LC1bA2 Three Encounters with number

THREE ENCOUNTERS WITH NUMBER
The encounters described here focus on approaches to teaching elementary mathematics. As a college student in the 1950’s and later as a computer systems engineer I had met the set theoretic formulation of the calculus and the need to do arithmetic in variously based number systems. The ‘New Math’ was not new to me when I met it in my son’s kindergarten and first grade classes. Issues surrounding the use of set theory as a basis of elementary mathematics mark the first encounter with number.

Entering graduate school at MIT after a decade in the commercial world, I developed some computer-based tools for elementary mathematics instruction. The next two encounters with number relate my son’s experience with those programs. The concluding discussion asks how studies such as this should affect our objectives for math education.

A FIRST ENCOUNTER
Imagine this scene: a kindergarten class, with the teacher and six children standing before the others, three girls on one side and three boys on the other. The teacher instructs the five year olds: “This is a set of three boys, and this is a set of three girls. The number of this set is three. [Pointing at the alternate group now.] What is the number of this set?… Three is the number of the set of three girls. When we make a union of the set of three boys and the set of three girls, we have a set of six children. What is the number of the set of six children?…” This unfortunate woman, an exemplary kindergarten teacher in other circumstances, was a captive of the content. She told me later that she thought the material was dreadful; it was impossible to connect it in any but the most superficial way to ideas meaningful to the children. When one day I joined my son’s first grade class, the lesson of the day was on the intersection of sets. (1).

The following weekend, I asked my son Robby, why he studied math at school. After a very long pause, he replied, “Well… I guess Mrs. Meanswell likes to teach math.” Even though this teacher had the best intentions, she was trapped by the curriculum and could not shape it to the children’s’ interests. When a few weeks later Rob decided to quit school because he already knew how to read and didn’t want to learn any more math, it was clear I had a problem. In first grade he had already concluded that school was a bore and to keep your mind alive you had to tune out the instruction. (2)

If my evaluation of this situation may seem a bit too engaged for objectivity, listen instead to Marvin Minsky, mathematician, computer scientist, and student of human intelligence, in his criticism of the ‘New Math’ (3):

“…By the ‘New Math’ I mean certain primary school attempts to imitate the formalistic methods of professional mathematicians. Precipitously adopted by many schools, in the wake of broad new concerns with early education, I think the approach is generally bad because of form-content displacements of several kinds….

Historically, the “set” approach used in the New Math comes from a formalist attempt to derive the intuitive properties of the continuum from a nearly-finite set-theory. They partly succeeded in this stunt… but in a manner so complex that one cannot talk seriously about the real numbers until well into High School, if one follows this model. The ideas of Topology are deferred until much later. But children, in their sixth year, already have well-developed geometric and topological ideas; only they have little ability to manipulate abstract symbols and definitions. We should build out from the child’s strong points, instead of undermining him by attempting to replace what he has by structures he cannot yet handle….”

The set theory is not as the logicians and publishers would have it, the only and true foundation of mathematics; it is a viewpoint that is pretty good for investigating the transfinite, but undistinguished for comprehending the real numbers, and quite substandard for learning about arithmetic, algebra and geometry….”

The set theory is ‘quite substandard for learning about arithmetic.’

The world famous Swiss psychologist and epistemologist Jean Piaget cautions didacticians about children’s development of mathematical concepts thus: (4)

“It is a great mistake to suppose that a child acquires the notion of number and other mathematical concepts just from teaching. On the contrary, to a remarkable degree he develops them himself, independently and spontaneously. When adults try to impose mathematical concepts on a child prematurely, his learning is merely verbal; true understanding of them comes only with his mental growth….”

Richard Feynman, nobel laureate in physics and an outstanding teacher, observes that practically everyone uses arithmetic, that hardly anyone uses set theory as do the ‘pure mathematicians’, that there is a growing class of users of branches of mathematics of a high form (for example, engineers, economists, business men), and that we should provide an early mathematical training that will encourage the type of thinking such people will later find most useful. In Feynman’s words: (5)

“What we have been doing in the past is teaching just one fixed way to do arithmetic problems, instead of teaching flexibility of mind — the various possible ways of writing down a problem, the possible ways of thinking about it, and the possible ways of getting at the problem….”

“The main change that is required is to remove the rigidity of thought found in the older arithmetic books….”

“Mathematical thinking… is a free, intuitive business, and we wish to maintain that spirit in the introduction of children to arithmetic from the very earliest time….”

I judge these goals Feynman proposes as essential: if we aim to help children be flexible in solving small problems today, it is partly in the hope that they will become productive and flexible citizens tomorrow. But how, specifically, can one hope to approach the goal of introducing mathematics in such a way as to foster the child’s flexibility of mind? And to do so even at the earliest stage, when children do not have a well developed concept of number. One method, proposed by Seymour Papert (6) is to provide even small children “a mathematical experience more like an engineer’s than like a bookkeeper’s.” The next two encounters I report tell of my son’s use of programs in our research laboratory, the Children’s Learning Lab, at the MIT Logo Project.

