Problem Solving and Mathematics Education

Maria Miller, seen above, has an interesting and useful blog on mathematics for home schoolers, as well as other web resources, not the least of which appears to be a recent web page called "Math word problems — the do's and dont's of teaching problem solving in math" that I've just started looking at today.

The most recent entries on her blog address this vital issue of teaching problem solving, via a couple of problems for 5th graders taken from a Canadian on-line resource.

The first post Maria made on these problems reads as follows:

I'd like for you to try solve these two 5th grade problems. They are from a very nice collection of word problems for kids by Canada's SchoolNet.

I'd like to use them to illustrate problem solving. But before I do that and give you the solutions, please try to solve them yourself. And don't post solutions here for now; I will have to reject such comments.

Solutions and discussion will follow in a few days.

THE FIRST PROBLEM IS:

* How many addition signs should be put between digits of the number 987654321 and where should we put them to get a total of 99?

THE SECOND IS:

* Divide the face of the clock into three parts with two lines so that the sum of the numbers in the three parts are equal.

I set about attacking these problems as soon as I saw them, both because they appealed to me and because I wanted to be ready for the ensuing discussion. You may want to take some time to think about the first one, for which solutions and comments appeared on her blog today.

Maria's comments can be read by clicking on that last link ("today") or going to her blog and finding the entry for 16 January 2008. You will also find a response from "mathmom" and one from me. By the time you read this, there will likely be others.

SPOILER WARNING: if you read past here, assuming you've not yet clicked on the last three links, you're going to see solutions to the plus sign problem.

Here is what "mathmom" had to say:

I approached this one a little more methodically. I first summed the 9 digits to get 45. I found that the required sum was 54 greater than this number.

I then thought about what would happen if I made a two-digit number. The digit in the ones place would be added into the sum just the same as before. But the new tens digit would count for 10 times as much as it did before. So if I "promoted" the digit D to a tens place, I would add 10D and subtract D (since it would no longer be counted as a ones digit anymore) for a net increase of 9D. So if I need to increase the sum by 54, I need 9D = 54 and D=6. Thus if I "promote" the 6 by making 6+5 into 65, I arrive at the required sum.

I didn't think about the fact that I might have promoted more than one digit and used more than one two-digit number, but my son came up with the second solution, where the two-digit numbers used are 42 and 21. Here he promoted two digits, a 4 and a 2, with the same net effect as promoting just the 6. He had computed that he needed to increase the sum by 54, and then found this solution by a guess-and-check method.

To show that these are the only possible solutions, consider the possible combinations of digits that sum to 6:

6
5 1
4 2
3 3
3 2 1

We can't promote the 1, since there's nothing after it in the row to serve as its ones digit. (If you add a zero to the end of the line, this admits the 5 1 solution, but not the 3 2 1 solution, because you can't promote both 3 and 2 at once.) And we can't promote 3 twice, so the only combinations that work are 6 alone, or 2 and 4.

Here is my comment:

One correction to mathmom's comment: in the second solution, the first two-digit number is 43, not 42.

I had done something similar to arrive at her first solution (and later formalized it algebraically, though in this case I'm not convinced I gained a lot by doing so), and I also missed the second solution by not thinking further about the problem, to my embarrassment. But in reading the third paragraph of her post, I noticed before reading further that since D = 6, using her notation, and the other solution involves using D1 = 4 and D2 = 2, with D1 + D2 = 6, I then wondered about using D1 = 5 and D2 = 1, where again D1 + D2 = 6. Before I could think through why that didn't work (duh!), I saw what mathmom had added and realized the issue, which she neatly pointed out could be solved by including 0 as a tenth digit.

So a great continuation/extension question would be (for students who have been exposed to negative integers or who wouldn't be troubled by thinking about them), can we find more solutions if we append more consecutive integers to the list on EITHER end (or both ends)? Why or why not?

