Math manipulatives are a BIG DEAL these days. Many math curriculums are built entirely around the idea that in order to learn mathematics best, a child must have within reach a set of objects specifically created for him to hold and handle in order to explore and understand mathematical relationships.
Well, I'm here to say I don't think they're really necessary. I don't think that the amount of money that is charged for them is worth the value that they provide. I think most children can learn math just fine the old-fashioned way — by memorizing facts and procedures.
I do think manipulatives are fun; don't get me wrong. It is fun to make a square of blocks that show a multiplication fact, or to put tangrams together like a puzzle, or to play with pie pieces that represent fractions. I get that. But I don't think they really do the job they claim to do.
The idea behind math manipulatives is that your child will learn to understand the WHY behind mathematical concepts if he can SEE them happening in front of him. Why does 2 x 3 make 6? Because if I set up three rows of 2 cubes each, there are a total of 6 blocks. How many 1/4's are in ½? Just stack these pie pieces on top of one another and you can see! How fun! In one manufacturer's own words, they “help make abstract ideas concrete” and they “make learning math interesting and enjoyable.”¹ This all sounds well and good. But I don't think it works that way.
First of all, in the early elementary years, the child is not capable of understanding abstract concepts, which is basically what math is made of. The brain does not develop the ability to think in the abstract until about middle school age. So in the younger years, trying to get the child to understand the WHY behind a mathematical concept is a futile task. Manipulatives will not magically make that happen.
Secondly, while math may be more fun with manipulatives, hopefully our goal is mastery of traditional math. Just because a child can move cubes around on the table to a prescribed formula does not mean he can do addition or subtraction when encountered on the printed page.
Case in point: EVERY time I tried to teach borrowing with manipulatives, the same thing happened. First I would show the child stacks of “10” bars and “1” cubes, representing the minuend of the subtraction problem. Then I would show how we would need to exchange a “10” bar for 10 cubes and combine them with the cubes already there — then we could do the subtraction. And every time, they were able to show me the procedure with the bars and cubes, exactly as it should be done. The next step, then, was to look at a subtraction problem on paper and use a pencil to complete the problem. But for some reason, the only way they could complete the problem on paper was by memorizing the procedure of crossing out the number in the tens column, writing one less, and adding 10 to the one's column. They did not understand WHY they were doing it, just that the only way to get the correct answer was to do it that way. They could not translate what they saw with the bars and cubes to what they needed to do with with the written numbers. If manipulatives were truly making a difference and giving them that “aha moment,” wouldn't they remember the meaning behind what they'd done? Wouldn't they be able to reproduce it for themselves on paper, explaining it as they went? I think they were just too young to make that kind of relationship happen in their brains.
A better use of the child's time than working with manipulatives ad infinitum would be to memorize addition and multiplication facts, measurements, etc. During the elementary years, this is what the brain is best suited to do. As the child masters those, then processes such as borrowing and long division can also be memorized — even without understanding what they mean abstractly. Frankly, it's often easier to memorize a procedure when you are NOT trying to understand the why.
And after those facts/procedures are memorized, an amazing thing happens. After the child has become so comfortable with all the facts and the procedures that he could do them in his sleep, then, when his brain is ready, that is when the understanding begins to kick in. “Oh, THIS is why we do it that way! I get that now!”
(This is actually the basis of the classical model of teaching. The first stage of classical education is called the grammar stage, where children learn lots of facts about just about everything – it is not until the dialectic stage, in the middle school years, that the child begins using those facts to understand the world around them. I think this model is particularly applicable to the study of mathematics.)
Another claim of manipulative users is that manipulatives “build student confidence.”¹ I think a child will gain more confidence in math from doing the work of memorizing, rather than messing about with manipulatives. It is not until those building blocks are cemented in the brain that higher learning can take place. Math builds on itself. You must know your addition facts to learn mulitplication; you must know your multiplication facts to learn division. To neglect the fact-building in favor of using manipulatives is to sow the seeds for LACK of confidence and LACK of understanding, in my humble opinion.
And who says memorizing can't be fun? At that age, kids LOVE to memorize things, and sing jingles, and time themselves. A lack of manipulatives need not equate to boring and dry math instruction.
