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The math classroom is a space full of decisions, interactions, and constant adaptations. In such a dynamic context, many teachers rely on materials that help us plan and manage sessions. Beyond books and worksheets, there is a fundamental and often under-analyzed resource: the teaching guide.
Teaching guides are a key tool for teachers: they not only help organize sessions but also connect curricular design with classroom reality. However, until recently, there was little research that clearly defined what a guide is. And even less that analyzed its quality from a competency-based perspective.
What we do know is that guides have a great impact on teaching practice: they can guide, train, and inspire. But they can also limit if they are too rigid or superficial. That is why it is essential to go further and begin to understand them as living documents with educational value.
This article is inspired by a line of research that seeks precisely that: to help us read guides more deeply, analyze them from a competency-based perspective, and enrich them to make them more powerful.
Let’s go!
What Is (and What Is Not) a Teaching Guide?
A teaching guide is much more than a set of instructions for carrying out a session or a sequence of classes.
From the field of mathematics education, various authors have worked to define what a teaching guide is. Today, we define it as a document that combines a resource (tasks or activities) with a usage scheme (proposals for its classroom management).
In short, it is a document that supports teaching with two main functions. On the one hand, it proposes specific activities to work on mathematical content and processes; on the other, it offers management guidelines, such as suggestions, questions to ask, adaptations, and decisions to make during the session.
Above all, and unlike textbooks, it is a resource that speaks to the teacher, designed to guide their classroom practice. Teaching guides explain the why and how, detailing the objectives of each session, proposing specific tasks, and providing classroom management recommendations.
We can say that a teaching guide is like a movie script. If the activities we experience in the classroom are the reality (changing and alive), the guide would be its representation: a way to anticipate, plan, and make sense of what will happen. But like a movie, it can be more or less faithful to reality; more or less complete; more or less rich.
What Do We Mean by Mathematical Richness?
The idea of richness in mathematical activities is a phrase you’ve probably heard us say often. It appears a lot in training sessions, in conferences, articles. But… what does it actually mean? When we talk about mathematical richness, it’s not just about complexity or difficulty. It is a broad concept that includes both the tasks proposed and how they are managed in the classroom.
“Mathematical richness is a combination of two main areas: the proposed tasks and the teacher’s management.”
These two blocks, when developed together, give rise to rich, meaningful, and competency-based mathematical activities that foster deep learning. Let’s explore how richness appears both in tasks and in management:
Richness in the Task
When a task is rich, it does not just reproduce procedures; rather, it truly invites thinking, exploring, and building deep mathematical knowledge. A rich task promotes mathematical processes: problem-solving, reasoning, connecting ideas, and communicating and representing in various formats.
Moreover, the content it addresses is rigorous, relevant, and connected to students’ reality. But it is also open and expandable, so it allows connections with prior and future learning.
Richness in the Management
Management is the other half of richness. Even if a task is potentially rich, if it is not well managed, it can lose its value. That’s why it is important to take care of the classroom environment and create a safe space where students feel comfortable to think, try, make mistakes, and share.
Of course, good management also creates opportunities for all students. Attending to diversity is key so that everyone can access the activity and, at the same time, feel they can go further. All of this contributes to building a positive identity towards mathematics.
How Can We Identify Mathematical Richness in a Teaching Guide
Identifying whether the guides we use daily are rich is quite a challenge. To this end, a tool has been developed for analysis and improvement, which allows studying the guides and observing to what extent they promote mathematical richness.
To do this, we must first know what is present and what is missing. The tool does not judge or classify guides as “good” or “bad.” But it helps make visible the key dimensions of richness: Does it talk about tasks or management? Does it promote processes? Does it offer guidance to deepen content and cognitive demand? Does it consider attention to diversity?
Moreover, the tool goes a step further and focuses specifically on a fundamental aspect of learning mathematics: connections, as a key dimension of enrichment. To identify the number of connections a session proposes, the tool is based on the Extended Theory of Connections (ETC), which classifies the different types of possible connections.
How Is the Tool Tested?
To test the tool, it was applied to a 4th-grade primary teaching guide by Innovamat, with the aim of seeing to what extent it supported deep mathematical learning. The analysis was done on two levels:
- First, a qualitative analysis was carried out. Through triangulation, the guide was divided into different parts (specific activities, modelings, resource proposals, etc.) and these were studied to assess whether they promoted richness and what types of connections they fostered.
- In parallel, a quantitative analysis was conducted, consisting of counting and classifying the parts according to the type of richness and detected connections. This provided a global view of the guide as a whole, identified trends, and detected possible improvements.
With all this information in hand, concrete improvement proposals were made to enrich the activities that showed shortcomings. For example, adding open-ended questions to promote reflection, changing the context of an exercise to make it more meaningful, or introducing new representations (such as a number line or a graph) to help establish connections between concepts.
The Result of a PhD
This research is the result of years of work by our reference and now PhD in mathematics education, Albert Vilalta. The future lines opened up by this research are as diverse as the classrooms we aim to transform.
Last year, Albert already presented part of it at ME47 in Auckland, an event focused on psychology for mathematics education. But just recently he made the definitive step: defending his thesis before the tribunal at the UAB and obtaining the title of doctor. A more than well-deserved recognition for such rigorous work. From the Innovamat didactic and research team, we were able to accompany Albert in presenting his thesis. What a moment! Congratulations!
At Innovamat, we remain convinced that research must go hand in hand with educational practice. We are committed to continuing to explore new paths, always in the service of education and the learning of mathematics.