Since I've just thought of this, I don't have an answer. But I raise the issue because I'm always looking for extensions, not just as challenges, but because extending and generalizing problems is one of the major activities of professional mathematicians. Doing so is one way to help develop new theorems, methods, and, sometimes, entirely new areas of mathematics. I look back on my own K-12 mathematics education with a great deal of regret, not the least of which grounded in the fact that I was never led to realize any of that aspect of mathematics. Like most Americans, I was taught in a way that made mathematics appear to be a dead, completely closed subject in which everything was already known (by the professionals), and where my job was to accept received knowledge and be able to show that I "got it" by solving textbook problems that were, for the most part, strictly computational in nature. By 9th grade, the approach simply didn't grab me any longer, and not because I didn't generally do extremely well in mathematics up to that point: in fact, I was an A student through the end of 8th grade. While there were other factors having little or nothing to do with the instruction or content that led to my decreasing interest and effort in math starting in 9th grade, I wish I'd been shown something along the lines of what I mention above: the idea of doing what might be something new and original, or finding creative solutions to problems that already had known solutions, may well have kept me "in the game" at a time when I was growing restless, impatient and bored when it came to math and science.

Now, since posting the above comment this afternoon, I've had a chance to play a bit with my extension questions, as you may already have done or are about to do. I will not post my additional findings right now, but will wait to let folks have fun with the extension question on their own.

It would also be great if anyone has other extension ideas for this. Don't feel limited to the alleged "5th grade" level of the original problem, but it would bear thinking about the target for any such extensions. The one I'd suggest right away is: What happens if we change 99 to other values?

"If math were a color. . ." and other crimes against humanity.

If you're a fan of the Math Wars, you've heard about the question in the first edition of the EVERDAY MATH K-5 curriculum, "If math were a color, it would probably be ______ because ________." This particular question has probably engendered more ridicule and garnered more notoriety than anything associated with the NCTM Standards of 1989 on, and the progressive reform curricula that emerged from NSF grants in the early 1990s. After Googling on the question, I've concluded that no one on the planet has the slightest idea what could possibly be the point of asking this question. It has resulted in so much negative feedback that it has been dropped from revisions and subsequent editions of EVERYDAY MATHMATICS. Indeed, as recently as last week, Andy Isaacs, an author of the program who works at the University of Chicago Mathematics Project (UCSMP) posted on math-teach at the Math Forum:

As for the "If math were a color, ..." question, it appeared in our
first edition in a Time to Reflect section at the end of the first
unit in fifth grade, a unit that focused on prime and composite
numbers, factoring, figurate numbers, and so on. The first edition of
fifth grade was published in 1996. The "If math were a color, ..."
question was deleted in the second edition, which was published in
2001, and the entire Time to Reflect feature was deleted from the
current edition, the third, so this question has not appeared for
some years in EM.

I suppose that since Andy, whom I've met and consider a friend, is washing his hands of the question and it's been dropped entirely from the series, I should let sleeping inquiries lie, even if the folks at Mathematically Correct, NYC-HOLD, and countless "parents-with-pitchforks" groups and web-sites (as well as right-wing pundits like Michelle Malkin continue to make "big" political points about how programs like EM are sending us and our children down the fast track to enslavement by those ever-inscrutable Asians (that Ms. Malkin is herself of Filipino extraction should in no way be taken as a mitigating circumstance in her willingness to help scare the James Jesus Angleton

out of hard-working Americans with the reliable threat of hordes of Asians (Chinese, Japanese, Singaporeans, and Indians, in particular) overtaking us in the constant battle to be #1 in anything and everything. Never mind that it's actually those sneaky Finns who top the latest list of countries that outperform the US in mathematics on an international test. No one is yet recommending that we adopt the Finnish national math curriculum and load up on glug and besides, the Finns themselves are not quite so sanguine about the situation. However, racist that I am, I just can't get sufficiently frightened by images of invading Finns. Asians, of course, are notorious for "teeming," and there's no question that they're going to overwhelm American any second, no doubt due to that famous "math gene" they're all born with.*

But let's get back to the point (sometimes my own digressions frighten even me). What about that "If math were a color" question? Am I seriously planning to defend it as not only "harmless" but actually sensible? Those of you who know me are no doubt sure that's exactly what I claim.

My argument is brief, but I believe it's quite reasonable:

It is increasingly common practice for some high school teachers and some professors of mathematics and mathematics education, especially those working with future or would-be future mathematics teachers, to ask their students for a brief "mathematical autobiography" at the beginning of the term. It's a way to get to know more about students and to find out their attitudes towards and experiences with mathematics (does that seem like a trivial thing to anyone? While I wouldn't be the least bit surprised if it did given some of the folks who teach mathematics, it's nonetheless the case that some educators think it's quite valuable to know how education students think AND feel about the subject(s) they propose to teach, and how students think and feel about a subject they are expected to learn).