A SECOND ENCOUNTER
ZOOM is a computer program which even pre-reading children may execute. My intention with ZOOM was to provide children with an engaging experience, wherein numbers would be seen, gradually, to play an important role through being useful to the child in meeting his own objectives. When the child presses a key on the computer terminal, the program generates a command in the Logo computer language to drive an output device, which we call a ‘turtle.’ ZOOM, proposed for use by pre-readers and novices visiting our project, was an interface permitting a newcomer to run the turtle very simply: keying a single letter, F, makes the turtle go forward a fixed number of units; an R or L makes the turtle turn through a different fixed number of degrees; U and D set the pen up or down. This simplification for the novice presents a uniform image on the child’s first encounter with the computer. (7) Even more, he can command actions without first having to comprehend the measures of the domain; this is especially important for those who may have in this system their first encounter with rotational measure, e.g. degrees. But most importantly, the child need not confront numbers at all until he himself generates a goal or objective which he is unable to achieve using the pre-set, default values of the operands. The child develops his own need-to-know before having to confront a new piece of knowledge.

Suppose, for example, the default value of rotation is set at 90 degrees, and the child wants to turn through a smaller angle. The child can modify the default rotation, but doing so requires his confronting numbers as a measure of action. For a pre-reader, this is the teacher’s opportunity to explore the child’s understanding of number and to work out with the child the way numbers are applied in this turtle world. The common technique in our lab called ‘playing turtle’ has the child and teacher, away from the computer, take turns pretending to be the turtle and acting out the directions the other gives. Thus the child has a chance to connect his knowledge of himself, his own body and its movement, with the new knowledge he is learning in the computer drawing world. He can ask and answer for himself the ways in which he and the turtle are similar and different. Let me recount now some details of Robby’s encounter with number through turtle geometry at the Children’s Learning Lab.

When Robby (aged 6 years 4 months) first entered the Children’s Learning Lab, he saw a logo-person about twice his age executing the Lunar Lander program. The visually striking aspects of that program were its drawing of a moonscape (with cliffs, a building, a canyon, etc.) and the use of the display turtle to represent the landing vehicle. Robby decided he wanted to make one of those. Upon starting the terminal session, I explained that we could run a program called ZOOM that would make it easy for him to draw pictures. We printed out the instruction; when I read them to him, Robby didn’t listen. We practiced a little with the basic commands, F [forward 20] and R [right turn 90 degrees], cleared the screen, and then Robby undertook drawing a moonscape.

But there was a problem right away: Robby wanted to start his picture at the left side of the screen; the turtle returns to center screen when the screen is cleared, and whenever it was moved an unwanted line appeared in the display. In solving this problem, Robby and I went to look at a floor turtle (8). On the floor turtle we looked at the pen mechanism, made the pen go up and down, made the turtle draw and then move without drawing. With this concrete example of the states of the pen, I was able to explain that the display turtle also had such a pen, which you could not see, but whose effect was the same.

The process this example exhibits is the following: a frustrated objective becomes a particular problem; solving that particular problem involves finding the most concrete form of the mechanism involved and showing how that mechanism works; finally, the situation in which the problem occurred is re-described in terms which permit the child to connect what he knows and sees with the concrete problem situation. Robby solved the initialization problem and began drawing quite happily from the left edge of the screen. He made a cliff, a landing site, a building… and developed a problem: the width of his building (20 units) was the smallest unit of distance he could command the turtle to move, so he could not make any windows. He let the problem slip by and went on to draw another cliff going down to a canyon. When Rob wanted to make some tiny rocks at the bottom of the canyon, his frustration became very clear. He asked me how to do it. I warned Robby that the explanation would be a long one and asked if he really wanted to know. He did.

Every child has a concrete mechanism for movement, his own body. We played turtle: “Forward,” I said. Robby took a step. “Forward”: another step. “Forward,” I said, “two steps.” “Do you have to tell the turtle how many steps to take?” We went to a different terminal with the floor turtle. I keyed ‘forward 1’ and the turtle twitched. “Did he move?” “Yes, Dad, but he didn’t go anywhere.” I keyed ‘forward 100.’ Robby came to the terminal and examined the printout. He noted that 100 steps didn’t take the turtle very far. The conclusion was clear that turtle steps were very tiny. Back at the display running under ZOOM, I explained to Robby that I had told the turtle to take 20 turtle steps every time the ‘F’ made him go forward. I asked Robby how far he could go in 20 steps. Robby paced off 20 steps and noted, from the hallway, that turtle steps are tiny. After I showed him how to alter the number of steps the turtle would move on a forward command, Robby proceeded to make a few rocks.

