I Love Math Manipulatives...But

I love math manipulatives. I really do. Manipulatives allow students to physically model mathematics concepts. But manipulatives are no panacea. Manipulatives have significant, often overlooked, limitations.

Mistaken Modeling

Many teachers view math instruction as teaching standard algorithms, that is, teaching students the conventional step-by step recipe for computing an answer. Thus teachers use manipulatives to model algorithms. However, teaching algorithms is not the same as teaching math. For example, the most common explanation for dividing fractions is to multiply by the reciprocal. Multiplying by the reciprocal works because something mathematical is going on. However, we usually teach the superficial procedure and ignore the mathematics. The purpose of manipulatives is to model the mathematics, not the algorithm. The difference is subtle, but crucial.

Manipulatives Cannot Model Everything

Math is far more powerful than physical manipulatives. Manipulatives are merely a bridge to that power. Manipulatives cannot model beyond three dimensions, but manipulatives can lead students to math beyond the three dimensions. Some Montessori schools have a manipulative that physically models a quadratic equation, Ax^2 + Bx + C. If the factors of the quadratic equation are equal to each other, the quadratic equation models a square. If the factors are unequal, the quadratic models a rectangle.

I first saw the intriguing quadratic equation model in a Montessori school in Japan where preschoolers were enthusiastically absorbing the geometry of the quadratic equation without resorting to pencil and paper. FOIL? Who needs it? The factors were perfectly obvious to them. Add a “height” factor to model three dimensions. If the height is “x,” we have a model of a third-degree equation. We have an “x-cubed.” Cubed! How cool is that? Can we build a model in of an equation in the fourth degree? Well, now we have bumped up against a limitation. Mathematical representations can express math much more powerfully than physical models.

The Training Curve

It can sometimes require substantial training in the symbolism and design of the manipulative before the child can use the manipulative. For some children, imagining that one thing stands for another can create an obstacle to the mathematics itself. It is an adult myth that children have superior imaginations. Children represent, pretend, or re-enact what they already know. They have trouble with pretending something they do not already know. Adults can manage with the incomplete sets of manipulatives often found in classrooms. Children may be stymied. Children especially have trouble with strings of representations. Dr. Kamii says manipulatives can end up being “abstractions of abstractions” rather than the concrete models usually intended. For example, a teacher might say “We do not have enough hundred-flats for every group to make their number. You can use a teddy bear to stand for a hundred-flat if you need to.” Such instructions only make things more perplexing for the kids.

Impractical for Problem Solving

If manipulatives are used as algorithm aids, students may not be able to solve problems when they have no manipulatives, like during a test. Constance Kamii, who researches the ways children learn math, found that when young children were given a problem for which they had received no instruction and free access to a variety of manipulatives, writing instruments and paper, children preferred their own constructions over those imposed by others. Children preferred to think their way through problems with pictures they draw themselves rather than with manipulatives.

Broken analogies

Math manipulatives are analogies. Every analogy breaks down at some point. Math manipulatives are no exception. Manipulatives have lots of features which may or may not be salient to the math. Children may have difficulty understanding which features to pay attention to and which to ignore. For example, Cuisenaire rods are different lengths. Each length is a different color, but the color is arbitrary and has nothing to do with the math. However, the colors are sure convenient because kids can use them to express math without numerals.

Too Much Fun

Perhaps the most dangerous limitation of manipulatives is the fun. Student teachers have often reported to me that their math methods courses were little more than a term's worth of “playing” with manipulatives. They loved their methods course, but when they got into a real classroom with real kids, they found to their chagrin that they were woefully ill-prepared to actually facilitate the acquisition of mathematics concepts. I have often observed teachers use manipulatives as a fun diversion without ever getting to the point of the mathematics involved. I have seen educators demonstrate the use of manipulatives without ever building the bridge to the concept.

Manipulatives cannot substitute for the teacher's own profound understanding of the fundamentals of mathematics (PUFM). Sadly, nearly every college of education has a version of the course “Principles of Mathematics for Elementary Teachers” because so many elementary education students lack PUFM.

The over exuberant adoption of manipulatives is yet one more instance of educational pendulum swinging. Good ideas get over-used and misapplied all the time, often turning what could have been promising strategies into just another education fad.

Patient vs. Impatient Problem Solving

According to Dan Meyer, the problem with a steady diet of TV sitcoms is students learn to expect easy problems resolved in twenty-two minutes “with a laugh track.” We have now raised several generations of “impatient” problem solvers, and typical math textbooks pander to the syndrome instead of challenging it.

Mr. Meyer has a prescription for what ails our math teaching.



