Ubiquitous Teaching

In the couple of months since my last post, I have been trying to incorporate small dabs of problem solving throughout my interactions with Suhana. These small doses are designed to be non-intrusive, yet, taken together, I believe they encourage thinking.

Suhana and I have created a cast of characters that appear in bed-time stories and in problems. The stars of our stories are two six-year old siblings, Ram and Chetty; Alpha, Bravo, Charlie, Delta, Echo, and Juliet (respectively, an alligator, a bee, a cat, a dog, an elephant, and a pheasant-tailed jacana); Sher Khan, Bhaloo, and Holly the Hippo. Consistently using the same characters has allowed a rich narrative to develop, including typical things these folks do. For instance, Ram and Chetty are farmers and they have to travel to sell the Bhindi they grow, and their travels and day-to-day problems form a natural backdrop for setting up arithmetical problems.

In a recent bed-time story, for example, Ram wanted to grow Bhindi as a birthday gift for that all important day, May 13, Suhana’s birthday. It happens that Bhindi takes two months to grow. So when should he plant it? This very tiny puzzle can be presented with zero intimidation. My actual words were: “Ram knew that Bhindi takes two months to grow. So he thought ‘When should I plant?’”, followed by a pause. In this case, Suhana counted backward and said “March”. If she had not said that, or ignored the pause, the story can still smoothly proceed to how Ram found the answer. Similarly, Ram knew that Bhindi should be planted two days before it rains, and the weather prediction is for rain on Sunday. So what day did he plant?

Every night, the bed-time story lasts about fifteen minutes and invariably involves some computation. Recently, Chetty had five toys but Ram only three. So Chetty gave some to Ram so that they had the same number. How many did she give him? I like this problem because the answer is not one of 5 - 3, 5 + 3, 5 * 3, or 5 / 3. Suhana came up with the right answer. Had she not (if she said “2”, for instance), the story could have smoothly continued with “yes, you know, that is what Chetty thought and so she gave Ram two toys and then counted both their toys. Do you know what she found?”. No right answer can be made to work even when there is  a right answer.

One final example from this morning to justify the “ubiquitous” in the title. I was working and Suhana kept pestering me. On a whim, I opened an excel stylesheet, her first exposure, and let her play around. My computer does not have a mouse, and she used the arrow keys to move the active cell around. We randomly typed stuff in various cells, and at one point she decreed that everything on the right was her territory (pointing to column F onward), and the left portion was mine, and that we should only change things in our area. When it was my turn, in cell C3, I typed “= G3”. Naturally, when she modified G3, her words appeared in C3 as well, and we had a mock fight. I then explained what I had done. Not much of my explanation was likely to stick, but the entire exercise made her more aware than before of the two-dimensional coordinate system, and we played a game of naming a cell (say, “F8”) and navigating there using arrow keys.

I then set up a stylesheet corresponding to the story “Ram had five toys, Chetty had three. Chetty asked her mother for more. Her mother said that she would get Ram the same toys she got Chetty. In the stylesheet, there was a column for Ram (with a sum at the bottom) and one for Chetty (with a sum), and the sums were 5 and 3 at the start. The stylesheet was set up so that any number added to Chetty’s column was copied to Ram’s column and no matter what Suhana tried, the sums of the two columns stayed two apart.

What did Suhana learn from this? Probably nothing that can be pin-pointed. However, even the realization (even the very wispy almost non-existant realization) of a coordinate system and the very vague realization of the possibility of using such tools to set up games or to solve problems, even if she has no clue how, are very very slightly useful, I think.  The other possibility of course is that I am just deluding myself.

In any case, all our stories share the common characteristic of being quite silly and yet just plausible in the right sort of imagined world given the right background story. We have plenty of fun along the way, so even if I am deluding myself regarding the edification I provide, not all is lost.

Cognitively Guided Instruction (CGI)

I have not forgotten my promise to talk about research — I just haven’t been able to figure out how to chop what I was reading into byte-sized chunks. Instead of summarizing an area of research, in this post, I will summarize a good book from this area, namely, “Children’s Mathematics: Cognitively Guided Instruction“ by Thomas Carpenter, et al.

The first sentence from the cover nicely captures the basic premiss of  Cognitively Guided Instruction (henceforth, CGI): “By the time they begin school, most children have already developed a sophisticated, informal understanding of basic mathematical concepts and problem-solving strategies.” Indeed, kids seem quite capable of using concrete modeling using fingers, tokens, and such to solve problems that, judging purely by the mathematical knowhow they posses, should be beyond them.