Rob then became puzzled again: he wanted to draw a line on a slope, but the turtle always turned square corners. How do you explain to a child so young what degrees are? Robby had seen the turtle turn through 90 degrees when he keyed a single R or L. We played turtle again, both making 90 degree turns when we said ‘right’ or ‘left.’ I told him those square corners were 90 degree turns and I said, when I turn right or left, that’s ‘how much of a turn’ that matters to me… but the turtle is different. Much as the turtle takes very tiny turtle steps, it makes very tiny turns, called degrees, much smaller than our turns. The number 90 meant very little to Robby. He could say the name and count that high, but this was the first time that he had seen it made special by being the number that gets you square corners when you want them. He asked what was a good number to use for making sloping lines. I told him 15 degrees. The number 15 is also special in rotation, for it’s a common divisor of 30, 45, 60, and 90 degree angles. The idea that there are domains with ‘good numbers’ was a definite enhancement of his previous concept of number.

This session continued for over two hours at Robby’s request. He finished his moonscape and was justified in feeling proud of himself. Subsequent sessions with ZOOM were interesting to him, but the content related less to number and more to ideas of controlling your procedures and planning your work and so will not be discussed here. The final number-related point that surfaces in the use of ZOOM (one which re-appears in the discussion of ADDVISOR) is the question of what a ‘correct answer’ is. Recall that 15 is a ‘good number’ of degrees for turning. There is no sense in which 15 can generally be considered a ‘correct’ number of degrees to turn. When the child’s own objective is achieved or missed, he can see that a ‘correct answer’ depends on the domain and his goals. It is important to learn that a ‘correct answer’ is not always the same thing as saying something that makes your teacher smile.

SUMMARY
The central points of Robby’s encounter with ZOOM I now find of interest are these:

– Robby saw his introduction to the computer as providing a new medium for an activity (drawing) he already much enjoyed.
– He came to the computer because it was involved with my work. This fact gave the experience a coloring of social seriousness.
– He developed an immediate objective — drawing a picture of a lunar landscape — by imitating the work of a more grownup kid, as well as he could comprehend that work.
– Because the ZOOM program is a tool whose adjustment is made by changing numbers, his own goals entrained him in problems wherein he had to augment his understanding of numbers.
– He was working independently in the sense that even though I had useful general knowledge, he alone imagined what he was doing and he had to choose those numbers that would solve his problem.

I consider Robby’s encounter with numbers through ZOOM the most successful of the three I describe, and there are good reasons it was so. I was with him as an individual tutor. The milieu was both personally and socially supportive. Robby was executing a skill of a kind he enjoyed (drawing), and he saw that as an appropriate activity for a child in that place. He found the experience sufficiently engaging that on his first encounter he worked at his project for about two hours of his own volition. I was pleased because this experience showed him that numbers are things you can use, not merely things to talk about.

A THIRD ENCOUNTER
Some few months after first being introduced to computers with the use of ZOOM, Robby underwent a considerably different experience with numbers, one more didactic and calculation-oriented than his use of numbers to control a drawing program. Since the program, named ADDVISOR, derived from ideas on left-right distinction appreciation and the kinds of errors children make in learning to add, I will describe the intellectual framework of the programs first and then present the observations on Robby’s encounter with it.

The Framework of Ideas
Teaching children reading is considered very important in our society. First graders spend hours each day at reading; and we teach them to read from left to right. When they approach the task of bi-columnar addition (usually for a short duration every day near the middle or end of second grade) we instruct them to proceed from right to left. Understanding the explanation for this change of direction, if such is ever offered, requires the child’s prior comprehension of addition in a place-value number representation.

Adding right to left is especially hard to understand when the most important numbers are always on the left. Where precision is not required, adults frequently round and estimate by the sum of the higher order digits. Some people retain their preference for adding from left to right. An example: at a nearby pastry store, I purchased two different cakes; one kind cost 54 cents and the other 32 cents. The saleswoman, a woman in her forties or fifties, calculated my bill thus:

50 - 4 
30 - 2 
-------- 
80 
     6 
-------- 
86 

The woman’s use of dashes indicated, of course, separation, not subtraction. This procedure is adequate for everyday use but becomes very cumbersome if you want to add larger numbers, for example, 7438 and 5753.

ADDVISOR
During the course of approximately six hours work, some of it with ADDVISOR, Robby progressed from single-digit sums to adding 7438 plus 5753. He understood that one could add either from the left or the right depending on his choice in various circumstances. He was also able to indicate why right addition with carries is preferred to left addition for large numbers.

ADDVISOR is a program designed to permit contrast between three methods for adding two two-digit numbers (computer scientists call these different methods ‘algorithms’). The contrast is made visible by drawing the numbers and signs on an video display scope connected to a computer (the video display screen of the turtle world). At any time, one or two contrasting forms of addition are displayed on the screen simultaneously. For example, adding left to right may be shown on one half of the screen while adding right to left with carries is shown for contrast on the other half of the screen. ADDVISOR controls, flexibly at the direction of the child, the sequence of algorithm steps. The ADDVISOR program performs NO addition at all; should an incorrect sum be entered by a child, that sum will be displayed and the program will be none the wiser.

