, the figure of the teacher and disciple, discussion, the representation of geometric ideas, formal demonstrations, etc.

During the Middle Ages in the West, mathematics began to be studied in monasteries and the first European universities, heavily influenced by the discoveries and contributions of Islamic culture, with arithmetic, geometry, and algebra as protagonists (Høyrup, 2014). From the Renaissance, mathematics education adopted an increasingly theoretical and abstract approach, with the introduction of algebraic notation and the analysis of functions. At the same time, education became more accessible to the general population, thanks to the invention of the printing press and the creation of the first public schools. The teaching approach focused on rigor and logic, with the development of analytical geometry and infinitesimal calculus.

In the 18th and 19th centuries, with the Industrial Revolution, education was oriented towards training future professionals. The emergence of applied mathematics and the formalization of statistics and probability fostered the creation of polytechnic schools and other specialized universities. And thus, we reach the 20th century, where the science of Math Learning was consolidated as a sister science but independent of mathematics itself. Today, we find experts in mathematics education at all stages, who contribute to the development of this science from various trends and approaches. Here, we take a look at three widely spread approaches:

• Transmission-based teaching and learning: This approach is based on the direct transfer of knowledge by the teacher. Generally, students listen and learn mechanically and by rote. Great importance is placed on repetitive practice of exercises to mechanize procedures. It is the predominant approach in the traditional educational system.
• Socioconstructivist teaching and learning: Constructivism holds that learning is an active process in which students build their own knowledge from their experiences and interactions with their environment, especially with peers (Arcavi, 1999). In this approach, the teacher acts as a guide and provides the necessary challenges to discover and formalize content. Often, when the focus is on guided inquiry, critical thinking, and the scientific method, it is also referred to as “inquiry-based learning.” Here, connections with project-based learning are common, which often integrates various subjects transversally.
• Problem-based teaching and learning: Emphasis is placed on the development of competency skills, with problem-solving at its core (Pólya, 1945). Within this approach, some view problems as any situation (whether realistic or abstract) for which there is no immediate obvious solution; others are more interested in real-life problem situations, in a more utilitarian approach that sees mathematics as a tool for other sciences.

As often happens in didactics, this classification is useful for understanding the differences between approaches, but it is not entirely realistic. When observing any classroom, it is likely that we will discover traditional moments, constructivist moments of inquiry and conversation, and moments focused on problem-solving. In reality, the underlying debate is simpler and lies in where we place the focus. Traditionally, the teacher was at the center: the teacher transmitted knowledge vertically. Today, we know that to develop mathematical competence, we need to put students at the center, make them agents of their learning, discover in a guided way, reason, argue, establish connections, and solve problems beyond what they have practiced in class (Niss and Højgaard, 2019).

And the debates on approach do not end here. For example, teaching styles can be discussed, which can be more or less formalist (advocating that the teaching of mathematics should follow an axiomatic and logical system, where students must understand and master rules and definitions precisely); more or less interventionist (advocating that the teacher should guide at all times). And once a particular approach is accepted, the nuances that open up are almost endless: How do we use manipulative materials? How do we understand assessment? How do we differentiate instruction? Etc.

Since mathematics is an exact science, at first glance we might think that its didactics must also be exact. This is not the case. Mathematics education, the didactics of mathematics, is a social science. Unlike pure sciences, such as mathematics or physics, social sciences focus on the study of human beings and their behaviors. This does not mean that anything goes or that everything is debatable: we have enough evidence to form a criterion and decide how we want our students to learn mathematics, as we will explore below.

A common perception is that mathematics is a cold, abstract, and difficult branch of knowledge, yet important, as Ernest (1995) described. The author explains that mathematics has the image of being “inaccessible, except to a privileged few.” Even many people who perform jobs that require a high level of competence in arithmetic, statistics, or computer science claim that they are not good at mathematics. In fact, in contrast to the social shame of not knowing enough history, geography, or literature, the lack of mathematical knowledge is not as badly viewed.

For others, mathematics is a source of anxiety. According to Menon, co-author of a study on the neurodevelopmental basis of anxiety (Young, Wu, and Menon, 2012), the part of the brain affected by math anxiety is the same part that is activated in dangerous situations, such as seeing a spider or a snake. Tracing the origin of this anxiety exhaustively is difficult, but it is clear that a large part comes from the approach traditionally given to it in school. Berkeley professor Edward Frenkel reflects on this in his bestseller Love and Math: The Heart of Hidden Reality (2013).

What if at school you had to take an “art class” in which you were only taught how to paint a fence? What if you were never shown the paintings of Leonardo da Vinci and Picasso? Would that make you appreciate art? Would you want to learn more about it? I doubt it. You would probably say something like this: “Learning art at school was a waste of my time. If I even need to have my fence painted, I’ll just hire people to do this for me.” Of course, this sounds ridiculous, but this is how math is taught, and so in the eyes of most of us it becomes the equivalent of watching paint dry. While the paintings of the great masters are readily available, the math of the great masters is locked away. (p. 1)

Let’s think about a language. If, for example, someone memorizes the entire German dictionary, can they be said to master the language? Obviously not. Besides the words, their meanings, grammar, and structure, it is also necessary to develop reading comprehension, writing, oratory, rhetoric, and even literature that makes the language loved. In this metaphor, vocabulary, grammar, and spelling are the mathematical content. Essential, yes, and activities or learning situations that cover the curriculum content must be designed, but that is not enough. Skills are also needed, a competence that allows using the content meaningfully to solve problems, reason mathematically, converse, argue, establish connections, communicate ideas, and represent concepts. Additionally, this fosters a school image of mathematics more connected to what it really is: a professional mathematician does not spend the day doing arithmetic calculations. Mathematical activity is not based on the repetition of algorithms but on observing patterns, formulating conjectures, and deducing. It is a much more creative and social job than it seems. And it precisely seems otherwise because that is what the traditional educational system conveys. Mathematics has value as a tool in the service of science, yes, but it is also valuable in itself, as philosophy, as a language, as a way of thinking, interpreting, ordering, and appreciating reality.

Foster (2023) explains that a mathematical task is any mathematical activity we ask our students to perform, and they are divided into “routine” tasks and “non-routine” tasks. Routine tasks, called “exercises,” are invitations for students to imitate a method they have been previously taught. Thus, a “non-routine” task or “problem” is “a task whose solution method is not known in advance” (NCTM, 2000). Any task for which the solver does not have an immediate strategy to find the solution (Schoenfeld, 2014); Liljedahl (2020) describes it in even simpler terms: “problem-solving is what we do when we don’t know what to do.”

Therefore, whether a task is an exercise or a problem depends mainly on whether the person facing it already has a resolution strategy or not. This distinction is independent of the effort required by the task. As Foster (2023) also explains: “A task can be routine but difficult. For example, multiplying two 5-digit numbers would be difficult but completely routine for someone well-versed in the multiplication algorithm.” This implies that most tasks that are problems for a student today will become exercises over time as they develop problem-solving skills.

The problem-solving process includes several phases: understanding and breaking down the statement, narrowing down the problem, planning and executing a resolution strategy, finding solutions, checking both the solutions and the resolution process, and finally, posing new related challenges. It is, therefore, the most core process of mathematical activity and, as Pólya (1945) pointed out, the one that provides the most suitable environment for developing overall mathematical competence.

Additional Sources for Further Exploration

• Hiebert, J., & Wearne, D. (2003). Developing understanding through problem solving. In Schoen, H. L. & Charles, R. I. (Eds.), Teaching mathematics through problem solving (pp. 6-12). Reston, VA: NCTM.
• Mason, J., Burton, L., & Stacey, K. (1982). Thinking Mathematically. Pearson Education Limited.
• Schoenfeld, A. H. (1992). Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics. In Grouws D. A. (Ed.), Handbook for Research on Mathematics Teaching and Learning. Macmillan Publishing Company.
• Schoenfeld, A. H. (2007). Problem solving in the United States, 1970-2008: Research and theory, practice and politics. ZDM, 39(5), 537-551. https://doi.org/10.1007/s118580070038-z

Through observation and the search for patterns and regularities, we develop the ability to question the reasons behind these patterns and the skill to formulate and test conjectures.

From the earliest mathematical experiences, it is important to help children understand that statements require reasoning to justify them: they must learn to justify their answers with evidence and arguments. As they progress in their learning, students should be able to understand and produce proofs or, more formally, mathematical demonstrations.

The investigation and development of conjectures are also fundamental skills. This involves making informed assumptions and testing them with generalization or rejecting them by reaching a contradiction (Lakatos, 1976; Pólya, 1954). Students should learn to make and investigate conjectures from a young age, using concrete materials, tools, and mathematical representations. Over time, they should be able to explore conjectures more systematically and justify more general arguments using more sophisticated mathematical knowledge and various types of reasoning (arithmetic, algebraic, geometric, proportional, probabilistic, statistical).

This process, therefore, focuses on the skills necessary to analyze a situation and, from there, formulate and test conjectures, make reasoned deductions, demonstrations, generalizations, and above all, argue or refute any statement made in the mathematics classroom.

Additional Sources for Further Exploration

• Jeannotte, D., & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96(1), 1-16.
• Knuth, E. J. (2002). Teachers’ Conceptions of Proof in the Context of Secondary School Mathematics. Journal of Mathematics Teacher Education, 5(1), 61-88. https://doi.org/10.1023/A1013838713648
• Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. The Journal of Mathematical Behavior, 15(1), 3-31. https://doi.org/10.1016/S073231239690036X
• Pólya, G. (1968, second edition). Mathematics and plausible reasoning. Princeton University Press.
• Stylianides, G. J. (2008). An Analytic Framework of Reasoning-and-Proving. For the Learning of Mathematics, 28(1), 9-16.

The connection between ideas and concepts is fundamental in the mathematical learning process, as it allows for a deeper and more integrated understanding of knowledge, as described by Sawyer (2008). Establishing links between different ideas enriches the way of thinking and promotes meaningful and interconnected learning.

This process includes all the relationships we find or establish between concepts and representations, which allow us to deepen our understanding of them. Sometimes, these connections occur within the realm of mathematics (between contents of the same domain or between different domains), and other times they occur between mathematics and the rest of the world, where mathematical modeling takes the spotlight, as explained by Meerschaert (2013):

Mathematical modeling is the link between mathematics and the rest of the world. You ask a question. You think a bit, and then you refine the question, phrasing it in precise mathematical terms. Once the question becomes a mathematics question, you use mathematics to find an answer. Then finally (and this is the part that too many people forget), you have to reverse the process, translating the mathematical solution back into a comprehensible, no–nonsense answer to the original question. Some people are fluent in English, and some people are fluent in calculus. We have plenty of each. We need more people who are fluent in both languages and are willing and able to translate. These are the people who will be influential in solving the problems of the future. (p. vii)

Additional Sources for Further Exploration

• De Lange, J. (1989). Trends and Barriers to Applications and Modelling in Mathematics Curricula. In W. Blum, M. Niss, I. Huntley, (Eds.), Modelling, applications and applied problem solving (pp. 196-204). Chichester: Ellis Horwood.
• Kilpatrick, Jeremy., Swafford, Jane., Findell, Bradford., & National Research Council (U.S.). Mathematics Learning Study Committee. (2001). Adding it up: Helping children learn mathematics. National Academy Press.
• Maaß, K. (2006). What are modeling competencies? ZDM, 38, 113-142. https://doi.org/10.1007/BF02655885
• Rodríguez-Nieto, C. A., Moll, V. F., Borji, V., & Rodríguez-Vásquez, F. M. (2021). Mathematical connections from a networking of theories between extended theory of mathematical connections and onto-semiotic approach. International Journal of Mathematical Education in Science and Technology, 53(9), 2364-2390. https://doi.org/10.1080/0020739x.2021.1875071

Communication is an essential part of mathematics and mathematics education, as described by NCTM (2000). It is fundamental that students, from a young age, learn to communicate their mathematical thinking coherently to peers and teachers. Developing these skills involves encouraging children to use precise mathematical language in their responses (Cooke and Buchholz, 2005).

On the other hand, representation can be understood as the vehicle through which mathematical ideas are expressed and communicated (Coulombe and Berenson, 2001). Preston and Garner (2003) view representations as vital tools for recording, analyzing, solving, and communicating data, problems, and mathematical ideas. These representations often need to be considered as strategies of cognitive activity, mechanisms for problem-solving, rather than just the resulting products of a task (Pape and Tchoshanov, 2001). In fact, a change in representation can be key to tackling a problem.

Communication and representation are, therefore, two sides of the same coin. This process addresses the skills related to the transmission of mathematical information, whether as senders or receivers (with indicators such as clarity, efficiency, or precision in vocabulary). It also emphasizes the importance of representation: from identifying and interpreting others’ representations to generating one’s own or being able to translate from one representation to another.

Additional Sources for Further Exploration

• Alsina, Á. (2016). Diseño, gestión y evaluación de actividades matemáticas competenciales en el aula. Epsilon: Revista de la Sociedad Andaluza de Educación Matemática «Thales», 92, 7-29.
• Brenner, M.E., Mayer, R.E., Moseley, B., Brar, T., Duran, R., Reed, B.S., & Webb, D. (1997). Learning by understanding: The role of multiple representations in learning algebra. American Educational Research Journal, 34, 663-689.
• Bruner, J. (1986). Actual minds, possible worlds. Harvard University Press.
• Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23, 2-33.
• Duval, R. (1999). Semiosis y pensamiento humano: Registros semióticos y aprendizajes intelectuales. Universidad del Valle.
• Greeno, J.G., & Hall, R.P. (1997). Practicing representation: Learning with and without representational forms. Phi Delta Kappan, 78, 361-367.
• Hiebert, J. (1988). A theory of developing competence with written mathematical symbols. Educational Studies in Mathematics, 19, 333-355.
• Mainali, B. (2021). Representation in Teaching and Learning Mathematics. International Journal of Education in Mathematics, Science and Technology, 9(1), 1-21.

Constructivism advocates the idea that learning is an active process where students construct their own knowledge through experiences and interaction with their environment, especially with peers (Arcavi, 1999). But then, what role do we, the teachers, play? Is the individual the only one who can design their own learning paths? Even constructivists like Arcavi reject this “anarchic tone” stance because they believe that most students cannot access the overall vision of a domain by themselves: “Without proper guidance, students will hardly know what to ask or what to be interested in” (p. 41). The educator acts as a mentor and presents the necessary challenges to facilitate guided discovery and structuring of content. Similarly, Vygotsky (1978) argues that cognitive skills develop through social interactions with family members, classmates, and especially teachers. Vygotsky expands this idea with the “Zone of Proximal Development (ZPD)”, where he defines the necessary support for students to advance to higher levels of understanding.

This approach, directly related to inquiry-based learning (Edelson et al., 1999; Banchi and Bell, 2008), focuses on guided research to develop critical thinking and the use of the scientific method. Therefore, without forgetting the subjective nature of learning processes, it is essential to propose a certain external structuring of teaching that serves the following functions (Arcavi, 1999):

1. Encourage students to start asking questions and inquiring.
2. Show and promote the meaningful use of the working tools specific to the domain (not just algorithmic procedures, but especially thinking strategies).
3. Propose models to transmit practices and ways of doing consistent with the spirit in which experts in the field act (as Schoenfeld also describes; 1992).

An accepted maxim of constructivism, and by extension, most of the current educational community, is that the goal of mathematics education should be for students to build understanding. And what does understanding mean? Bishop (1985), for example, argues that understanding a mathematical idea is connecting the meaning of that idea with the meaning of another mathematical idea, another domain of knowledge, or everyday life. Bettencourt (1991) and Schoenfeld (1992) add that understanding also involves connecting these ideas and circumscribing them within a specific community, with its practices, thinking strategies, beliefs, discourse, and ways of doing. Therefore, in the mathematics classroom, this type of understanding should be promoted, and interactions similar to those that would occur in a community of mathematicians should be stimulated: lots of conversation and reflection, both individual and collective.