As an example, this morning, Suhana solved this problem: “Suhana had 15 cookies that she distributed equally to Anna, Betty, and Calvin. How many cookies did each get?” Suhana is unfamiliar with division. Quite possibly, she has never heard the word used in the mathematical sense. But using tokens (poker chips, in case you are curious), she distributed fifteen tokens into three rows (she put three in each row and then distributed the rest), and could easily tell that each will get 5 cookies. Just because grown ups reach for division for this problem it does not follow that division is needed for this problem. Carpenter et al. are ascribing just such inventiveness to children, and have over two decades of classroom observation to back them up. Far from unusual, what Suhana did is par for the course.

The authors lay the groundwork in the first chapter, “Children’s Mathematical Thinking”. It begins with three problems that, to adults, are all merely instances of “eight minus three” but to kids are quite distinct. The distinct strategies children routinely use to solve these “identical” problems lays bare their differences.

1. Eliz had 8 cookies. She ate 3 of them. How many cookies does she have left?
2. Eliz has 3 dollars to buy cookies. How many dollars does she need to earn to have 8 dollars?
3. Eliz has 3 dollars. Tom has 8 dollars. How many more dollars does Tom have than Eliz?

One first grader solved these three thus, respectively:

1. She layed out 8 tokens, removed 3, and counted what was left.
2. She layed out 3 tokens, added tokens until 8 tokens were laid out, and counted the new tokens.
3. She laid out two rows of tokens side by side, 3 and 8, and counted how many more the longer row had.

A teacher using CGI listens patiently to the students solving problems and to their explanations. Students are invited to describe their strategies, and they learn from each others strategies. Over time, richer strategies develop and the need to use concrete objects gives way to more efficient paper based strategies which in turn give way to using numbers and to “derived number facts” (that is, recognizing that 6 + 8 is the same as 7 + 7, which the child may already know is 14).

The point to emphasize here is that “[children] do not need to be taught specific derived facts. In an environment that encourages children to use procedures that are meaningful to them, they will construct these strategies for themselves.”. Different strategies are suggested by different problems, and exposing children to a range of problems is beneficial. Chapters 2 and 4 lay out one way to classify addition/subtraction and multiplication/division problems, respectively. Here, I will merely list one example of each class of addition/subtraction:

• Join
• (Result unknown) Connie had 5 marbles. Juan gave her 8 more. How many marbles does Connie have altogether?
• (Change unknown) Connie had 5 marbles. How many more does she need to have 13 marbles altogether?
• (Start  unknown) Connie had some marbles. Juan gave her 5 more. Now she has 13. How many did she have initially?
• Separate
• (Result unknown) Connie had 13 marbles. She gave 5 to Juan. How many marbles did she have left?
• (Change unknown) Connie had 13 marbles. She gave some to Juan. Now she has 5 left. How many did she give Juan?
• (Start unknown) Connie had some marbles. She gave 5 to Juan. Now she has 8 marbles left. How many did she have to start with?
• Compare
• (Difference unknown) Connie had 13 marbles. Juan had 5 marbles. How many more marbles did Connie have than Juan?
• (Compare quantity unknown) Juan had 5 marbles. Connie had 8 more than Juan. How many marbles did Connie have?
• (Referent unknown) Connie had 13 marbles. She has 5 more than Juan. How many marbles does Juan have?
• Part-Part-Whole
• (Whole unknown) Connie has 5 red marbles and 8 blue marbles. How many marbles does she have?
• (Part unknown) Connie had 13 marbles. 5 are red and the rest are blue. How many blue marbles does Connie have?

Chapter 3 details  the strategies children routinely invent for addition/subtraction problems, and chapter 4 does the same for multiplication/division problems.

Chapter 5 is a defense of such modeling strategies used by children, and gives examples of seemingly far more complex problems that very young children can solve using modeling: “19 children are taking the minibus to the zoo. They will have to sit either 2 or 3 to a seat. The bus has 7 seats. How many children will have to sit 3 to a seat?”. The authors deplore that “It may seem obvious that children would attempt to perform the action in a problem if they have no other way to solve it. Many older children, however, appear to approach word problems by looking for superficial clues to decide whether to add, subtract, multiply, or divide. In other words, many students seem to abandon a fundamentally sound and powerful general problem-solving approach for the mechanical application of arithmetic skills.”

Subsequent chapters deal with multidigit concepts and furnish suggestions for starting to use CGI in a classroom. The appendix provides concrete numbers demonstrating the effectiveness of CGI in classrooms.