According to Mr. Meyer, there are two kinds of mathematics: computation, or “the step you forgot” and math reasoning. Within computation, there are a lot of tricks and gimmicks, like counting decimal places. The tricks work because of the underlying math reasoning. We teach the tricks, the non-math, and call it math. Good grades for non-math amount to “congratulating students for following the smooth path and stepping over the cracks.” No wonder our students display symptoms of impatient problem solving syndrome:

Lack of Initiative,
Lack of Perseverance,
Lack of Retention,
Aversion to Word Problems, and
Eagerness for Formulas.

The older your students, the more likely you can be teaching math reasoning well and still encounter not only the symptoms, but also resistance to the cure. Your students have been so conditioned by previous experience, that like chemical tolerance, they do not believe they can function mathematically any other way. It might be a good idea to show this video the first day of class to shock their systems into even entertaining the idea that math could be different.

His description of his presentation of the water tank problem is very like the way Japanese elementary teachers have been teaching math for decades (that I know about). They can easily spend a whole period on a single problem, but they actually save time, because they are not wasting it practicing a forgettable blind procedure on twenty problems. They invest the time it requires to think about math, for as Mr. Meyer says, “Math is the vocabulary for your own intuition.”

Mr. Meyers suggests a five-part prescription:

Use Multimedia,
Encourage Student Intuition,
Ask the Shortest Possible Question,
Let Students Build the Problem, and
Be Less Helpful.

Teachers ignore many features of a problem as irrelevant without discussion as if we expect students to figure it out on their own. Many do, some do not. Asking what matters, says Mr. Meyer, is probably the most underrepresented question in math curriculum.

After, and only after, students have acquired the math reasoning should we give them shortcuts, tricks and mnemonics.This video is an excellent example of a math teacher receiving accolades for teaching non-math.




And finally, just for fun.

Review: Zeroing In On Numbers and Operations

Anne Collins and Linda Dacey. (2010). Zeroing In on Number and Operations: Key Ideas and Misconceptions. Portland, ME: Stenhouse Publishers.

This is a set of four books (Grades1-2, Grades 3-4, Grades 5-6, Grades 7-8 ) formatted as spiral-bound flip charts with each page longer than the one before.

According to the publisher's description,

(Each book of the set) provides thirty research-based, classroom-tested modules that focus on the key mathematical strategies and concepts... while highlighting the importance of teacher language in the development of those skills. The flipchart format makes it easy to access the key resources: summaries that identify the mathematical focus and associated challenges and misconceptions; instructional strategies and activities that develop conceptual understanding and computation skills; activities and ideas for adjusting the activities to meet individual needs; reproducibles for instructional use; and resources for further reading.

First, A Digression

Primary math can and should anticipate algebra.

One reason students have trouble with algebra is that teachers typically lead students to believe it to be a new and harder topic, a different math than arithmetic. Algebra should be taught as a natural problem-solving strategy, even in first grade. On the page entitled “Join and Separate” from the book “Grades 1-2” (there are no page numbers), the authors present this story problem: “Jake had 5 knights for his toy castle. His sister, Emma, gave him some more knights for his birthday. Now Jake has 11 knights. How many knights did Emma give Jake?”

How do we usually teach children to approach these kinds of story problems? We drill them on math fact families so they will recognize a math fact buried in words. Now I believe children should memorize their math facts, BUT I also think that language can be an ally rather than the enemy it usually becomes. The natural progressive analysis for this problem starts “5 knights (gave more) ?knights (so now) 11 knights,” proceeding to “5 knights plus ? knights equals 11 knights,” and finally “5 + ? (or box) = 11.” The “?” (or box) is what algebra calls “a variable” and if replacing the “?” or box with a lower case letter makes the expression look like algebra. Even first graders can set up the problem with a manipulative like the Algebra Gear by putting a turquoise piece (standing for the unknown) and five yellow cubes on one side of the equivalence mat and eleven yellow cubes on the other side. The child removes five yellow cubes from both sides to isolate the turquoise piece on one side, and voila! There are six yellow cubes on the other side. A big advantage is the lack of reliance on numerals.

We do not give children enough credit for their ability to think. We discourage thinking by presenting mathematics not as something that can be reasoned about, but as something that must be memorized and accurately recalled. If they do not remember, they have no recourse. If they fail to remember often enough, they soon conclude wrongly that they are bad at math. Math-phobia is just one more short step away.

The text's idea that join and separate problems both share the start-change-end structure is helpful, but the graphic organizer is not obvious or intuitive to children. The teacher would be better off going straight to the algebra gear which requires a lot less instruction and makes more intuitive sense. Then the problems can be “played” like a game.

