Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

Where To Start?

May 1999

By the commonly held view, "Math is hard". But there is one thing that may be even harder: math education. The views on the right way to teach mathematics and the current state of affairs are highly polarized and the discourse is sometimes caustic, sometimes acrimonious. See, for example a review of an algebra text adopted by the Davis School District in California and the author's response. In the words of Steven Pinker, director of the Center for Cognitive Neuroscience at MIT,

The ascendant philosophy of mathematical education in the United States is constructivism, a mixture of Piaget's psychology with counterculture and postmodernist ideology. Children must actively construct mathematical knowledge for themselves in a social enterprise driven by disagreements about the meanings of concepts. The teacher provides the materials and the social milieu but does not lecture or guide the discussion. Drill and practice, the routes to automaticity, are called 'mechanistic' and seen as detrimental to understanding.

That children should construct their own knowledge is indeed mentioned in the NCTM Standards: "Research findings from psychology indicate that learning does not occur by passive absorption alone. Instead, in many situations individuals approach a new task with prior knowledge, assimilate new information, and construct their own meanings."

I wonder. Given the difficulty with which even the best minds accepted some mathematical concepts (irrational, negative, complex numbers to name a few), the notion that children may construct by themselves the knowledge they are supposed to acquire in school seems rather obviously preposterous. Paul Erdös is said [Schechter] to have discovered negative numbers at the age of 4. But Erdös who later authored or co-authored more than 1400 mathematical papers - more than anybody else in the history of mathematics - was not your average child.

Constructivism is not new in educational psychology. In [Ginsburg] we find the following remark: "The view that mathematics is a 'sense-making activity' in which children 'construct their own meanings' is not very different from the early theories of such progressive educators such as John Dewey (1895) and even earlier figures such as Pestalozzi." And further: "The new contribution from psychological research is a body of empirical data and theory concerning both the nature of mathematical knowledge and the methods of assessing this knowledge." Unfortunately, the empirical data is quite recent. There is no evidence that the earlier theories had any recorded influence on math education.

On the other hand, the current wave of constructivism originates with the teaching of Jean Piaget from about 50 years ago. According to Piaget [Dehaene], "logical and mathematical abilities are progressively constructed in the baby's mind by observing, internalizing, and abstracting regularities about the external world. At birth, the brain is a blank page devoid of any conceptual knowledge." This aspect of Piaget's theory was discredited by the neuropsychological research of the past two decades. It is now an acceptable fact that children are born with some number sense.

Being thus loaded, the term "constructivism" may not be a good descriptive for the current pedagogical thinking. As I understand, the NCTM Standards attempt to promote education based on acquisition of conceptual knowledge as opposed to rote memorization. An assumption made in the K-4 chapters suggests that "Teachers need to create an environment that encourages children to explore, develop, test, discuss, and apply ideas. They need to listen carefully to children and to guide the development of their ideas." Perhaps "learning by mediated conceptualization" reflects better on the intention of the Standards. The intention may be laudable. How successful will its realization be depends on several factors.

First, the new trend requires extensive re-education of teachers [Ginsburg], in the subject matter, instructional practices, and child psychology. In a context where many teachers are poorly prepared to begin with, this is a tall order. However, support for the current ideas is far from being overwhelming. Many teachers are simply turned off by the constructivist thesis. Others have no incentive to adopt the new trend that requires considerable time investment beyond the needs of the current practices.

Second, there exists a considerable gap between education research and education practice [Renninger] as "... education-related studies in developmental psychology are often more distal than proximal for educators, meaning that the connections between the research question being addressed and classroom practice are not obvious. Even in cases where researchers specify implications for practice, these often take the form of suggesting that practitioners promote metacognition, for example, rather than specifying how one might actually implement this suggestion in the classroom."

A delightful sample of classroom practices can be found at Marilyn Burns' Math Solutions® web site. I was especially taken with the Addition Facts and Strategies, second grade class notes. The lesson demonstrates how the addition table - something that usually falls under the rubric of rote memorization - can be mastered actively by students with teacher mediation.

At the other end, no amount of mediation will help with an example by the Northwest Regional Educational Laboratory:

... students can use geometric construction software to investigate the relationship between the circumference and diameter of a circle. They can measure several round objects and record the circumference and diameter (numerical representation). They can plot the values and estimate a "best fit" (graphical representation). Students can then determine the best fit equation (symbolic representation).

As a result of this activity, stidents may or may not discover that the diameter and the circumference of the circle stand in a linear relationship. In any event, I doubt that an average student will realize that the relationship is an immediate consequence of the fact that all circles are similar. Measuring round objects in this context is a deplorable waste of time that rather misrepresents what mathematics is about. Measurements, tabulation and graphing can be used to investigate similarity in general. Subsequently, students may draw conclusions with regard to the similarity of circles.

