If kids are going to be prepared for the careers of tomorrow, learning mathematics is essential. Math forms a critical foundation for work in high-tech and research careers. Even in our work on Cognitive Daily, Greta and I have found that our math knowledge has been stretched to the absolute limit (I've considered taking a brush-up statistics course to help me with some of the more numbers-heavy papers I read -- and these come primarily from the *social* end of psychology research).

So when kids learn the most basic concepts in mathematics, starting in elementary school and moving beyond, teachers seek out the methods that are most effective. One approach that has met with considerable success is using manipulatives. Students use physical objects such as blocks to represent numerical concepts. Elementary-school kids love the idea of using real objects they can touch and handle. Teachers enjoy changing up the usual chalk-and-talk lessons. But do students learn more?

While the research isn't completely consistent, many studies do suggest that manipulatives improve both long- and short-term retention of math. Here's an abstract that's typical of the research:

Two third-grade classes with 26 students each were selected to participate in the study. [...] A 2-week geometry unit from the Silver Burdett textbook was administered in both classes. The experimental group teacher used mathematics manipulatives to teach the concepts presented in the unit, and the control group teacher used only drawings and diagrams to teach concepts. Analysis of covariance revealed that the experimental group using mathematics manipulatives scored significantly higher in mathematics achievement on the posttest scores than the control group.

Obviously this is a small study, but there's a lot of additional research out there on manipulatives. This wiki entry offers a good basic summary. The strongest effects are in affect: children *like* math more when using manipulatives. But many studies have also found that when manipulatives are correctly used, both short- and long-term retention is improved.

Unfortunately, many classrooms don't have manipulatives because there simply isn't the funding. Some schools only have partial sets, or sets that must be shared among several different classrooms. That's why one of the fundraisers in our Donors Choose challenge will give a teacher a full set of manipulatives for third- through fifth-graders. Ninety percent of the students in this school are in poverty, meaning their income is less than $23,000 per year for a family of four. These are the kids that can benefit the most from a little extra help, and whose families are least able to offer it.

Let's see if we can fully fund this challenge by October 12! If our readers get us close, Greta and I will donate the rest out of our promised $600 donation to make sure this important project gets funded.

One other thing: Seed Magazine will be matching gifts up to $15,000 in total donations, so whatever you donate now will not only be supported by us, but matched by Seed!

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As a former math teacher I agree with you 100%. The problem now is the NCLB. In the elementary grades, when addition and multiplication are being taught, it takes longer for a teacher to teach students to UNDERSTAND why addition or multiplication give the answers they do, and WHY the algorithms work, than it does to just try to get the students to try to memorize HOW to use the algorithms. And they are under pressure to teach (not necessarily to get students to learn) a certain amount of material in each year. I could write pages more on this topic but I will stop here.

I could not agree with you more Karl. Also, I think that affect should never be discounted as a learning and memory tool. We all remember best those moments associated with strong emotions. Money is a strong motivation for learning math, and my sons wanted to read so that they could understand their video game screens. In fact, I think a test of making change at a grocery store and following the directions on a recipe would make a better standardized test than the current NCLB stuff they have now.

As a student in Israel, where the manipulatives have been used for more than ten years, I disagree completely.

In fact, during the period the manipulatives were used, Israel dropped from near the top of the scores in the world kids math test (whatever it's called) to a very low score near the end of the bunch.

In further, extensive (this system was mandatory throughout Israel), research it has been shown to be the least effective method of all those tested.

I'm sorry to sound emotional, but I love math, I was very good at it in high school, and I was a very weak student with those blocks in elementary school.

And I still get emotional about those horrid orange blocks, nearly twenty years later.

By and large the literature on manipulatives in the last decades shows that they provide no *long-term* benefits and may in fact retard development by putting off the inevitable transition into symbolic reasoning.

I'm sorry, but you're way off on this one, and schools don't need more money for yet another educational boondoggle.

I have no first hand experience in K-12 teaching, so take my opinion with a grain of salt. It seems students may enjoy learning some of the elementary topics better and even faster if some real life objects are used, but it may hinder the inevitable transition into abstract concepts that are necessary for higher level math. For each new concept being taught, it is good to use some real life examples that are easy to visualize, but it might be better to switch to symbol based math soon after.

I think it would be more valuable if the studies are done to measure the effect of each method on students' grasp of mathematical topics taught 5-10 years later, instead of just after the immediate few weeks.

It's interesting that Eyal and Cog Scientist have such different opinions about manipulatives from Ann and Karl. Most of the literature I've found about manipulatives is quite positive, but there are some negative results. I'd be interested to see some research that attempts to uncover why there is so much variety in the results. My guess -- and this is just a hunch -- is that when teachers are excited about manipulatives, they can get kids excited, and that's where learning occurs. In a top-down system, where all teachers are required to use manipulatives, it may not be as successful. And certainly there are some cases when the approach is probably not effective.

Cog Scientist, I'd be interested to know where you get the impression that "by and large" the literature doesn't support the effectiveness of manipulatives. Can you cite a literature review or meta-analysis?

Interesting... I was introduced to algebra using a manipulatives set called 'Hands on Equations' in elementary school. I used the same program recently to introduce my little brother to algebra. He had as much fun with it as much as I remembered having. Somehow, the idea of balancing equations seems to come naturally when you use a scale.