Of course there is room for some use of visual aids – but it is not necessary to spend a fortune on manipulatives. Dried beans can be used for counters. Glue the beans on a popsicle stick to make a bar of 10. Graph paper cut into squares can be used to illustrate all sorts of relationships. But don't expect the child to understand the concepts or be able to explain them. That will come later on, when his brain is able to put two and two together and come up with with a sentence.
Yes, math can be a difficult subject for many kids. But I do not believe manipulatives are the answer to that. Let's get back to the basics and stop expecting the unrealistic and unnecessary.
Do you agree or disagree? I don't mind opposing views, as long as they are expressed politely. Feel free to add your two cents in the comments, or come to my Facebook page and let me know what you think!
¹http://www.scholastic.com/parents/resources/article/more-homework-help/math-manipulatives
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Ann has written Cure the Fear of Homeschooling High School: A Step-by-Step Manual for Research and Planning, Save Your Sanity While Homeschooling High School: Practical Principles for a Firm Foundation, and recently Taming the Transcript: The Essential Guide to Creating Your Teen's Homeschool Transcript from Scratch (without overwhelm). She also founded the popular Facebook groups It's Not that Hard to Homeschool High School and It's Not Hard to Homeschool K-8, and in addition she voices the It's Not That Hard to Homeschool High School Podcast.
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I don’t think that manipulatives should replace memorization, but used to supplement it. And real world examples to me are better than special toys or tools. Also, isn’t the goal of common core curriculum for the kids to understand the WHY behind math even in early elementary school? If so, that is interesting that they are expecting developmentally inappropriate achievement.
Hi Sara! Yes, I agree with you that manipulatives can, when used properly, supplement memorization — but unfortunately that is often not how they are used. And even then, I’m not sure that they really do help the child understand the WHY. I love your idea of using real world examples, though! As far as the common core requirements, I’m not convinced they are what is most beneficial to students. That is one of the reasons we homeschool. :-) Great comment!! Thanks for stopping by. :-)
We have never been big into using manipulatives. I used beans some for counting, grouping and adding and subtracting. And, we used fraction circles and a clock for telling time. Those are it. My kids are in 4th and 2nd grade and I never bought the big manipulative kits. I thought I was alone :)
No, you are not alone! Maybe we should form a lobbyist group… ;-)
I am visiting from The Hip Homeschool Hop. I must admit that I have never purchased math manipulatives myself and have never bought into the idea that they were a necessity. From time to time if I have had a child that has struggled with a problem we have used coins or beans for counters but that is the extent of it for us.
Thanks for coming by, Stephanie! I’m so glad you agree with me; I was a little hesitant to write this post, because it’s tough to go against the crowd. It’s so nice to know I have friends, lol. :-)
Interesting. I just found this because I am now homeschooling my children. I DID buy some manipulatives and my thoughts were “Wow. I really wish these would have been around when I was a kid.”
I struggled with math when I was younger. I have an above average IQ and did well in other subjects. I read several grade levels ahead throughout grade school. I excelled in life sciences and got abstract concepts such as atoms and matter, yet math was my weak subject.
I am not sure that they are necessary, either. People have been learning equations well before a plastic base ten set was ever around. However, I find them to be a great way to keep my kids interested while we repeat, together, facts to learn. If they simple add some counters to each other and then hastily write down an answer, it certainly only teaches them to do addition in that manner. However, if we work together all the while repeating “5 plus 4 equals 9”, they are practicing memorizing the equations.
I know the basics of Classical education. I am not sure I agree with it. In fact, I have never met a child over the age of 4 who isn’t able to grasp some abstract concepts. We hear this “fact” stated repeatedly, and perhaps it is true. However, without scientific studies in front of me, I am hesitant to believe it due to my own anecdotal experiences.
Two other things I thought of-
(I know, just let it go, right?? :) ) I think the reason manipulatives fail so miserably in public schools is because there is too few teachers to students. When students are left unattended to “explore math”, they play with them and are not focused on the repetition needed to really ingrain those concepts.
Secondly, on the abstract ideas concept, I find that my Sunday school classes are sometimes better at understanding the trinity ( a very abstract idea) than my adult class is!