Now, if you're teaching K-5, especially if you're teaching K-2, how would you propose to get students to relate their beliefs about and attitudes towards mathematics? Direct questioning, like direct instruction, is not always effective, which is why many child psychologists work with their clients through playing games and other indirect methods for getting insight into experiences and feelings about them. The question"If math were a color. . .?" is a perfectly imaginative and reasonable way to get children to explore and reveal some of their feelings and beliefs about mathematics. The important thing is not, of course, to elicit a particular response or "right answer" (and isn't that a radical concept?), but to help students to find a non-threatening, non-intellectualized way to talk about math. Heaven forfend that child psychology should enter into any elementary school teacher's classroom or pedagogical repertoire, of course. Sensible, rational, intellectual questions only, most particularly in math.


*For the irony-impaired, that's satire.

Book: OUT OF THE LABYRINTH: Setting Mathematics Free

Ellen & Robert Kaplan and The Math Circle


If you only read one book about mathematics teaching and learning this coming season, let me suggest that it should be OUT OF THE LABYRINTH: Setting Mathematics Free by Robert and Ellen Kaplan, founders of The Math Circle in Cambridge, MA.

I had the distinct privilege several years ago of attending a session Bob and Ellen led at Northwestern University's Math Club. We were taken through a short series of problems that led to the main question of the day: is it possible to cover a particular rectangle with non-congruent squares? The way things were led was truly masterful, perhaps the best teaching I've ever seen or experienced. The preliminary questions helped scaffold the main one, but once we entered into trying to solve the main problem, for which most in attendance seemed to believe the answer was "No," very few comments or questions were offered to help us. And yet those "hints" that were forthcoming seemed to be just the right ones needed to steer us out of ruts or stimulate our best thinking to move forward, and within about an hour, the students (who included undergraduate math majors, graduate students in mathematics, and some faculty members from Northwestern's mathematics department, none of whom seemed to be familiar with this problem or area of math) had successfully found a correct solution. The entire process was beautiful to see.

Afterwards, I introduced myself to the Kaplans and commented on how impressed I was with the way in which the lesson had been taught with such minimal but incisive questions and comments from them. Bob stated that they did the same problem recently with a group of 5th graders in Cambridge. Before I could express any skepticism or incredulity, he added, "Of course, it took about twelve weeks!" (The problem was presented as part of an on-going Math Circles course, and a great deal of time was spent building up the necessary tools so that the students could tackle the bigger question. I had no doubt, however, that the basic approach to helping these students was the same as what I experienced in Chicago.

When I asked Bob about who else taught in the Math Circle classes, he said that it was no problem finding people who knew the requisite mathematics. The difficulty was in finding such people who could also keep their mouths shut. And indeed, that is not only a problem in Cambridge, but nationally, including in my own teaching. I struggle mightily to resist the temptation to bail students out prematurely, to succumb to the pressure, both internal and external, to show what I know and to relieve the students of responsibility for thinking and learning mathematics the only way that really makes a difference, in my view: by doing it themselves. The cliche "Mathematics is not a spectator sport" is clever, but it is also very true. It's difficult to really "own" a piece of mathematics without struggling to wrap one's brain around it. This is not to say that we can gain nothing at all from a lecture, of course. But when the money is on the line and there's no one there to help us solve a challenging problem (where "challenging" is, of course, a very personal and relative term), will we do best by recalling a step we saw someone else use, seemingly pulled effortlessly from thin air, or by calling on methods we've used ourselves and built up sufficient experience with to sense that one might be of particular value in the current case?

The story the Kaplans have to tell in their new book is far more powerful than any argument I could offer here as to why it is so crucial for students to be led to believe correctly that they CAN do it themselves and, perhaps more importantly, that they SHOULD do it themselves, collaboratively, as a community of learners, without the usual competition to be the first or the best, but out of a collective passion to know.

I will say in all honesty that what I saw in Chicago had me nearly weeping with pleasure and frustration. Pleasure at the wonderful way in which the Kaplans taught, but frustration at the rarity of the experience and my knowledge that far too few Americans ever get to see and learn mathematics (or much of anything) in this powerful way. Reading their new book, I continue to experience these strong emotions. If you care about mathematics and really empowering people to DO it, go to the Math Circles web site: check into their books, look at some of the notes you can download from recent courses, look at the photos, and generally get a taste of what they do. You'll be very glad you did.

Finding The LCD, or Why Does My Math Teacher Insist On Making Me Hate Math?

