Concrete models in math can help your students develop a deep understanding of math for years to come. Don’t underestimate the power of concrete models in math. 

In this episode we’re chatting about how concrete models in math can help build your students’ math intuition.

In this episode you will chat about: 

• the meaning of math intuition 
• how manipulatives can encourage math intuition 
• 4 ways to support students with using concrete models in math

Links Mentioned in the Episode:  

🤍Guide to Engaging Math Discussions

Books I love & mentioned often: 

📗Adding it Up https://amzn.to/3FzM4as 

📘Children’s Mathematics Cognitively Guided Instruction  https://amzn.to/3FzLMQU 

📙Principles to Action: Ensuring Mathematical Success for All https://amzn.to/42o816h 

📕5 Practices for Orchestrating Math Discussions https://amzn.to/3zagEEl 

Leave a Review!

If you love listening to Honest Math Chat, please take a few seconds to leave a review so other teachers know this is a valuable place to spend a few minutes each week. 

Want to Read More About Concrete Models in Math?

What role do manipulatives play in your students’ math experience?

I know that many of our curriculums come with a set of manipulatives. Then, if you’re anything like me, we put them into our favorite colorful bins and add a cute label. Display them on a bookshelf and it looks inviting. However, most of the time I didn’t disturb them. If the lesson asked for rulers or the use of pattern blocks I just kinda skipped over that part… thinking about how it would be a management nightmare. And really, how much are they really going to even help my students? 

Somewhere in the back of our minds… we know that’s not accurate and it probably is worth the time, but we just have so many things to cover before the end of the year and we know that manipulatives are going to slow us down. 

What are concrete manipulatives? 

They are math “tools” that help students solve by representing their thinking or manipulating the math. 

Quote: “Intuitive experiences must be acquired by the student through his/her own activities- they cannot be learned through verbal instruction” -Erich Wittman “The Complementary Roles of Intuitive and Reflective Thinking Mathematics Teaching” 

So what this quote is saying is to build intuition with mathematics a student must do the thinking, reasoning, and experiencing themselves. That we can’t just tell them- i do (you watch) we do it (together walking through the steps) now you try it doesn’t work. We can’t take the opportunity away from our students to actually gain intuition.

Does that make you feel uncomfortable? For me it does because that means the entire (and I mean every single math class I took) was teaching me wrong. But it makes sense, because I really started to learn math… intuitively when I started to play around with it IN FRONT OF MY STUDENTS. 

So what is math intuition anyways? It definitely isn’t a gene or a way you were born. Math is not innate or natural to some more than others. But it is “that having a feel for math” that comes from experience and practice.

“Burton 1999 says, ” My intuitions are based on my knowledge and my experiences. The more I have, the more robust my intuitions are likely to be.” 

That means as teachers we can help our students DEVELOP math intuition. 

So how do manipulatives come into play with math intuition? 

Mathematician Reuben Hersh argued that manipulating objects, symbols, and representation and mental images is HOW we develop math intuition. He said, “intuition is the effect in the mind of manipulating concrete objections- at a later stage, of making marks on paper, and still later, manipulative mental images. This experience leaves a trace, an effect, in the mind. That trace of manipulative experience is your representation of the natural numbers. We have intuition because we have mental representations of mathematical objects. We acquire those representations, not mainly by memorizing formulas, but by repeated experiences. (Hersh 1997) 

This experience leaves a trace in the mind. WOW! So that means we have to be thoughtful about the experiences we give our students with manipulatives in order to help them build math intuition. 

How can you support students with concrete models? 

So how can you support your students with concrete models while also not going crazy with managing their behavior and it taking WAY too much time. Let’s talk about 4 key things to do with manipulatives to make them meaningful. 

1. Build a culture of mathematics. One where students feel safe to take risks, make mistakes, and tinker with math. But how does this relate to manipulatives? I believe it is the basis for everything. Our classroom culture is our foundation. So therefore we have to start there. Ensuring every single student FEELS like a mathematician. 
2. Expectations are high. I mean this for all parts of math. Meaning you expect students to use the manipulatives as a TOOL not toy. Meaning tey will use those math tools to solve problems, represent their thinking and explain and justify that thinking to their peers. THe expectations are that we all do math here because we are all mathematicians. And that extends to the manipulatives. 
3. Make it normal to use manipulatives. When I taught 6th grade and we were learning about angles and finding the area of triangles, a manipulative we used often was tracing paper. Now, if I kept the tracing paper on the shelf in the corner of the room… NO ONE would get up to get the tracing paper they needed to solve the problems. A few might ask me to get it for them. You know… fear of embarrassment of walking across the room to get something that might be perceived as whatever it was perceived as… you know 6th graders– draw NO attention to myself kinda vibe?
   When I kept the tracing paper cut into smaller rectangles in small bins on their tables every single day… students used it, helped each other and it became a tool we referred to consistently in our discussions.
   And not to mention it really helped students understand the formula for finding the area of a triangle. 

So create the classroom where every single day there is a choice of manipulatives. Think about creating bins of manipulatives to live right on students’ tables. They can go get them at math time and are welcome to use them at any time during math. When it is every once in a while we use manipulatives that feel different, when it’s normal, routine… students are more likely to incorporate that into the way they “do math” 

4. Highlight the use of manipulatives. Watch and listen to your students as their model their thinking with manipulatives. You’ll see that students OFTEN use manipulatives in ways you weren’t expecting. This is important to note. You should also ask students to describe their thinking to their peers. Giving students ample time for focused, academically productive discussions helps move their math understanding. Remember our initial quote from Erich Wittman…  “Intuitive experiences must be acquired by the student through his/her own activities- they cannot be learned through verbal instruction” You can tell them all you want on HOW to use the manipulative to help them understand… but until they do it, watch the person sitting next to them do it or even see it demonstrated by their crush under the elmo (haha sorry still in middle school teacher life) they won’t get it. Give your students opportunities to experience and share those experiences, then respond to those experiences of others in the math discussion. 

So let’s review… 

Manipulatives are the way we can support our students in building their math intuition through experiences. Manipulating those concrete objects helps students later make connections to visual and symbolic representations. 
First build a strong culture of mathematics in your classroom where students are comfortable taking risks and explaining their mistakes. Then, set high expectations– for problem solving & the use of manipulatives. Also, You’ll want to make this a normal situation in your class– one that uses manipulatives everyday to build that math intuition. Finally, SHOW students how you see manipulatives being used around the classroom. Better yet amplify th invoices of your students by having THEM share how they are using the manipulatives. 

Your Guide to Engaging Math Discussions