Anyway, there’s a different perspective for you. Have a good evening.
Yes, I heartily agree that students will not use manipulatives effectively if not guided to do so. I maintain that even at home, using manipulatives encourages too much reliance on them to explain/explore abstract concepts, and then there is not enough emphasis on memorization of math facts needed to move forward successfully and understand those concepts when it is more natural to do so. Just my two cents! :-)
I think that’s a very reasonable way to use manipulatives — to practice memorization facts. I think not knowing math facts will make math much harder for kids than it needs to be, so you’re preaching to the choir when you talk about that, lol. But can younger kids learn the abstract concepts that manipulatives are usually used to teach? I personally don’t think so. I agree with you that they can understand “some,” but I don’t think manipulatives are going to help them understand better or more. It’s definitely a soapbox of mine, and I don’t expect everyone to agree with me. :-)
While memorization is an important part of being a good problem-solver, there is a reason why teachers have, over the past few decades, come to rely on using manipulatives to connect the abstract, memorized procedures to the concrete.
It’s because people forget. All. The. Time.
In fact, it is a widely recognized problem: we go through years of mandatory math education and still tons of perfectly intelligent adults say that they’re “bad at math.” As a country, we’ve pondered over why we are so behind in math education compared to other countries, and have learned from them that a memorization-only curriculum doesn’t work.
As a former high school teacher, I can attest to that. Simple procedures are gone by the time kids are in 9th grade. Extremely motivated kids struggle because every procedure they thought they had memorized was no longer remembered the way they were taught. They had nothing concrete to tie these procedures to, so they couldn’t even tell that they’d forgotten.
Some of the problems you describe with using manipulatives can result from a child’s incomplete understanding of how procedures tie to concepts. That’s a natural part of learning mathematics, and it is up to the teacher to help them make these connections over time. I wouldn’t give up on using manipulatives altogether.
Your article is an excellent balance to the current trends that are often times developmentally inappropriate. I appreciate your 2 cents :)!
Thanks, Heather!
You say that, “in the early elementary years, the child is not capable of understanding abstract concepts, which is basically what math is made of. The brain does not develop the ability to think in the abstract until about middle school age. So in the younger years, trying to get the child to understand the WHY behind a mathematical concept is a futile task.” This is EXACTLY the reason we use manipulatives–to make the abstract more concrete. For example, to make it possible to see that the abstract concept of regrouping, or borrowing as you call it, is not a magical process but is simply the regrouping of the same number in different ways. So that 152 can be made with 1 hundred, 5 tens, or 2 ones, but it can also be made with 152 ones, or 15 tens and 2 ones. As a veteran elementary teacher, I can tell you that many children are absolutely able to understand this, and manipulatives are a great tool to push their conceptual understanding and number sense.
I think manipulatives can be helpful but shouldn’t be relied upon. I think sometimes they make it harder for the child to understand. Many children ARE able to understand concrete representations of some concepts — but many find them distracting or confusing. Especially when the concepts get more complex and more (you guessed it) abstract.
I was coming here to post exactly the same sentiment. Manipulatives are used to make ideas more concrete for students. We can not stay at the concrete level forever, so we have students transfer their learning to pictorial representations and abstract representation (symbols). Manipulatives are not a panacea for math instruction but a tool. Effective use of tools will assist in transfer learning.
Hi! I’m wondering if you could share the educational research that supports your ideas about children not being able to think abstractly or make abstract connections in math until middle school age, as well as any research supporting the memorization of math facts (versus the learning of math facts).
Thanks!
I am basing my opinion about these things on Piaget’s theory of the four stages of cognitive development, which is explained here: https://www.medicalnewstoday.com/articles/325030.php. This is well-known among educators, and while it is only a theory, my own experience as a math teacher has seemed to justify it to a certain extent. I’ve known kids who were amazing at arithmetic (general math) try to move to Algebra and be absolutely flummoxed. I believe their brains just weren’t ready to process the abstract concepts that Algebra introduces. After a year or two they were much more able to comprehend the coursework. As far as research supporting the memorization of math facts vs. just learning the facts, I haven’t tried to find any. I’m not attempting to write a scholarly work here, just present my opinion based on my own education and experience. Most of the time you can find research to back up whatever position you want. I just know that most kids who don’t know their math facts well struggle more both with computation and with comprehension than kids who do. In my opinion, memorization helps their brain set the facts aside to be recalled when needed, so that it can focus on learning the more challenging concepts that come after. I write more about this here: https://www.notthathardtohomeschool.com/hate-math-multiplication-facts/. Thanks for the question.