This, then, was a brief taste of what CGI is. Standard disclaimers such as “I am not a practitioner, perhaps I am even an outsider to education” apply. I liked the book — and if you are trying to teach math to young children, I whole-heartedly recommend this book.

A small measure of fun

On each of the last two days, Suhana and I played an ad-hoc game using a small dry-erase board, some markers, and a standard die. We wrote the numbers 1 through 6 along the left edge and each roll was recorded with a dot in the appropriate row. Nothing fancy. There was no target concept to be taught — just plain frolic and pointless activity of the sort Suhana enjoys.

Despite the non-targetedness (or should I say “because of the non-targetedness?”) plenty of opportunities showed up to show Suhana some very tiny bit of math.

• At one point on the first day, the distribution of the numbers was nearly even — each of the six numbers had four or five dots. And we talked and wondered about it.
• A bit later, naturally, it was much more uneven. And we talked and wondered about that.
• Midway through, it was decreed that the number that reaches 13 dots would be the winner. I drew a vertical line to indicate “5 dots” and a second vertical line for “10 dots”. At one point, Suhana counted the number of dots in front of 4 — 8 dots. I then showed her how the vertical line speeded things up — we already know that there are five dots to its left, and it is easy to count up from there. Her face lit up, and she used this strategy several times.
• Today, Suhana insisted on playing the game again. She set up the board, and I was surprised that she drew the two vertical lines and labeled them “5” and “10”. Relaxed playing turns up opportunities to demonstrate “best practices”.
• Today, “Suhana’s number” — she is crazily in love with 4 because it is her age — was not winning. “2” had 11 dots, whereas “4” merely had 9. Suhana was in tears. A toy horse with a twisted torso happened to be lying around, and I tried to show her the concept of a “loaded horse” by repeatedly tossing it and showing that the same side tended to be on top. Although the point was entirely lost on her, her crying stopped.
• I confess that I then cheated and made 4 win.
• Along the way, I set up a puzzle. I hid two dice and told her that their top faces had a combined total of 7 dots and that one of them had 3 dots. How many dots were face up on the other? She could not figure this out and was getting frustrated at her inability and at this intrusion.
• One of these days, I will set up an analogous 2-dice game where the sum is noted after each role. The puzzle should turn out to be a lot easier in that context. Plus, if Suhana warms up to that game, she will get plenty of practice of small number addition. The relative rarity of 2 and 12 and the many ways of reaching 7 can perhaps be seen. I should get dice of two different colors.

To summarize, we had a great time!

Unreflectingly Applying Known Steps

Encouraged by the non-zero readership of the previous post, here is the second chapter in the adventure. Thanks for the comments, and thanks, CJH, for the title!

Hypothesis About One Cause of Misperception

I will use the blanket term “misperception” to refer to a hasty misinterpretation of a problem and applying a familiar — habitual — operation to reach an answer. A motivating example is Suhana’s response “8” to the problem “There were five birds in a tree and three flew away. How many are now present?”

I hazard a statistical argument for why this might happen. In a nutshell, I argue that this strategy of quickly judging a problem is quite efficient for the tasks the student typically faces.

The mind-numbing homogeneity in problems is nicely demonstrated by the Kumon’s 80-page magnum opus “My  Book of Simple Addition”. I urge you to briefly pause and guess what the 700 addition problems  might be. It turns out that the problems range from “1 + 1” all the way through “28 + 2”. That’s right: the second number is always “1” or “2”. You don’t even see “2 + 28”. Mechanical strategies are perfectly adequate here! I haven’t read “My Book of Simple Multiplication”, but hope that it’s problems do not range from 1 * 0 to 28 * 1.

From elementary education, let’s turn to something a bit higher — Grade X in Maharashtra, India, where I went to high school. There is a well-defined pattern for what questions are asked. Question I has eight parts of which you may answer any six. One subquestion usually concerns finding the determinant of a 2x2 matrix, another subquestion is to simplify a poynomial ratio (where the denominator has degree 2), and a third is to find the HCF or LCM of two small-degree polynomials. Recent question papers can be found here. Who needs to understand math to excel at such an exam?

The homogeneity also extends to certain word problems. It makes possible tricks for guessing what the operation to be done is — multiply or add or subtract? An example from a random website:

The third step is to look for “key” words. Certain words indicate certain mathematical operations. Below is a partial list. Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

Addition

increased by more than combined, together total of sum added to

Really? “increased by” means addition? How about “I had a few balls. My collection increased by 3. Now I have 7. How many did I have?”. The answer ain’t 10. I don’t even have to use such esoteric examples as apparent magnitudes of planets “increasing by 1” — magnitude is a log scale where smaller values represent brighter objects. Such triggers only work in a limited context, but if one is exposed primarily to that context the trigger words appear quite potent.