What Does “Doing Mathematics” Mean?

This famous quote from a great mathematician like G. H. Hardy (1992) very well conveys how mathematicians perceive their work:

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. (p. 13)

The quote reflects Hardy’s vision and that of many other mathematicians over the centuries: mathematics is not a collection of formulas and utilitarian techniques but rather a creative process that involves the discovery and deductive exploration of new timeless, universal, and inherently interesting ideas and patterns. Unlike physical objects, mathematical ideas endure and can be discovered and appreciated by anyone, anywhere, at any time. It is not surprising that many of the most famous mathematicians of antiquity were also considered philosophers. In general, quotes like Hardy’s reflect the process behind mathematical activity, understood as a fundamental and universal form of human expression, as a surprisingly humanistic discipline.

Therefore, every time a concept is explained masterfully, although it may seem the quickest way to transmit knowledge, an opportunity to discover a pattern, to deduce and jointly build an idea, is lost. Moreover, this gives an image of mathematics far removed from Hardy’s conception. It gives the image that mathematics is an opaque science, filled with rote procedures, arbitrary facts, and individualism that stifles collective creativity. On the other hand, from the socio-constructivist framework, designing activities that promote investigation and guided discovery (Edelson et al., 1999; Banchi and Bell, 2008), with conversations guided by good questions, generates more meaningful and lasting learning. The classroom should be buzzing, as peer conversation, well-guided, is an excellent engine of learning (Sfard et al., 1998). If interactions similar to those that would occur in a community of people using mathematical reasoning are to be stimulated, students must invent, discuss, argue, compare ideas, ways of doing, and solutions. This vision of the classroom fully aligns with the problem-solving environment described by Pólya (1945): when the focus on mathematics is from a problem-solving perspective, with the freedom to express ideas and the flexibility to make mistakes and learn from them, it is essential to discuss, listen to students, question their statements, and guide them in finding the necessary criteria to choose the most efficient strategy at each moment. These approaches also have another benefit: they provide students with tools that transcend the mathematics classroom, such as the ability to conjecture, argue a point of view, refute a statement made by a peer, or listen actively, among others.

In any case, perhaps due to the abstract nature of the concepts it addresses, mathematical activity is not an easy task, no matter how beautiful it may be. In an article in Nature, Hardy (1942) himself writes:

Mathematics is a serious and often demanding discipline, and anyone who wants to master it, even a little, must be prepared to face a certain number of uncomfortable hours. No one can learn anything worthwhile in mathematics without a certain degree of intellectual maturity and, above all, the willingness to think for oneself. (p. 674)

This difficulty, however, from a didactic perspective, can be leveraged to favor learning. The idea that effort or struggle is essential for intellectual growth is neither new nor exclusive to mathematics. Dewey (1933, p. 12) spoke almost 100 years ago about involving students “from some perplexity, confusion, or doubt” as key to building deep understanding. Piaget (1960) described this effort of learners as a process of restructuring their disequilibrium, necessary for constructing new learning. More recently, in the mathematical field, Hiebert and Wearne (2003) affirm that “all students must struggle and face difficult problems if they are to learn mathematics deeply” (p. 6). Clearly, to avoid frustration, this difficulty must be productive, must be within the so-called zone of proximal development of Vygotsky (1978), and must ultimately lead somewhere: students must feel that they are progressing despite the difficulty. In later works, Hiebert and Grouws (2007) suggest that struggling to make sense of mathematics is a necessary component for learning from understanding. Thus, far from avoiding them, teaching mathematics should often involve moments of challenge and effort, uncomfortable problems that further foster discussion in search of a good strategy and convey that this difficulty is an inherent part of mathematical activity. In fact, anyone who has faced a difficult problem knows that the satisfaction felt in solving it, a sensation difficult to characterize and hardly comparable to any other human experience, is proportional to the effort required to solve it.

The Use of Manipulative Materials to Reach Abstraction

According to Moyer (2001), manipulative materials are “objects designed to explicitly and concretely represent abstract mathematical ideas, with visual and tactile appeal.” Over the past few decades, several factors have contributed to the popularization of manipulative materials in mathematics teaching. Following Moyer’s bibliographic analysis, one can start with the works of the renowned pedagogue Piaget (1952), who more than 70 years ago suggested that children do not have the mental maturity to understand abstract mathematical concepts presented only in words or symbols, and that they need many experiences with concrete materials and drawings. Dienes (1969) also explained the necessity of various representations of a concept, the so-called “multiple embodiments,” to support students’ understanding of mathematics. Skemp’s theories (1987) describe that students’ early experiences and interactions with certain physical objects form the basis for later, more abstract learning. Based on Vygotsky’s (1978) theories of cognition and social construction of knowledge, Cobb’s research (1995) explores cultural tools, such as the 100-chart, and shows the close relationship between manipulative materials and the sociocultural aspect of a specific territory. Thompson (1992), for his part, states that students’ abstract thinking depends on their perceptions of the world, so active manipulation of materials allows them to develop a repertoire of images they can later use in the mental manipulation of abstract concepts. Beyond the benefits of manipulation in building abstract concepts in early stages, the use of materials in the mathematics classroom is also a motivation driver for students that must be considered when planning activities.

In any case, it is important to note that the manipulation of materials is only a scaffold that will need to be removed later, an intermediate step on the path to abstraction. As Bruner (1960, 1986) describes, children can show understanding through three stages of representation: the enactive stage (physical objects), the iconic stage (images and drawings), and the symbolic stage (schemes, symbols, formulas, and words). In fact, it is easy to detect parallels between these three stages and those that make up the CPA cycle of the well-known Singapore method (Singapore Math Inc., 2023): the concrete stage, the pictorial stage, and the abstract stage. Although the transition between stages is diffuse and can occur at different rates depending on the concept explored and the child, the use of physical manipulative materials is limited to the first stage (and their representations, to the second). This is why it is so important to choose the best manipulative materials for each occasion and to know when to remove them at the right time. Precisely to be able to remove them, it is also essential that the representation that replaces each material is close to the material and easy for the student to reproduce.

Several authors, from a theoretical perspective, point out some indicators, principles, and recommendations that can serve as a reference framework for choosing good materials. Laski et al. (2015), for example, describe four principles to maximize the effectiveness of manipulative materials and relate them to the Montessori method. These four principles can be stated as follows:

1. Use the material consistently, for the necessary time.
2. Start with concrete, transparent representations and gradually move to more abstract representations over time.
3. Avoid materials that resemble everyday objects or have irrelevant distracting features.
4. Explicitly explain the relationship between the materials and the concept they represent.

From these principles, several interesting reflections can arise. According to the first principle, the coherence between the materials and the concepts they represent is important; this coherence should be sought even among the various materials and over time. The second principle speaks of transparency, which is the quality of a tool to explain itself and be easy to understand. The third principle warns us of the dangers of presenting materials with too much meaning for students, as they may become distracted from the mathematical activity. Finally, the fourth principle recommends being explicit with students when connecting materials and concepts, especially with mature students.

McNeil and Jarvin (2007) also pointed out two recommendations that reinforce these principles. The first is to minimize the use of materials that are (a) highly concrete and rich in perceptual details and/or (b) too familiar to children in non-school contexts (e.g., toys). These materials risk distracting students from learning objectives. Precisely, as the authors point out, many common manipulative materials (e.g., base-10 blocks) are quite the opposite: they are not rich in perceptual details nor do they arouse much interest as objects in themselves. The second recommendation is to take the time to show students the connections between the informal constructs created when using materials and the symbolic formalization of concepts. This way, the risk of students relying excessively on the informal representations inherent in the manipulative material is reduced, and the path to abstraction within the “manipulate-represent-abstract” cycle is promoted.

Finally, at a time when screens and devices are present in many classrooms and homes, virtual manipulative materials should also be considered as an appendix deserving differentiated treatment. A dynamic virtual environment that simulates physical materials can facilitate sharing or allow all children to interact with a material that the school does not have, among other things. Depending on the knowledge area, it is easier or harder to find bibliography analyzing the impact of using virtual manipulative materials in the classroom. In the field of geometry or algebra, for example, applets programmed with GeoGebra enjoy an excellent reputation among teachers and researchers (Saha et al., 2010; Barahona Avecilla et al., 2015). From a more global perspective, authors like Moyer-Packenham and Westenskow (2013) also state that virtual manipulation can have positive effects on students’ performance, depending on how it is used in the classroom.

As observed, many authors have explored the practical benefits of using manipulative materials in the mathematics classroom. Similarly, several authors (often the same ones describing the benefits) also warn of the risks of indiscriminately using manipulative materials and relying on them blindly. In this vein, Ball points out that materials “are not magical” and that “although kinesthetic experience can enhance perception and thinking, understanding does not travel up through the fingertips, arm” (Ball, 1992, p. 47). Sometimes, Hiebert and Wearne (1992) warn, students learn to use manipulative materials mechanically, with little or no learning of the mathematical concepts behind the procedures. This is why it is not enough to use materials: they must be chosen with didactic criteria, used appropriately, and knowing when it is time to leave them behind to move towards abstraction. In this sense, the meta-analysis by Carbonneau et al. (2012) is particularly interesting. Their quantitative research indicates that using manipulative materials, compared to instruction that directly uses abstract symbols, produces a small to medium positive effect on learning. Most interestingly, as expected, the strength of this effect depends on instructional variables such as “the perceptual richness of the object, the level of guidance provided during the learning process, or the students’ developmental state” (p. 396). Again, it is evident that the teacher’s classroom management with the materials is crucial.

Examples of the First Principle at Innovamat

Guided Discovery

Many sessions start with a challenge, an open question that invites experimentation. Similarly, many written activities ask students, “What do you observe?” so they go beyond the concrete result and explore patterns and relationships between mathematical objects. In any case, we avoid the direct transmission of content and ensure that everything students learn comes from deduction and understanding. Examples include the multiple strategies we propose for tackling arithmetic operations beyond the traditional algorithm to encourage visualization and the creation of mental models. More paradigmatic, however, are examples of how we discover or deduce concepts and properties traditionally taught directly, such as the formula for the area of a circle, the multiplication of fractions, or the powers of zero exponent, as seen below:

In the example, we guide the student to observe the pattern based on what they know about powers: as the exponent decreases by 1, the result is divided by 3. This way, we can deduce that 3⁰ = 1. And, by extension, that any power with an exponent of zero is equal to 1. In fact, in subsequent sessions, we extend this pattern to also discover negative exponent powers. These are just a couple of examples of how we can use mathematical skills such as deduction or pattern observation to discover properties or curricular content in a guided manner.

Conversation at the Center

Any session, whether in the early childhood, elementary, or secondary program, places peer conversation (guided by the teacher) at the forefront. In the teaching guides, we recommend how to group students at each moment to foster times of conversation in large or small groups. We also include tips and question suggestions to help teachers manage these conversations. For example, in this 6th-grade challenge, we start with a dynamic called “Two Truths and a Lie” to discuss areas and fractions:

The example illustrates how any guide proposes using conversation as the engine of the activity. In this specific case, various learning opportunities can be generated through conversation to talk about areas, fractions, and algebra while testing competency skills such as reasoning or argumentation.

The Manipulative Materials Box

The proposal includes a manipulative materials box for each class, with a carefully selected range of materials to meet the needs of each grade. Some examples are base-10 blocks to represent quantities in early stages, positive and negative chips to represent integers in secondary school, bead necklaces to construct the first versions of the number line, number cards, stackable cubes, 3D shapes to work on spatial geometry, etc. In many sessions, knowledge is built from the material, which serves as a scaffold to reach abstraction. The didactic guides detail when we recommend using each material and when we recommend removing it.

In this excerpt from a 4th-grade guide, we see how cubes are used to represent the process of division, paying attention to the relationship between dividend, divisor, and remainder:

When discussing “sequence of activities,” it refers to an ordered list of activities that, when appropriately presented to students, serve to build the construction of knowledge. The various activities in a sequence must fit together like puzzle pieces, like gears that activate one another to provide a global perspective on learning. Adding the adjective “rich” opens the door to the “competency-based” learning of these contents (Vilalta, Deulofeu, and Morera, 2024). To delve into this principle, it is necessary to define richness and develop the various learning trajectories for each concept.

Mathematical Richness

Many authors have defined their conception of what it means to enrich mathematical activity in the classroom. Thus, the term “rich mathematical task” is widely used. Teachers and experts in the field, such as Finkel (Math For Love, 2022), have adopted this terminology: “Every student deserves the opportunity to solve problems and engage in genuine mathematical thinking. Rich tasks are designed to make these rich learning experiences possible.” Similarly, curriculum developers like Gojak (2017) also use the term: “A rich task presents a high level of cognitive demand and requires students to think abstractly to make connections within and between mathematical concepts” (p. 3). Government agencies like the Virginia Department of Education (2023) also speak of it: “Rich mathematical tasks engage students in making sense through deep learning that requires high levels of thinking, reasoning, and problem-solving.” Piggott (2007, 2011), in the context of the Nrich initiative at the University of Cambridge, proposes a framework for richness that comprises two factors: content and teaching. According to Piggott, rich activities require content presented in tasks that foster the development and use of strategies and mathematical thinking. These are what other authors describe as high-level or high cognitive demand tasks (Stein et al., 1996; Henningsen and Stein, 1997; Smith et al., 2008). Besides tasks, Piggott (2011) also talks about a teaching approach (i.e., classroom management) that fosters an open and flexible environment where cooperative work, exploration, and communication are promoted, and where differences are leveraged as a learning tool. The two factors Piggott describes clearly relate to the elements of a “rich mathematical activity” described by Vilalta, Deulofeu, and Morera (2024): “A task, or a set of tasks, combined with the teacher’s management as they interact with the students, is what we understand as a mathematical activity, which can be more or less rich.” Additionally, the authors propose five components of a rich activity:

Components of a Rich Activity

1. Processes — The task promotes one or more of the four processes of mathematical competence: problem-solving, reasoning and proof, connections, communication and representation.
2. Contents — The task fosters content that is contextualized, rigorous and extensible.
3. Cognitive Demand — The task requires reflecting rather than relating & relating rather than reproducing.
4. Differentiated Instruction — The management considers access and challenge for all students.
5. Environment — The management fosters engagement and autonomy, develops math identity, promotes peer learning, etc.

In this list, beyond the components referring to the task, the components addressing the teacher’s management may stand out. Poor management can impoverish a potentially rich task, and vice versa, good management can enrich a poor task. According to Bartell et al. (2017), teaching practices without explicit attention to equity are inevitably doomed to fail. Indeed, management is not entirely rich if it does not consider the diversity of students. This means that through classroom management, efforts must be made to ensure that all students (both those showing difficulties and the more talented) participate in the same activity. This goal is possible, as described by Nrich (2013, 2017) based on Papert’s ideas (1993), if the activity is low-threshold (allows all students to enter) and high-ceiling (allows for many connections and extends the activity to go further with new questions). Additionally, Resnick and Silverman (2005) add to the metaphor the wide walls (the activity admits various approaches and solution strategies).