Place value is a central concept in our representation of the natural numbers. Children encounter this fact as a problem for the first time in bi-columnar additions for those cases where the unit’s sum exceeds 9. Many children add from left to right; most, I aver, proceed initially on the assumption that the columns of numbers can be added independently of each other. Contrast these sums:

   I         II        III         IV  
  20         23         24         24  
 +20        +26        +26        +26  
----       ----        ----       ----  
  40         49        410         50 

In sum III above, the columns of numbers previously manipulated adequately with the assumption of independence are now seen to interact. Advancing from Sum III to Sum IV may be achieved in two ways. First, the child may accept as true, on expert authority, that following the standard algorithm, which he can be taught, produces a correct result. A better goal is to render explicit and accessible to the novice arithmetician the idea of place value and the interdependence of columns in vertical addition. Such is the design objective of ADDVISOR.

That objective is approached by visibly focusing on the elements that are involved in each step of whatever addition algorithm is chosen by the child. That focus is achieved by placing a box around the digits to be added and the places where the partial sum must go. For examples, (see Figure 1). The visual display manipulation capability of the computer makes possible the easy control of the placement and removal of these boxes in preset locations in a sequence determined by the selected algorithm. This implementation provides a crisp and non-confusing focus on the appropriate single-digit sums. With no intruding corrections of occasional single-digit errors, and with the sequence of operations visibly and rigidly defined, the child is able to elevate his perspective on the process of adding. He can see and contrast the processes as the addition advances from left to right or right to left at a pace he controls by advancing one addition or the other one step at a time.

C&C Fig 2.1a

FIGURE 1 A: Adding from Left to Right

The implementation uses these means to place the maximum of control in the child’s hands. He can pick out whatever numbers he wants to add between 1 and 99. In practice, since children don’t see much reason for choosing one number rather than another, they will be quite flexible in negotiating with some human advisor the numbers to be added. The second important fact is that the child himself does all the addition. Who wants a machine to tell him he made a mistake? As a child comes to see the power of even this arithmetic, he will want to require of himself precision in low level computations so that he may depend upon his result. Parents and teachers should not require that precision as a pre-requisite for learning a different type of knowledge.

C&C Fig.2.1b

FIGURE 1 B: Adding from Right to Left

The child’s step-wise control of the addition algorithms is represented in Figure 1. In the left-to-right addition, procedure boxes are placed around each single-digit addend pair in its turn. All those sums which are the sum of two digits have the shape of a reversed ‘L’. This is a formal requirement, for the sum need not have two digits. Those sums which derive from a single digit (the bringing down of a digit) are enclosed in columnar boxes, for there is no case in which a single digit will generate two in a sum. These simple, formal arguments are demonstrable to a child. Once they are understood, the child can then follow or, as he gains confidence, predict the course the addition algorithm will take on the basis of these local constraints and his earlier decision to add from left to right or right to left. Contrasting the working out of the various algorithms, step by step, makes clear the relative complexity of the three processes. To be able to specify what to do next and to know the consequences of a global decision (which addition procedure will I choose?) constitute understanding a process. You may understand a process even if you make occasional errors in its application. The working out of these algorithms is more like a demonstration than a telling. The computer is used not so much as an ‘aid’ to the student as a medium of construction in which flexible examples can be built to serve as a local domain for exploration by the child.

The execution of ADDVISOR provides examples of three procedures whose sequence is determined by the formal characteristics of the data and not by the variable values of the data items. It further permits a formal definition of an ‘answer’: an answer is the content derived from applying a procedure over and over till you can’t apply it anymore. It is the data content of a state, reached by the re-iteration of a procedure, which prohibits application of the procedure. This sounds like fancy talk; more simply, as a child would see it in Figure 1, an answer is when you have all the numbers in a line. Why do we usually add from right to left? A direct answer is that right to left addition with carries is the simplest algorithm of tolerable internal complexity. Rephrased: it lets us put our numbers on one line right away. ADDVISOR exhibits these ideas in a form accessible to first graders. Let us now proceed to recount Robby’s experience with ADDVISOR.

At the age of six years and seven months, Robby visited the Children’s Learning Lab over a period of four successive days. His arithmetic education in the public schools had progressed uncertainly through most of the single-digit sums; as a secondary procedure, for use when he could not recall a sum, he would resort to representing the individual numbers as hash marks and counting the total to determine the sum. He inclined to estimate the sum of two two-digit addends as ten times the sum of the leftmost digits. (He said later that this idea was his own invention.) I asked if anyone had ever pointed out to him how the sum you reached in one column of numbers might affect the sum in another. He said that no one had done so. Whether his recollection was accurate or not, it was clear that he had not been able to absorb and comprehend that fact. I will report Robby’s encounters with ADDVISOR as seven episodes (four involving use of the computer and three not so). For convenience, I will label them by the day and time of day of their occurrence.