This late in this thread this is probably just for Dave. As I said above, I can go on for pages on this topic - so here's a little more. I have a BA in math ed and an MS in math. I taught math for 15 years and for the last five I was a curriculum coordinator for elementary and junior high. My experience is that the problem is the teachers. Just as they do not understand why the algorithms work, (what it MEANS to "carry" in addition, or "borrow" in subtraction, etc.) they do not understand the point of the manipulatives. So they are teaching manipulation and not understanding. All of math needs to be taught from concrete to abstract. And use of symbols and algorithms is abstract. Students need to begin with concrete examples and stay with them until they understand, for example, that when you accumulate ten "one" size rods, you can trade them in for one "ten" size rod. Then, when they are taught the addition algorithm they will understand what "carrying" a "1" really means - rather than just "this is what you do".

There are pages more of this explanation if anyone wants it.

The research-based Number Worlds program ( http://clarku.edu/numberworlds/nw_Overview.htm ) seems to have proven to be very effective in teaching basic math skills. It's basic premise is that children acquire an intuitive number sense (numerical cognition) well before entrance into elementary school. Due primarily to differences in learning experiences before entering school, children come into a math learning environment with varying levels of readiness; especially disadvantaged are those from lower ses environments who are shown to have less of an intuitive number sense. The Number Worlds has identified five basic skills necessary to acquire important numerical cognition skills. These skills are identified as "Number Worlds" and are best described as follows:

"There are actually five different worlds, or "lands," which kindergarten children must explore and connect if they are to develop a good number sense. The first, Object Land, is the world of real objects. The second, Picture Land, is the world in which numbers are depicted in a stylized form, as groups of dots on a page. The third is the world in which numbers appear as numerals along a path; this is Line Land. The fourth is Sky Land -- the world in which numerals appear at the side of a device such as a thermometer or a ruler, and are used to measure some continuous quantity. Such devices are normally oriented vertically, with higher representing "more." Finally, the fifth world is one in which numbers appear in a circular array, as in clocks and dials. This is Circle Land. A more detailed description of each land is provided in Chapter V."

From this definition, it would appear that a strict use of manipulatives would appear to teach an understanding of numbers as they pertain to the Object Land World. A full teaching of number sense would require teaching the remaining 4 worlds, which would seem to connect to a more abstract understanding of numbers.

So I suspect the differences in performance concerning the use of manipulates must have something to do with not teaching all 5 important Number Worlds from concrete to abstract.

Liz Spelke's lab has published some recent work on the short-comings of manipulatives, and from there you can trace back a body of studies showing no benefits or showing decrements in long-term acquisition of abstract/symbolic reasoning.

Most of the studies that indicate some benefit typically report small effect sizes. You don't reform a curriculum by spending a lot more money to produce a small increase in performance, which is the best case scenario with manipulatives.

One of the few exceptions I've seen is Art Glenberg's recent work with manipulatives and language comprehension. His work seems a bit different in my mind because it seems to be highly specified in the requirements for using manipulatives. At the same time he hasn't provided a systematic account for why his technique works but others don't.

I should think the effects would be dominated (indeed, beaten up ;-) ) by individual differences in the students' learning styles and cognitive profiles. In any given class, there will be kids who respond well to abstractions and explanations, others who favor graphical representations, still others who need to touch and feel things.

The real problem is the idea that any one method can, or

should, be "best" forallstudents, regardless of their individual differences. The standardized tests don't help either....Cog Scientist:

"Liz Spelke's lab has published some recent work on the short-comings of manipulatives"

I would like to read it (them). Can you specify? Is it available on-line?

I googled "Spelke" and got lots of articles but they seem not to be pertinent to this question.

David, I agree with you that no "one" way is going to be "the" way to teach mathematics. Children do have individual learning styles and they need to be treated as individuals instead of data points in a standardized program. I also agree with Tony's points that there is a developmental progression in mathematical thinking ability just as there is in other cognitive skills. One important skill in language, for example, is being able to talk about a person or an object that is not there. The only way to do that is to understand that words are symbolic representations of objects. Children also go through a stage where they say numbers and letters, and then they learn that they represent concepts. Knowing what a concept represents is vital to being able to use it in abstraction.

Being able to read a paragraph aloud perfectly and not understand its meaning is not true reading. Being able to solve a math problem according to a rule without understanding how it works is not really comprehending mathematics.

However, a child needs to be able to sound out words and to memorize math rules.

Perhaps the research is so confusing because different children who learn in different ways are chosen for these studies. Both methods need to be made available to children so that they can learn the way they know best.

In fact, is a double-blind, "majority rules" test the best way to answer questions of this kind?

manipulatives are hot! everybody who is anybody says that they use manipulatives. the question is do they know how to use manipulatives in a structured way that allows students (of all learning styles) to grasp the concept and move from a concrete understanding of reality to the abstract understanding of the mathematical, symbolic representation of that reality and master the procedures as a part of that understanding?

I didn't think so! I think that a lot of teachers may feel that the use of manipulatives is mostly useful for engaging the interest of the student, but may not have an understanding of how to get from concrete to abstract.

For those interested, check out Making Math Real http://www.makingmathreal.org/, a program being used by an educational therapist who is helping my son with a math learning disability. I'm really good at math myself, and I couldn't figure out how to teach him, and neither could 8 general ed teachers or 5 resource specialists, many of whom used manipulatives and were well-versed in the "learning style" lingo.

It's not the blocks. It's a good teacher who is effective at delivering a good methodology.