One of the critisism of the Algebra text I mentioned in the beginning of the column is directed against the book's attitude towards formula memorization. One example is really startling. The authors instruct teachers to discourage students from memorizing the formula for the sum of the integers from 1 through n, although the book does lead students to its discovery. My personal feeling is that it is simply impossible to understand or conceptualize the process of summation without remembering the famous formula. To any one who understood the process, the formula serves as the iconic representation of the concept of pairing the members of the series starting from both ends. Here is the author's response:

Our book is used in grades 7-9, mostly, and in some cases in "remedial" classes. Do those students need to know formulas for arithmetic and geometric sequences and series? I doubt it. Will students who have done the work on sequences in ATTC (the book's title, AB) in grades 7-9 be able to grasp it better in Algebra 2 and learn those formulas then, with understanding? In my experience, yes.

I must admit that this approach also makes sense. Teach students in such a way as to not only impart some specific knowledge but also give them a glimpse of what they will be learning in the next course, or even the one after. Such an approach, in a sense, would answer a remark by Ralph P. Boas, Jr.

... the students always have to be taught what they should have learned in the preceding course. (We, the teachers, were of course exceptions; it is consequently hard for us to understand the deficiencies of our students.) The average student does not really learn to add fractions in arithmetic class; but by the time he has survived a course in algebra he can add numerical fractions. He does not learn algebra in the algebra course; he learns it in calculus, when he is forced to use it. He does not learn calculus in the calculus course, either; but if he goes on to differential equations he may have a pretty good grasp of elementary calculus when he gets through. And so on through the hierarchy of courses; the most advanced course, naturally, is learned only by teaching it.

The idea is very tempting. What if in every course students take, they learn solving problems from the previous course in an easier or more general manner? Of course they practice what they learn with the problems modeled on those they will encounter later on. Two birds are killed with one stone. First, students learn what they are supposed to learn and when they are expected to do so. Second, mathematics acquires an aura of self-justification: one studies not only in order to know more but also in order to know better.

Let's have a look at a simple problem:

The sum of the ages of two brothers is 17, the difference is 3. How old are the brothers?

This is a word problem whose solution includes solving a system of two linear equations. Either way one looks, this is an undeniably algebraic problem. Now, the problem can be solved by pure arithmetic. The applet below is supposed to help you out with this task. All numbers are clickable. They may be increased or decreased by clicking a little off their center lines. Check that this is indeed the way to manipulate the numbers, by setting the right sides to 17 and 3, respectively.

Please try switching to IE 11 (Windows) or Safari (Mac), for no other browser nowadays runs Java applets. If asked whether to allow the applet to load, click Yes - the applet is signed with a security certificate from a trusted company. But, regardless, there is an explanation below.

What if applet does not run?

The problem can be solved by counting. Take any two numbers, say 7 and 5. Their sum is 12 which is short of 17; their difference is 2 which is a little short of 3. Note that increasing the first number increases both their sum and their difference. Think of what happens when the number is decreased or when the second number changes.

At the beginning, both the sum (12) and the difference (2) are short of their corresponding targets (17 and 3). Increase the first number by 1. The second line shows an equality. We should now modify the first line without disturbing the equality of the second. Change both numbers up by the same amount.

The NCTM Standards are intentionally quite general [Ginsburg]. They allow good and bad pedagogy under the same banner. This is one of their weak points. The other one is that they set goals only for K-12 math education. The Standards absolutely leave out early education, math education of children up to 5 years of age. One reason, of course, is that NCTM is a teacher organization and the first mandatory schooling children receive occures in the kindergarten. The second reason might be in deference to Piaget's constructivist developmental theory. What do you teach preschoolers anyway?

According to research findings [Ginsburg, pp 423-427], preschoolers from middle-class families outperform poor and minority children, especially in all kinds of verbal arithmetic tasks. One possible explanation is that parental involvement with children is an important factor in early math education.

There is also a comparison in performance between American and Asian (Chinese, Korean, Japanese) children that indicates that the latter are developmentally more advanced. In Chinese, Korean and Japanese languages, words for double digit numerals are strictly rule-based: ten-one, ten-two, and so on. It must be easier to memorize them than their equivalents in European languages. Reseachers also point out to other difference in culture and schooling that contribute to the superior performance of Asian children. One that is mentioned in passing is a widely spread use of the abacus.

What may be good about the abacus? By removing the need to memorize number words, abacus helps children develop counting concepts before or in parallel with acquisition of language. Abacus is perfect for learning addition and grouping of objects and the place-value systems in general - without the necessity of knowing all the possible number words.

Children perceive small numbers (1-5) directly by a kind of instant recognition which is called subitizing (from the Italian "subito" meaning "immediate".) Why not promote counting by grouping of small numbers of objects? Say, a child learned to count to 5. 11 is "two 5 and 1" and 7 is "5 and 2". There is no need to wait till he masters the number words up to eleven. An abacus with 5 beads per wire will do quite nicely. "Two 5 and 1" and "5 and 2" is "three 5 and 3". The numeration can be learned along the way. In their classic Mathematics and the Imagination, E.Kasner and J.Newman wrote: "Learning to compare is learning to count. Numbers come much later; they are an artificiality, an abstraction."

