Teaching with Manipulatives in Math

The research about math manipulatives really makes us reflect on HOW we are using these tools in the classroom. Pull a chair up to our teacher table and let’s chat about this topic together!

Topics Discussed

  • What are math manipulatives?
  • Why are they effective in the middle school math classroom?
  • How to use them effectively (and not use them effectively)
  • Math teacher preparation programs
  • The role of exploration and demonstration

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Brittany 0:00

Ellie, do you enjoy research?

Ellie 0:03

I do. I love researching topics. And I could just get caught for days reading and learning about different topics, especially math related. Do you enjoy research?

Brittany 0:17

I often get hooked by the Google black hole. Yeah.

Welcome to the teaching Toolbox Podcast. I'm Brittany, and I'm here with Ellie.

Ellie 0:30


Brittany 0:32

And in this podcast, we'll share some of the recent research we found about using manipulatives in the math classroom, along with a few ideas for how you can use manipulatives in your middle school math class. Now, there's a lot of research and examples out there. So today, we'll touch on a few different ideas that might help you get thinking and headed in the right direction. Let's get started.

Ellie 1:00

So let's start by actually defining math manipulatives. So we're all on the same page. Math manipulatives are any concrete object that allows students to explore a math concept with a hands on approach. They can be used to model math concepts and to represent and solve mathematical problems. manipulatives can include things like base 10 blocks, fraction tiles, counters, algebra tiles, and even things like beans, money, popsicle sticks, or pictures of objects. They also might be protractors, rulers, Dominos, dice, two sided color counters, pattern blocks, and so much more.

Brittany 1:43

As we've entered the digital age, manipulatives can even be digital tools that students can manipulate. An article by the American Federation of Teachers cites research over the last decade, that indicates that computer based virtual manipulatives usually help students just as much as the physical ones. This is great to keep in mind if you don't have access to a lot of different physical manipulatives. Or if you hesitate to use manipulatives very often because it takes too long to pass them out and collect them. Or if you think your students might play with them, throw them not take them seriously things like that. So in cases like that, using digital manipulatives might be a great alternatives.

Ellie 2:31

Yeah, sounds great.

Brittany 2:32

So Ellie, why are manipulatives effective in the middle school classroom?

Ellie 2:38

Well, middle school students are still developing their abstract thinking skills. And many students might still need more development with some foundational concepts from elementary school, even as they're now studying more complex math concepts like fraction operations with larger numbers or ratios and proportions and different algebraic concepts. So using manipulatives can help make math concepts more accessible to learners. And while they are especially helpful for students who struggle with concepts and those students that need more development, they can also provide opportunities to challenge students who understand the concepts quickly. The site hand to mind.com cites research indicating that over the past four decades, studies done at all different grade levels and in several different countries indicate that math achievement increases when manipulatives are put to good use. With long term use of manipulatives in math, educators have found that students make gains in the following general areas: verbalizing math thinking,

Brittany 3:44

discussing mathematical ideas and concepts.

Ellie 3:47

And frankly, I have to say I love discussion in the math classroom side note. The manipulative use also helps them relate real world situations to mathematical symbolism,

Brittany 3:59

working collaboratively,

Ellie 4:01

thinking divergently to find a variety of ways to solve problems,

Brittany 4:07

expressing those problems and solutions using a variety of mathematical symbols,

Ellie 4:13

making presentations of what they've done with their manipulatives, and

Brittany 4:17

then taking ownership of their learning experiences.

And finally, this article says that the use of manipulatives helps students gain confidence in their abilities to find solutions to mathematical problems using methods that they come up with themselves without relying on directions from the teacher.

Manipulatives also help us as both teachers and students to meet several mathematical practices and principles that show what good mathematicians do. Some manipulatives help you to make sense of problems and persevere in solving them. They help you to reason abstractly and quantitatively. You can model with mathematics And manipulatives and use the appropriate tools strategically while attending to precision like making sure you're reading your ruler or your protractor exactly. And then look for and make use of structure with those manipulatives. An EdSurge article explains that while it was initially thought that students learn math concepts, or even how to use manipulatives, linearly, progressing neatly from understanding a concept to using it to having a procedural set of rules, research over the last three to five years show that it's much more complicated than that. Students have to keep working things out again and again and again, picking up pieces of the concept and fluency as they go. And it varies for every student. So you can't just teach the kids about a concept or how to use the manipulatives one time, we need to go over it again, and again. We need to practice with them repeatedly. Manipulatives need to come out all the time. So once your students have been taught how to use different manipulatives, and they've practiced repeatedly and done all that work over and over, they should be able to be presented with a basket of manipulatives and be able to decipher which of those manipulatives are valuable for a particular problem, which of them are useless, and which might work maybe part of the way. So this involves some great critical thinking skills.

Ellie 6:33

However, when it comes to manipulatives, it's critical that the teachers using them understand how to use them well. Several different articles indicate that while students struggle with math concepts, sometimes they're teachers do to making it difficult for them to incorporate manipulatives in a way that really benefits students. According to an Ed surg.com article, and Associate Professor of Education at the University of Southern California, whose research focuses on teacher education, says we're not preparing teachers well in math, especially at the elementary level, which then causes you know, different issues for kids as they move to middle school in high school. She's found that many teachers have high levels of math anxiety, and she has commonly heard statements like oh, you know, I decided to be an elementary school teacher because I don't want to teach math. And she says, When teachers don't like the subject like this, it hinders student achievement.

Brittany 7:34

Yeah, kids can sense your anxiety, your panic, your fear of the subject.