Adding with ADDVISOR
Sunday Evening:
On the drive to Cambridge, I told Robby I had written a program to help him learn about adding. At Logo, we started up the computer and read in the programs. Robby chose to add 50 and 65. He could read the words ‘LEFT’ and ‘RIGHT’ displayed above the columns of the addends and understood when I told him he could start adding either with the left or right column of numbers. He chose to add left to right. Pushing start caused the display of a reversed ‘L’ shaped box around the digits of the ten’s column of the numbers (see ‘A’ of Figure 1 for an example). I explained the purpose of the box: first, to show the digits to be added next; secondly, to make a place for the sum of those two digits. With a selected example, on scratch paper, Robby showed himself how adding two digits could sometimes lead to a two-digit sum but that sometimes the sum was only one digit. At this point, I used ADDVISOR’s ignorance of how to add single-digit sums as an explanation of why it always allocates two digit positions for the result of a two-digit sum. (9)

After he was satisfied that he understood why the L-block was used, Robby pushed start, thus directing the computer to ‘do the next thing.’ The computer printed on the terminal log ‘5 + 6 = ?’. Responding ’11’ and carriage return posted his sum to the display screen in the allocated open spaces of the L-block and put slashes through the numbers 5 and 6. Pressing start and ‘doing the next thing’ removed the L-block from the screen. These three steps, then, comprise a complete column addition: box placement, digit addition, box removal.

Upon completing the ten’s column addition, I asked what we would add next. Clearly, we would add the unit’s column. Well, then, where will the box go? Since the addition result might have two digits, ADDVISOR must make room for two digits while avoiding overlaying the rightmost digit of the intermediate sum from the ten’s column (see the placement in ‘B’ of Figure 1). When the unit’s column addition was completed, I asked whether we had an answer, whether or not we were finished. The visible test is that the sum for all the columns are on the same line. Following such steps and arguments we continued till we reached an answer after three addition passes from left to right.

Since the unit’s column sum of 50 and 65 was less than 10, no inter-column interaction occurred. I asked Robby if we could add from left to right again. He agreed and let me pick two numbers, 76 and 25. While re-initializing the program, I asked Robby if he could figure out what the answer might be. His guess was that it must be in the nineties because 7 plus 2 equals 9. He thereupon wrote down the addends and proceeded, as in Sum III above, to produce the sum 911. He noted that it could not be correct because 900 is ‘way too big.’ Oh, it’s 91. From this state of uncomfortable speculation we proceeded to adding from right to left.

The unit’s column addition proceeded with no complication. However, when I asked what sort of block would be applied to the ten’s column and where it would go, Robby said, “Oh, it’s impossible. I don’t want to do it.” This was the situation.

   7 6          7 6 
  +2 5         +2 5 
  -----       ----- 
   1 1          9 
                1 1
              ----- 
              1 0 1 
   R-L         L-R

We started up an addition on the second side of the screen and performed addition using the same numbers (76 and 25) from left to right. We worked through the three addition passes from the left and reached the sum ‘101’. Robby expressed surprise that the sum was not in the nineties. When I asked why it was not, he looked back at the display and said that you had to add the ‘1’ from the ’11’ and that 9 and 1 make 10.

It was hard to interest Rob in returning to the first side of the screen. Once I forced the issue by going on, he became caught up in the details of the process, predicting where the next box would go, whether it would be L-shaped or columnar, performing the additions of the single digits. Robby was quite surprised again on coming to the end when he realized “it’s going to be the same.” He apparently had made the natural assumption that if you manipulate numbers differently you get different answers. Having contrasted the adding of the same set of numbers, we were able to see one result and judge the relative difficulty of the two procedures. Robby, although admitting that left to right addition was harder, said he preferred that method at the end of our Sunday session.

Monday Morning
We negotiated the numbers for adding. Robby picked 24 and 25 but willingly accepted 27 as a substitute for 24. He either failed to see the coming complications or did not care about them. He freely chose to begin adding from the right but would not venture a guess as to what decade the answer would fall in.

Robby worked directly through to the sum 52. Before we began adding the same numbers from left to right, he predicted the answer would be different. Thereafter, with both forms before him, he understood in this case why the answers were the same. I asked him why there was a ‘4’ (the intermediate result of the ten’s column addition) in the left-right addition while no ‘4’ appeared in the right-left addition. He explained, pointing to the right to left form, “We’ve already added in the 1 over here.” Pursuing the point, I asked whether the answers should be the same or different. Robby said, “They should be different; we did it different ways.” I objected, “But the numbers are the same, 27 and 25. Should they always add to the same sum?” Robby agreed, “It has to be the same.” I continued, probing to raise the idea of checking an answer: “What does it mean if you did it different ways and got different answers?” Robby responded, “It has to be a different problem.” Robby had become restive at this point and turned his attention to drawing a steamroller with ZOOM.

Monday Afternoon
Some six hours after the morning session, I put the numbers 46 and 27 on the chalkboard in my office and asked Robby if he could perform the addition. I continued with my other work.