I learned something from reading your article. I must admit, I had a strong reaction when you said children can’t learn abstract math concepts until middle school. I absolutely believe they start picking up abstract concepts well before middle school even if they cannot articulate it. I remember learning with those blocks myself. It did more than just reinforce memorization, it gave me a way to build something on my own that I could use to check my answers when learning a procedure. That being said, I learned from your article that they cannot replace facts and procedures. In my own experience as a student they were only used occasionally as a special lesson. The main focus was still procedures. And I understand the timing is important for day to day lessons to avoid distractions. I was also unaware they were expensive. Drawing dots and circles can work just as well for multiplication. Visuals were a major help for me at every age. Building and drawing patterns inspires curiosity. I think physical representations of math problems are essential. But I now understand the majority of lesson time should be dedicated to facts and procedures first.
I have to add for honesty, I only used the blocks once. Most of the time I drew examples with pencil and paper, no special tools! And I still learned multiplication tables. That common core stuff might be over the top. I just want to mention because I really think concrete examples help and particularly concrete examples that the student can recreate on their own.
I am half way with you. I think we have to be careful not to overload children and manipulatives are yet another thing to deal with. My daughter, who struggles with maths, finds the sheer amount of information hard that she is expected to remember. I am also a believer in rote learning in maths. Practice definately makes perfect in the early years. In my experience children have to do a lot of repetitions with manipulatives to handle them confidently, so we have think hard about how to use the limited time available.
On the other hand, my daughter just could not understand why 5 cent could never be counted as 1 cent (it is one coin after all). So I took coins, drew dots on them which corresponded with the coins’ values (5 dots on 5 cents, 1 on cent etc) and went through them with her. I made her find out all the ways to make 1 cent, then 2 cents, then 3 cents etc. We did that at first with the dots visible, so she could count them, and when she was secure in the values, we turned them around so the dots became invisible. After practicing for a while, she got it and used coins intuitively. So I think there are situations where manipulatives can be extremely helpful.
I think coins are a manipulative that any kid can understand better by handling, for sure. Thanks for the comment!
Loved this article! Math has always been easy for me, but I believe it comes so easy for me is because of my memorization of the math facts from elementary years. I’ve felt pressured to use manipulative items with my children and I sometimes do. But I avoid using curriculums that pushes manipulates for almost every lesson. Not that I’m really against using them, but it doesn’t work with my personality. And my kids seem to think the same. They love working on worksheets instead of taking time with manipulatives. My son once told me to “stop using these things (manipulative items) to try to make school fun. Let’s just get to the sheets and save the FUN for my real toys.” Ha!
What I do want to teach my children is two things, besides memorization the math facts. 1) The ten-frame, which can be easily taught on worksheets without manipulatives. I wasn’t ever taught with the 10-frame. However, I believe I was really good at math was because my mind automatically thought with the 10-frame, naturally. I got lucky with that. I assumed that all children thought this way naturally, but I’ve realized that I was wrong. So, I want make sure my sure children have this idea entered into their minds (which again can be taught on worksheets alone). 2) Instead of using manipulatives with numbers to visualize, I was taught the “dots” method in 1st grade. Every number 1-9 had dots attached to them. It was implemented into my brain so deep that I still sometimes use it. 1 in the middle of the number 1. 2 dots attached to number 2. 3 dots attached to number 3, and so on. Those DOTS were memorized per number. I’m honestly surprised zero homeschooling curriculum teach the dot method. A kid may be doing a math problem and have to use their fingers to solve a problem, just as my son was doing for awhile, which took longer. So, I introduced him the DOT method (that I can find very little information on it online), and it’s done wonders. His math has grown faster and less distracted (by not looking away from the sheet and concentrating on his fingers and then being lost where he’s supposed to be on his sheet). So I love the dot method and plan to teach it to each of my children.