Spiral Learning Trajectories

The mathematical content brought to the classroom must be relevant to the student, must make mathematical sense (i.e., be rigorous and appropriate to the student’s level), and must be connected with prior and future knowledge. This involves designing and implementing learning trajectories (Clements and Sarama, 2004; Siemon et al., 2019), which are pathways meant to guide students’ learning through a series of objectives that consider the interdependencies between different areas of knowledge. But what knowledge is included, and who decides this? Each region, influenced by its educational traditions, has its own list of knowledge topics and organizes them into blocks. These blocks do not always align with pedagogical criteria. Beyond divergences regarding which grade is best to introduce each concept, the content changes little between official curricula worldwide. Broadly speaking, they can be organized into five blocks:

• Numeration and Calculation / Numbers and Operations (recognition of different types of numbers, counting, ordering, calculating arithmetic operations, etc.)
• Geometry / Spatial Sense (2D, 3D, coordinate axes, transformations, etc.)
• Algebraic Thinking / Relationships and Change (patterns, functions, equations, etc.)
• Measurement (lengths, angles, areas, volumes, etc.)
• Probability and Statistics (data representation in tables or graphs, measures of central tendency, chance, etc.)

Each block has its own learning trajectories, which are more hierarchical or less so, depending on the content. Numeration and calculation, for example, form a very hierarchical block: before learning to multiply, one must learn to add; and before adding, one must know how to count. Other blocks, such as geometry, allow for less strict learning trajectories: one can start with 3D geometry and move to plane geometry through the study of the faces or projections of three-dimensional shapes, or vice versa. In any case, planning is key to covering all knowledge. Indeed, several studies speak of the success of learning trajectories as a formative tool for teachers (Wilson et al., 2013; Sarama et al., 2017), as a tool to improve students’ mathematical learning (Clements et al., 2011), and as a cornerstone of a research-based teaching framework (Sztajn et al., 2012).

To understand the artificiality involved in analyzing mathematical knowledge this way, one can think of a swimming pool. From outside the water, as a teacher, it is possible to see the lanes of the pool and differentiate them. From inside the pool, however, as a student or person developing mathematical activity, everything is the same water. When one uses a ruler to measure a length, they are within the measurement block but also use numeration. When trigonometry is used to find the side of a triangle, one can talk about geometry but also measurement, algebraic thinking, and numeration. Knowledge intermixes, feeds off each other, and mutually reinforces to form a cohesive whole: mathematics.

Some researchers have referred to this as relational learning—where mathematics is presented as a series of connections that are made explicit and expanded as we learn—in contrast to more classical instrumental learning (Skemp, 1976):

The kind of learning which leads to instrumental mathematics consists of the learning of an increasing number of fixed plans, by which pupils can find their way from particular starting points (the data) to required finishing points (the answers to the questions). The plan tells them what to do at each choice point. […] There is no awareness of the overall relationship between successive stages, and the final goal. And in both cases, the learner is dependent on outside guidance for learning each new ‘way to get there’.

In contrast, learning relational mathematics consists of building up a conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point. (I say ‘in principle’ because of course some of these paths will be much harder to construct than others.) (p. 25)

Based on this idea, it seems clear that designing sequences for isolated topics or teaching units, without considering the relationships between content, makes little sense, especially in early childhood and primary education. Therefore, it is better to design trajectories, curricula, in a “spiral” or “helical” fashion, a term proposed by Bruner in the 1960s (Bruner, 1960) and explored by more authors (Harden and Stamper, 1999). A spiral curriculum introduces various concepts incrementally and revisits them over time, building on prior knowledge and fostering connections. This approach leverages what is known in cognitive psychology as the “spaced memory effect” (Vlach, Sandhofer, and Kornell, 2008; Carpenter, 2017), a well-researched phenomenon: distributing learning over time leads to more robust long-term retention compared to massed learning (i.e., learning and practicing each concept thoroughly before moving on), which may seem effective in the short term but is less so in the long term (as demonstrated by experiments such as Rawson and Kintsch’s (2005) in reading, or Rohrer and Taylor’s (2007) in mathematics).

A spiral curriculum contrasts with the traditional unit-based curriculum, where efforts focus on mastering specific content before moving on to the next topic. This spiral is like a winding staircase that ascends through increasingly deep and complex content but revisits the same ideas repeatedly. A simplistic example: instead of doing “Topic 1. Addition” and dedicating several sessions to constructing the concept of addition before moving on to “Topic 2. Subtraction,” it is preferable to untangle the existing interrelations in additive thinking (which includes both addition and subtraction) by having one day for a rich activity on constructing addition, another day for constructing subtraction, followed by practicing addition, then practicing subtraction, and finally a session that connects addition with subtraction, etc. A well-designed spiral facilitates a gradual increase in complexity and ensures that new learning builds and connects with previously learned material. In mathematics, this reinforces a more cohesive understanding of content, where each idea is seen in relation to others, potentially reducing boredom and fostering a deeper, interconnected understanding of the subject. Designing spiral sequences also better addresses the developmental differences of students: at a particular moment, some students might not be cognitively ready to understand a specific concept. If a topic ends and is not revisited until the following year, these students are left behind. However, if the concept is revisited (in a spiral) throughout the year, these students will have more opportunities to catch up. At the same time, a spiral design that starts from richness makes it easier to highlight connections between knowledge and offers a much more realistic picture of what it means to do integrated mathematics. As abstraction increases and therefore the necessary exposure time to work on certain content, the cycles of the spiral can be extended, always reinforcing the connections between blocks.

Returning to the learning trajectories of the different blocks: countless sources and references can help design them for the various stages with the academic rigor they deserve. The most important point is the coherence of the whole, not only between the different sessions within and across grades, from early childhood to secondary education. A good sequence must consider where the students come from and where they are going within each trajectory to promote the connected construction of all blocks. This is what Ball calls the mathematical horizon (Ball, 1993). To teach mathematics effectively, teachers need not only specific knowledge about the topics they teach but also a broader vision of how these topics fit within the larger framework of mathematics.

Here are some relevant sources for designing these trajectories within each domain:

Numeration and Calculation

• Van Den Heuvel-Panhuzien, M. (2008). Children Learn Mathematics. A Learning-Teaching Trajectory with Intermediate Attainment Targets for Calculation with Whole Numbers in Primary School. Dutch Design in Mathematics Education, V 1. Utrecht: Freudenthal Institute, Sense Publishers.
• Klein, A. S., Beishuizen, M., & Treffers, A. (1998). The Empty Number Line in Dutch Second Grades: Realistic Versus Gradual Program Design. Journal for Research in Mathematics Education, 29(4), 443-464. https://doi.org/10.5951/jresematheduc.29.4.0443

Geometry

• Castelnuovo, E., Arzarello, F., & Bussi, M. G. B. (2017). Didattica della matematica. UTET Università.
• Van Den Heuvel-Panhuizen, M., & Buys, K. (2008). Young Children Learn Measurement and Geometry. BRILL.

Measurement

• Van Den Heuvel-Panhuizen, M., & Buys, K. (2008). Young Children Learn Measurement and Geometry. BRILL.
• Szilágyi, J., Clements, D. H., & Sarama, J. (2013). Young Children’s Understandings of Length Measurement: Evaluating a Learning Trajectory. Journal for Research in Mathematics Education, 44(3), 581-620. https://doi.org/10.5951/jresematheduc.44.3.0581

Algebraic Thinking

• Ontario Ministry of Education. (2007). A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 3: Patterning and Algebra. Ontario Ministry of Education.
• Canals, M. A. (2022). Lógica a todas las edades (Los dossiers de Maria Antonia Canals). Associació de Mestres Rosa Sensat.

Statistics and Probability

• Curcio, F. R. (1987). Comprehension of mathematical relationships expressed in graphs. Journal for Research in Mathematics Education, 18(5), 382-393. https://doi.org/10.5951/jresematheduc.18.5.0382
• Curcio, F. R. (1989). Developing graph comprehension: elementary and middle school activities. https://eric.ed.gov/?id=ED314245 Reston, VA: NCTM.
• Friel, S. N., Curcio, F. R., & Bright, G. W. (2001). Making sense of graphs: Critical factors influencing comprehension and instructional implications. Journal for Research in Mathematics Education, 32(2), 1-24. https://doi.org/10.2307/749671
• Ontario Ministry of Education. (2007). A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 3: Data Management and Probability. Ontario Ministry of Education.

Various Knowledge

• Calvo Pesce, C., Deulofeu Piquet, J., Jareño Ruiz, J., Morera Úbeda, L. (2016). Aprender a enseñar matemáticas en la educación secundaria obligatoria. Síntesis.
• Albarracín, L., Badillo, E., Gíménez, J., Vanegas, Y., Vilella, X. (2018). Aprender a enseñar matemáticas en la educación primaria. Síntesis.

Examples of the Second Principle at Innovamat

Rich Activities to Develop Processes

Rich activities are the backbone of our proposal: they are the best way to work on competency-based learning. Our vision of richness in mathematics draws from various cited sources, and we have presented it at several local, national, and even international mathematics education conferences, such as the ICME (International Congress of Mathematical Education), as well as in articles published in specialized research journals.

Therefore, any session serves to see our competency-based approach to mathematics through rich activities. The didactic guides are filled with management advice related to the development of processes in the classroom. We have developed a color code for the four processes, consistent with the four fictional characters that represent them (referring to the fictional storyline that contextualizes some of the primary school sessions in the form of videos, images, etc.). This color code appears throughout all the guides and aims to train teachers’ competency-based perspective to help them be more sensitive to processes.

Below we see how, in green, we mark the “problem-solving” process in the guide of a workshop for early childhood on the number 7: The example shows that even from the early stages, we aim for students to develop processes. In this case, in a numeration context where we work with the number 7 and its additive decompositions with two addends, the questions are designed to awaken the need to follow an ordered strategy to find all combinations, a key skill in the problem-solving process.

Spiral Sequences

One of the didactic features that initially surprises teachers the most is that we avoid thematic units. The best way to see this is to take a look at the session table of the 4th-grade laboratory, for example, where just by looking at the first 10 sessions, it is clear that consecutive sessions do not necessarily address the same content:

At the same time, we ensure didactic coherence between courses and stages, not just within a specific course. For example, jumps on the number line to perform additive calculations, which form one of the central strategies in the early elementary grades, could almost disappear in the middle primary cycle, where multiplicative thinking takes center stage. However, knowing that later in the upper primary and secondary cycles, we will revisit the number line to find divisors, locate rationals, understand negative numbers, or represent coordinates, we consciously decide to keep the model alive in 3rd and 4th grades.

A Map of Mathematical Learning

To ensure that, regardless of the particularities of each territory where we are, our learning trajectories are coherent across courses and cover all curricular content, we have developed a map of mathematical learning based on various sources as cited. This map includes all the content that constitutes school mathematics, from early childhood to the end of secondary education, with hierarchical relationships and progressions among them. This allows us to a coherent sequence of sessions and provide students with a tailored learning path through our self-adaptive practice app.

Interest and curiosity are key factors in finding motivation for meaningful learning (Hindi & Renniger, 2019). This interest, as explained by Ruiz Martín (2023) citing Schiefele (2009), can be more personal for each student (and thus conditioned by their experience, successes and failures, environment, etc.) or more situational (provoked by the learning situation presented to them). In recent years, there has been a trend to revolve all educational approaches around students’ personal interests. This poses a risk, as explained by Ruiz Martín (2023):

Focusing on the personal interests of students might mean we are dedicating time to what they already know how to do and missing the opportunity for them to learn what they need to learn, as they have not yet mastered it. […] Therefore, suggesting that the topics covered in class are limited to those that coincide with the personal interests of students is very delicate, as it risks distorting the role of school as a tool for social equity that facilitates access to facts, ideas, and skills that would otherwise be out of reach for some students. In other words, limiting oneself to students’ interests may deny them the opportunity to broaden their horizons, which further disadvantages those with fewer opportunities outside of school. (p. 145)

The key to offering relevant contexts to students is not just to think about their interests, but to propose situations that foster curiosity through challenges (Litman, 2019), with constant connections to other areas. Fortunately, the gateways to mathematics are diverse. While keeping in mind that mathematics is a relevant science in itself, and it is necessary to foster the satisfaction of solving problems without excuses, it is also important to find contexts that foster creativity through all kinds of connections (with painting, music, sports, history, games, or everyday life), to break down false barriers between different areas of knowledge.

Connections with Other Fields

Mathematics is an essential tool in solving problems in fields such as physics, economics, chemistry, engineering, biology, medicine, or computer science, among others. These connections are the origin of the STEM disciplines (Science, Technology, Engineering, and Mathematics) (Bybee, 2013). Couso (2017) describes STEM literacy as “being able to identify and apply both key knowledge and ways of doing, thinking, speaking, and feeling in science, engineering, and mathematics.” Therefore, the choice of contexts through which to develop connected learning of mathematics must be sensitive to these interrelationships across sciences. It is important to prepare students for a world where mathematicians work in teams with experts from other areas to solve complex problems that require skills from different disciplines. Moreover, when we put mathematics at the service of solving concrete tangible problems, we provide students who do not yet enjoy mathematics for its own sake with other reasons to see its relevance.

However, STEM disciplines do not constitute a learning methodology: one can approach this perspective from both a project and a simple one-hour class activity. Generally, a task is considered fully STEM if, through context, it generates the need to build or apply knowledge about science, technology, engineering, and mathematics in an integrated way and especially if it also generates opportunities to develop skills inherent to the practice of these disciplines. Often, there are examples of self-proclaimed STEM activities that consider content from all branches but only skills from science or engineering, because they see mathematics only as a tool for the others. From a mathematical perspective, it is important to emphasize that the “M” is also capitalized: it must have its own entity and dialogue equally with the other disciplines. The beauty of STEM projects is that the lines across the involved areas become blurred because everything is well integrated.

Additionally, seeking and highlighting connections with other fields, even beyond the sciences, stimulates creativity. These stimuli work in both directions: artistic disciplines such as painting, sculpture, cinema, or literature, for example, are enriched when viewed with a mathematical perspective, and mathematics in turn feeds on new ideas and perspectives when observing art.

History of Mathematics and Narrative in the Classroom

Recently, many authors have advocated for the importance of including the history of mathematics in the classroom and have theorized about how to do so. In the well-known ICMI study (International Commission on Mathematical Instruction), which includes various authors with established careers in didactics, five areas are described in which mathematics education can be “enriched and improved by integrating the history of mathematics into the educational process”:

The learning of mathematics; the development of views on the nature of mathematics and mathematical activity; the didactical background of teachers and their pedagogical repertoire; the affective predisposition towards mathematics; and the appreciation of mathematics as a cultural-human endeavor. (Fauvel & Maanen, 2006, p. 203)

Thus, teachers must make an effort to master the history of mathematics and communicate it to students in an integrated manner with the curriculum. Indeed, several territories include the history of mathematics in official documents as a mechanism to humanize it and as a learning objective. In Fauvel and Maanen (2006), the authors talk from different perspectives about the importance of incorporating history into the classroom:

At first glance mathematics would appear to be the curriculum area in which students from all diverse backgrounds have common ground. After all, except few minor differences in the algorithms of basic skills (such as the way in which calculations are written down on paper), computational skills throughout the world are much alike. None the less, students globally learn mathematics based on their familiar linguistic and cultural patterns […]. Many study school mathematics without understanding the use for it; many dislike school mathematics; many more exhibit great mathematics anxiety. Minority students appear to suffer the worst.

One way in which curriculum specialists encourage teachers to make their content area meaningful is by humanising the subject. What better way can mathematics be humanised then by the use of mathematics history in the classroom? Besides being entertaining, the history of mathematics provides the student with information about the global roots of mathematics. Mathematics history helps students realise that mathematics is not just the invention of the dominant school culture. Rather, it helps students realise that mathematics evolved from many sources and in many places. (p. 186)

It seems evident that history, well introduced into the classroom, can become a wonderful gateway for students who do not enjoy mathematics from more abstract or disconnected approaches from other human activities. Additionally, multicultural approaches and strategies to address mathematical problems in the classroom enrich their resolution:

This approach […] allows students and teachers to think of mathematics as a discipline of continuous reflection and action influenced by thoughtfulness, reasoning, known procedures, intuitiveness, experimentation, and application to practical situations. (Fauvel & Maanen, 2006, p. 46)

Furinghetti (2000), one of the pioneers in theorizing the use of history in the classroom, in a work prior to ICMI’s reflections, explains that “the history of mathematics can be an effective element to provide students with flexibility, open-mindedness, and motivation towards mathematics.” She adds a historicist perspective that must be considered:

History is a kind of magnifying glass for the conceptual nodes of a certain theory, a means to identify critical points, to stop at difficult concepts and to analyze them through the words of the authors of the past. The basic idea in this use of the history of mathematics is to avoid the presentation of a polished theory, and to build it together with the students following the path of those who struggled with the mathematical problems underlying this theory. (p. 43)

This way of viewing mathematics as a living science, evolving alongside humanity, presenting difficulty and even error as sources of knowledge, conveys a realistic image of what it means to do mathematics.