Sprinkling Variety

To the extent that homogeneity is the cause of these ills, variety is the cure. If there is enough variety, the argument goes, then no automatic maladaptive strategy will get sufficiently reinforced and habituated.

I have started doing word problems with Suhana. I print these out, and making use of her recently increased fluency in reading, let her read the problems. The problems are tailored to include things in her world — real and imagined. I have been focusing on asking problems that force her to pay attention to what words mean.

Here is an example: “Suhana had three balls. Reesha had two bats. How many balls are there?” Can’t get simpler than that! Other examples I tried initially included “Three princesses had to go to the ball. They had seven socks in the closet. Each princess wore two, so how many socks are left?” and “Baba had two cats. They had a few kittens. Now Baba has seven cats. How many kittens were born?”

Teething Troubles

I have nothing insightful to share about “attention spans are short, distractions are many, I don’t want to push” except to note that it appears to my biased eyes that excitement is contagious and that doing things with a parent is a draw.

Some Success

Today, Suhana solved, with no help, this: “Suhana took six toys to the park. Reesha took four toys. Sadly, the girls broke five toys and brought the rest back home. How many did they bring back home?” She did not use the words add or subtract — those words are not really needed.

That's all for tonight, folks!

Teaching Suhana Math

How does one go about teaching math to a 4-year old? Last week, when asked “There were five birds in a tree and three flew away. How many are now present?”, Suhana gave the understandable answer “8”. How can this tendency to do things mechanically be countered? I have too often seen such jumping-to-conclusions in adults — “I think we should add those probabilities, or I think I should just take the product of probabilities” and I have seen similar guesswork from programmers with impressive resumes working on  algorithmic problems during job interviews, and doubtless I have blundered similarly on countless occasions. Being able to see things clearly and having the patience and discipline to not “solve-problems-by-coincidence”, is I think a necessity for deeply understanding math. How does one teach carefulness?

I haven’t have a clue, but figured that now was a good time to start teaching Suhana and to see what works. I anticipate an exhilarating ride and I am unsure if that anticipation of exhilaration bodes well or bodes ill for this endeavor. I plan to document as I go along. The depth of my ignorance of teaching has started to show in my early attempts, as I will cover in the next post.

My partner-in-crime is my wife Shweta, an Assistant Professor of Math Education at Kean University. She has been watching my early forays with bemusement, and has started giving me literature from the field — empirical studies and multi-decade field work from the trenches. This, too, I shall talk about in the next post.

“Weighted” Pigeonhole Principle

One of the greatest joys in problem solving is to solve a puzzle and realize that the solution is an unexpected variant of something very familiar. I solved a puzzle yesterday whose solution’s unexpectedness left a nice aftertaste.

The Puzzle

You are given four identical right triangles. You will now repeatedly pick one of the triangles and split it into two triangles by dropping a perpendicular from the right angle to the hypotenuse (see figure). Note that both these smaller triangles are also right triangles and are in fact similar to the original triangle.

ΔABC is split into two: ΔABD and ΔACD

As we repeatedly pick one of the triangles and chop it into two, it turns out—and this is what we want to prove—that no matter what order you go about picking triangles to split, you will always have two triangles that are of the same size.

Starting with Three Triangles is Not Enough

Since all triangles we will ever produce are similar, two triangles are congruent if their hypotenuse is the same length. Let's say that each of the three triangles we start with has hypotenuse 1, and splitting it results in triangles with hypotenuse x and y.

Let's call the three triangles we start with A, B, and C. They have hypotenuse 1 each.
Split B into two, and C into two. Also split both the parts generated from C into two. After these 4 splittings, the seven triangles left have hypotenuse 1(A), x and y (parts of B), and x², xy, xy, and y² (parts of parts of C). If we finally split one of the xy triangles, we are left with eight triangles no two of which are congruent.

Starting with four triangles, however, it turns out, we are guarenteed to always have at least two congruent triangles.

Smacks of Pigeonhole

Features of problems alert us to possibly techniques that might work. For example, a problem that asks us to “show that for all integers such and such as true” can often be tackled using mathematical induction. Similarly, the phrasing “show that there are guaranteed to be two entities with such and such properties” suggests to me the use of the pigeonhole principle.