Ellie 7:40

The article also states that teachers seem to struggle with the conceptual understanding of math themselves due to the broader cultural anxiety about math and inadequate teacher development. And this need for math teachers to have a deeper conceptual knowledge was also shared in the book, knowing and teaching elementary mathematics, in which the author completed research comparing elementary math teachers in China, with teachers in the US. Now interestingly, the elementary math teachers in China typically had less formal education than the United States teachers. In China, they completed ninth grade and then two to three years of normal school, while teachers in the US have 16 to 18 years of formal education. But the teachers in China had a much deeper conceptual understanding of elementary math topics. And her study focused on subtraction with regrouping, multiplication, division by fractions, and the relationship between area and perimeter, which we also study and see struggles with in middle school.

Brittany 8:45

This study was completed in 1999, I believe, so it's interesting to see that the newer articles are still pointing to this much needed knowledge base for US elementary teachers. One thing I thought was very interesting in the book knowing and teaching elementary mathematics is the idea that in some cases, teachers would explain their use of manipulatives. But their use of the manipulatives didn't really lend itself to helping students understand the actual concept. The author states that the direction students go with manipulatives depends largely on the steering of their teacher. So another article on the American Federation of Teachers site says research in the last few decades, has complicated our view of manipulatives. Yes, they often help children understand complex ideas, but their effectiveness depends on the nature of the manipulative and how the teacher encourages its use. When these are not handled in the right way. manipulatives can actually make it harder for children to learn. So this leads us the idea that while using manipulatives well can result in deeper understanding for students, we need to be sure that the way we're using them is correct and beneficial for the students. So let's look at a couple of examples of using manipulatives.


Right. This this research is so interesting just because you think about manipulatives, and pull them out and do something with them, but sometimes we might not be thinking about the fact that if we're not using them very well, we might not be helping students as much as we would hope.

So first, we'll look at an example of not using manipulatives well. In knowing and teaching elementary mathmatics, the author talks about a problem like 23 minus 17, where the concept being taught was regrouping. So some teachers in that study said they might use 23 beans and show taking 17 of them away from 23 or have a picture of 23 items and cross off 17. But using the manipulatives in that way, does not actually develop the idea of subtracting with regrouping it shows some subtracting, but it doesn't show the regrouping. Instead, she says a better way to use manipulatives for a problem like 23 minus 17 would be to have bundles of something like 10 popsicle sticks, bundled together to represent 1 ten and then show that if you unbundle one of the bundles, you would have 1 ten and 13 ones. And then you can subtract the seven ones in 17 from the 13 ones. This use of manipulatives shows decomposing of that number or renaming the number 23 as 10 and 13. Now I know that's an elementary example. And we're thinking middle school here. But if we think about regrouping with subtracting mixed numbers, which we often need to do in middle school and high school, we can apply a similar approach with the manipulatives for a problem like three and a half minus three fourths and try to kind of visualize that since you're not looking at it right now, we could use fraction strips to show three and a half with three whole fraction strips, each divided into fourths, so we're showing the fourths on each of those holes. And then the half can be shown as two fourths, so we can use fraction strips to show those, then students can see that if we take one of the fraction strips and count it as fourths, instead of a whole, we would have to whole and six fourths, and students can see that they can take three fourths from the six fourths and have three fourths left, and then two whole minus zero is two. So it's again, kind of decomposing that three and a half, or renaming that three and a half into two and six fourths.


Another use of manipulatives is to use algebra tiles to help students write algebraic expressions, and solve algebraic expressions. So you might have squares to represent your x's or long bars to represent your numbers, things like that. We can use manipulatives to complete directed explorations. For example, you can use sets of straws that are cut to certain lengths, and ask students if they can create triangles from those straws. And when one set of straws doesn't work, and the other set does, you can then head the class into a discussion about why and develop a more solid understanding of the triangle inequality theorem.


If we're studying something like area and perimeter, and the difference between the two, we can use cubes or blocks of different sizes to create squares or rectangles of different sizes, and then they can explore what is the area in that case? What is the perimeter? How are they different? Once students understand that difference, we could present them with a question like true or false: as the perimeter of a closed figure increases, so does the area and ask them to use their manipulatives to create shapes with different dimensions and find the area and perimeter of each to try to determine whether or not that statement is true. In this case, students are exploring with the manipulatives, but they have a very specific objective in mind.


And then follow up an exploration like that, with a discussion of students results in discoveries, true false or always, sometimes, never questions are far better for developing critical thinking skills than straight problems with numbers that ask you to find an answer. In our Pi Day episode, we talked about using manipulatives like cookies, donuts, pies if you're able to use food, or objects like coins, Frisbee, CDs, to explore the concepts of circumference, diameter and pi by measuring those items and discussing the results. This would be another example of using manipulatives to further explore and do a math investigation.

One of the most important aspects of using manipulatives is the class discussion to reinforce the concepts being explored or reinforced.


All right, well, that does it for today. So we've got research and we've got a few different ideas. We could go on and on with the research and examples of using manipulatives. But it's time for us to go now. If you aren't using manipulatives. In the classroom, think about how you can incorporate them either physically or digitally. If you'd like to hear more about this topic, or here's some more ways or ideas and examples of how to use manipulatives. Please let us know in the comments on Instagram or Facebook.


If you're getting some great ideas from this podcast, please go ahead and leave us a review so other middle school teachers can find us more easily. We hope this episode has provided a new tool for your teaching toolbox. We'll catch you next week for a new episode. Goodbye.


Have a great day.

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