In a later work, Jankvist (2009) proposes three ways to use history in teaching mathematics, either as a tool or as a learning objective:

a) Learning history, by the provision of direct historical information. b) Learning mathematical topics, by following a teaching and learning approach inspired by history. c) Developing deeper awareness, both of mathematics itself and of the social and cultural contexts in which mathematics has been done. (Jankvist, 2009, p. 247)

This last point, which sees history as a catalyst that opens the door to the more humanistic and cultural aspects of mathematics, perfectly aligns with one of the central themes of the NCTM (National Council of Teachers of Mathematics) 2022 conference:

Our personal stories have never been more important for giving us a sense of identity and belonging in our fragile and changing world. They have become needed catalysts and levers to reexamine the purposes of learning mathematics for all our students. Sessions in this strand will not only support the critical emergence of anti-racist math education but will also examine the universal and historical themes of joy and wonder that have crossed all cultures, civilizations, and socioeconomic situations. Mathematics must be seen through a prism that refracts all its possibilities and enchantments. Through a powerful narrative, we can ensure that all our students find their unique voice and purpose in learning mathematics. (Flynn et al., 2021, p. 8)

When used well in the classroom, history can act as a mirror that reinforces students’ mathematical identity and the feeling of belonging to a society that has faced difficulties and concerns they can relate to. This is especially relevant for historically discriminated groups based on gender (Henrion, 1997), race, disability, or socioeconomic status (Bartell et al., 2017): a sensitive treatment of history can open the door to enriching debates that otherwise rarely occur in mathematics classrooms and give voice to those students who often feel less invited to express their ideas.

At the end of the previous quote, the NCTM scientific committee speaks of narrative as a powerful tool for students to develop their purpose in the mathematics classroom. However, the use of narrative in education is not a new idea. In fact, many preschool and primary school teachers are aware that it is relatively easy, at these stages, to find mathematical resources that revolve around storytelling. Nonetheless, this practice is not common in later stages. And yet, as early as the 1990s, various authors highlighted the power of narrative as an educational tool. Witherell and Noddings (1991) stated that narrative and dialogue are fundamental to education. Weber (1993) goes further, emphasizing the role that narrative can play even in teacher training:

Each retelling of a narrative anecdote provokes further reflection and raises questions, becoming part of the research process and a form of pedagogical theorizing. Through an illustrative telling of anecdotes, it is suggested that the discerning use of the narrative is a way in which we can construct and assimilate our knowledge base in teacher education. (p. 71)

Zazkis and Liljedahl (2009), in a more recent study, present storytelling as a means to create a classroom where mathematics is understood and enjoyed, demonstrating how students’ mathematical activity can improve through storytelling. Additionally, they present a framework for creating new stories, ideas for using and enriching existing ones, as well as various techniques for telling stories that make storytelling more interactive and engaging for students.

Socio-emotional skills

The socio-emotional aspect of students also significantly influences the teaching and learning processes: if a student is not willing to learn, it is difficult for them to take advantage of the opportunities presented to them (McCombs and Marzano, 1990). This is why transferable skills such as perseverance, accepting and learning from mistakes, cooperation, autonomy, and responsibility are important. A paradigm sequence of activities must promote the development of these skills: they need to be put into practice and frequently discussed with students (Schulz, 2008).

To define socio-emotional skills, we start with the skills that cover the five major categories of personality in the school context, patterns of thought, feeling, and behavior that reflect the tendency of individuals to respond in a certain way in specific circumstances (Roberts, 2009). These general categories, described by Costa and McCrae (1992), are also used by the OECD (2017): openness to experience, conscientiousness, emotional regulation, extraversion, and cooperation; each subdivided into more specific skills. When analyzing these categories from a mathematics didactics perspective, certain nuances must be added:

1. Responsibility: working consistently, persevering in the face of difficulties, respecting rules, etc.
2. Cooperation: listening and incorporating what others say, participating in consensual decision-making, facilitating conflict resolution, recognizing one’s own responsibilities and those of others, etc.
3. Autonomy and Initiative: contributing ideas, asking questions when in doubt, making decisions, convincing peers, defending a position with arguments, etc.
4. Emotional Management: conveying joy, showing calm under pressure, adapting behavior to the situation, accepting failure and errors as sources of learning, enjoying moments when learning mathematics, etc.

Thanks to the definition of behaviors associated with each of the four socio-emotional skills (responsibility, cooperation, autonomy and initiative, emotional management), it is possible to foster contexts where students feel most engaged.

Examples of the Third Principle at Innovamat

The Challenge, the Best Context to Motivate

A good way to engage students is to structure sessions as challenges that test their skills while building curricular content. In the proposal, any session takes the form of a challenge, a concept closely linked to richness. When the content allows, this challenge aligns more with students’ usual interests, such as classifying animals in preschool, the adventure of the Nim game in primary school, or the percussion session that allows us to work on divisibility in secondary school through music. Other times, the challenge takes the form of an investigable initial question, exclusively mathematical, such as “How many times does the letter V appear when writing from 1 to 100 in Roman numerals?” or “How many different triangles can we build on a 3×3 geoboard?”.

These kinds of examples highlight that challenges, to be motivating, do not need to present falsely realistic application contexts to students, such as the traditional “going to the market to shop”. It is enough to pose investigable questions so that students feel challenged and begin to think.

The Mathematical Perspective on Other Fields

Our program is exclusively about Math. However, there are several moments when we open the door to other fields, to art and the rest of the so-called STEM disciplines, always with a mathematical perspective.

In the following case, for example, we see the header of a preschool workshop that deals with geometry in Kandinsky’s art:

In primary school, there are even some sessions explicitly called “STEM Challenges” that propose activities of this nature, such as the project to build a zip line to transport a cup of water that appears in 6th grade.

In secondary school, we also find several challenges framed in a context very favorable to these connections with other fields. In 8th grade, for example, one of the challenges proposes working on concepts of geometry and proportionality by studying the perimeter of the Earth’s maximum circle experimentally. In 8th grade, we find another good example in the challenge called “Symmetry with Friezes”, which proposes analyzing the patterns that occur in dance and relating them to the patterns that can form friezes in a mosaic, and all this allows us to delve into the concept of symmetry:

Narrative in the Service of Mathematical Learning

We are human because we tell stories. Fiction is inseparable from the history of human beings; it has always helped us rewrite our experience in the world and understand it better. We are used to seeing how fictional stories enter the classrooms of the early and primary stages, but upon reaching secondary school, fiction is often relegated to the field of literature. At Innovamat, we believe in the power of fiction in all stages of education: students need to explain and understand the world through stories that can also serve as a gateway to mathematics, especially for students with a more humanistic inclination. Therefore, fictional stories accompany Innovamat’s proposal from preschool to secondary school.

In Preschool Education, we find the Cricks, insects and arachnids with anthropomorphic qualities that represent each of the mathematical senses: the spider Byte, numeration and calculation; the grasshopper Palm, measurement; the bee Geo, the sense of space and form; the ladybug Dice, the sense of statistics and chance; and the butterfly Feebie, the sense of relationships and change. Throughout the course, teachers have access to fiction stories starring the Cricks. In them, we address socio-emotional issues linked to the contexts of the stage, such as adaptation to school, the joy of a new discovery, the fear of the unknown, or the difficulty of sharing.

In Primary Education, the Adventure sessions are led by the Bmath, a group of five friends with names linked to important mathematicians or scientists in history (Newt for Isaac Newton; Tina for Valentina Tereshkova; Ada for Ada Lovelace; Leo and Gali for Galileo). The adventures the Bmath live lead students through a mathematical world full of humor, with many references to pop culture. We should remember that in math classrooms, incorporating diverse cultural perspectives can enrich and broaden students’ understanding of cultural references and traditions.

Moreover, the Bmath are accompanied by four characters representing the four mathematical processes: Logos, who represents reasoning and proof; Solver, problem solving; Link, connections; and Graf, communication and representation. The four processes also accompany students in solving challenges and allow them to develop more self-awareness of what we are working on at any given moment.

In middle and high school education, we have made a strong commitment to incorporating the History of Mathematics into the curriculum, as it is standard in many regions. We do this through two fiction series: “Sam’s Journey” for 7th and 8th grades, and “L.O.V.E.” for 9th grade. The two series follow the same four precepts: first, to serve as a gateway to Mathematics for students who usually do not enter through the doors of manipulation, challenge, or abstraction. Second, to build a canon of historical mathematicians that shows, on the one hand, that Mathematics has been built by people capable of asking good questions, and on the other hand, that the history of mathematics is diverse: we find women and people from places and cultures outside the Western world. The Sumerians, Liu Hui, Brahmagupta, Al-Khwarizmi, Évariste Galois, Sophie Germain, Florence Nightingale, or Maryam Mirzakhani are just some of the characters that appear in the two series.

Third, as with the Bmath, the two Secondary series remind us that the classroom can be an excellent context for enriching students’ cultural universe, and we have filled them with cinematic and literary references, intending to take their knowledge of the world and themselves further. And fourth, having human and therefore fallible references helps students work on their socio-emotional skills. For example, knowing that the Indian mathematician Brahmagupta died wrong, thinking that zero divided by zero is equal to zero, and knowing that he is still one of the most important mathematicians in history, mirrors students and paves the way for planting the idea that the mathematics classroom is a place where error is not only welcome but is the scaffold for future successes.

Repetition until memorization is not a guarantee of learning the content, and even less so a guarantee of giving it meaning. But then, within the competency-based paradigm, is it never necessary to practice? Is it never necessary to memorize? That’s not the case either. It’s important to find a balance (Schoenfeld, 1992). After discovering the concepts, after constructing them with manipulative materials and understanding them, it’s essential to find moments to practice the procedures used most often and develop precision, efficiency, and flexibility until they are automated or even memorized. This practice contributes to a deeper understanding of the fundamentals, necessary for grasping more advanced concepts, while also providing self-sufficiency, building confidence, and especially helping develop the ability to estimate and verify the reasonableness of results. In fact, numerous studies explain that numerical sense and arithmetic fluency, more than other cognitive factors, are predictors of students’ mathematical success in the short and long term (Geary, 2011; Jordan et al., 2009; Locuniak and Jordan, 2008; Mazzocco and Thompson, 2005).

Automating vs. Memorizing

Memory is not the only way to evoke a result fluently: it can also be automated. Van Den Heuvel-Panhuizen (2008) differentiates these two concepts as follows:

Automatization concerns the virtually automatic carrying out of arithmetical operations, in contrast to memorization, which involves learning arithmetical facts by heart. A student is said to have automatized a problem like 9 × 12 if he or she can quickly give the answer by working out, for example, 9 × 10 and 9 × 2, followed by adding these results. (p. 89)

As external observers, the line between automation and memorization is blurry. In any case, our task as teachers is to provide different strategies to deduce and quickly recall the most common results (also called “facts”). And to offer space and time to practice them until they are automated or memorized, that is, to develop fluency.

What is Fluency?

Arithmetic fluency can be defined as the ability to work with numbers, operations, and procedures with precision, efficiency, and flexibility, all while developing numerical sense. Bay-Williams and SanGiovanni (2021, 2022) describe that fluency has three levels of depth:

• At the first level, they describe factual fluency, which is the ability to recall known or derived facts through memorization or automation. Evidently, the facts that students are expected to recall fluently differ at each stage: in 1st grade, 7 + 3 = 10 should be a known fact, while the 5 times table should be a known fact from 3rd or 4th grade onward.

To achieve this factual fluency, authors like Baroody (2006) define three phases that must be passed to master a fact automatically or even from memory:

• Phase 1. Counting (by ones, by twos, by fives, by tens, etc.). For example, solving 6 + 9 by counting up from 6.
• Phase 2. Deriving (deducing facts from others using reasoning strategies). For example, solving 6 + 9 by knowing that 6 + 10 is 16.
• Phase 3. Mastering (automating or even memorizing).

• At the second level, which includes the first, is computational fluency. This is the ability to perform operations (compute) beyond known simple facts. Again, the operations in which students are expected to show fluency differ in each grade.
• At the third level, which includes the other two, is procedural fluency. This is the ability to follow procedures that go beyond a basic operation and also depend on each stage.

Figure 2. Types of Fluency. Source: Own elaboration based on Bay-Williams and SanGiovanni (2022).

According to the same authors, procedural fluency, which includes the others, is made up of three components and six related actions:

Figure 3. Components of Procedural Fluency. Source: Own elaboration based on Bay-Williams and SanGiovanni (2022), citing Spangler and Wanko (2017).

Although the work of these authors has focused on arithmetic fluency, within numeracy and calculation, these ideas could be extended further to discuss fluency in other branches, such as geometry, measurement, or statistics.

Developing Arithmetic Fluency in the Classroom

Various studies point out that the best way to start learning basic facts is through explicit teaching of strategies (Baroody et al., 2016; Purpura et al., 2016; Henry and Brown, 2008). That is, classroom time should be dedicated to explicitly discussing the different strategies that allow us to solve basic operations, comparing them in terms of efficiency and convenience. From there, the key is to practice them in various contexts.

Although school tradition tends to focus only on one type of practice, reproductive practice, didactics distinguish between two (Van Den Heuvel-Panhuizen, 2008): productive and reproductive. Both have advantages that make them more or less suitable depending on the teaching objectives. For example: using natural numbers between 1 and 60, without repeating any, form a chain of 10 numbers, so that each number is a multiple or divisor of the adjacent numbers. In this case, with the goal of “producing” number chains, the student practices calculating multiples and divisors. The advantage of posing productive practice sessions like this is that they are self-adaptive: if the student masters the procedure, a few iterations will suffice to produce an answer and they can pose new questions; if they have difficulties, the nature of the task will force them to spend more time searching for multiples and divisors, which is precisely what needs to be practiced. A variant to consider within productive practice, which almost deserves a separate chapter, is play. Playing board games presents an engaging context for learning (Bayeck, 2020). Mathematics and logic are at the root of most games where one must think of a strategy to win, making them a fantastic environment to develop mathematical processes such as problem-solving or reasoning. Additionally, well-chosen games can present an appropriate context for practicing and developing arithmetic fluency (Bay-Williams and Kling, 2019).

On the other hand, reproductive practice, the most conventional, involves reproducing a procedure multiple times without any investigation or production involved. Traditionally, this type of practice is done through worksheets with repetitive sections. The main drawback is that if the student has understood the procedure, they will get bored and lose motivation after a few repetitions; if they haven’t understood it, it is unlikely that they will grasp it through simple repetition. In either case, it conveys an image of mathematics as repetitive.

Today, however, thanks to digital platforms, there are alternative reproductive practices to the traditional worksheet that have the advantage of allowing personalized learning. These platforms can, for example, control the time variable, limit the response time, measure how long a student spends on a specific section, and act accordingly to minimize reproductive repetition. At the same time, they can provide visual aids that are difficult to manage with the whole group in front of a conventional worksheet. They also offer the possibility to practice content that is difficult to present on paper, such as 3D visualization, angle estimation, or locating integers on the number line. Additionally, if the digital platform corrects activities automatically, it saves correction time for teachers, which can be used for more formative assessment and providing almost immediate feedback to the student.