Recently, there's been an on-going conversation/argument on math-teach@mathforum.org, unusual in its frequently bordering on civility and even some degree of agreement among antagonists, regarding the role of teaching/learning/using the Lowest Common Denominator (LCD) or Least Common Multiple (LCM), which are effectively the same thing, when working with, say, addition and subtraction of fractions. It should be noted that similar issues arise later in working with rational algebraic expressions, particularly when there are polynomials in one or more of the numerators or denominators that can be factored in some of the standard ways that students have worked with previously. But my concern is with something else, specifically the arbitrary way in which many K-5 teachers insist that to add/subtract fractions one MUST find the LCD. Failure to do so often results in all sorts of negative consequences, from stern glances to loss of credit on classwork and tests. On my view, this is a classic example of abuse of teacher authority, as well as an indication in many instances of mathematical ignorance on the part of some teachers.

It is false to state that in order to add or subtract fractions (or rational algebraic expressions, for that matter) that one must first find the LCD. What is needed is only to find A common denominator and create equivalent fractions for the numbers or expressions one wishes to add or subtract that have the same denominator. The rest is fairly trivial.

Since one can ALWAYS find a common denominator by simply multiplying together the denominators of any two (or more) fractions with unlike denominators, the issue becomes a matter of taste, to some extent, but also convenience, speed, and perhaps elegance or aesthetic qualities as well. But it suffices to find ANY common denominator if one so chooses. If the goal is to find the sum or difference, we just need equivalent fractions with the same denominator.

Some people seem to believe, however, that there is a huge need to put students through the grind of doing lots of examples in which they factor a bunch of denominators and possibly numerators (not just with numbers, but in rational algebraic expressions). They argue that this is essential stuff for algebra and above. But of course that is true only to the extent that textbook authors join in the conspiracy to present cooked problems that always submit to just the techniques of factoring that we make the heart and soul of Algebra 1 courses. It's hard to deny that if you rig the game you can guarantee the outcome. If in the real world, we knew that most problems people have to solve were going to have convenient factorizations in them, then by all means it would be worthwhile to obsess on this sort of thing. But that's not really the case, no matter what math textbook authors might mislead us into believing. And so all this focus on finding the simplifications, while mathematically nice, is not necessarily what people will want or need to do outside of math class: they're going to be more interested in getting the numbers crunched efficiently and accurately and then doing what actually matters: making sense of the results. Elegance and "oh, isn't that slick" mathematical aesthetics are indulgences for those who have the time and inclination. Nothing wrong with that at all, but do we really think misleading students that such will be the situation most of the time when they need to deal with fractions or rational algebraic expressions? Do we really want them to think that most people don't grab the calculator, possible with a CAS system, or sit down at their computer? Please. Pull the other leg, it's shorter.

Yet I read people arguing with very straight faces that we have to teach this and teach it thoroughly and that it is essential for algebraic readiness (so to not make finding LCDs a MAJOR emphasis is to deprive students of their futures. Seriously. You can't make this stuff up).

When I and others retort that the main goal should be to understand what fractions are and how their basic arithmetic operations work, and that methods and techniques, while worth exploring, are not in and of themselves valuable without understanding, and that further, it's possible to learn to do an algorithm by following steps but really not having much or even any understanding of the symbolic manipulations one is doing, I'm told that being able to repeat steps IS an indication of understanding, though maybe not of a deep and thorough kind. I find this truly bizarre.

I think my preference would be to say that when students make choices about how and what to do and can intelligently explain their reasons for the choices they've made, then they show understanding. Following cookbook steps to solve cooked-up, convenient problems may be little or nothing more than donkey-like behavior, what Douglas Hofstadter has described as "sphexishness."

I'd prefer not to focus specifically on procedures/algorithms, but not to exclude them from consideration, either. That's because as students move through elementary mathematics, it is easy to forget that there are deep levels of understanding relative to their degree of mathematical maturity and experience that can easily be forgotten, overlooked, or consciously ignored by overly-focusing on procedures. Not only does making algorithms for calculation THE primary focus (not that it isn't a part of what should be attended to) of early mathematics instruction do a disservice to the students, it readily can hide and distort important information about who actually has a very good mathematical head on his/her shoulders AND which students may have significant difficulties despite seeming success at following steps they don't really understand at all.