The pigeonhole principle, stated by itself, is exceedingly obvious and it is hard to believe that it can be a powerful technique. Here is what it says: if you stuff eleven pigeons into ten holes, some hole must contain at least two pigeons. What could be more elementary?

Yet, it can be used to prove such things as: “if you pick any 21 integers from among integers between 1 and 40, you are bound to have chosen two numbers such that one is a multiple of the other” or even that there must be two people in the world who have read exactly the same number of words in their lifetimes. Here, I will present the proof of the first of these since it gives the flavor of how the principle is in practice used.

We will distribute the 40 integers between 1 and 40 into 20 “holes”. These holes are: {1, 2, 4, 8, 16, 32}, {3, 6, 12, 24}, {5, 10, 20, 40}, {7, 14, 28}, …, {39}. That is: for the first hole, we started with 1 and included everything obtained by repeatedly multiplying by two. In the next hole, we started at 3 and did the same thing. In the next, we started at 5. The 20 holes correspond to each odd number less than 40. Notice that if we select two numbers from a single hole, one must be a multiple of the other. Moreover, since we pick 21 numbers and there are only 20 holes, by the pigeonhole principle, no matter which actual numbers we pick, we will necessarily have picked two numbers from some hole, and this completes the proof that one of the numbers we picked is a multiple of another number we picked.

The Weighted Pigeonhole Principle

This solution to the puzzle involving triangles can be expressed using a “weighted” generalization of the pigeonhole principle. Imagine that you have ten pigeons with girths g1, g2, …, g10. Further, you have a twenty holes that have entrance sizes e1, e2, …, e20, where the entrance size is the largest girth of a pigeon that may enter. If it is true that no more than one pigeon is present in a hole, then the sum of the girths (that is, ∑gi ) can be at most the sum of entrance sizes (that is ∑ei).

Conversely, if the sum of girths is greater than the sum of entrance sizes, some hole must have more than one pigeon in it.

Solution to the Puzzle

Some observations. Since all the triangles generated by splitting are similar to each other, the length of the hypotenuse identifies the triangle. If the initial hypotenuse is 1, and the other two sides are x and y, then splitting one of the original triangles will result in two triangles that have x and y as the hypotenuse. Splitting the triangle with hypotenuse x will result in two triangles that have hypotenuse x² and xy. Similarly, splitting the triangle with hypotenuse y will result in two triangles that have hypotenuse y² and xy.

Defining the Holes

We define holes corresponding to every possible hypotenuse size, that is, x^ky^l, for every pair of integers k and l. We shall put a pigeon with a particular hypotenuse (say, xy) into the corresponding hole. Thus, if there are two pigeons in a given hole, they have the same hypotenuse and are congruent.

Entrance Sizes

 A hypotenuse x^ky^l can be reached after splitting the original triangle k+l times, and we define its entrance size to be (½)^k+l. That is, the entrance size for the initial hole is 1, and that for x and y is ½, and for x², xy, and y² it is ¼.

Girths

The girths of the pigeons is initially 1 each (therefore totaling 4 since there are four triangles). Each time a triangle is split, it results in two triangles that both have half the girth. Thus, the total girths remain unchanged after each step.

Sum of Entrance Sizes

What about the sum of the entrance sizes? There is one hole of size one, two of size half, three of size one by four, four of size one by eight, and so forth. Up to any finite depth, this sums to just shy of 4.

Applying the Weighted Pigeonhole Principle

Since the sum of the girths (this is 4) is greater than the sum of the entrance sizes (which is just a bit less than four), by the pigeonhole principle, some hole must have more than one pigeon, which is to say, two triangles must have the same size. This completes the proof.

Understandable, but not acceptable

Today's entry concerns dealing with the behavior of my two-year-old daughter. Don't get me wrong—for a two-year-old, she is an extremely well behaved kid. I am talking here not about biting, kicking, screaming, and such—which she does not do—but about comparatively minor issues such as getting angry quickly—which she does.

In my response to her outburst, I believe that I have been confusing two very different things. Often, I understand exactly what she is irritated about. I empathize, and I let the behavior go unchallenged. It is as if understanding why she is acting out makes acting out acceptable.

No, what I just said is a half-truth. Understanding why she does something makes me believe that it is not her fault, and she should not be scolded for it. What I had not realized was that there are possibilities beyond scolding and doing nothing, and these include explaining, even if she won’t fully comprehend.

Sometimes, it is appropriate and necessary to call out behavior that you understand but do not condone. Teaching kids to cope—including coping with today’s emerging problems such as stuck YouTube videos—is part of our duty as parents.