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How can we ensure our students see mathematics as something more than just a list of rules to memorize? This was the question that led us to talk with Howie Hua, a leading figure in math education in the United States, a teacher educator in California, and a well-known social media influencer.
Alongside Albert Vilalta, a member of the Learning team, we had the chance to speak with Howie during the training series on interpreting the California Math Framework. In this article, we bring you rich activities and strategies for kindergarten, elementary, and secondary levels to develop number sense. Let’s go!
About Howie Hua: “Everyone is a math person”
Howie Hua is a math instructor at California State University, Fresno, where he trains future teachers. His motto is clear: “Everyone is a math person.”
After being named Outstanding Lecturer in 2019 and receiving the 2024 Provost Award for Outstanding Lecturer, Howie has gone beyond the classroom. On social media, he has become a reference point that democratizes access to math education and proves that mathematics is a living and fascinating language to describe patterns, connect ideas, and solve problems creatively.
Warm-up Activity: How do you count what you see?
Before diving into the 5 main activities, Howie proposed a challenge to warm up: count the squares in a figure with a hole in the center.
The key question wasn’t “how many squares are there?” but “how did you count them?” And the responses were diverse:
Some saw four 3×5 rectangles
Others counted concentric rings
And one participant suggested mentally moving the corners to the center, creating a perfect 5×5 square
This activity perfectly illustrates the concept of composing and decomposing numbers, visualizing groups, and using properties to facilitate calculation.
The 5 Rich Activities to Develop Number Sense
1. Subitizing and “Dot Talks”
Subitizing and dot talks help children develop the ability to perceive quantities without counting each individual element. This skill begins to develop from age 3 and is progressively consolidated during Kindergarten and the first years of Elementary school (up to 2nd grade).
Through the observation of dots, images, and small quantities, children learn to recognize how many elements there are at a single glance, to explain how they see it, and to build a deeper understanding of number in a visual, natural, and meaningful way.
Howie Hua exemplified this for us with activities where an image with repeated and grouped elements is shown (like the holes in a manhole cover) and a few seconds are given to provide an answer. The goal is to find efficient strategies without having to count the holes one by one.
From there, the question is launched: “How many do you see? How did you see it?”, inviting us to explain our strategies and highlighting different ways of looking: “I see 3 columns of 4 and 2 columns of 2…” or “I see a 4×4 square.”
This type of activity helps students understand that numbers can be decomposed and represented in many ways. Each answer anticipates algebraic thinking. These routines, based on rich questions and classroom conversation, favor the development of flexibility, an essential element of fluency.
“It is imperative that students implement a standard method only after they have fully developed an understanding of the operation, can connect previous strategies and representations to the steps of the algorithm, and make sense of this abstract process.” California Math Framework, page 15.
2. Managing the Conversation in the Use of Numeracy Strategies: The Number Line and “Constant Difference”
Starting in the second cycle of elementary school, many students get stuck with the vertical subtraction algorithm, especially when “borrowing” or “regrouping” appears.
That’s why it’s important to have different strategies for solving basic operations, as it helps develop judgment and flexibility.
A very powerful way to do this is to think with the number line and talk about jumping strategies, noting that jumps can be organized in different ways without losing the relationship between the numbers…
For example, to solve 41-18, Howie invited us to consider the different properties and strategies of jumps:
The first way is friendly jumps, the jumps that are easiest for us to make, such as jumping by 10s.
The second strategy is called friendly checkpoints; we look at which number is easiest to “land on” after the jump. In the example, 20 is a good number to land on and jump from there; therefore, we make a jump of 2 to get to 20.
The third strategy is called “same distance, same difference” or ‘constant difference,’ in which we see and understand the concept of difference. A good example is using the age analogy: if I am 18 today and my uncle is 41, when I am 20 the age gap will remain the same and my uncle will be 43.
Working on these strategies in conversational dynamics allows for comparing procedures, arguing why they work, and reflecting on which is more efficient in each case. Thus, students don’t just learn to “do subtractions,” but build understanding, develop flexibility, choose with judgment, and consolidate subtraction as a meaningful operation, beyond mechanical steps.
3. Where’s that decimal? Estimation with Decimals
In 6th grade and 7th–8th grade (1st–2nd of ESO), a common problem with decimals is that many students place the decimal point by counting places mechanically, without understanding what the number represents or if the result “makes sense.”
To break this routine, Howie proposed working on estimation before calculating:
He presented different operations with decimals and asked us, first, to make a reasonable approximation to develop estimation skills and number sense.
The idea is to prompt students to think about the meaning of operations and use approximate calculation to develop fluency and judgment.
Approximating leads us to practice productively, as we start from known facts to approximate the answer. For example, in the first operation, our known fact is that 5×6=30, so we can deduce that the decimal point will have to be placed after the 36.
This type of activity builds a sense of magnitude, reinforces number sense, and prevents major conceptual errors because it forces the validation of the result through mathematical logic and not just through a procedure.
4. The strategy for multiplying doesn’t have to change regardless of what we are multiplying – Decomposition Strategy
When working on multiplication, starting from 3rd grade and moving up to 8th grade (2nd of ESO), the strategy shouldn’t change based on the type of numbers we are working with; instead, it should be maintained and the student should gain fluency as we progress.
Howie Hua connected this very well by showing that if students understand 13 × 12 through decomposition and a geometric representation of areas, that same way of thinking can later be transferred to multiplying fractions or even polynomials.
Area visualization allows one to “see” the product as the sum of parts: for example, when decomposing in an area model, regions like x·x = x², 3·x = 3x, 2·x = 2x, and 3·2 = 6 appear, so the total is naturally constructed as x² + 5x + 6.
However, we must keep in mind a warning Howie gave us. Teaching “shortcuts” or tricks without understanding creates massive confusion in the long run and disconnects us from the meaning of the operation; instead, with strategies based on decomposition and geometry, deep connections between numeracy and algebra are built that sustain multiplication in higher grades.
A Final Challenge for Productive Practice
We also had the opportunity to present an activity from our curriculum proposal to Howie to discuss the importance of productive practice within a problem-solving context.
Our idea was to contrast “reproductive” practice (doing, for example, 20 mechanical subtractions like 7−2 or 5−1), which often becomes repetitive and can generate anxiety, with practice where the exact same skill is trained but with purpose and decision-making. The example we shared was the Difference Race:
Two dice are thrown
The difference is calculated
The corresponding racer moves
The question is asked: “Which racer will win?”
The result is that students solve a lot of subtractions mentally while analyzing patterns and playing with probability.
Before finishing… 5 Key Ideas to bring to your classroom
Redefine Fluency
Fluency has three components:
Efficiency (not getting lost in unnecessary steps)
Accuracy (getting to the right answer)
Flexibility (multiple tools for one problem)
Conversation is the Engine
Mathematics should not be a silent experience. Asking “how did you do it?” is vital.
Strategies before Procedures
Encourage decomposition, constant difference, and friendly numbers before jumping to the standard algorithm.
The CRA Approach
Concrete: objects, photos
Representational: drawings, area models
Abstract: formulas, algorithms
Skipping the first two steps generates learning gaps.
Beware of Shortcuts
Proceduralizing without a solid foundation can lead to understanding problems in later grades, such as applying the cross-multiplication algorithm for fractions without understanding what we are doing.
In Conclusion
We especially enjoyed this conversation with Howie Hua because he showed us different strategies and activities to understand and develop number sense, rather than just repeating mindlessly.
We hope these activities inspire you to bring more conversation, more visuals, and more flexible strategies to your classes.
