Published: February 5, 2026
I've learned over the years that children don't learn mathematics by being told; they learn by doing, exploring, and making sense of ideas for themselves. When students use manipulatives and models, something wonderful happens: abstract mathematical ideas become concrete experiences they can see, touch, and think about. A kindergartner arranging pattern blocks discovers that two trapezoids make a hexagon. A fourth grader using base-ten blocks suddenly realizes why regrouping works. An eighth grader folding paper to explore fractions has an "aha!" moment about equivalent ratios. All Learners Network believes that these aren't just activities to keep students busy; they're essential mathematical experiences that build understanding from the inside out. Without manipulatives and models, we're asking children to work with symbols and procedures that have no real meaning to them, like asking someone to have a conversation in a language they've never heard spoken. With manipulatives, students construct their own understanding, test their thinking, and develop the kind of deep mathematical reasoning that lasts far beyond the lesson at hand.
But what exactly are models, and why do they matter so much? Models are more than just objects for students to touch and move around. They're tools to think with that give students a way to manipulate mathematical ideas that would otherwise remain frustratingly abstract. A good model helps students understand how mathematics behaves. ALN has identified models that we believe provide long lasting impactful benefits for students throughout their school journey. When a first grader joins three blocks with two blocks and sees that the total is five blocks, she's not just counting objects. She's experiencing a physical model that mirrors the behavior of additive reasoning itself. The blocks become a stand-in for the mathematics, creating a concrete connection to abstract numerical relationships. This is what makes models so powerful: they don't just represent mathematical ideas; they actually embody the same relationships and structures. A fraction bar that can be divided into equal parts doesn't just show fractions. It behaves like fractions behave, allowing students to physically explore equivalence, comparison, and operations in ways that match the underlying mathematics. The model becomes an extension of their thinking, helping them make sense of mathematics in ways that words and symbols alone never could.
When we talk about models in mathematics, we're really talking about three different levels of visual representation, each serving an important purpose. Concrete models are the physical materials students can actually hold: the unifix cubes, base-ten blocks, fraction circles, and algebra tiles that give students tangible, hands-on experience with mathematical relationships. Representational models are the pictures, diagrams, and drawings that students create to stand for those concrete experiences: the circles drawn to represent counters, the rectangles sketched to show base-ten blocks, the number lines that capture the idea of a measuring tool. These representations are a critical bridge, helping students move from the physical world toward more abstract thinking while still maintaining a visual connection to the mathematics. Finally, abstract models are the symbols, numbers, and equations themselves, the mathematical notation that represents ideas in their most efficient, generalized form. The power lies not in choosing one type over the others, but in recognizing how students need to move fluidly between them. A student might solve a multiplication problem with blocks, draw a picture to record their thinking, and then write an equation to capture the relationship symbolically. Each type of model supports the others, and together they create multiple pathways for students to access, explore, and express mathematical understanding.
This is where the real magic happens: when students use models to grapple with genuine problems, they experience productive struggle that leads to those wonderful "aha!" moments we all hope for. Real mathematical understanding doesn't come from practicing procedures before understanding them—it emerges from these moments of conceptual insight, when something suddenly clicks into place and makes sense. I've watched it happen countless times: a student working with fraction strips, trying to figure out why three-fourths is greater than two-thirds, suddenly sees the relationship and lights up with understanding. That insight is more likely to occur when students are in a relaxed state, not anxious about getting the right answer quickly or worried about making mistakes. In classrooms where productive struggle is routine—where students expect to wrestle with problems, use models to explore possibilities, and learn from their attempts—anxiety usually decreases. Students learn that confusion is part of learning, that models are there to help them think, and that understanding is the goal, not speed. When we create these conditions and put powerful models in students' hands to solve meaningful problems, we're not just teaching mathematics. We're helping students become confident mathematical thinkers.
The models identified on the HLC maps and embedded throughout the HLC Progressions provide essential scaffolding that gives all students, especially those who struggle, genuine access to mathematical thinking. Too often, when students have difficulty with mathematics, our instinct is to break things down into smaller steps, to simplify, to focus on procedures and memorization. We think we're helping, but what we're actually doing is taking away the very tools these students need most. Students with learning challenges don't need less mathematically rich experiences; they need more. They need models that give them something to think with, a way to make sense of relationships and ideas that seem impossibly abstract when presented only through symbols and rules. A student who can't remember the steps for multi-digit subtraction might find real understanding through base-ten blocks, watching what actually happens when you decompose a ten into ones. A student who struggles with fraction operations might finally grasp why you need a common denominator by working with fraction strips and seeing that you can't combine pieces that aren't the same size. The model becomes the scaffold that supports their thinking, not a crutch to be taken away, but a legitimate mathematical tool that makes reasoning possible. When we deny certain students access to models and instead drill them on disconnected steps, we're essentially telling them that real mathematical thinking isn't for them. That's not just ineffective; it's fundamentally unfair.
Models help students develop the kind of flexible, generalizable thinking that is at the heart of real mathematical understanding. Without models to ground their learning, students can only solve problems that look exactly like the ones they've practiced. They become dependent on recognizing problem types and applying memorized procedures, and when they encounter something even slightly different, they're stuck. But when students build their understanding through working with models, something fundamentally different happens. They begin to generalize what they know, seeing the underlying mathematical relationships that apply across different situations. A student who has used area models to understand multiplication doesn't just know how to multiply; they understand what multiplication means and can apply that understanding whether they're finding the area of a rectangle, calculating equal groups, or scaling a recipe. The model has given them access to the concept itself, not just a procedure for specific problem types. This kind of understanding leads to flexible thinking, the ability to approach problems in multiple ways and to choose strategies that make sense for the particular situation. Students who have worked extensively with models tackle unfamiliar problems with confidence, drawing on their conceptual understanding to figure out what to do rather than searching their memory for which rule to apply. They make connections between mathematical ideas because the models have helped them see how ideas relate: how multiplication connects to area, how fractions relate to division, how place value underlies all of our work with whole numbers and decimals.
When we think about models and manipulatives in mathematics education, we're really thinking about what it means to truly understand mathematics and how we can help every student get there. Models aren't extras or enrichment activities for when we have time left over. They're essential tools that transform mathematics from a collection of rules to be memorized into a coherent system of ideas to be explored and understood. They are an essential component of our HLC resources. When we give students concrete materials to work with, teach them to create representations of their thinking, and help them connect these experiences to abstract symbols, we're doing more than making math more engaging. We're building the foundation for genuine mathematical reasoning, creating opportunities for insight and understanding, and ensuring that all students—not just those who easily memorize procedures—have access to mathematical ideas. We're helping students develop strategies that emerge from their own problem-solving experiences, fostering the kind of flexible thinking that allows them to tackle unfamiliar problems with confidence, and building connections between mathematical ideas that will serve them throughout their lives. The question we need to ask ourselves isn't whether we can afford the time to use models in our teaching. The real question is: can we afford not to? Our students deserve mathematics instruction that makes sense, and models are how we make that happen.
If you'd like to learn more about ALN models and manipulatives, here are some useful resources: