## Devlin's Angle |

Two roads diverged in a yellow wood, And sorry I could not travel both, And be one traveler, long I stood, And looked down one as far as I could, To where it bent in the undergrowth;-- Robert Frost, "Road Not Taken"

Then took the other, as just as fair, And having perhaps the better claim, Because it was grassy and wanted wear; Though as for that, the passing there, Had worn them really about the same,

And both that morning equally lay, In leaves no step had trodden black, Oh, I kept the first for another day! Yet knowing how way leads on to way, I doubted if I should ever come back.

I shall be telling this with a sigh, Somewhere ages and ages hence: two roads diverged in a wood, and I -- I took the one less traveled by, And that has made all the difference.

I began my last month's column with the famous quotation by the German mathematician Leopold Kronecker (1823-1891): "God made the integers; all else is the work of man." I ended the essay with a number of questions about the way we teach beginning students mathematics, and promised to say something about an alternative approach to the one prevalent in the US.

This month's column begins where my last left off. To avoid repeating myself, I shall assume readers have read what I wrote last month. In particular, I provided evidence in support of my thesis (advanced by others in addition to myself) that, whereas numbers and perhaps other elements of basic, K-8 mathematics are abstracted from everyday experience, more advanced parts of the subject are created and learned as rule-specified, and often initially meaningless, "symbol games." The former can be learned by the formation of a real-world-grounded chain of cognitive metaphors that at each stage provide an understanding of the new in terms of what is already familiar. The latter must be learned in much the same way we learn to play chess: first merely following the rules, with little comprehension, then, with practice, reaching a level of play where meaning and understanding emerge. (Lakoff and Nunez describe the former process in their book *Where Mathematics Comes From*. Most of us can recall that the latter was the way that we learned calculus - an observation that appears to run counter to - and which I think actually does refute - Lakoff and Nunez's claim that the metaphor-construction process they describe yields all of pure mathematics.)

If indeed there are these two, essentially different kinds of mathematical thinking, that must be (or at least are best) learned in very different ways, then a natural question is where, in the traditional K-university curriculum, the one ends and the other starts. And make no mistake about it, the two forms of learning I am talking about are very different. In the first, meaning gives rise to rules; in the second, rules eventually yield meaning. Somewhere between the acquisition of the (whole) number concept and calculus, the process of learning changes from one of abstraction to linguistic creation.

Note that both can generate mathematics that has meaning in the world and may be applied in the world. The difference is that in the former, the real-world connection precedes the new mathematics, in the latter the new mathematics must be "cognitively bootstrapped" before real-world connections can be understood and applications made.

Before I go any further, I should point out that, since I am talking about human cognition here, my simplistic classification into two categories is precisely that: a simplistic classification, convenient as a basis for making the general points I wish to convey. As always when people are concerned, the world is not black-and-white, but a continuous spectrum where there are many shades of gray between the two extremes. If my monthly email inbox is anything to go by, mathematicians, as a breed, seem particularly prone to trying to view everything in binary fashion. (So was I until I found myself, first a department chair and then a dean, when I had to deal with people and university politics on a daily basis!)

In particular, it may in principle be possible for a student, with guidance, to learn all of mathematics in the iterated-metaphor fashion described by Lakoff and Nunez, where each step is one of both understanding and competence (of performance). But in practice it would take far too long to reach most of contemporary mathematics. What makes it possible to learn advanced math fairly quickly is that the human brain is capable of learning to follow a given set of rules without understanding them, and apply them in an intelligent and useful fashion. Given sufficient practice, the brain eventually discovers (or creates) meaning in what began as a meaningless game, but it is in general not necessary to reach that stage in order to apply the rules effectively. An obvious example can be seen every year, when first-year university physics and engineering students learn and apply advanced methods of differential equations, say, without understanding them - a feat that takes the mathematics majors (where the goal very definitely *is* understanding) four years of struggle to achieve.

Backing up from university level now, where the approach of rapidly achieving *procedural competence* is effective for students who need to use various mathematical techniques, what is the best way to teach beginning mathematics to students in the early grades of the schools? Given the ability of young children to learn to play games, often highly complicated fantasy games, and the high level of skill they exhibit in videogames, many of which have a complexity level that taxes most adults - and if you don't believe me, go ahead and try one for yourself (I have and they can be very hard to master) - I guess it might be possible they could learn elementary math that way. But I'm not aware that this approach has ever been tried, and it is not clear to me it would work. In fact, I suspect it would not. One thing we want our children to learn is how to apply mathematics to the everyday world, and that may well depend upon grounding the subject in that real world. After all, a university student who learns how to use differential equations in a rule-based fashion approaches the task with a more mature mind and an awful lot of prior knowledge and experience in using mathematics. In other words, the effectiveness of the rule-based, fast-track to procedural competence for older children and adults may well depend upon an initial grounding where the beginning math student *abstracts* the first basic concepts of (say) number and arithmetic from his or her everyday experience.

That, after all, is - as far as we know - how our ancestors first started out on the mathematical path many thousands of years ago. I caveated that last assertion with an "as far as we know" because, of course, all we have to go on is the archeological evidence of the artifacts they left behind. We don't know how they actually thought about their world.

It is also clear from the archeological evidence that our early mathematically-capable forebears developed systems of measurement, both of length and of area, in order to measure land, plant crops, and eventually to design and erect buildings. From a present-day perspective, this looks awfully like the beginnings of the real number system, though just when that activity became *numerical* to an extent we would recognize today is not clear.

Today's US mathematics curriculum starts with the positive whole numbers and addition, and builds on to in a fairly linear fashion, through negative numbers and rationals, until it reaches the real number system as the culmination. That approach can give rise to the assumption or even the belief that the natural numbers are somehow more basic or more natural than the reals. But that is not how things unfolded historically. True, if you try to build up the real numbers, starting with the natural numbers, you are faced with a long and complicated process that took mathematicians some two thousand years of effort to figure out, completing the task as recently as the end of the nineteenth century. But that does not mean that the real numbers are a cognitively more difficult concept to acquire than the natural numbers, or that one builds cognitively on the other. Humans have not only a natural ability to abstract discrete counting numbers from our everyday experience (sizes of collections of discrete objects) but also have a natural sense of continuous quantities such as length and volume (area seems less natural), and abstraction in that domain leads to positive real numbers.

In other words, from a cognitive viewpoint (as opposed to a mathematical one), the natural numbers are neither more fundamental nor more natural than the real numbers. They both arise directly from our experiences in the everyday world. Moreover, they appear to arise in parallel, out of different cognitive processes, used for different purposes, with neither dependent on the other. In fact, what little evidence there is from present-day brain research suggests that from a neurophysiological viewpoint, the real numbers - our sense of continuous number - is more basic than the natural numbers, which appear to build upon the continuous *number sense* by way of our language capacity. (See the recent books and articles of researchers such as Stanislaw Dehaene or Brian Butterworth for details.)

It seems then, that when we guide our children along the first steps of the long path to mathematical thinking, assuming we want to ground those key first steps in everyday experience and build upon natural human cognitive capacities, we have two possible ways to begin: the discrete world of assessing the sizes of collections and the continuous world of assessing lengths and volumes. The first leads to the natural numbers and counting, the second to the real numbers and measurement.

In the Unites States and many other countries, the choice was made - perhaps unreflectively - long ago to take our facility for counting as the starting point, and thus to start the mathematical journey with the natural numbers. But there has been at least one serious attempt to build an entire mathematical curriculum on the other approach, and that is the focus of the remainder of this essay. Not because I think one is intrinsically better than the other - though that may be the case. Rather because, whichever approach we adopt, I think it is highly likely we will do a better job, and understand better what we are doing as teachers, if we are aware of an(y) alternative approach.

Indeed, knowledge of another approach may help us guide our students through particularly tricky areas such as the multiplication concept, the topic of some of my previous columns. As Piaget observed, and others have written on extensively, helping students achieve a good understanding of multiplication in the counting-first curriculum is extremely difficult. In a measuring-first curriculum, in contrast, some of the more thorny subtleties of multiplication that plague the counting-first progression simply do not arise. Maybe the way forward to greater success in early mathematics education is to adopt a hybrid approach that builds simultaneously on both human intuitions? (Arguably this occurs anyway to some extent. US children in a counting-first curriculum use lengths, volumes, and other real-number measures in their everyday lives, and children in the real-numbers-first curriculum I am about to describe can surely count, and possibly add and subtract natural numbers, before they get to school. But I am not aware of a formal school curriculum that tries to combine both approaches.)

Whichever of the two approaches we adopt, the expressed primary goal of current K-12 mathematics education is the same the world over: to equip future citizens with an understanding of, and procedural fluency with, the real number system. In the US school system, this is done progressively, with the first stages (natural numbers, integers, rationals) taught under the name "arithmetic" and the real numbers going under the banner "algebra". (Until relatively recently, geometry and trigonometry were part of a typical school curriculum, bringing elements of the measuring-first approach into the classroom, but that was, as we all know, abandoned, though not without a fight by its proponents.)

It is interesting to note that coverage of the real numbers as "algebra" in the US approach ensures an entirely procedural treatment, avoiding the enormous difficulties involved in constructing the *concept* of real numbers starting from the rationals. Eventually, even our counting-first approach has to rely on our intuitions of, and everyday experience with, continuous measurement, even if it does not start with them.

In a series of studies of the development of primates, children, and traditional peoples, Vygotsky observed that cognitive development occurs when a problem is encountered for which previous methods of solution are inadequate (Vygotsky & Luria, 1993). The Davydov mathematics curriculum is built on top of this observation, and consists of a series of carefully sequenced problems that require progressively more powerful insights and methods for their solution. This is of course quite different from the instructional approach adopted by most US teachers, which consists of an instructional lecture, with worked examples, followed by a set of exercises focused on repeated practice of the particular skill the instructor has demonstrated in class.

But that is just the first of several differences between the two approaches. Whereas the US K-12 mathematics curriculum has an understanding of and computational facility with the real numbers system as the declared end-point, the first several years are taken up with the progression through positive whole numbers, fractions, and negative integers/rationals, with the real number system covered in the later grades, primarily under the name "algebra". In contrast, the Davydov curriculum sets its sights squarely on the real number system from the getgo. Davydov believed that starting with specific numbers (the counting numbers) leads to difficulties later on when the students work with rational and real numbers or do algebra.

I'll come back to the focus on the real number system momentarily, but first I need to introduce another distinquishing feature of Davydov's approach.

Davydov took account of Vygotsky's distinction between what he called *spontaneous* concepts and *scientific* concepts. The former arise when children abstract properties from everyday experiences or from specific instances; the latter develop from formal experiences with the properties themselves.

This distinction is more or less (but not entirely) the same as the one I discussed last month between mathematics we learn by abstraction from the world and mathematics we learn in a rule-based fashion the same way we learn to play chess. For example, children who learn about the positive integers by counting collections of objects thereby acquire a spontaneous concept. Learning to play chess leads to a "scientific" understanding of the game. The point I made earlier was that in my experience, both as a learner and a teacher of advanced mathematics, the scientific approach is the most efficient, and perhaps the only way, to learn a highly abstract subject such as calculus.

In last month's column I asked where the abstract-it-from-the-world kind of mathematics (spontaneous concepts) ends and learn-it-by-the-rules kind (scientific concepts) starts. As I have noted, that question is a naive one that obscures the fact that there is most likely a continuous spectrum of change rather than a break point. A more usefully phrased question from an educational perspective is, which parts of mathematics should we teach in a spontaneous-concepts fashion and which in a scientific-concepts way?

The accepted wisdom in the US is that the spontaneous approach is the way to go at least all the way through K-8, and maybe all the way up to grade 12. (Adopting the approach all the way to grade 12 tends to force a presentation of calculus as "a method to calculate slopes", which I personally dislike because it reduces one of the greatest ever achievements of human intellect to a bag of procedural tricks. But that is another issue for another time.)

The Davydov curriculum adopts the scientific-concepts approach from day 1. Davydov believed that learning mathematics using a general-to-specific, "scientific" approach leads to better mathematical understanding and performance in the long run than does the spontaneous approach. His reasoning was that if very young children begin their mathematics learning with abstractions, they will be better prepared to use formal abstractions in later school years, and their thinking will develop in a way that can support the capacity to handle more complex mathematics.

He wrote (Davydov 1966), "there is nothing about the intellectual capabilities of primary schoolchildren to hinder the algebraization of elementary mathematics. In fact, such an approach helps to bring and to increase these very capabilities children have for learning mathematics."

I should stress that Davydov's adoption of the "scientific-concepts" approach is not at all the same as teaching mathematics in an abstract, axiomatic fashion. (This is where my analogy with learning to play chess breaks down, as do all analogies sooner or later, no matter how helpful they may be at the start; which reminds me, did I ever mention the problems that can result from introducing multiplication as repeated addition?) The Davydov approach is grounded firmly in real-world experience, and lots of it. Indeed, students spend more time at the start doing nothing but real-world activities (before doing any explicit mathematics) than is the case in the US curriculum. But when the actual mathematical concepts are introduced, it is in a scientific fashion. The students are able to link the scientific concept to their real world experience not because that concept arose spontaneously out of that experience (it did not), but because they had been guided through sufficiently rich, preparatory real-world experiences that they are able to at once see how the concept applies to the real world. (In terms of metaphors, the metaphor mapping is contructed back from the new to the old, not the other way round as in the Lakoff and Nunez framework for learning.)

In Grade 1, the pupils are asked to describe and define physical attributes of objects that can be compared. As I hinted a moment ago, the intention is to provide a context for the children to explore relationships, both equality and comparative. Six-year-olds typically compare physically lengths, volumes, and masses of objects, and describe their findings with statements like

where H and B are unspecified *quantities* being compared, not objects. (At this stage the unspecified quantities are not numbers.) Notice this immediate focus on abstractions. The physical context and the act of recording mean that the elements of "abstract" algebra are introduced in a meaningful way, and are not seen by the children as abstract.

For instance, the pupils are asked how to make unequal quantities equal or how to make equal quantities unequal by adding or subtracting an amount. Starting from a volume situation recorded as H < B, the children could achieve equality by adding to volume H or subtracting from volume B. They observe that whichever action they choose, the amount added or subtracted is the same. They are told it is called the difference.

Only after they have mastered this pre-numeric understanding of size and of part-whole relationships are they presented with tasks that require quantification. For example, if they have been working with mass and have noticed that mass Y is the whole and masses A and Q are the parts that make up the whole, which they may be encouraged to express by means of a simple inverted-V diagram like this:

This sets the stage for putting specific numerical values for the "variables" in order to solve equations that arise from real-world problems. (Numbers, that is, real numbers, are introduced in the second half of the first grade, as abstract measures of lengths, volumes, masses, and the like.) As a result, the pupils do not have to learn rules for solving algebraic equations; rather they become sophisticated in reasoning directly about part-whole relationships.

When the pupils get to multiplication and division, Davydov's curriculum requires that they connect the new actions of multiplication and division with their prior knowledge of measurement and place value, as well as addition and subtraction, and to apply them to problems involving the metric system, number systems in other bases (studied in grade 1), area and perimeter, and the solution of more complex equations. In other words, the new operations come both with real world grounding and their connections to previously learned mathematics. Thee pupils have to explore the two new operations and their systemic interrelationships with previously learned concepts. They are constantly presented with problems that require them to forge connections to prior knowledge. Every new problem is different in some significant way from its predecessors and its successors. (Contrast this to the US approach where problems are presented in sets, with each set focusing on a single procedure.) As a result, the pupils must continuously think about what they are doing in order that it makes sense to them. By working through many problems designed so make them create connections between the new actions of multiplying and dividing and their previous knowledge of addition, subtraction, positional systems, and equations, they integrate their knowledge into a single conceptual system.

Thus, the Davydov curriculum is grounded in the real world, but the starting point is the continuous world of measurement rather than the discrete world of counting. I don't know about you, but measurement and counting both seem to me to offer pretty concrete starting points for the mathematical journey. Humans are born with a capacity to make judgments and to reason about length, area, volume, etc. as well as a capacity to compare sizes of collections. Each capacity leads directly to a number concept, but to different ones: real numbers and counting numbers, respectively.

Clearly, the Davydov approach has no such difficulties. With the real number system as basic, integers and rational numbers are just particular points on the real number line.

Another possible advantage of the Davydov approach is that the more troublesome problems about how to successfully introduce multiplication and division that plague the start-with-counting approach to learning math - the focus of three of my columns last year - simply do not arise, since multiplication and division are natural concepts in the world of lengths, volumes, masses, etc. and part-whole relationships between them.

One feature of the Davydov approach that I personally (as a mathematician, remember, not a teacher or an expert in mathematics education, which I am not) find worrying is the absence of exercise sets focused on specific skills. Chunking and the acquisition of procedural fluency are crucial requirements for making progress in mathematics, and I don't know of any way to achieve that than through repetitive practice. While a math curriculum that consists of little else than repetitive exercises would surely turn many more students off math than it would produce skilled numbers people, their absence seems to me just as problematic. A colleague in mathematics education tells me that Russian teachers sometimes (often?) do get their pupils to work through focused, repetitive exercise sets, and I wonder if success with a more strict Davydov curriculum might depend at least in part on parents working on repetitive exercises with their children at home.

Whether one approach is, overall, inherently better than the other, however, I simply do not know. Absent lots of evidence, no one knows. Unfortunately - and that's a mild word to use given the high stakes of the math ed business in today's world - there haven't been anything like enough comparative studies to settle the matter.

One of the few US-based studies I am aware of involved an implementation of the entire three years of Davydov's elementary mathematics curriculum in a New York school. The study was led by Jean Schmittau of the State University of New York at Binghamton. Schmittau (2004, p.20) reports that "the children in the study found the continual necessity to problem solve a considerable - even daunting - challenge, which required virtually a year to meet, as they gradually developed the ability to sustain the concentration and intense focus necessary for success. However, upon completion of the curriculum, they were able to solve problems normally given only to US high school students."

Countering commonly made claims that in the era of cheap electronic calculators, there is no need for children to learn how to compute, and that time spent on computation actually hinders conceptual mathematical learning (for instance, you'll find these claims made repeatedly in the *1998 NCTM Yearbook*), Schmittau writes (2004, p.40), "In light of the results presented [in her paper], it is impossible to subscribe to the contention that conceptualization and the ability to solve difficult problems, are compromised by learning to compute. Not only did the children using Davydov's curriculum attain high levels of both procedural competence and mathematical understanding, they were able to analyze and solve problems that are typically difficult for US high school students. They did not use calculators, and they resolved every computational error *conceptually,* without ever appealing to a "rule". In addition, developing computational proficiency required of them both mathematical thinking and the establishing of new connections - the sine qua non of meaningful learning."

Here again, I find myself worrying about the balance between, on the one hand, deep conceptual understanding and the ability to reason from first principles - highly important features of doing math - and, on the other hand, the need for rule-based, algorithmic methods that are practiced to the point of automatic fluency in order to progress further in the subject. The continued popularity - with parents if not their children - of commercially-offered, Saturday morning, math-skills classes suggests that I am not alone in valuing basic skills acquisition (procedural fluency), and as I mentioned once already, I often wonder if the success of some curricular experiments does not depend in part on unreported activities outside the classroom.

Another US atudy was carried out at the same time at two schools in Hawai'i by Barbara J. Dougherty and Hannah Slovin of the University of Hawai'i, and there too the researchers reported a successful outcome. They write (2004, p.301), "Student solution methods strongly suggest that young children are capable of using algebraic symbols and generalized diagrams to solve problems. The diagrams and associated symbols can represent the structure of a mathematical situation and may be applied across a variety of settings." (The students used algebraic symbols coupled with diagrammatic representations like the inverted-V diagram shown above. The children in the study referred to in this quote were in the third grade.)

In fact, if we pursue that last observation a bit, we get to what I suspect is the really important factor here: teachers who have a deep understanding of basic mathematics. Hmmm, now where have I heard (and read) that before? Liping Ma, anyone?

In fact, in the context of this country, bedeviled by the incessant math wars and the intense politicization of mathematics education that drives them, my view is that debate about the curriculum and the educational theory that drives it is a distraction best avoided (at least for now). To me the real issue facing us is a starkly simple one: Teacher education. No matter what the curriculum, and regardless of the psychological and educational theory it is built upon, teaching comes down to one human being interacting with a number of (usually) younger, other human beings. If that teacher does not love what he or she is teaching, and does not understand it, deeply and profoundly, then the results are simply not going to come. The solution? Attract the best and the brightest to become mathematics teachers, teach them well, pay them at a level commensurate with their training, skills, and responsibilities, and provide them with opportunities for continuous professional development. Just what we do in (for example) the medical or engineering professions. It's that simple.

L. P. Steffe, (Ed.), *Children's capacity for learning mathematics. Soviet Studies in the Psychology of Learning and Teaching Mathematics, Vol. VII*, Chicago: University of Chicago. Specific articles in that volume are listed below.

My brief summary of the Davydov approach is based primarily on Dougherty & Slovin 2004 and on Schmittau 2004.

The Dougherty & Slovin article describes a US-based (Hawaii) research and development project called *Measure Up* that uses the Davydov approach to introduce mathematics through measurement and algebra in grades 1-3.

Davydov, V.V. (1966). Logical and psychological problems of elementary mathematics as an academic subject. From D. B. Elkonin & V. V. Davydov (eds.), *Learning Capacity and Age Level: Primary Grades,* (pp. 54-103). Moscow: Prosveshchenie.

Davydov, V.V. (1975a). Logical and psychological problems of elementary mathematics as an academic subject. In L. P. Steffe, (Ed.), *Children's capacity for learning mathematics. Soviet Studies in the Psychology of Learning and Teaching Mathematics, Vol. VII* (pp.55-107). University of Chicago.

Davydov, V.V. (1975b). The psychological characteristics of the "prenumerical" period of mathematics instruction. In L. P. Steffe, (Ed.), *Children's capacity for learning mathematics. Soviet Studies in the Psychology of Learning and Teaching Mathematics, Vol. VII* (pp.109-205). University of Chicago.

Davydov, V. V., Gorbov, S., Mukulina, T., Savelyeva, M., & Tabachnikova, N. (1999). *Mathematics. * Moscow Press.

Dehaene, S. (1997). *The Number Sense: How the Mind Creates Mathematics,* Oxford University Press.

Devlin, K. (2000). *The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip*, Basic Books.

Dougherty, B. & Slovin, H. Generalized diagrams as a tool for young children's problem solving. *Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 2004, Vol 2* (pp.295-302). PME: Capetown, South Africa.

Ma, Liping, (1999). *Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States,* Lawrence Erlbaum: Studies in Mathematical Thinking and Learning.

Morrow, L.J. & M.J. Kenney, M.J. (Eds,) (1998), * NCTM Yearbook: The teaching and learning of algorithms in school mathematics.* Reston, VA: National
Council of Teachers of Mathematics.

Schmittau, J. Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy. *European Journal of Psychology of Education, 2004, Vol.XIX, No 1* (pp.19-43). Instituto Superior de Psicologia Aplicada : Lisbon, Spain.

Vygotsky, L. (1978). *Mind in society: The development of higher psychological processes.* Harvard Press.

Devlin's Angle is updated at the beginning of each month. Devlin's most recent book is The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern, published by Basic Books.

Mathematician Keith Devlin (email: devlin@stanford.edu) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR's Weekend Edition.