Examples of the Fourth Principle in Innovamat

The diversity of strategies

One key to developing fluency, as we have seen, is knowing various ways to approach basic arithmetic operations. Let’s focus on addition and subtraction, for example. Sometimes, it’s useful to use the written algorithm. Other times, we prefer to do mental calculations by imagining jumps on the number line. There are also times when we can deduce the result from facts we already know. Sometimes, we even want to use a calculator. Knowing this variety and knowing how to choose is what makes us efficient when calculating.

Within our proposal, there are countless sessions dedicated to building and practicing these strategies or procedures, and also to developing the criteria for choosing the best one in each case. In the following set, we can see examples of initial cycle sessions where various ways of solving additive operations appear, for example:

It is through the various ways we have of visualizing and executing operations that we promote a deep understanding of them. And this is one of the keys to developing fluency, even before we start practicing.

Moments of Productive Practice

If we leave aside reproductive practice, which is presented through the app once a week, we could say that almost all practice moments in the classroom are productive practice.

In the early childhood stage, we find various activities of this type that allow us to practice the first counting or additive thinking skills, such as in the workshop on the number 10 where students look for all the ways to sum this amount with two cards, thus productively practicing the addition of two one-digit numbers:

In primary school, it is also common for the practice of basic arithmetic operations to be framed in a productive context. For example, in the following activity, we propose practicing the multiplication tables through an investigation into the concept of “multiplicative persistence” of two-digit numbers:

And middle school is no exception. We find activities like “In Search of 0” where, to practice adding and subtracting integers, we lead an investigation to see how many jumps on the line it takes, following certain rules, to reach 0.

Reproductive Practice in the App

At Innovamat, the bulk of reproductive practice for the procedures we want to consolidate is done through the app we have specifically designed for this purpose. Students use the app once a week, between 15 minutes (in early childhood) and an hour (in primary and secondary).

This practice is presented in the form of small applications (applets) that pose activities in an individualized itinerary based on the knowledge the student shows. Personalization allows us to adapt to each student’s needs and offer well-thought-out didactic aids. A good example of this is found in this applet where additive boxes are worked on:

At first, the additive box is presented, which has been worked on in class before appearing in the app. If the student asks for help or makes a mistake, they are first reminded of what to do:

If they still don’t succeed, the application proposes an aid that favors visual counting with egg cartons:

We can still ask for help or make another mistake, and the app reorganizes the egg cartons to facilitate counting or interpreting the quantity:

If we make a final mistake, an animation shows that the task is to count the eggs, and counts them one by one until the answer is found. Then, a new box is presented to see if the student improves. On the other hand, if the student correctly answers several boxes in a row, the app understands that the content has been understood and proposes new activities. This also avoids unnecessary repetitions.

As the example shows, it is clear that we understand the role of technology as a vehicle to bring practice into the classroom goes far beyond a change of format: it is not simply about dressing up repetitive lists to make them more digestible on screen. It is about leveraging the advantages of technology to improve the student’s individualized practice experience. It also improves the teacher’s experience: thanks to weekly progress reports, it saves the correction of these more repetitive activities and provides an overview of which contents and procedures students dominate more or in which they have more difficulties. All of this should serve mainly to better plan the following sessions and think about how to introduce new content.

The right to inclusive education is a fundamental right for all students, regardless of their needs or abilities. This means that all students must have access to quality education, and necessary measures must be taken to ensure that this education is inclusive and tailored to individual needs.

In a traditional lecture-based class, the challenge of addressing this diversity can be masked by teaching strategies: if the only thing the teacher does is explain and write on the board, it is irrelevant whether the audience is one student or one hundred fifty. Conversely, in classrooms driven by inquiry, where students spend more time manipulating materials and discussing with each other than listening to lectures, diversity emerges in all its splendor and complexity. Class sizes make it nearly impossible to personalize learning for everyone all the time. The first solution, then, seems simple: reduce class sizes. Either have smaller classes or more than one teacher per class. In most cases, however, schools do not have the necessary resources to achieve this reduction. Moreover, overly reducing class sizes has a drawback: it is not always beneficial to be in a small class if there are no dissenting opinions to foster argumentation and debate. So, what can be done?

The Three P’s: Presence, Participation, and Progress

The framework of the three P’s by Booth and Ainscow (2002) is a tool that can be used to assess whether a school or educational system is providing inclusive education. This framework is based on three dimensions:

• Presence: Refers to the physical inclusion of students in the classroom and school. This means taking necessary measures to ensure that any student can access the classroom and school, regardless of their abilities or special needs.
• Participation: Refers to the social and emotional inclusion of students in the classroom and school. This involves taking necessary measures to ensure that all students can actively participate in learning and school life, regardless of their abilities.
• Progress: Refers to the academic inclusion of students. This means taking necessary measures to ensure that all students can make progress in learning and develop their skills and abilities, regardless of their difficulties or limitations.

To ensure that the three P’s are met and that all children can access mathematical education, it is advisable to follow the Universal Design for Learning (UDL) framework, described below.

Universal Design for Learning (UDL)

UDL is based on guidelines that began as a project of the National Center on Accessing the General Curriculum (NCAC) of the U.S. Department of Education (Rose, Meyer, and Hitchcock, 2005; Rose and Meyer, 2006; Meyer, Gordon, and Rose, 2015). Essentially, these guidelines focus, more than on deficits, on environmental barriers, which can be cognitive, physical, sensory, structural, emotional, communicative, or curricular.

These barriers, which do not affect everyone equally, prevent all children from learning the same. Based on the characterization of these barriers, UDL proposes three fundamental principles:

1. Provide multiple representations of content: This means presenting information in different ways to accommodate students’ diverse learning and information processing styles. This includes, among others, graphics, videos, audio, and texts.
2. Provide multiple ways of acting and expressing knowledge: This involves offering students different options to demonstrate what they have learned, such as oral presentations, written texts, videos, use of manipulatives, etc.
3. Provide multiple forms of engagement: This means offering students different ways to interact with the content, such as collaborative activities, games, simulations, and other rich activities.

Through UDL, teachers can design activities and materials that are flexible and adaptable to students’ needs, offering multiple ways to learn and demonstrate understanding. This approach has proven effective for students with functional diversity and other specific needs (Fovet, 2021). In mathematics, UDL can help address learning difficulties some children experience. For example, it has been shown that offering multiple representations of content helps students better understand mathematical concepts (Murawski and Scott, 2019). This may include the use of diagrams, graphs, and other visual images that help represent mathematical ideas.

UDL can also be useful for students experiencing specific difficulties in numerical processing and problem-solving (Fovet, 2021). Providing multiple ways of acting and expressing knowledge allows students to approach mathematical concepts from different angles and through different methods, such as physical manipulation of materials, visual representation, and group problem-solving.

In conclusion, universal design for learning is a pedagogical approach that seeks to create an inclusive and accessible learning environment for all students, regardless of their individual differences or specific needs. By implementing the three fundamental principles of UDL, teachers can create activities tailored to each child’s needs. Similar to universal architectural design, UDL ensures that learning materials are accessible to everyone.

When discussing this universal design in mathematics, it is also important to consider the ideas described in Nrich (2013, 2017), based on the ideas of Papert (1993) and Resnick and Silverman (2005), already explained in the second principle: if the activity is low floor (allows entry to all students), high ceiling (allows many connections and extends the activity to go further with new questions), and wide walls (admits various approaches and strategies for resolution), it means that there are no entry barriers and any student can start thinking about the problem. When activities of this kind are posed, attention to diversity occurs very organically, as most students can start working, and those who are faster can continue exploring new ideas.

Personalization of Learning

Apart from universal design, research suggests that individualization can have a positive impact on student outcomes (Graham and Perin, 2007; Pane et al., 2014). Therefore, especially in reproductive practice, it is essential to adapt to the level and needs of each student at all times.

In terms of educational technology, this functionality is considered an Intelligent Tutoring System (ITS). These systems use artificial intelligence models to provide personalized and adaptive feedback to students based on their responses, analysis of their strengths and weaknesses, and their performance in learning. ITS are considered one of the most advanced technologies in individualizing learning, providing a highly personalized and adaptive learning experience. According to various studies (e.g., Kulik and Fletcher, 2016; Ma et al., 2014; Van Lehn, 2011), ITS have proven effective in improving students’ academic performance in various areas, including mathematics, sciences, and language.

To improve performance in mathematics, however, it is not enough to individualize learning and make it accessible to everyone; in some cases, it is also necessary to extend instruction time through educational interventions that analyze the cognitive effects and prior knowledge influencing mathematics learning. In some countries, like the United States or Finland, Multi-Tiered Systems of Support (MTSS) have been implemented (Harn et al., 2011; Sugai and Horner, 2009), offering the level and intensity of instruction needed for students with greater difficulties to achieve adequate levels.

To determine the level of mastery of knowledge for each student, Curriculum-Based Measures (CBM) are conducted (Deno, 1985; Espin et al., 2012). These constant measures allow teachers and support teams to have a comprehensive idea of each student’s knowledge at any given time. From there, educational interventions are adapted to each one’s needs, based on their cognitive skills and knowledge — what is called Data-Based Interventions (DBI) (Fuchs, 2004). Recent studies show that DBI are advisable, as they improve student outcomes with a moderate effect size (Jung et al., 2017).

Intervention

Both UDL, the three P’s framework, and personalization foster an ecosystem where learning difficulties are minimized, but it is unlikely that these will completely disappear: there will always be children with difficulties requiring intervention by teachers and specialists. Fuchs et al. (2021), through the U.S. Department of Education’s Institute of Education Sciences, propose several recommendations for carrying out these interventions in mathematics, especially in early stages, such as being explicit in the systematic instruction of ideas and strategies, teaching clear and concise mathematical language, or using a well-chosen set of representations that support abstract ideas (with a special focus on the number line), among others.

In addition to these general recommendations, it is crucial to identify children with cognitive difficulties in understanding basic numeration concepts early on in first and second grade. To consider the possible detection and diagnosis of dyscalculia and other related difficulties, a variation of MTSS called the “Response to Intervention” (RTI) model can be implemented in these grades (Fletcher and Vaughn, 2009; D. Fuchs and Fuchs, 2006). RTI begins by measuring performance (arithmetic fluency, in this case) at the start of schooling and then administering an early intervention to all students who show low performance enough to be considered at risk of not reaching the expected minimum level. This intervention typically lasts between 12 and 18 weeks and can have two outcomes: on one hand, a significant percentage of children will respond correctly to the intervention, improving mathematical performance to normal levels and, therefore, following the regular curriculum with the class. On the other hand, some children will not respond correctly to the intervention. These will be candidates for a diagnosis of dyscalculia or other learning difficulties.

Examples of the Fifth Principle at Innovamat

Low Threshold, High Ceiling, and Wide Walls

When we design activities and propose them through the teaching guides, we aim to keep the UDL framework in mind, especially the concept of “low threshold, high ceiling, and wide walls.” We apply these ideas at different scales: looking at an entire course, we see that the initial sessions tend to be simpler (though they always offer the possibility to go further); in a specific class session, we see how it begins with a warm-up accessible to everyone to awaken prior knowledge and set the stage, followed by activities of increasing difficulty; finally, looking at each specific activity, we see that many start with an open premise, inviting everyone to begin exploring and allowing further questions to delve deeper as desired. A good example of this is “WODB” (an acronym for “Which One Doesn’t Belong”):

As we see, any student can start thinking and find that the one that doesn’t belong out is the lower right triangle because it is the only one that is not purple (it is green). This way, the student enters the activity with confidence (low thereshold), and through conversations with classmates, they can go further. At the same time, students with more knowledge can find arguments involving more advanced content (high ceiling), such as identifying the green triangle as the odd one out because it is the only one with no line of symmetry. Moreover, there are various ways to find the doesn’t belonging one out, depending on the perspective and criteria chosen for analysis (wide walls), such as, in this example, the size of the triangle sides, the amplitude of their angles, their orientation on the plane, etc.

Support and Extension Resources

Beyond universal design, we strive to fill the guides with tips to address diversity, both for students who show difficulties and those who can go further. Here are a couple of examples. The first, marked with an “S” for “support”, suggests a manipulative way to help students with difficulties find one of the factors in a multiplication in a 4th-grade activity:

The second is an example of an “extension” tip for 6th grade, where varying a restriction is proposed to find new symmetrical solutions:

Additionally, many sessions include a version of the student’s support notebook, where the same content is worked on but in a simpler way and with visual aids, and an extension version, where the same content is addressed with more complexity and abstraction. This is how we explain it in 2nd grade, for example:

In middle school, where students already show more autonomy, we offer three levels of daily paper practice of increasing difficulty, named “X”, “Y”, and “Z”, respectively. We advise that the students themselves choose the difficulty level they feel they should face that day, depending on how they feel about the content. Here is an example from 7th grade:

The differences in increasing difficulty are evident when attempting to solve the tasks. At level x, the purple spots in task 1 are easier to isolate than at level z, where greater mastery of the procedure and the priority of combined operations is required. The increasing difficulty is also noted in the amount of data and restrictions that appear in task 2.

Response to Intervention (RTI) Program

At Innovamat, we have developed our own tool to carry out RTI through the digital practice app. This tool, designed for 1st and 2nd graders (where this approach has been most effective), includes an initial assessment, a 16-week support intervention for students struggling the most (about 20 minutes each day), and a follow-up test to measure each student’s progress.

Those who do not improve with the intervention may have learning difficulties, and we can detect them as early as possible, inform the teachers, and help them confirm and address the issue. For more details on the tool’s characteristics and validation, you can read the article published by our research group (Correig-Fraga et al., 2024).

For many teachers, and even more so for students, evaluation is a headache. This anxiety stems, among other reasons, from a biased conception that equates evaluation with “giving grades.” As we show in this principle, evaluation is much more than testing and grading: it should be a reflective process that fosters learning and provides satisfaction to both students and teachers.

What Does Evaluating Mean?

When discussing evaluation, the two main purposes often get mixed up:

• Formative and Forming Evaluation: Evaluation becomes a means to regulate learning, identify successes, difficulties, and errors, and find ways to progress. This information can reach the student as feedback from the teacher or through a peer. In both cases, we talk about formative evaluation. Simultaneously, we also expect the student to reflect on their own learning process and develop the ability to self-regulate; in this case, we refer to it as forming evaluation.
• Summative Evaluation: Evaluation serves as a means to check what has been learned afterward and takes the form of a number or qualitative comment. This is the type of evaluation referred to when thinking about grading for report cards. Grades allow for classification (e.g., for university admission) or certification that a level has been acquired (e.g., in official language exams).

Although for most students, teachers, and society in general, the primary concern is numerical grading, a truly paradigmatic evaluation process must consider both purposes and especially enhance the first one (Sanmartí, 2020). Sanmartí (2020) explains:

In implementing a competency-based curriculum, evaluation plays a critical role in the entire learning process. It allows for the adaptation of teaching strategies to meet student needs and tracks their progress as they learn. Evaluation helps teachers assess how well students are achieving basic competencies and make necessary adjustments to their teaching methods. For students, evaluation is crucial for learning, as those who can see their progress and self-regulate are better equipped to advance and continue learning. Therefore, it’s important to involve students in the evaluation process, making them active participants in their learning journey. Additionally, sharing evaluation criteria with other teachers and families ensures consistency across subjects, projects, and school activities. (p. 4)

Boaler, in the same vein, describes it as follows (2020):

One problem […] is that many teachers use summative assessment formatively; that is, they give students an end score or grade when they are still learning the materials. […] In the weeks and months that students are learning in a course, it is very important to assess formatively, not summatively. In Assessment for Learning (AFL), students become knowledgeable about what they know, what they need to know, and ways to close the gap between the two. (p. 224)

According to Boaler, AFL helps teachers make their instruction more effective and students learn as much as possible. Teachers who use AFL spend less time telling students about their achievements and more time empowering them to take charge of their learning paths.