C&C fig. 2.2

Robby started at the left, writing 6 in the ten’s column. To perform the unit’s column addition, he spontaneously wrote an L-shaped box on the chalkboard, then filled in the intermediate result, 13. He continued by erasing the box, drawing a second horizontal line and summed the 6 and 13 to 73. I asked if that was the correct answer. When he vacillated, I asked if he could add the sum the other way. Robby started on the second addition of the same numbers from the right. He used an L-shaped block and placed the intermediate sum 13 in the appropriate place. I erased that L-block for him because the lines were very close to all the digits. He placed another L-block in the ten’s column, then rewrote the ten’s column addends as a horizontal equation: 4 + 2 + 1 = 7. Then he wrote the sum, 7, in the L-block in the hundred’s position. I asked, “Is that right?” No answer. “Not quite, Robby, put the 7 over there.” Robby replied, “I don’t want to do it anymore.” To my question why, he replied, “I know what it’s going to be, 73.” I asked, “Is that right?” He answered, “Yes.”

Monday Night
Out to a late dinner with him and a friend, I asked Robby to explain what we had been doing that day. I wrote 48 and 27 on a 3 x 5 card. Robby added left to right, using a single L-block for guidance in the unit’s column addition. (He found it necessary to count hash marks to sum 7 plus 8; this is not surprising since he had been up and active for fifteen hours at this time.) When I asked if his answer was correct, he said he was pretty certain.

I then re-introduced 76 and 25. Abandoning the previous method, Robby wrote 9 under the ten’s column and on a lower horizontal line, both digits of the 11 under the unit’s column. Then in his second-pass addition, with that alignment, his answer came to 911. His perplexity was immense. Robby recalled then the work of the morning and his answer ‘101’. When asked to show why he thought 101 was the correct answer, Robby started marking down hash marks. He kept going till 76 (more or less). We asked if there was some easier way to show us. Robby then explained that the answer would be in the nineties except that we had to add the one from the eleven and that made the nineties a hundred.

Tuesday
After the demanding day that Monday had been, Robby showed resistance to using ADDVISOR. He agreed to add 16 and 18 from the left and 25 plus 46 from the right. When I asked him which method he preferred, adding from the left or right, he replied ” any one”. Robby refused to work anymore on addition. Failing to engage his interest in the evening session, I held him to the task only long enough to show him one rapid demonstration of adding right to left with carries.

Wednesday Morning
Engaging Robby’s interest remained a problem. He explained he was mad at me because he wanted to go to the Children’s Museum and that would be more fun than doing sums. After agreeing to take him there later, I proposed as candidate addends 97 and 56. Robby balked but agreed to go on with 25 substituted for 56. Although the procedure for adding with carries, which Robby chose to execute first, is shorter than the others, we made slow progress. We dealt explicitly with the fact that only columnar blocks enclosed the column to be added, the fact that lack of space for the second digit, the ten’s digit of the intermediate result, required use of something called a ‘carry’. When Robby keyed ’12’ as the sum of ‘7 + 5 = ?’, only the 2 appeared on the screen in the unit’s position. Where did the 1 go? I explained that the 1 was not used by ADDVISOR, but that Robby could use it to answer the question he now confronted (did he have a carry?), for the carry is the number left over, the digit you don’t have room for in the line where the answer will be. Robby said he wanted a carry. We went on together and concluded with a correct answer. Robby wanted to stop and not perform the contrasting right to left addition because ‘it will be the same.’ When I asked if he could prove that, he agreed to humor me. At our conclusion with the same answer, 122, I asked Robby which was the best way to add. He replied ‘carries’, but when I asked him why, he was non-committal. He said he understood carries now and wanted to stop. With that short exposure, I doubted his understanding and asked if he could add with carries on the chalkboard. “Sure.”

At the Chalkboard
Robby picked the numbers 96 and 25 for adding and wrote them in the vertical form. He placed a columnar block around the unit’s column, wrote a 1 in the unit’s column and a carry over the 9. He went back to review the work on the display screen (I believe to avoid having to figure out the sum of 9 + 2 + carry) and then wrote a 12 on the answer line. I asked if he had a carry there, in the ten’s column. After some pause, Robby said, “Yes, but it doesn’t matter; we would write it down in the answer anyway… and am I done now?”

I said we should do one more sum with carries and wrote on the chalkboard 9768 plus 5844. Robby laughed and said nobody could do that. I said we could together, that I would write down the answer and carries if he would do the adding. We proceeded directly to our answer. Robby was astounded: we had made a number over 15 thousand. This was clearly at the upper reaches of his imagination of magnitude. He asked if we could do another. I chose 856 plus 376. After we completed this sum together, I said he should work out the next one all on his own: 746 plus 365. He asked for another when done; I selected 857 and 424 (these addends have no carry out of the ten’s column) which he also added correctly. In all these additions, Robby used columnar blocks to keep his work organized and aligned properly. He was obviously delighted in witnessing the power of what he had learned and went to find a friend, Danny Hillis, another Logo lab member, to tell him of his discovery.

Later in the day, I asked Robby if he remembered the addition that made more than 15 thousand; could it be added starting from the left? He replied that it could and when I asked how long it would take, said, “Probably a year.”

Conclusions
Two days after the computer sessions at the Children’s Learning Lab, I asked Robby to add 7438 and 5753 on a chalkboard while I did some bookwork. When he finished, each column was enclosed in a columnar block, the carries were marked in the standard places, and the answer was correct.