That has long been my gripe with how many people talk about mathematics at the K-8 level or so: as if computational proficiency suffices to be mathematically on track. Lack of such proficiency MAY indicate serious lack of understanding, or it may only indicate a student who has not yet developed specific knowledge, yet still has clear ideas about the concepts involved (e.g., a student may very clearly understand what addition means and can recognize when to add in a given situation, but may not have mastered all the facts yet, or may not have them at finger-tip recall. My son took his time in that regard, but without flash cards or empty drills, managed to get them all eventually. Some of his teachers doubted his mathematical abilities because he didn't seem overly concerned about such immediate recall: he had various compensatory strategies (none of which involved calculators or counting on his fingers, etc.) that worked, generally within a few seconds, and I thought they were fine. He didn't obsess about passing those medieval 100 problems in 60 seconds tests that have become widespread in this part of Michigan, regardless of whether the elementary math curriculum was EM, INVESTIGATIONS, or something more traditional. Such tests drove his very intelligent half-sister to tears. He just told me that he took his time, got most of them right, and would know them all, quickly, sooner or later. And he was, of course, completely right. I was very gratified at his recent parent-teacher-student conference to have the principal make a point of being there to tell us what a great kid he is, how friendly and open he is with both adults and kids, and what a great student he is. I reminded him and his mother that in second grade, we were told by his teacher, the principal, and several other experts that he likely wouldn't be ready socially or intellectually for middle school (he's in 7th grade and got straight A's first quarter). Without wanting to draw any definitive conclusions from n = 1, I find it telling that it's easier for some teachers to focus on arbitrary and shallow tasks like 100 facts in 60 seconds than to recognize that a student actually THINKS mathematically and likes doing so.

Further, I wouldn't conclude that his situation is proof that we should stop breathing down the throats of kids who are working at a slightly different pace at fact-mastery than might be mandated by some arbitrary scope and sequence guide, or local, state, or national curriculum or norm. We're never going to establish such proofs to the satisfaction of the skeptics, and we're unlikely to ever reach the point where we generally tolerate AND support such differences, or recognize when a student is doing just fine (as my son truly was) or really is in trouble even if s/he can calculate or do recall on demand. As long as we put the focus on an extremely narrow band of skills and call that mathematical proficiency for elementary school, we're going to screw up a lot of kids. We've been doing a lovely job of that in schools when it comes to mathematics and other subjects for more than a century, by and large. It's one reason I fully empathize with those who choose to home school, even if I think that, too, has drawbacks, and even if I don't agree with all the reasons that some people believe they should home school their kids. It's a free country, though, and as Kirby Urner says, there's room for intellectual competition in how we educate and what we educate about.

The Book Gods?

Something that always blows me away is how teachers will follow books blindly in the face of what should be big warnings from what they know about their own students. All too few teachers are immune to book worship, having been led to believe by their own experience as kids and education students: the math textbook (and its magical authors) knows more than any regular old K-6 teacher.

Two cases in point: I was coaching upper elementary teachers in math at a low-performing K-6 school in a district near Detroit a few years ago. They were using the Everyday Math program for the first time. I was asked to guest teach some lessons on fractions in a couple of the 4th and 5th grade classrooms. I noticed that in one lesson, involving pattern blocks, there were three problems for classroom discussion. The first one was clearly needed to establish the relationships amongst the smaller shapes (triangles, parallelograms, trapezoids, that could be fit together to make a hexagon (if you have the standard pattern blocks, the relationships were that two triangles formed a parallelogram; three triangles made a trapezoid; six triangles made a hexagon, which in the first problem was the "whole" or "1." Also, you could make the trapezoid from a triangle and a parallelogram; and you could make the hexagon from two trapezoids or from three parallelograms, etc.



In the first problem, therefore, students establish that the triangle is 1/6 of the unit hexagon (which is an outline drawing in the book that you cover with these various combinations); the parallelogram is 1/3 of the unit hexagon; and the trapezoid is 1/2 of the unit hexagon.

While some students had difficulty with this, most did pretty well as I had figured. But when I saw the second problem, I sensed danger (interestingly, the third problem was much easier, and relatively few students struggled with it). #2 was an outline drawing of TWO of the hexagons with an adjacent edge, but the line where the edges met was erased, so you had a double hexagon as the new unit. The idea was that students would cover this new figure and see it as a new "unit"; it would take, for instance, 12 triangles to cover it, and so the triangle would now represent 1/12 of this unit, and so on. Each figure would represent a fraction half its previous size, since the new unit was double the old unit.