With this conception in mind, and to simplify, we understand that evaluation consists of three phases, as described by Fernández and Morales (2022). The difficulty lies in answering the questions posed by each phase:

1. Collecting evidence of progress (When? About what? How do we collect it?)
2. Analyzing it (Based on what reference? For what purpose?)
3. Making decisions to help progress (How do we record it? How do we adapt our work? How do we give feedback to the student? How do we convey this information to families and the administration?)

What Do We Evaluate in Mathematics?

Traditionally, mathematics teachers focused on mastering content: Can the student add? Do they know what an isosceles triangle is? Today, as we have explained, most experts in mathematics education, and increasingly more institutions, advocate for learning (and thus evaluation) that considers not only content (numeration, measurement, geometry, etc.) but also the four processes of mathematical competence (problem-solving, reasoning and proof, connections, communication, and representation; described in chapter 2). For example, what we can evaluate regarding problem-solving is whether the student can be thorough and find all solutions in an activity involving additive thinking within the range of 1–20. It makes no sense to separate processes from content, as they are almost always a necessary condition. However, as was done in more traditional approaches, it is possible to evaluate content acquisition independently of the processes involved.

Moreover, as authors like Boaler (2020) and Johnson (2022) argue, in addition to processes and content, we must also consider the evaluation of socio-emotional skills (responsibility, autonomy and initiative, cooperation, emotional management; described in the first principle of chapter 3). In fact, a socio-emotional problem can create fundamental barriers that prevent learning content or processes. Therefore, teachers must be very aware of this third layer: sometimes learning difficulties are not conceptual or competency-based but emotional.

Ultimately, the most important thing is to be clear about what we want to evaluate and act accordingly. Evaluation decisions, no matter how well-founded, will always be biased by the teacher’s perspective: evaluation is inherently subjective, depending as much on the observer as on the observed. Not even standardized tests are objective: the choice of questions in the test is subjective; the scoring and weighting of each question are subjective; even the correction is subjective because, more or less, and with the exception of multiple-choice tests, it depends on each corrector and the moment they correct it, etc. Admitting this subjectivity does not imply evaluating without a basis nor does it diminish the value of finding evidence to justify it. On the contrary, it is the teachers, the experts in the subject, who can choose how to collect evidence in the best way at each moment and decide what is best for each student to foster their progress.

When Do We Evaluate?

Always. Every time a doubt is resolved, every time a student is guided, every time a question is asked… evaluation is happening. Teachers must take advantage of spending the entire course with students to continuously gather various types of evidence. To ideally evaluate, though, one should listen, observe, and collect evidence from students while they perform all tasks every day of the course and throughout all sessions. Obviously, this is impossible in a classroom with more than 25 students. And even more so when trying to evaluate content, processes, and socio-emotional skills. Traditionally, teachers feel much more comfortable determining whether the student can add, multiply, or solve quadratic equations, for example. However, they find it much more challenging to determine whether the student is competent in the reasoning and proof process. Why? Because the current generation of teachers has received an education much more focused on content than on processes: they master addition, multiplication, or quadratic equations much more than the reasoning and proof process. That is why teacher training is so important (Baumert et al., 2010). Only when teachers can think about processes in an integrated and natural way will they be able to evaluate them with ease and comfort daily. In fact, there is no need to wait for a specific evaluation session to gather evidence: it should be a continuous attitude (Leanda et al., 2021). Sometimes the evidence will be physical and remain (from responses in the notebook to written tests, if applicable), and other times it will be ephemeral (from an oral intervention or gesture to a representation built with manipulative material).

Working this way reduces student anxiety about tests because they are not betting everything on one card (Boaler, 2020). It also reduces the pressure teachers feel when the evaluation period arrives. Some voices argue that continuous evaluation is unfair because it does not focus on the same thing for all students or at the same time. To respond to this, first, we must remember that we want an evaluation that accompanies the journey and promotes autonomy, not a simple exam that captures a snapshot of a specific moment. And second, if well-planned, continuous evidence collection admits various formats and favors all students having many opportunities to show their progress in terms of content, processes, and socio-emotional skills. In fact, one of these pieces of evidence can be a traditional written test. Ruiz Martín (2021) explains that traditional tests can aid learning because they rely on recall: if designed well, they can be an excellent tool. However, he also warns that if these tests are given too much weight in students’ grades, the anxiety they cause may outweigh their educational benefits.

Examples of the Sixth Principle at Innovamat

Resources and Opportunities for Formative Evaluation

In the Innovamat classroom, there are numerous opportunities for teachers to continuously gather evidence of learning.

In early childhood education, the teaching guides for Workshops include a section titled “As a teacher, what can I observe?” where different evaluation indicators are detailed, classified by mathematical process. Additionally, at the end of each workshop, there is space for students to express their reflections and learnings related to the session, both in group conversations and individually, using worksheets and blank pages. The observation tables in the Spaces also include evaluation indicators in the sections “What can I observe?” and “What can I ask?” aimed at helping teachers focus on the competency-based learning objectives. Finally, the so-called “check-in tests,” designed to evaluate students approximately every five weeks, are accompanied by a teacher’s guide with detailed objectives, assessed standards, answer keys, and recommendations for formative evaluation.

In primary education, we also include multiple opportunities for formative evaluation. The evaluable tasks throughout the course, highlighted in the guides for both the Laboratory and Adventures, are particularly suitable for evaluating fundamental content or specific processes. These tasks include evaluation guidelines detailing objectives, indicators, and observations, with examples of real student responses. The evaluation indicator tables, available in editable and printable format, also offer specific indicators based on the task. Additionally, the notebook that the student completes at the end of each Laboratory session provides the opportunity for observations and formative feedback. Lastly, just as in early childhood education, the “check-in tests” also allow for student evaluation every five weeks and contribute to formative assessment.

Secondary education is no exception. The guides also include various tips for formative assessment. For example, in the section “As a teacher, what can I observe?”, we highlight opportunities to evaluate processes. This is also indicated in the class slides, so the teacher keeps it in mind while in front of the students. Individual practice moments also present various opportunities for formative assessment, in this case more focused on content. Finally, at the end of each section, we also propose an evaluation test, with the corresponding indicator tables that help analyze the development of processes in each task.

Resources and Opportunities for Summative Evaluation

In the Innovamat classroom, various resources are also considered for summative, qualifying evaluation. Since summative evaluation is closely tied to the regulations of each region and school, we leave it to the teachers to decide how to use these resources based on their criteria.

In early childhood education, the individualized practice that students perform weekly using the practice app generates a detailed report that compiles the percentages of correctness in different activities. This report aggregates the information for each of the course standards and allows for analyzing the individual progress of each student. Additionally, the “check-in tests”, despite being initially designed for formative evaluation, can be used as a summative tool at the teacher’s discretion.

In primary education, summative evaluation benefits from the same resources used in early childhood, with some particularities. The individualized practice with the app also generates detailed weekly reports that allow analyzing the progress of each student. The evaluable tasks from the Laboratory of Numbers have variations that teachers can use as a summative evaluation tool to verify the level of content and process acquisition worked on in a session. The evaluation indicator tables can also be used as a summative evaluation tool, as they include observation, conversation, and written indicators. Each indicator offers a gradation of three performance levels to facilitate the collection of evidence, allowing them to be converted into a grade. Like in early childhood, the “check-in tests” can also be used as a summative tool, according to the teacher’s criteria.

In secondary education, we also have weekly digital practice through the app, which generates detailed reports that allow knowing the progress of each student and assessing them summatively. Lastly, the evaluation test at the end of each section, with the indicator tables, is also a tool that allows summative assessment.

Although a pencil or a calculator, two common objects in a mathematics classroom, can be considered “technology,” it is evident that when we talk about educational technological tools today, we mainly refer to screens (computers, tablets, smartphones, and similar devices). The use of these electronic devices in the classroom sparks a heated debate with opposing views on their impact on children’s education. But what does educational research say about the use of this technology in classrooms?

The first thing to consider is that the presence of screens in everyday environments today is almost inevitable. In this sense, the school emerges as the ideal setting where children can become familiar with the educational uses of screens and their responsible management, which various authors and official texts refer to as “digital competence” (Pettersson, 2017). In any case, the most decisive factor is not the incorporation of technology itself, but how we use it, as Azoulay, UNESCO’s Director-General, explains (2023):

The digital revolution holds immeasurable potential but, just as warnings have been voiced for how it should be regulated in society, similar attention must be paid to the way it is used in education. Its use must be for enhanced learning experiences and for the well-being of students and teachers, not to their detriment.

Benefits and Risks of Technology in the Classroom

Beyond making children aware of the responsible use of screens, the presence of such technologies in the mathematics classroom can bring a series of advantages and disadvantages worth exploring. In a meta-analysis that examines over 90 different studies on the use of digital tools in the classroom, Hillmayr et al. (2020) explain that integrating these technologies into education benefits student learning. According to the authors, adopting digital tools in teaching mathematics and science significantly improves students’ academic performance, especially due to two of their most widespread uses.

On one hand, they highlight tutoring and practice systems through a digital platform, which can adapt to each child’s pace within certain parameters and provide evidence of their progress. Many teachers recognize that it is not always possible to successfully manage the different paces that exist in a group of 25 or 30 students. These platforms have the potential to present individualized practice activities for each child, regulating the time factor and providing both immediate feedback and help in case of difficulty. Feedback, especially when it answers the questions “Where am I going?”, “How am I doing?”, and “What comes next?”, is key to student development, as described by Hattie and Timperley (2007). When the activities also appear mixed and not grouped by thematic blocks, students’ learning improves significantly in the medium and long term (Rohrer and Taylor, 2007). Moreover, many of these digital practice tools communicate with the teacher in some way so that they can make more informed decisions, which can contribute to better continuous assessment.

On the other hand, Hillmayr et al. (2020) also highlight simulators that allow for dynamic visualization of mathematical concepts: technology offers the possibility of working on content that is difficult to present on paper, such as 3D visualization, angle estimation, or locating rational numbers on the number line, to name three highly relevant specific examples in a mathematics classroom.

In any case, this positive effect is reinforced when teachers receive the appropriate training to maximize the use of these tools, and even more so when their use is combined with non-technological instructional methods, meaning that technology acts as a complement and not a substitute for the teacher.

Other authors, while acknowledging the potential benefits, warn that these technologies alone are never transformative (Selwyn et al., 2020) and emphasize the need to reflect on how to integrate them effectively. In fact, according to these authors, it is necessary to consider not only the potential of technology to enrich learning but also how it is perceived socially and how it will be used in the school context. Again, as Azoulay (2023) said, the authors focus the debate on how we use technology rather than its intrinsic nature.

When seeking reasons against incorporating screens in the classroom, the focus is often on their potentially harmful effects on children’s health. Concerns are raised about the impact on concentration, myopia, sleep disorders, addictions, or language disorders (Ruiz Martín, 2023). However, according to Ruiz Martín (2023), citing studies from various fields, there is no conclusive evidence that the educational use of screens causes any of these consequences. For example, we know that fixing the gaze on a close point for reading increases the risk of developing myopia (more than playing outdoors), but no differences have been observed between reading on paper or on a screen (Lanca et al., 2022). Screen use (including television) just before bedtime can inhibit melatonin secretion (Higuchi, S., et al., 2003) or create an emotional state that makes it difficult to fall asleep, but this is unrelated to their use in school, hours before bedtime. Screens do not generate addiction by themselves, as evidenced by the fact that no one becomes addicted to a word processor or spreadsheet despite spending all day on them. What can generate dependency are those applications that “continuously promise new information that the brain deems relevant, such as social networks,” as Ruiz Martín (2023) explains, citing Bhanji & Delgado (2014) and Kilford et al. (2016), “in any case, we should avoid talking about addiction, except in exceptional cases, since addictions are very serious behavioral disorders that go beyond simple excessive or dysfunctional use, and using the term in this way only contributes to its trivialization.” Regarding the possible harmful effect of screens on language development, there is also no conclusive evidence. Madigan et al. (2020) warn that the absence of linguistic and social stimuli that young children need to develop language (such as hearing adult conversations, being read to, talking with them, etc.) can negatively affect them. Thus, the authors focus on the risk for young children of not interacting with peers and adults rather than on the use of screens per se. And, in any case, they refer to preschool-aged children.

Therefore, it seems clear that digital technologies entering the classroom are neither inherently good nor bad; it all depends on how they are used. They have the potential to complement teaching and serve as support for proposing practice moments, gathering evidence, addressing diversity, etc. In any case, their presence in the classroom must provide added value that justifies the investment, both economically and in terms of time. As Ruiz Martín (2023) concludes:

There is no doubt that we need to be critical of the use of digital devices in classrooms. First of all, we need to ask in which cases they will add value in achieving educational objectives. This implies being aware that their usefulness and convenience will not be the same at all educational levels, as learning objectives and students’ skills will differ. It also means recognizing that using technology to do exactly the same things that were done without it makes no sense. If the technology used does not bring a differential value that contributes to the teaching and learning processes in a way that would be unthinkable without it, then integrating it into the classroom is hardly justified. (p. 366)

Examples of the Seventh Principle at Innovamat

The Digital Practice App

At Innovamat, we are very clear that for technology to enter the classroom, it must provide a differential value. That is why we have invested considerable efforts, both didactic and technical, in developing our digital practice app. This involves connecting students to an individual device for a few minutes a week (from 15 in early childhood to 60 in secondary education). During this time, students have a personalized itinerary that depends on their responses and progress up to that point, with real-time corrections and individual reports for the teacher. Each digital activity aims to achieve a didactic objective: the statement, graphic representations, animations, step-by-step solutions, numerical range, etc., all point towards achieving this objective. At the same time, the system offers specific aids for each activity, as seen in the examples of the fourth principle, where we discussed practice. The app also controls the time variable, allowing for response time limits or measuring how long a student spends on a particular section and acting accordingly. Moreover, the motivational effect it has on most students enhances their concentration. This has evident consequences on the time they spend practicing and, at the same time, allows us, taking advantage of the fact that the rest of the students are focused, to sit with those who have shown more difficulties throughout the week to help them individually.

The Classroom Manager and Manipulative Environment

When we think about using technology in the classroom, we tend to focus on the student, but what about the teacher? At Innovamat, we have developed a technological tool designed for the daily use of teachers who use our proposal: the classroom manager.

The manager is a web-access platform that gathers all available resources in digital format. Thus, within a course, we find all the sessions with one click, with the digitization of the session’s teaching guide and the slides, videos, and interactive simulators needed to carry it out in front of the children. Additionally, complementary downloadable resources can also be found, such as support or extension resources to address diversity or evaluation tips.

The sessions in the manager also allow the functionality to mark them as “completed.” By doing so, the system recognizes the constructed content and enables the possibility of practicing it in the students’ digital app. This ensures that the applets presented to students for practice never get ahead of what has been built in class. Similarly, the manager allows tracking this practice through the corresponding tabs: in “digital practice,” we find the list of all applets, in case we want to try them or project them for group work; in “my classes,” we can manage the list of students and check their weekly and overall progress reports.

The digital format also allowed us to create the so-called “manipulative environment,” a tool that gathers the digital version of many manipulative materials (but does not aim to replace them). This digital version of base 10 blocks, the number line, constructions with cubes, or the geoboard, among many other materials, allows simulating aspects difficult to visualize on paper (such as views of a 3D figure), substitutes physical materials when there are not enough for everyone, and above all, enables group discussions visible to all students in the classroom, who can practice for when they encounter these models and representations during digital practice.