I am convinced Robby experienced a conceptual breakthrough. The evidence is both from performance (getting the answers correct consistently in all algorithms) and from explanation (he knows that there are at least three variations on two addend adding and in what circumstances he prefers to use one form over another).

To what extent is this experience a special case? Let me first summarize Robby’s prior developmental position:
– he was capable of adding single digit sums, though he was uncertain of results beyond 7 + 7; for larger numbers he used the alternate procedure of counting hash marks.
– he could count beyond one hundred and easily read three digit numbers; with difficulty, he could read numbers in the thousands.
– he showed a useful sense of magnitude; for example, he could recognize that ‘911’ was the wrong sum for 76 and 25 because it was far different from his estimate of ‘in the nineties.’ Without this last perception, he would not have recognized the inter-column interaction which is central to our representation of the number system.

I believe that a child with these three capabilities is ripe for the kind of experience Robby had at the Children’s Learning Lab, and in addition that this developmental configuration is common, although the age at which it is reached may vary. Consequently, I see nothing special in Robby that would make him specifically suited to having such a conceptual breakthrough.

With regard to the circumstances of this work, there are four factors that one might examine for significance:
– as Robby’s father, I have a special influence over him (as his father, it is clear to me how over-estimated this influence may be).
– the Logo project is a small sub-culture where numbers are much more important than they are in the world at large.
– Robby’s encounter with the material was intense and compressed in time.
– the implementation of the ideas was computer based.
I believe factor 2 is the critical one, that Logo is a place where numbers have meaning through applicability and are important enough to worry about. You need numbers to make the turtle turn or to move forward. You need numbers to draw on the display screen or to play computer games. Big numbers are as common as little numbers, and big people use numbers a lot. Logo is home ground for the natural numbers and a place where Robby feels at home. His sense of belonging at Logo may derive from my involvement there but it is broadly supported by other people’s interest in him and in what he thinks. The intensity and time compression of the experience are an accident of his spending only a few days at the Children’s Learning Lab and simply made it harder for him to endure. The method of teaching addition embodied in ADDVISOR is not bound to computers but would be much harder to put into effect without them.

If this encounter, as the one with ZOOM, represents a beginning mathematics education in an ideal environment, why did Robby occasionally show resistance to using the resources of the Children’s Learning Lab? I could argue that we’re all cranky sometimes and so on, but there’s more to it than that. There’s a common dilemma with new ideas introduced in school: one can’t really appreciate the value of an idea until it’s understood; but why would anyone want to bother with an idea unless one can appreciate its value? Social support, such as the community of the Logo research lab offers, is one answer to this problem of education, but such a community is accessible to only some of the people some of the time.

Back in School
After his encounter with ADDVISOR at the Children’s Learning Lab, Robby returned to his school. During ‘show and tell’ the following Monday, he explained to his classmates how to add two three-digit addends (he noted that he only gave two examples because he didn’t want to take up all the time). Commended for his skill, Robby was then informed that “in first grade we never add any numbers with a sum greater than 12.” The message was clear to him. Robby later told me that if I really wanted to help him with math I should teach him to do single-digit additions quick so he could score points for his team of boys in their competition against the girls. In his everyday school world, his development was denigrated and his skill was made to appear of no use whatever.

Discussion
One can marshal arguments in favor of teaching a limited set of single-digit additions in first grade:
– it exemplifies an incremental approach.
– it provides a basis of well known facts from which the child can build up his ideas of addition.
– one must be able to add ‘8 + 7′ before one can add ’87 + 78’.

On the other hand, limiting the computation problems one poses to children by the sum ’12’ exhibits a profound miscomprehension of the character of arithmetic, in this case addition. My contrasting view follows.

One basic process is computing single-digit sums, e.g. 2 + 6 or 8 + 9.

A second process is integrating results from these computations. The fact that a child, or all children, can make occasional errors in single-digit sums does not restrict the ability to comprehend or execute the second process. Would anyone argue that the error in this sum, 468 + 357 = 823, is any worse than the error in this, 8 + 7 = 13 ?

The central fact of our natural number system is the multi-digit representation of numbers. One might argue against computing sums of value greater than 9 because this might entrain you in explaining why one needs two digits to represent the number 10 (understanding that fact may require sophistication beyond the reach of most six year old children; but most seem to accept without too much argument the representational fact). Twelve, as an upper limit, is hard to defend unless seen as the sum of fingers and ears (or some other such absurdity). In contrast, consider this sum:

     3 4 6 9 2 7 5 
   + 8 5 3 0 4 1 4 
   ----------------- 

Because there is no column interaction (except at the left where it doesn’t matter), any child who can sum the columns can compute the sum correctly. Notice also that the child can add left to right or right to left as he wishes. Why do we radically restrict the computational range of what we teach? It is precisely that range that expresses the power of the hindu-arabic notation. Recall Robby’s delight in finding that he could add sums as great as he could read. We can easily teach a few-step, direction-insensitive addition algorithm, thus:

     3   4   6    9    2   7   5 
   + 5   5   5    5   5   5   5 
  ------------------------------- 
     8   9  11   14   7  12  10 