I would like to say that I brilliantly smelled a rat, but I didn't. Or that is to say, I smelled it, but I didn't trust my reactions sufficiently. Then again, I can plead lamely that I was not experienced with this book (though the mathematics wasn't a problem for me) or teaching kids this age. I was there because I had some previous experience as a coach, and because I knew the math well, and because I am very quick to adapt to new ideas and approaches).

In any event, I went ahead with the first group and had them do the problems in the order given by the book. As I had sensed but failed to act upon, the students struggled mightily with the second problem. They couldn't wrap their minds around the shift in the unit. The picture, I suspect, was perceived by most of them not as a new "unit" but simply as "two"; "obviously" it WAS equal to two of the OLD units connected to one another. They likely were visually filling in the removed boundary line where the adjacent edges met. It was very difficult to get a lot of them to make the leap to seeing this as a new "one." Even when they did the individual tasks ("cover this new shape with the triangles. How many triangles does it take?") and got the correct numbers, they were not going to budge from the notion that if a little green triangle was 1/6th in the first problem, then it was still 1/6th in the second problem. When I asked about why it now took 12 triangles, not 6, to cover the figure, they just said in essence, "Well, sure: there are two hexagons there."

I hope I have made this clear enough without the exact problem drawings that would likely made it more transparent.

So, when I guest-taught the same lesson in another class, I changed the order of the problems. I got much better results, overall. And I warned them that the problem just described, now their last problem, was challenging and might prove upsetting until they played with it a while. Interestingly, some of the metaphors I tried that bombed the previous time worked well here (e.g., a quarter is a half of a half-dollar, but it's only 1/4th of a whole dollar, etc.). Was it really that simple? Just change the order, build their confidence, warn them of quicksand, and things would go better? I'm not sure.

But the teacher in the second class was truly SHOCKED that I changed the order, and after class, when we debriefed, it took a lot to convince her that I had made an informed choice based on the previous class and my own previously ignored intuition that there was too great a leap in that second problem to go to it directly from the first one and without offering some warning bells. Her feeling was that the textbooks authors knew more than she did and must have a good reason for the order of the problems.

I assured her that SHE was the expert on her students in her class, and no author would dare usurp that position. I was making an educated guess, and had the experience of the previous class to back me up. (I know one of the main authors of EM personally, I told the teacher I would be e-mailing him with a summary of the experiences I had that day. I told her that I had no doubt at all he would understand my decision. May he might suggest a change in the next edition as a result. This was all amazing to her. And she was not a rookie teacher.

In another example from the same school and book, I chose to omit entirely a couple of problems that introduced mixed numbers into a situation where they were NOT the main point of the lesson. I anticipated (and this time trusted my instinct) that throwing mixed numbers into the fray was an error that would distract students from the real mathematical residue I wanted them to take away. Again, the classroom teacher (not the same one), was surprised by my choice and not confident that she was allowed to make a decision to remove problems, temporarily or permanently (I made clear in class to the students that we would come back to them another day).

I think this is tragic. And I think it extends even to home schooling parents, even though they know their own child(ren). They can make pedagogical choices based on that knowledge, as well as other factors. There's no textbook that can anticipate the needs of each kid, and a good teacher must intervene to make the best choices s/he can given the limitations of either whole-class or individual instruction. Clearly, teaching one child or only a few is a huge advantage and allows great flexibility. Not being under the thumb of a district or even the state or NCLB makes things even better. But the god of the book is intimidating. What if you're wrong?

But I must add this caveat: if math isn't "your thing," you need to think carefully about your choices and you are going to make some definite mistakes. Few of them will be fatal (I'm talking about the sorts of pedagogical choices already described, not mathematical errors, which are a separate issue entirely). There is what is called "pedagogical content knowledge": an understanding of both the subject and effective ways to teach it. If you're weak in content, it's hard to be strong in this area, but if you are strong in content, you may not be strong in it anyway. (And college math departments prove this daily throughout the land). It takes a lot of thinking, before, during, and after teaching lessons to be effective. It takes a willingness to really anticipate how a reasonable-seeming problem could be a mine-field for your student(s), to think about where your student(s) may go astray in the task and/or topic at hand, based on their strengths and weaknesses AND your knowledge of the mathematics (obvious example: kids learning fractions will be prone to do addition by adding the numerators and adding the denominators. It's a good idea to be prepared with an example like 2/3 + 1/5 so that you can ask the student(s) if it makes sense that the answer would be 3/8, since 2/3 is already greater than 1/2, but 3/8 is less than 1/2).