The decimal system is a great tool for communication because it is now an integral part of most languages and cultures. It is also a convenient tool for carrying out arithmetic operations. However, the latter property is shared by all positional systems. The base 5 seems far more natural than 10 for young children. With just ten fingers a child can count to 30!

The ancient Mayans perceived the need and invented a symbol for zero independently of the Hindus. Their numeration is usually described (with some caveats) as being in base 20. They had only 3 symbols - dot, bar, and a symbol for 0 - to compose numbers up to 20. Patti Weeg, an activist educator, put together a site to accompany her recent book Kids@work. In Chapter 2 there we learn about the Mayan arithmetic [see also Beckmann]. It's a pure base 5 system for the numbers up to 20.

As in the case of Pestalozzi and Dewey, there is no empirical data for us to judge how successful was Mayan math education. Something tells me that if that were possible, the Mayans would not mind taking up our kids in any K-4 math competition.

Friedrich Fröbel (1782-1852) was the originator of kindergartens and manipulatives. Baroness von Marenholtz Bülow recollects:

A tall, spare man with long grey hair was leading a troop of village children between the ages of three and eight, most of them barefooted, up a hill where they played and sang.

The loving patience of the man, the whole bearing brought tears to my eyes and to those of my companion.

The play being ended I approached the man with the words," I see you are occupied with the education of the people."

He agreed and said ," the other [older] people will not come unless we educate them, so we must begin with the children."

Fröbel showed some interest in geometry and designed a series of tools (Fröbel blocks) and activities (paper perforating, paper cutting and folding, interlacing, weaving, drawing, clay modeling). It is said [Brosterman] that Fröbel invented Kindergarten in 1830 "to teach young children about art, design, mathematics, and natural history." Fröbel's vision was to stimulate an appreciation and love for children, to provide a new but small world for children to play with their age group and experience their first gentle taste of independence. Brosterman traces the influence of Fröbel's ideas and kindergarten education on modern art and architecture. He connects the creations of Kandinsky, Frank Lloyd Wright, Le Corbusier, Klee, Mondrian to their childhood exposure to the then revolutionary educational activities.

I am unaware of any research linking achievement in mathematics to kindergarten education. I will appreciate any pointers. In Pictured Knowledge (p 899), the philosophy of Fröbel's kindergarten was described thus: "We do not teach arithmetic in Kindergarten. A knowledge of mathematics is not necessary to a four year old. Any premature instruction arrests development." I find this quote confusing. It sounds like a farfetched idea.

Children like counting. The modern research indicates [Ginsburg] that counting for young children is a very much conceptual process. Which jibes nicely with the goals annunciated in the NCTM Standrads. Technology is also available. Ginsburg (p 77) refers to a curious event:

According to the Wall Street Journal (1976), a prominent mathematical publication, there is still hope for informal methods of calculation. A competition was held between a United States executive using a calculator and an Asian businessman using the ancient abacus. While the American poked around trying to find the plus button, the Asian smoothly manipulated the wooden beans and water buffalo horn pegs. It was no contest; the abacus was quicker almost every time.

And remember what Fröbel said: "... so we must begin with the children."

(The famous R. Feynman, too, had once to compete with an abacus salesman. As the luck would have it, he had to compute the cubic root of an intersting number, which he happened to know by heart due to his professional interests.)


  1. P.Beckmann, The History of π, St. Martin's Press, 1971
  2. R.P.Boas Jr., "If This Be Treason ...", Monthly 64(1957), 247-249. Reprinted in A Century of Mathematics Through the Eyes of the Monthly, E.H.Ewing (ed), MAA, 1994
  3. N. Brosterman, Inventing Kindergarten : Nineteenth Century Children, Harry N Abrams, 1997
  4. S. Dehaene, The Number Sense, Oxford University Press, 1999
  5. H.P.Ginsburg et al, The Development of Children's Mathematical Thinking: Connecting Research with Practice, in Handbook of Child Psychology, v4, I.E.Sigel, K.A.Renninger (eds), John Wiley & Sons, 1998
  6. H.P.Ginsburg, Children's Arithmetic, II edition, pro-ed, 1989
  7. E.Kasner and J.Newman, Mathematics and the Imagination, Simon and Schuster, Sixteenth Printing, 1958
  8. Pictured Knowledge, Marshall Hughes Co., several editions 1916-1927
  9. S.Pinker, How the Mind Works, W.W.Norton & Company, 1997
  10. K.A.Renninger, Developmental Psychology and Instruction: Issues From and For Practice, in Handbook of Child Psychology, v4, I.E.Sigel, K.A.Renninger (eds), John Wiley & Sons, 1998
  11. B.Schechter, My Brain Is Open, Simon & Schuster, 1998 (This book was kindly sent to me by Tom Haynes, a math teacher from Tennessee.)

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