In other tabs or sections of the manager, one can find training videos (the so-called “capsules”), resources to share with families, mathematical games, planning documents, etc.

In any case, the manager is a living tool that allows us to constantly evolve to respond to the feedback from users, the teachers who use Innovamat in their daily work.

As we saw in the first principle, within the framework of constructivist competency-based learning, the teacher’s role is even more crucial, as it goes beyond the traditional role of simply delivering information. The teacher becomes a guide who facilitates learning and encourages students to explore, share, and discover for themselves while developing skills such as problem-solving and critical thinking. At the same time, the teacher must know when to intervene to set the path and consolidate these discoveries. To understand this role, however, it is necessary first to define what we mean by “inquiry-based learning.”

Inquiry-Based Learning

When we talk about inquiry-based learning, we must be very careful. Some have taken the paradigm to the extreme, expecting children to discover all curricular content freely and autonomously. Let’s be clear: it is unlikely that a child will discover or construct all this knowledge independently, knowledge that humanity has taken thousands of years to consolidate (Kirschner et al., 2006). No one would propose learning how to manage a nuclear power plant through inquiry-based learning, to use an exaggerated example. As Ruiz Martín (2023) explains, citing Geary (2012), it is crucial to differentiate between biologically primary skills (such as walking or speaking the mother tongue), which are learned in a very deductive manner with little guidance, and secondary skills (which include most mathematical skills, literacy, etc.), which require an appropriate context:

From a very young age, we know how to count to 3 (even up to 4) and perform basic arithmetic operations with these numbers (addition and subtraction). We also have some knowledge of geometry. However, learning more advanced mathematics is unlikely without the help of teaching. (…) In fact, we have seen that much of our knowledge is human conventions (such as the decimal number system), so it is almost impossible to share it without teaching. (pp. 52–53)

However, the teaching reality is full of nuances, shades of gray halfway between one extreme (the student discovers everything independently) and the other (the teacher explaining everything directly). Many support guided inquiry or discovery learning (Edelson et al., 1999; Banchi & Bell, 2008). In this paradigm, the teacher sets up learning situations to foster discovery but does not leave students to face them alone. Instead, the teacher intervenes by asking appropriate questions or providing hints, offering examples and counterexamples, validating or refuting observations, consolidating learnings, etc. — in other words, acting as a guide. As Hmelo-Silver et al. (2007) explain in a study responding to the critiques by Kirschner et al. (2006) of inquiry-based learning:

Kirschner and colleagues have indiscriminately lumped together several distinct pedagogical approaches—constructivist, discovery, problem-based, experiential, and inquiry-based—under the category of minimally guided instruction. We argue here that at least some of these approaches, in particular, problem-based learning (PBL) and inquiry learning (IL), are not minimally guided instructional approaches but rather provide extensive scaffolding and guidance to facilitate student learning. (p. 99)

With the teacher’s help, who must accompany and set the appropriate milestones, the student is quite capable of discovering a significant part of the content individually or collectively, leading to more meaningful and profound learning. Banchi and Bell (2008), references in inquiry-based learning, describe four distinct levels:

1. Confirmation Inquiry: Students confirm a known principle (content) through an activity where the results are known in advance. This level is useful for introducing students to inquiry and helping them understand the scientific method.
2. Structured Inquiry: Students investigate a question posed by the teacher through a pre-established procedure. This level allows students to practice designing and conducting investigations.
3. Guided Inquiry: Students investigate a question posed by the teacher through a procedure designed by the students themselves. This level allows students to practice more independent thinking and decision-making.
4. Open Inquiry: Students investigate questions they pose through procedures they also design. This final level is achieved when students are independent in their investigations.

These four levels are approached cyclically: every time a new fundamental concept is to be introduced, activities and lessons are designed so that the student, under the teacher’s guidance, progresses through the inquiry phases until consolidating the learnings. Then new questions are posed, and the cycle starts again. These learnings we seek in students, by opening the doors to a competency-based environment, go beyond content that could be taught transmissively. We refer to the mathematical processes seen in Chapter 3 and more, as Hmelo-Silver et al. (2007) describe, citing Bereiter and Scardamalia (2006):

It requires one to also consider the goals of education—including not only learning content but also learning “softer skills” (Bereiter & Scardamalia, 2006) such as epistemic practices, self-directed learning, and collaboration that are not measured on achievement tests but are important for being lifelong learners and citizens in a knowledge society. (p. 105)

According to the authors, guided inquiry learning facilitates the development of these personal competencies (also known as soft skills). If the teacher can create favorable conditions for inquiry, they will awaken curiosity in students. Curiosity acts as a driver for learning, as satisfying it is rewarding, as Perlovsky et al. (2010) explain: “The acquisition of knowledge is emotionally rewarding. Satisfying curiosity through obtaining knowledge produces pleasure” (p. 3).

The Moment of Mathematical Glory: Emotion and Passion

Anton Aubanell, a professor of Teaching of Mathematics at the University of Barcelona and a reference in mathematics education in Spain, speaks of the “moment of mathematical glory” as a right that should be guaranteed to each student. Even those who participate less deserve a moment of discovery to celebrate with their peers. When the teacher gives the floor, asks questions, decides groupings, repeats a student’s contributions to the class, celebrates successes (and errors), etc., they must always keep in mind that their decisions condition the moments of joy for one student or another, thus shaping the child’s “mathematical identity.” As Gutiérrez (2013) explains:

All mathematics teachers are identity workers, regardless of whether they consider themselves as such or not. They contribute to the identities students con-struct as well as constantly reproduce—what mathematics is and how people might relate to it (or not). (p. 11)

It is essential to find the balance and foster these moments that, for many, will become a beautiful mathematical memory of their time in the classroom. This does not necessarily mean awakening a vocation in all students. Instead, it means ensuring that mathematics does not become an insurmountable obstacle for anyone, regardless of their gender, race, or social status, so that no one stops studying what they like “because of mathematics.” Cross (2004) cites this quote from Professor Wheeler, which summarizes it well:

I do not expect, nor do I want, all students to find mathematics captivating or a discipline to which they want to dedicate themselves. Only a few will find mathematics seductive enough to maintain a long-term commitment. What I would like is for all students to experience at some moments the power and excitement of mathematics, so that at the end of their formal education, they at least know what it is like and whether it is an activity that fits into their future. (p. 4)

Teacher Knowledge and Continuous Training

To be a good guide, it is not enough to master the subject; one must know the learning path in-depth, knowing all its twists, shortcuts, clearings, and steep slopes, and knowing how to position each student considering their developmental stage. This is why continuous training in the specific didactics of the subjects to be taught is essential (Siegle & McCoach, 2007; Harris & Sass, 2011). Schulman (1986) was a pioneer in theorizing about this; it is what he calls “pedagogical content knowledge” (PCK):

Within the category of pedagogical content knowledge, I include, for the most regularly taught topics in a subject, the most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations—in a word, the ways of representing and formulating the subject that make it comprehensible to others. (…) It also includes an understanding of what makes the learning of specific topics easy or difficult: the conceptions and preconceptions that students of different ages and backgrounds bring with them (…). (p. 9)

According to Schulman (1986) and many authors who have continued to develop his ideas (Brousseau, 1997; Ball et al., 2005; Rowland et al., 2005; Park & Oliver, 2007; Kleickmann et al., 2012; or Depaepe et al., 2013, among others), pedagogical content knowledge is essential because it enables teachers to adapt their teaching strategies to the various ways students learn and process information.

The well-known theoretical framework of the “knowledge quartet” (Rowland et al., 2005) helps us analyze how teachers’ mathematical knowledge affects what happens in the classroom. This knowledge is divided into four main axes:

• Foundation: Includes the knowledge and beliefs acquired during teachers’ academic training.
• Transformation: It is the knowledge in action in the classroom; that is, the ability of teachers to transform their content knowledge into pedagogically effective actions through activities, analogies, examples, demonstrations, etc.
• Connection: This dimension focuses on the coherence between planned activities, ensuring that the content is sequenced logically and appropriately for students’ cognitive development.
• Contingency: Refers to the ability to adapt to unforeseen events in the classroom, in response to students’ ideas, and the skill to adjust strategies and plans as needed.

These four axes provide a solid structure for reflecting on teaching practices and developing continuous training strategies that address both mathematical knowledge and didactic and pedagogical skills.

Rowland himself was part of an ICMI (International Commission on Mathematical Instruction) study authored by Liljedahl et al. (2009). This study delves into the three types of knowledge that a mathematics teacher must possess, as described by Schulman (1987):

• Content Knowledge (SMK, from English Subject Matter Knowledge): This includes a deep understanding of mathematical concepts, operations, and structures. For example, in a 5th-grade context, this implies a detailed understanding of the concept of a fraction, its equivalence, the addition and subtraction of fractions, etc., as well as the ability to explain why the procedures work as they do.
• Pedagogical Knowledge (PK, from English Pedagogical Knowledge): This refers to general teaching and classroom management techniques. A teacher should know how to structure and conduct sessions to keep students engaged.
• Pedagogical Content Knowledge (PCK, from English Pedagogical Content Knowledge): This involves the ability to present and represent mathematical concepts in ways that are understandable and digestible for students. Returning to the 5th-grade example, this might include using manipulatives to teach fractions, employing appropriate metaphors and analogies, and implementing activities that help students build a deep understanding.

Furthermore, the authors also emphasize the importance of teachers’ beliefs or preconceived ideas about teaching and learning mathematics, which often influence their classroom practices and can also be addressed through training. Ball et al. (2008), in similar terms, describe the dimensions of “Mathematical Knowledge for Teaching” (MKT), proposing a theoretical framework for analyzing the skills of mathematics teachers that is now quite widespread among the research community:

Figure 4. Domains of MKT. Source: Ball et al. (2008).

In any case, research by Ball et al. (2005), Hill et al. (2008), or Jacob et al. (2017) confirms the evidence: students who have better-trained teachers achieve better results. This evidence highlights the need for continuous training that not only focuses on mathematical content knowledge but also on the best pedagogical and didactic practices to facilitate meaningful learning.

In short, continuous training in mathematics didactics involves not only improving content knowledge but also understanding children’s learning processes, identifying common difficulties, and knowing how to relate content across different educational levels. Mastering these competencies is essential for effective teaching that promotes both the development of mathematical skills and children’s personal competencies.

Examples of the Eighth Principle at Innovamat

Personalized Guidance

Whether the teaching ideas and strategies we propose take teachers out of their comfort zone or not, implementing them requires a certain effort. In many cases, no matter how good they are, resources and training alone are not enough. Specific support is needed that, week after week, is sensitive to difficulties, resolves doubts (whether logistical or didactic), adapts the use of resources to the reality of the school, and, ultimately, does everything possible for an appropriate implementation of Innovamat.

Thus, the figure of the Math Coach is born, a person with good didactic knowledge and an expert in the proposal who supports schools. Each school has the same advisor throughout the year. In this way, we strive to build a relationship of trust, where constant communication allows for open and constructive dialogue. This continuous guidance ensures that each teacher feels supported in their adaptation process, facilitates real-time problem solving, and allows for adjusting strategies to the specific needs of each educational context.

Professional Development in Various Formats

At Innovamat, we believe continuous training is crucial. Learning new strategies, sharing experiences among teachers, listening to or talking with didactics experts, etc., are vital attitudes for professional development. Since we do not intend or seek in any way for teachers to apply our resources without thinking, it is essential to provide them with tools and criteria so they can make them their own and adapt them to their classroom reality.

This training intent begins from the moment we write the teaching guides, filled with justifications that allow understanding the didactic reason for what we propose at any given moment. For example, in the first session of the 1st-grade Lab, right from the start, we explain the reason behind the proposal to accompany acoustic counting with the body:

Beyond these small comments in the guides, from primary school onward, the proposal includes various capsules in written and video format. In these capsules, one of our experts explains in detail the didactics behind a specific content:

Additionally, each year we offer various occasions for in-person (when circumstances allow) or virtual training. Sometimes, these are large-scale training sessions, such as the “Initial Training Session,” the “ConfiMaths” during the pandemic, or the more recent “ForMaths,” which also serve to network and foster synergies among schools in a territory. Other times, they are specific training sessions at the school to meet the concrete needs of the staff at a given time. In any case, the goal is always to empower teachers in mathematics didactics, beyond our particular proposal.

One of the keys to the success of any project is for all the stakeholders involved to work towards the same goal. In a school, the ultimate project is the development of the students, and the stakeholders that make up the educational community are teachers and administrators, the students themselves, and their families. When the goal is to initiate a shift in perspective, as argued in these principles, likely encountering resistance and inertia largely rooted in tradition, cohesion among stakeholders becomes even more crucial. As we will see below, cohesion must begin within the school, with a teaching staff and a management team aligned in terms of educational project, priorities, and methodologies. Then, it should be extended and shared with families, who often want to be involved but, in many cases, lack the resources (time, academic training, money) to do so in a way that benefits the student most.

The So-called “Instructional Program Coherence”

As mentioned, it is essential for the school to have a coherent educational project across grades and stages; what is popularly known as having a “school line.” Newmann et al. (2001), pioneers in addressing this issue transversally (based on preliminary ideas from Hill and Celio, 1998; and Sergiovanni, 1994, among others), refer to it as “instructional program coherence.” This means “as a set of interrelated programs for students and staff that are guided by a common framework for curriculum, instruction, assessment, and learning climate and that are pursued over a sustained period” (p. 297). The authors explain that school projects incorporating this coherence as an element to consider “are more likely to advance student achievement than multiple, unrelated efforts”, no matter how innovative they may be. According to the authors, strong coherence is evident when three major conditions are met within the school:

1. A common instructional framework guides curriculum, teaching, assessment, and learning climate. The framework combines specific expectations for student learning, with specific strategies and materials to guide teaching and assessment. […] A common instructional framework would mean the following:
2. Curriculum, instructional strategies, and assessments of students are coordinated among teachers within a grade level.
3. Curriculum and assessments of students proceed logically from one grade level to the next and offer a progression of increasingly complex subject matter rather than repeating rudimentary material previously taught.
4. Key student support programs, such as tutoring, remedial instruction, parent education, and opportunities for parent involvement focus consistently on the school’s instructional framework.
5. Staff working conditions support implementation of the framework:
6. Administrators and teachers expect one another to implement the framework.
7. Criteria for recruiting and hiring teachers emphasize commitment to and competence in executing the framework.
8. Teachers are evaluated and held accountable largely on the basis of how effectively they use the common instructional framework.
9. Professional development opportunities for staff are focused on the common instructional framework, and professional development on complex topics is pursued over a sustained period.
10. The school allocates resources such as funding, materials, time, and staff assignments to advance the school’s common instructional framework and to avoid diffuse, scattered improvement efforts, with the following results:
11. Curriculum and student assessments remain stable over time.
12. Teachers’ professional assignments are stable enough that teachers have sustained opportunities to learn how to teach well in their specific roles. (pp. 298–299)

Suppose a school manages to meet the three conditions and thus achieves the desired instructional program coherence. Why should this contribute to student learning? On the one hand, according to Newmann et al. (2001), it would help teachers work more effectively, and on the other, it would directly increase student engagement.

Achieving strong coherence, however, is difficult. A context of instability with high teacher turnover, differences among resource providers who rarely coordinate with each other (rather, they compete), diverse interests at play, etc., are just some of the factors that foster incoherence. Therefore, as the sources of incoherence come from both inside and outside the school, strengthening the instructional program coherence requires a simultaneous effort from all stakeholders.

Family Participation

Education is a compulsory stage for children and young people up to a certain age. This implies that, in some way, it is also a compulsory stage for their families. Some families want and can get involved, others cannot, or are not interested; some children tell their parents what they do in class, others rarely share; some schools are transparent with families, while others prefer to keep them as uninvolved as possible; etc. In a context with so many different realities, it is complex to determine the right measure that favors the student’s progress.