On page A14 of Grades 1-2, there are three examples of sentences where all three components (start-change-end) are left blank. The idea is to play around with providing any two out of three. The authors rightly note, “leaving the initial state blank is the most challenging, as many students are uncertain where to begin.” The algebra approach addresses and eliminates this uncertainty. Students simply use the turquoise cube to stand for the blank and march on.

These Books Are Necessary

Teachers need a resource that explicitly addresses the common misconceptions children (and their teachers) hold about math. Sometimes teachers deliberately teach misconceptions because they do not know any better.

The set is comprised of four very slim volumes of fifteen informational pages and about fifteen pages of problems and exercises for a total of thirty pages printed on both side of the paper. Thus the entire set is about 120 pages. The list price per single book is fifteen dollars and sixty dollars for the set of four. In a strategic marketing maneuver, by dividing what normally would be one book into four, the publishers may be able to capture more income. Teachers are likely to be most interested in the information pertaining to the particular grades they teach. A teacher might not be inclined to pay sixty dollars for largely “irrelevant” material (although that point could be argued), but may willingly spend fifteen dollars for grade-specific content. The books could also be useful to education students.

I intended to read a sampling of pages from each book very carefully and peruse the rest. I wanted to get a feel for the quality of the information across the scope and sequence. I ended up reading all four books line-by-line, analyzing the references and working the problems. I made copious notes on every page.

A Selection of Some Glittering Gems

Grade 1-2, Counting by Tens and Ones: I like the concept of “counting the tens and the leftover ones.” In fact, I like to rename the “ones” place the “leftovers” because they are not in a group.

Grade 1-2, Writing Numbers: Children can learn a lot of math without numerals. This page has a good strategy for using cards to illustrate digit positions.

Grade 1-2, Equivalent Representations: This is a valuable trading exercise, all the better because it is done on the overhead, avoiding the possible “magic” of computer trading exercises. Computer simulations of physical activities often look like magic to students. They resign themselves to taking the teachers word instead of understanding for themselves.

I would only caution the teacher to make sure the students very intentionally see all aspects of the trade. Students need to be certain that the teacher added nothing nor removed anything. I actually prefer a manipulative like Digi-Blocks Each box can hold only ten units. Dumping the box to simulate “borrowing” makes the trade crystal clear. I have seen even junior high students incredulous that after dumping the box, the total number of packed and unpacked units did not change. These students have done trading activities with base ten flats, rods, and cubes without ever acquiring true conservation of number.

I also like the use of the term “equivalent” as opposed to “equal” because the form of 3 tens 2 ones is not identical to the form 2 tens 12 ones. Carefully distinguishing the difference implicitly anticipates “equivalent” fractions, where two fractions of differing appearance are equivalent because the underlying value is equal. The use of “equivalent” helps build consistency, for example, in geometry, when students must differentiate equal measure as opposed to identical and/or congruent. Perhaps the “equal” sign should be renamed the “equivalent” sign because equivalent is what we usually mean.

Another valuable way to exploit differing representations is to use different ways to record the model. What the authors call equivalent “representation” is actually equivalent variations of the model, in this case, base ten blocks. There are a number of ways to represent the model, drawing a picture or coloring preprinted diagrams of rods and cubes, using written words, numerals and symbolic language like 3r2c = 2r12c, where r stands for rods and c stands for cubes. Older students also benefit from using various types of representation.

Grades 1-2, Subtraction Is More Than Take Away: I like the discussion of the different meanings of subtraction. In keeping with the importance of language precision, teachers should say “three plus five equals eight,” not “three and five are (or is) eight.” Even better,”three plus five is equivalent to eight.”

Grades 1-2, Modeling Addition and Subtraction: Of course I like this page if only for the reference to Digi-Blocks. The Win 300 and Lose 299 activities are gratifyingly similar to my Chocolate Factory activity, inspired by an I Love Lucy episode



Grades 3-4, Helping Facts: Students who are acquiring profound understanding of fundamental mathematics still need fluency with facts. This page contains useful tips for recalling and reconstructing multiplication facts.

Grades 3-4, Meaning of Division is a good explanation of the various types of division. The authors did a good job with Remainders, even providing a nice segue into bases. I also liked the Multiplication Menu, and the discussion of the meaninglessness of “gozinta” and misconceptions inherent in the long division algorithm.



Grades 3-4: Number Lines and Benchmark Fractions: I like the emphasis on kids sharing and explaining their strategies to each other, but instead of singling a child out as the authors so often do, let the children work in groups and have a group spokesman present the group's findings to the class. The authors often state that “a student” did this or that, showing the individualistic bent of American education, as opposed to, for example, Japanese elementary schools, where math activities are generally group activities.