So I don't advocate just tossing things out because they look unappealing or you don't like the topic or something like that. Or because you read somewhere that a particular method isn't good. You want to think it through and in terms of the student(s). Then, you do what you can, being prepared to change course if necessary. No shame in that. Indeed, it's a wise teacher who can admit error, doubt, and change course. Too bad some people in Washington, DC seem to lack that wisdom.

Mastery of What?

A lot of conversation on one of the lists I read about spiraling vs. mastery in mathematics curricula. Nothing new about that. Just another issue that strikes me as adding more confusion than clarity by creating false dichotomies instead of seeing that most of these things go hand in hand. Frankly, I'm hard-pressed to see how it would be possible to teach or study something as enormous as merely the tiny slice of mathematics we want all students to learn and be able to use in K-12 education (which brings us up to pretty much nothing invented in the field as recently as the 17th century and still excludes enormous amounts of what was already known when Newton and Leibniz were inventing the differential and integral calculus), without doing a reasonable amount of "spiraling" (which is to say that we must revisit already-explored ideas again when students have more sets of numbers to look at, say, or have developed sufficient mathematical maturity to delve deeper into things that many of us mistakenly think of as simple, elementary, "easy," basic, etc. At the same time, it's hard to move forward (at least in the linear way we teach the subject in the US), if you haven't attained some facility with the procedures (if not the actual ideas behind them) that are often referred to as "the basics" (that's the stuff educational conservatives keep telling us we need desperately to "get back to," as if we've somehow been skipping that and teaching partial differential equations, differential geometry, and category theory to elementary students while they weren't watching us crazy progressives carefully enough, and now we must return to sensible arithmetic, a taste of geometry, some faux algebra, etc.) However, interesting questions arise about whether it is absolutely mandatory to shove every child through the same narrow funnel, and whether much wouldn't be gained by giving kids (and teachers) a lot more options about what route(s) they want to explore on their ways up and around in the tree of mathematics. (see Dan Kennedy's provocative "Climbing Around In The Tree of Mathematics"

In any event, here's some of what I've been thinking about regarding the whole "mastery" thing that so many people worry about (or say that they do):

People talk a great deal about "mastery" in K-12 math, as if mathematics was somehow like typing. You practice and attain mastery.

But that part of mathematics, while important up to a point, isn't really what mathematics is except for kids (and then, only a small piece of it).

Consider the following quotation:

"There is something odd about the way we teach mathematics. We teach it as if assuming our students will themselves never have occasion to make new mathematics. We do not teach language that way. . . the nature of mathematics instruction is such that when a teacher assigns a theorem to prove, the student ordinarily assumes that the theorem is true and that a proof can be found. This constitutes a kind of satire on the nature of mathematical thinking and the way new mathematics is made. The central activity in the making of new mathematics lies in making and testing conjectures." (Judah I. Schwartz and Michal Yerushalmy, quoted in "Geometer's Sketchpad in the Classroom" by Tim Garry, in GEOMETRY TURNED ON, p. 55).

You could readily change a few words above and talk about problem-solving as well. We pose problems to students the solutions for which are well-known. Students cry "Foul!" when confronted by: 1) problems they haven't specifically been trained to do. That is, they think it's dirty pool to be asked to solve problems that aren't identical to others the teacher/book has explicitly worked through with them, or nearly so. These, of course, aren't problems, but rather they are exercises, just like typing drills. I show you how to use the keys with your right index finger, then drill you on that, and so forth; b) problems for which the solution pushes them beyond the immediate topic, perhaps calling on general strategies they've learned, things they've worked on, but also a bit more, stretching their minds, asking them to reach a bit, speculate, imagine; and c) any sort of problem that has no known solution (or maybe just no solution with the methods or numbers they're familiar with) just so they learn that math never ends, mathematics is always being extended and invented, and part of what drives that is unsolved problems (along with new problems, new math, and so on). It could be something as simple as asking a student who hasn't learned about negative numbers what 6 - 8 equals, or someone who hasn't learned about complex numbers to consider the equations x ^2 + 1 = 0. Or it could be having students explore accessible but unsolved problems like Fermat's Last Theorem (when it was still unsolved) or the Goldbach Conjecture. Or something like the three utilities problem or Konigsberg Bridge problem which led Euler (see photo above) to invent new mathematics (graph theory). Fun stuff, really, but many kids think all problems in math class should be trivial exercises, not real problems for them to speculate about and experiment with.