In fact, as Boonk et al. (2018) explain, several studies question the generalized assumption that the more family involvement, the greater the academic achievement, because not all forms of involvement have a positive influence. Moreover, this correlation is not the same for all ethnic or racial groups. In the cited study, the most consistent relationship between academic achievement and family involvement is observed when parents or relatives have high expectations for the student, rather than when they intervene directly. Daniel et al. (2016), in another study focusing on younger children, explain that parental involvement in school in early stages predicts success in children’s reading ability but does not seem to predict success in numeracy. The meta-analysis by Castro et al. (2015) analyzes nearly 40 studies on the subject, including early childhood, primary, and secondary education. Their conclusion is that, indeed, family involvement can have a positive, albeit moderate, impact on students’ academic achievement. Again, the authors state that the strongest relationship between involvement and achievement is observed when family members have high academic expectations for their children, maintain fluent communication with them about school activities, and promote the development of reading habits. According to the authors, however, “other family behaviors such as supervising homework or attending certain school activities do not seem to be related to children’s academic achievement.” Hill and Tyson (2009) are also skeptical about the importance of helping children with homework: “assistance with homework and its supervision or review is the only type of involvement that is not consistently related to academic achievement.” Wilder (2013) reaches similar conclusions:

The results indicated that the relationship between parental involvement and academic achievement was positive, regardless of a definition of parental involvement or measure of achievement. Furthermore, the findings revealed that this relationship was strongest if parental involvement was defined as parental expectations for academic achievement of their children. However, the impact of parental involvement on student academic achievement was weakest if parental involvement was defined as homework assistance. (p. 377)

Similarly, the NCTM Principles and Standards (2000) argue that the role of families is crucial in setting expectations for both children in general and learning mathematics in particular:

When parents understand and support the schools’ mathematics program, they can be invaluable in convincing their daughters and sons of the need to learn mathematics and to take schooling seriously. Families become advocates for education standards when they understand the importance of a high-quality mathematics education for their children. (p. 378)

In any case, one of the keys to this family involvement in the school is communication with the teachers. Kraft and Rogers (2015) focus on this communication and are surprised by how families generally perceive that there is not enough communication. They even propose an intervention that invited teachers to send a weekly one-sentence message to each family. Messages that emphasized areas for improvement had a greater impact. According to the authors, this reduced the percentage of students failing summer courses from 16% to 9%.

When analyzing the various ways families can get involved in their children’s academic reality, a distinction arises between involvement at home (day-to-day conversations, homework monitoring, etc.) and involvement at the school (meetings with teachers, participation in workshops or family days, etc.). Reynolds (1992) argues that getting involved at school is better than doing so at home. In this line, there is an entire branch of research that explores how this family participation in school should be integrated, especially concerning mathematics education, by leveraging what naturally happens at home, breaking down barriers. Andrade et al. (2001), for example, explore how to “understand the mathematical potential of households and ‘mathematize’ domestic practices.” According to the authors, “while other areas of knowledge (language, social studies, etc.) can directly integrate household knowledge, mathematical knowledge tends not to be so easily incorporated.” Civil and Bernier (2006), specializing in integrating the Latino community in U.S. schools, propose a framework that starts with the so-called funds of knowledge (the set of practical and cultural knowledge that communities have, resulting from their life experiences) and translates into workshops with groups of parents at the school:

We build on the concept of dialogic learning and on the characterization of parents as intellectual resources and present a model for parental involvement in mathematics in which parents engage as (a) parents, (b) learners, (c) facilitators, and (d) leaders. In particular, in this article, we focus on the third component—parents as facilitators of mathematics workshops for the community at large—centering on some of the challenges as parents and teachers engage in this type of collaboration. (p. 115)

In this way, in line with the cited research sources, the school can convey its vision of mathematics education to families while valuing these funds of knowledge and thus improving their expectations.

The Role of Homework

Research on the effects of homework on student development is inconclusive because many variables influence it: the grade level, the subject, the difficulty, family help or supervision, etc. Fan et al. (2017), in a meta-analysis encompassing numerous studies over 30 years, identify up to five variables: the weekly time dedicated to homework, frequency, completion rate (the percentage of homework done), the effort required to do it, and finally, its evaluation. The authors explain, for example, how Cooper et al. (1998) found that the completion rate is more important for academic achievement than the time spent on it. In line with many of the studies they cite (Cooper et al., 1998; Flunger et al., 2015; Trautwein, 2007; Xu, 2011, 2012, among others), Fan et al. (2017) conclude that the completion rate of homework, how it is evaluated, and the effort required are more influential indicators of its effectiveness than frequency or time spent on it. In any case, they explain that there is a globally positive but small relationship (less than what many teachers and families likely think) between homework and academic achievement in mathematics or science. From all this, it seems more effective to assign little homework that allows practicing what has already been learned in class to consolidate it and spend time evaluating it formatively rather than demanding it too frequently and requiring too much time.

In fact, there are also studies describing the “mathematics trauma” that homework can generate (Lange and Meaney, 2011):

If procedural mathematics is to be done as homework, there is potential for students to experience emotional trauma, especially if they have difficulties with mathematics. This trauma arises when students are unable to do the calculations that they need to practise and is compounded when the support that the parents provide is in conflict with what the child has learnt at school or is beyond what the child is capable of following. There is also potential for the children to suffer mathematical trauma when school beliefs about mathematics learning are limited to practising multiplication tables or doing calculations. (…) By placing parents in the role of the teachers, their relationships with their children can be affected. (pp. 48–49)

And others that explain how homework can be a generator of inequalities (Rønning, 2011):

The test score gap is larger in classes where everybody gets homework than in classes where nobody gets homework. More precisely pupils belonging to the upper part of the socioeconomic scale perform better when homework is given, whereas pupils from the lowest part are unaffected. At the same time more disadvantaged children get less help from their parents with their homework. Homework can therefore amplify existing inequalities through complementarities with home inputs. (p. 55)

Other authors, such as Maltese et al. (2012), focus on analyzing homework as a source of practice to better face exams and other standardized tests in secondary school. According to the authors, “there is no consistent significant relationship between the time spent on homework and grades, but there is a consistent positive significant relationship between homework and performance on standardized tests.”

Šilinskas and Kikas (2017) analyze how family involvement in homework affects academic performance and children’s motivation. Their results show how low self-esteem in mathematics predicts an increase in parental control, which is related to lower performance, worse persistence, and even worse self-esteem: “boys’ motivation to do homework (i.e., persistence during homework and self-esteem) decreases if they perceive their parents as controlling their efforts.” From sixth grade or first year of secondary school, where the amount of homework traditionally increases, the authors provide some advice:

Based on the results of the present study, parents should be advised to pay a special attention to how their children perceive and interpret parental help with homework. This is crucially important because it is neither parental intentions nor the frequency of actual involvement, but rather children’s perceptions of such involvement, that matter for the subsequent development of math performance, homework task persistence, and math self-concept. Another educational implication that follows the results of the present study suggests that quality of parental involvement with homework matters. The growing body of literature suggests that the quality of parental homework assistance can be improved by parent training programs. Finally, strong school-family partnerships and cooperation is an important factor improving the quality of parental involvement in children’s homework. This way, through cooperating with families, teachers can guide parents on how to assist their children with homework so that it does not trigger negativity in homework situations. (p. 17)

Other authors are much more critical of homework. For example, Liljedahl (2020), in his best-seller Building Thinking Classrooms, explains that:

Homework, in its current institutionalized normative form as daily iterative practice to be done at home, doesn’t work. Almost every teacher I have interviewed says the same thing—the students who need to do their homework don’t, and the ones who do their homework are the ones who don’t really need to do it. It is a broken construct that long ago lost the good intention under which it was conceived. (p. 119)

According to the author, the underlying problem is that the purpose of homework is not the same for everyone:

If I ask a teacher what homework is for, I almost always get the same answer—homework is a chance for students to test their understanding, to learn from their mistakes, and to find what they need more help with. These are lofty objectives. When I ask students what homework is for, they tell me that it is for marks. When I ask them who homework is for, they tell me it is for the teacher. And when I ask them why their teacher gives them homework, they tell me it is for practice. (p. 119)

As an alternative to traditional homework, Liljedahl (2020) proposes a framework to unify criteria with students and do this consolidation or checking of learning through so-called “check-your-understanding questions,” which serve for self-assessment and which students themselves can choose whether or not to do. To promote autonomy, the author recommends that teachers neither review nor evaluate these questions but instead provide the answer immediately and the written solution step-by-step the next day.

Similarly, Vatterott (2010) focuses on analyzing the characteristics that a task must have to be considered “good homework.” First, the task must have a clear academic purpose, such as practice, verifying understanding, or applying knowledge or skills. Second, the task must allow the student to effectively show their learning. Third, the task must enable the student to make it their own, offering various paths and being personally relevant. Fourth, the task must instill a sense of competence: the student must be able to complete it successfully without help. Finally, especially with younger children, the task must be aesthetically pleasing; it should look fun and interesting. According to the author, when a task meets these five aspects, it has all the ingredients to be effective as homework if they are to be assigned.

Examples of the Ninth Principle at Innovamat

Resources for Families

One of the initiatives that has been well received by both teachers and families with children using Innovamat is the letters or magazines for families, which briefly explain what students will do that year. For example, below is a fragment of the 3rd-grade letter for the 23/24 school year:

In preschool, where families tend to be more present, the “traveling folder” allows teachers who wish to communicate with them. As the name suggests, the folder can travel from school to home so that children can occasionally show and explain some work or blank page to their families.

We also often offer talks (online or even in-person) aimed at explaining ourselves better to families, who always appreciate it:

Beyond this, our website also includes resources for families interested in mathematics education. The Innovamat blog, filled with articles addressing both teachers and families, the newsletter, and the 2-minute strategy videos, where we explain the keys behind the main arithmetic strategies, are just a few examples.

Although Innovamat does not actively encourage homework, we understand that assigning it is a decision that rests with the school. Therefore, several of our resources (such as the support versions of the Laboratory notebook in primary school or the XYZ practice in secondary school) can be used as tasks to do at home, as long as they serve to practice knowledge already built in class.

Finally, we have also launched initiatives to reach the general public of families with children, regardless of whether they use Innovamat at school or not. One example is the open version of our digital app, called “bmath,” which can be found in major mobile app stores. Another example that has also been well received is the “Enigmath” booklets for primary school, filled with challenges that engage mathematical ingenuity, with questions and reflections for family discussion.

Presence in Contests

A good way to foster a love for mathematics in the community is to participate in local competitions, such as the Kangaroo tests or mathematical photography contests, to name just a couple of examples. Within the proposal, we find several sessions inspired by these types of contests, which serve to introduce them and encourage participation.

The tenth and final principle is the natural culmination of the reflective and rigorous efforts proposed in the previous principles. It has become evident the importance of thinking and designing didactic resources based on research evidence, theory, and the reality of the classroom. These resources should reflect a deep understanding of mathematical concepts and strategies proven effective in educational practice, be coherent, and be sensitive, above all, to the reality of the teachers who implement them in the classroom. As Bruner (1960) would say when talking about curriculum design (understanding “curriculum” as a proposal or program, not as educational law):

A curriculum is more for the teachers than for the students. If it cannot change, move, disturb, inform the teachers, it will have no effect on those they teach. It must be, first and foremost, a curriculum for the teachers. If it has any effect on the students, it will be by virtue of having an effect on the teachers. (p. xv)

Foster et al. (2021), in the same line on curriculum design, talk about the difficulty of achieving this coherence:

An effective curriculum cannot simply be a consensus curriculum, where experts come together and each contribute their own preferred tasks and approaches. This would be almost guaranteed not to achieve a coherent whole. Instead, difficult issues and choices need to be surfaced, considered and resolved; some good lesson resources will find no place, because, however strong, they do not fit coherently within an overall agreed trajectory. Ideas and approaches that might be effective for one standalone lesson, but which fit less well into an overall progression, will need to be rejected. So, hard decisions need to be made about directions to take. (p. 627)

They propose up to five guidelines to ensure the didactic quality of mathematics resources:

1. Harness and develop the skills and expertise of teachers.
2. Balance the teaching of fluency, reasoning and problem solving.
3. Give explicit attention to important errors and misconceptions.
4. Compare and contrast alternative methods.
5. Engineer coherence through strategic use of consistent representations and contexts. (pp. 628–653)

All five pieces of advice harmonize with the principles explored throughout the text. When designing resources, having all stakeholders follow the same principles facilitates coherence and cohesion. By methodically integrating the conclusions derived from each principle—from the application of socio-constructivist techniques and the development of competencies to the conscious use of technology—and then testing and validating the resources in the classroom, we aim to promote meaningful and motivating mathematical learning.

In this continuous innovation process, it is essential to recognize that educational science is constantly evolving. As Swan and Burkhardt (2014) say, “high-quality coherent curricular resources must be developed through cycles of iterative testing and refinements, following detailed classroom feedback.” Researchers and teachers work together to discover increasingly efficient techniques, adapting to the emerging changes and needs of education. This dynamic nature of knowledge requires our didactic strategies to be continuously reviewed and updated. Therefore, reflections on how to improve mathematics teaching never end; there are always new perspectives to explore and new approaches to experiment with. We must remain attentive to developments in both the scientific and educational communities to ensure our approaches remain relevant and effective.

Examples of the Tenth Principle at Innovamat

Feedback for Continuous Improvement

Within Innovamat’s Learning Team, many active teachers test the activities in their classrooms. This allows us to evaluate how they work, if they can be enriched, or if we can explain them better in the guides, among other aspects.

Moreover, the feedback we recurrently receive from the thousands of teachers using the proposal allows us to iterate it to make it increasingly useful, without losing rigor or richness. Beyond quarterly surveys and direct consultation, a good example of how we collect feedback from teachers is the “rate the session” feature in the manager:

Research at the Service of Learning: Vertex Schools

In Innovamat’s didactic team, we have always thought, designed, and written our activity proposals based on a powerful combination: the sum of years of experience in research and university teaching with decades of classroom experience. In other words: we start from what research tells us and combine it with what we have tried for years in school. We read, investigate, discuss, teach, test, and, ultimately, learn together in a privileged ecosystem. This is where our educational resources come from, and that is why we often say that Innovamat is a project of teachers for teachers, based on research.

But that is not enough. We have a responsibility that must go further: we must take advantage of knowing and working together with the thousands of teachers who are bringing our ideas to the classroom. In fact, we must leverage this force to close the loop: the implementation of our resources in the classroom must contribute to the research from which we started to create these same resources. That is why we started the Pyramid Group, our research group linked to several universities, which aims to conduct studies that contribute to mathematics didactics in general and measure the impact of Innovamat’s proposal in particular, to improve it.

The “Vertex Schools” project is the cornerstone of the research we do at Pyramid. These schools form a community that, regardless of whether they use Innovamat or not, have decided to be part of a leading group of schools committed to scientific rigor. From there, each center decides which studies to participate in each year. These centers benefit directly from the research we conduct through detailed reports for each study, helping them gain more information about their students. We, of course, also learn: we publish in indexed journals and present at local, national, or international conferences, rethink aspects to iterate resources, etc. Ultimately, it is a symbiosis that also benefits mathematics didactics as a science.

This document is a work created by Innovamat Education. Therefore, in accordance with the Intellectual Property Law (Royal Legislative Decree 1/1996, of April 12, approving the revised text of the Intellectual Property Law, which regularizes, clarifies, and harmonizes the current legal provisions on the matter), no third party may use it, claim moral rights, or exploit any of the proprietary rights of this work except when expressly permitted by Innovamat Education.