Grades 3-4: Finding Parts and Making Wholes contains a nice list of misconceptions.

Grades 3-4: Parts of a Group: American egg cartons are very useful for modeling fractions. Instead of putting any old counters in the egg cups, it is better to use plastic eggs in up to six colors. Then the ribbons are unnecessary and the egg cartons can be used to play fraction games with even first and second graders. As an aside, Japanese egg cartons hold ten eggs, making them ideal for place value lessons.

Grades 5-6 Greatest Common Factors and Least common Multiples:I like the Venn diagram for finding common factors.

The authors really shine when it comes to fractions, I liked six pages in a row:
Fractions on the Number Line “Fractions are used in three distinct ways: (1) as numbers, (2) as ratios, (3) as division.”
Adding and Subtracting Fractions with Pattern Blocks, good explanations and activities.
Modeling Multiplication of Fractions, good activities.
Modeling Division of Fractions with Pattern Blocks, avoids multiplying by the reciprocal.
Dividing Fractions with Area Model
Posing Problems With Fractions

Grades 5-6, Estimating Decimals: I like the emphasis on the significance of zero “placeholders” as indicators of precision because of the connection to measurement and data recording in science. The authors also point out the problems with “context-free” computation. Real math occurs in a context. Real math always has a story. Numbers have referents.

Grades 7-8, Analyzing Change: The story graphs nicely anticipate the early topics of physics.

Some of the Quibbles and Errors

Grades 1-2, Connecting Representations: I would have liked the confusion over the difference between number and numeral or other representations explicitly stated, however this major misconception is implied in the text and diagram. Whenever I show my Japanese rulers (which have no numerals) to kids, they wonder how it is possible to measure anything with such strange rulers. Letting them figure it out for themselves is quite a worthwhile group activity.

Grades 1-2, Counting by Tens and Ones: Students are asked to count the strawberries on page A6 of the appendix. There are forty-two strawberries arranged in a 7-by-6 array. Then students are asked where they see the 4 of forty-two in the strawberries, and then where they see the 2. I could understand if students had been asked to circle groups of ten, so I am not sure what the authors had in mind.

Grades 1-2, Along the Line and Open Number Line: Although the authors correctly describe the integer “2” as being units units away from zero, the origin (positive direction understood), they abandon origin two paragraphs later. Nearly every presentation of the number line in all four books fails to start from zero. It is important to emphasize that 3 + 5 does NOT mean “start at 3.” It means start at 0, and go 3 units in the positive direction, and then five more units in the positive direction. Maintaining a sense of origin helps students to understand absolute value later.

Digi-Blocks points out,

Note that with drawn number lines like this one, you are supposed to count the steps. Here we see that there are 3 steps between 0 and 3. But often times children try to count the hash marks. This becomes confusing. Do they count three hash marks or 4? With the Digi-Block number lines, it is entirely clear that there are 3 blocks.

I actually prefer Cuisenaire's “Rod Track” over Digi-Block's number line because you can turn one track vertical to model not only multiplication arrays but also distributive property and quadratic equations. Cuisenaire used to have a rod track for modeling negative numbers (I have one), but it is apparently no longer available. More's the pity.

Grades 3-4, Mental Computation: It is not true that when adding 56 +6, it requires greater skill to mentally perform the standard algorithm than to first add 56+4 = 60, then 60 + 2 = 62. Furthermore, the arrow code on this page adds an necessary layer of complexity. Finally, if the authors are worried that on a standard hundred chart, bigger numbers are below smaller numbers, try rewriting the chart with 1-10 at the bottom instead of at the top.

Grades 3-4, Column Addition: The authors present a trick for adding a column of numbers. Tricks work because of the math behind them, but they are no substitute for understanding the math. It is a neat trick, but should be introduced after scaffolding.

Grades 5-6, Fact Practice: There is indeed a difference between practice and drill, but practice is not “doing mathematics.” It is doing procedures that work because of the mathematics behind them. We have to constantly remind ourselves of the difference between procedure and concept. It becomes more clear when we remember that a procedural explanation is by no means mathematical. There is no math in telling a student to move the decimal two places to the left when multiplying by 0.01, regardless of the presence of numbers in the explanation.

Grades 7-8 Integers on the Cartesian Coordinate Plane: The Cartesian plane models the multiplication of variously signed integers only if ground rules are arbitrarily established first. The authors do not develop a rationale for the first and third quadrants containing positive products, and the second and fourth quadrants containing negative products.

A Sampling of Editorial Issues and Typos

Grades 1-2, Equality: The string has two “9”s, with one superscripted in a box. I suspect a misprint.