So, I guess the dot method is using manipulative in a way, to help visualize the number, withOUT using any items. And I honestly feel that this dot method is better than using manipulatives.
HOWEVER, although using the dot method has helped my son dramatically, it didn’t help him to memorize the math facts. And I know it’s the most important thing for him. So, he has been using a computer program to practice his math facts and it has done the best for him. I do believe memorizing the math facts is the most important thing to ever be good in math.
TBH, with no disrespect to the author or the other commentors, but this is why this is a blog and not an evidence based factual article professionals refer to. Knowing that ALL children learn differently, and ALL teachers are teach differently, it is because of that fact that we have actually factual information that uses proven evidence to justify why manipulatives are needed, especially in math, and why children as young a 1 are capable of developing concrete and abstract thinking. It requires a lot of hard work and dedication to teach those concepts to young children, which is why some might give up and accept that their child/students are not capable, but it can be done, and manipulatives strengthen that. Furthermore, it is thoughts like the authors and commentors that ignores groups such as children with disabilities and low SES that needs that extra help in their learning. If people believe opinions like these, theres a difference between fact and opinion, and facts do not care about “feelings”, then they would have a harder time accepting that Kindergartens in PA are learning how to read, write and add numbers, all of which are concrete/abstract forms of thinking many people in the past believed those ages were “too young to learn.” It is because concepts like those and many more that are introduced to children so late in their lives that their reading, math, and testing scores suffer later on in their lives. The younger you start with the child the better, and that all depends on how the child learns, how the teacher educates, how the parent is involved, and how the information is presented. I am not saying that manipulatives should be used ALL the time, but they are crucial to children learning and there are MANY different forms of manipulatives and how they should be implemented. Regardless of how you perceive it, abstract thinking of any form cannot be developed/build up without seeing how it works first. Think about it this way, people do not come out of the womb knowing manners, they have to be taught them and as to why they are important, some type of manipulative was more than likely used, so why should learning in any domain (math, science, history, ELA, art, STEM, etc.) be any different.
FYI its 2023 not 1987 and education has changed dramatically since then and for GOOD reason. I suggest getting with the times and putting aside personal feelings because it’s not just about you (in general) and your (in general) education.
Older article, and I didn’t take the time to read through every comment on it, so maybe what I’m about to say has already been said, but I’ll give my two cents. For background, I was homeschooled, have a masters, am a former teacher, and am homeschooling my children now. I don’t know everything, no one does, but I also am not inexperienced here.
I’ll agree that most manipulatives are overpriced. You can use toys, legos, beans, or just about anything in your house for that. I’ll also agree that memorization of facts is necessary, but saying that we shouldn’t bother teaching the why is an extreme. Memorization is for speed and ease; the why is for understanding, utility, life long learning, and growth. The why will take a child further than memorization ever will. That is true in any subject, not just math. And they can learn the why. Children love to know how and why. What parent hasn’t been bombarded with question after question by their young child?
First, every child is different. A child who learns best with hands on or visuals will benefit from manipulatives more than a child who is more of an oral/auditory learner (btw, there’s lots of research on teaching through all three methods of learning regardless of preference if you want to look that up). Second, just because you gave your children manipulatives and they didn’t get it doesn’t mean that they don’t work. Any time we teach a child something, not just math, and they don’t get it, we teach it again in a different way. We work with them on it. We don’t say “you just can’t understand this.” Or at least we shouldn’t be doing that to them. Third, math is less abstract than people are led to believe. We as adults do not count, add, multiply, ect. in a void. When is the last time you did a page of math problems just because? Not doing bills, not because you’re an accountant, not because you were helping your child- for no purpose other than to just do it. Unless it is because you just happen to love math to the extreme, I’m willing to bet you haven’t done it since it was required by a class. But do you still use math? Of course you do! We count off our grocery list, we add up costs, balance our accounts, calculate miles, measure our homes to see if that new couch will fit. You get the idea. Sadly, math is not usually taught to children in a practical way. Yet math is tangable, and it is useful. When you ignore that and teach it as an abstract subject that can only be memorized and not understood… well, no wonder so many children hate and struggle with math.