Now, one must start at the right and check for ‘bugs’. If we have a two-digit sum in any column, that’s a ‘bug’; the ‘1’ must be added to the next left column. Thus we get:

      8   9  11  14   7  12  10 
          1   1       1   1  
    ---------------------------
      8  10   2   4   8   3   0

And here, we see, we have to check for ‘bugs’ again; so our answer is 9,024,830. (10) This proposal, at the level of introductory arithmetic, admits that humans operate with imperfect procedures, the results of which are subsequently fixed up by correcting errors. Another way of expressing this idea is that one should add starting with the most important numbers and making the best estimate you can while ignoring interactions of terms, then correct those estimates by
accounting for the interactions caused by the place value representation of our number system. The correctness and precision of this computation can be seen as a process of getting closer to the ‘correct answer’ by successive refinements. A richer appreciation of this process notices that it embodies the assumption of linearity — that complex processes can be broken up into separable parts for better understanding. This assumption is basic to much scientific reasoning, even of the most sophisticated kind.

These last few comments make it sound as though one must know a lot of mathematics or computer science to help children deal with numbers. Not so; not so at all in this case. The essential idea is doing what comes naturally… adding from left to right and correcting the flaws of that procedure when they occur. Remember the lady who sold me the pastry. She added from left to right and did it well. Remember Robby’s estimate that 76 + 25 must be in the nineties because 7 + 2 equals 9. The former shows the usefulness, the latter holds a key. A mathematician has a strong result if he can place bounds on a computation. For a child who has been working on 5 + 6, to realize that he can bound a seven digit sum may be an engaging result. For example:

    5 7 9 6 5 2 7 
  + 6 6 1 4 7 2 3 
 -----------------

The sum will be more than (5 + 6) million and less than (6 + 7) million. The sense of power a child can get from controlling, even approximately, large numbers, may inspire him to improve his single-digit computational skills leading him eventually to refine his estimates to perfect results.

GENERAL CONCLUSIONS
The conclusions I draw from the three encounters with numbers I’ve described are of these sorts:
– even granting that the earlier attempts at curriculum reform have failed does not imply that one should go ‘back to the basics’ if those ‘basics’ do no more than constrain the range of computation within which children practice rote memorization. The desirable goal, even in the earliest training, is to foster a flexible, results-oriented approach to calculation.
– a good way to introduce children to numbers is to provide them a medium of expression, within which, as their objectives grow more discriminating, their involvement with numbers must increase.
– one should distinguish between the need to memorize single-digit
additions and the child’s ability to comprehend and execute the assembly of those single-digit additions to multi-digit sums. There is no reason, as Robby’s rapid success with ADDVISOR shows, to demand that children should be capable of the former before confronting the latter.
– a natural way to introduce children to multi-digit addition is through estimating the bounds of impressively large sums; a second step is to introduce left to right recursive addition as the basis of primary school computation; the intention of such an approach would be to give children a sense of the power of being able to deal with large numbers and a sense that one discovers ‘correct answers’ by a process of refining first estimates.

But finally, the rigid reception Rob encountered on his return to school is a simple indication of how much inertia exists. It may be important to take seriously Piaget’s assertion that children learn about number by themselves more than they are taught by would-be teachers. What those circumstances of natural learning are like is our next theme.

Footnotes:
(1) For example, a pea is in the set of ‘all green things’ and more or less, in the set of ‘all round things’; we say these two sets, ‘all green things’ and ‘all round things’, intersect because they have in common at least one member, as exemplified by a pea. That’s all there is to it.
(2) My sympathy with Robby was that of fellow sufferer as much as father. Despite my entering Caltech as a freshman with a perfect score on the ETS Advanced Math Test, I had difficulty with the set theoretic formulation of calculus that was then being introduced to college freshmen. This new math had gone from college material to public school kindergarten fare with no increase in success. I believe an abstract approach to mathematics was a disservice to me as a student and has been so to many another besides. Yet I was eighteen and capable of formal thought. What a worse disservice to subject children, whose mental capabilities have yet to master formal operations, to a mass of incomprehensible abstractions.
(3) The following excerpts are all drawn from a more extensive discussion in “Form and Content in Computer Science” (Minsky, 1969):
(4) “How Children Form Mathematical Concepts”, Piaget, 1953.
(5) These comments are drawn from “New Textbooks for the ‘New’ Mathematics”, Feynman, 1965.
(6) The founder of the MIT Logo project. Papert’s early vision was set out in three papers. See Papert 1971a, 1971b, and 1973.
(7) More specifically, each action is represented by a single keystroke. Further notes on such single key interfaces appear in Lawler, 1982.
(8) A device controllable by the same commands which moves and draws on the floor instead of on a video display.
(9) In this case, the unnecessary stupidity of a computer-based tool is the justification for making an educational point.
(10) This proposal embodies the idea that left-right addition is a recursive procedure, a technical observation made by Howard Peele, Professor at the University of Massachusetts at Amherst, in a discussion of an earlier version of this paper.

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