Grades 3-4, Two-Digit Multipliers: “Where are the 24 square feet for cucumbers...” should say “42 square feet.”

Grades 3-4, Problem Solving with All Operations: delete “this teacher read” in the phrase, “... an article this teacher read by Kim ...” so it reads “...an article by Kim...”

Grades 7-8, Finding Factors With Square Roots: This whole page is done completely wrong. I do not believe the challenge to “find a prime factor of a number that is greater than its square root” was ever issued by the teacher in the story. I do not believe the students looked all week without finding a single one, when there are millions of examples. It would not take them a minute to figure out that 5, a prime factor of 15, is greater than its square root of 3.87. It makes me wonder how many other stories are fabricated and do not represent the experience of real students at all. The page overlooks that pairs of factors align on both sides of the square root. The reason students have only to check the prime factors up to the integer of the square root, and that every prime so checked will pop out its corresponding friend on the other side of the integer of the square root. Some of these friends might also be primes greater than the square root. Also, there is a mention of an author named Zany, but no citation.

Grades 7-8, Unit Rates: The purported student quote under the table does not make sense in light of the data in the table. The student would not have said what he is reported as saying.

Grades 7-8, Exponents, (A16) Answer: A step is missing from the proof of Josh's conjecture in problem #5. The way it is presented, there is no obvious reason to add the exponents.
Solving Problems with Ratios (A30): Problem number 3 needs to be rewritten from scratch or deleted. It is nearly incomprehensible to junior high as is.
Making Rate Tables (A31): The answer in the back does not correspond to the first story problem. Also, there are some additions I would make to the graph designs to help anticipate graphing data in science classes.
Answers in the Back A28-2: Second sentence is the wrong reason.
A29: ¾ does NOT equal 2/3

Conclusion
Overall, the books could have used some serious pre-publication editing. There are some sparkling gems of insight sprinkled throughout. The authors' strong suit is clearly fractions. However, there are too many outright errors and too many missed fundamental misconceptions. The authors' use of number lines consistently overlooks the importance of starting at zero. Even though there are references to algebra, the books often miss opportunities to anticipate advanced material. Furthermore, the authors inconsistently evaluate the math skills of their target audience, elementary and junior high math teachers. The authors note that many misconceptions are shared by student and teacher alike, yet write as if these same weak teachers will be able to follow the many oblique references to specific math concepts. In “Grades 3-4: Adding Numbers in the Thousands,” the authors allude to the main problem with the spiral curriculum, but do nothing to challenge it. Sadly, the spiral curriculum is a major factor in students moving from grade to grade without learning the subject matter. Although the authors often mention the mistake of emphasizing procedure, “what is most important,” they write, “is that students develop a reliable technique...” Perhaps the authors are being practical. A reliable procedure is better than nothing, I guess.

The authors seem more at home with upper elementary math topics and a bit at sea with primary math topics and middle school topics. Since misconceptions, once acquired are difficult to unlearn, I would have preferred the strongest treatment of primary math, where foundations are laid, for better or for worse. The authors overlooked some important researchers. Jean Piaget and Constance Kamii come immediately to mind. On the other hand, there seems to be an implied rule: avoid references from pre-turn of the century, as if all important work is relatively recent. In-text citations are often missing. Although the teaching ideas are billed as being research-based, most of them look to be anecdotal accounts of one or another teacher's favorite lesson. Researchers are fond of denigrating “unscientific” research teachers do every day, forgetting that teachers do not have time to wait for the verdicts from “the ivory tower.” Lessons need to work immediately. Good teachers are constantly customizing, adjusting and refining.

I would like the appealing color scheme and the flip chart design of the books more if all the pages were the same length. Each page is about a quarter longer than the one before with the page topic as the footer on each page. The design is attractive and convenient because all the topics have the appearance of being tabbed on the first page. However, the design also guarantees some topics get short shrift simply because the page is half as long as other pages. I do wish the content pages were numbered. The many obligatory nods to what may turn out to be educational fads annoy me. Furthermore, it should not be necessary to explicitly market to RTI or promise standards alignment. The suggestions for calculator use add nothing. In fact, there is no evidence that calculators enhance number reasoning skills in the early grades, NCTM claims to the contrary notwithstanding. To their credit, the authors acknowledge the value of Montessori materials.

To summarize, the books could be useful resources for the novice teacher, but they are too expensive, and the novice teacher will likely not have enough experience to recognize the flaws. Even so, I might be willing to recommend the books if they were four or five dollars each instead of fifteen dollars. I had such high hopes for this material, and I regret I cannot give it a stellar recommendation.