I think much of the error we make in putting so much emphasis on mastery lies in cheating students of knowing what it means to think mathematically, even though they are quite capable of doing so. There is a body of work out there that suggests kids can do much more mathematical thinking than we give them credit for. But for most, by the time they get to do some, they hate mathematics (even though they actually don't know what it is, really).

Just a little late-night food for thought.

Language, division, and calculators.

Since several related issues are floating around on various math education lists I'm reading these days, I thought the following problem and what I observed recently with some African-American students would be worth sharing.

This problem appeared on an actual ACT exam; it should be noted that it was only #5 of 60 questions in a section of a test which allows 60 minutes total time for those problems:

The oxygen saturation level of a river is found by dividing the amount of dissolved oxygen the river water currently has per liter by the dissolved oxygen capacity per liter of the water and then converting to a percent. If the river currently 7.3 milligrams of dissolved oxygen per liter of water and the dissolved oxygen capacity is 9.8 milligrams per liter, what is the oxygen saturation level, to the nearest percent?

A) 34% B) 70% C) 73% D) 74% E) 98%

First, I want to observe that there are quite a lot of words in the above problem, most of which make it difficult to read and which don't flow terribly smoothly. The actual mathematics is hardly at the high end of difficulty for high school students, if they get to it, or at least we would hope that to be the case. But the language seems quite "high end" for where this problem appears in the section, and that would potentially be an obstacle for students who are not good readers, not native English speakers, or who may be thrown by a science situation with which they are unfamiliar. Is it a good idea to embed this particular mathematical task in language like this? What, exactly, is being tested? Here in Michigan, where the ACT is now the official "exit" exam for high school students, these concerns are not trivial.

That said, let's look first at how I went over this problem with some students, once they had it set up as 7.3/9.8 or agreed that such was a reasonable way to begin. Of course, the ACT allows calculators (but unlike the SAT, does not permit the TI-89, which has a computer algebra system included). But I always start with the assumption that it might be quicker and safer to do mental math, and that knowing how to do the problems several ways is worthwhile. (That's also a personal holdover from the period prior to 1995, when calculators first were allowed on the SAT. I learned how to excel on these timed tests by studying them starting in 1979. Mental math, estimation, and knowing that fractions are often the easiest form in which to do quick arithmetic has stayed with me and I still encourage students to think that way on timed tests. I rarely use calculators for doing them, though it's not a bad idea to have one, if you know how to use it intelligently.)

First, I pointed out that we could multiply by 10/10, getting the equivalent expression, 73/98. Then I asked about 34%, hoping that students would see that it was far too low, 73 being clearly less than 50% of 98. Most students agreed. I suggested further that 98% was unlikely, as 73 is not "almost all of" 98. Again, this line of reasoning was amenable to most of the students. I next asked about 73%. Some students saw that for that to be correct, the denominator would have to be 100. That left 70 or 74, and most (though not all, of course) students at this point saw that if the denominator was less than 100, then 73 would comprise more than 73%, so that 74% was more reasonable than 70% would be.

The advantage of this sort of estimating and "number sense" approach avoids some of the errors that I found occurred with alarming frequency among those who attacked the problem with more traditional methods or with calculators. One student in particular ran into interesting mistakes, not unique to him. Without my help, he correctly set up 73/98. He articulated that he should divide, but then proceeded to start dividing 98 by 73. Since this yields approximately 1.34, it's not a coincidence that one of the choices is 34%: students simply assume that they should "drop" the "1" and round. Rather than considering that they've converted a fraction considerably less than one half into a percentage that initially (before the groundless dropping of the 1) represented more than 100%. Would they be likely to do this with the mental math approach?

But this student took it further. He grabbed his low-end calculator (non-graphing), pressed some buttons, and told me that he had been right to begin with, because he still got "34%" as the answer. Of course, he'd failed to realize or recall that when entering 73/98 into a calculator, his "wrong" impulse in the hand calculation was now the correct order (and such would be the case as far as the numbers are concerned were he using a HP or other Reverse Polish Notation device).

What I find provocative is that students can wind up going wrong on the same problem in so many ways and for reasons that aren't strictly mathematical. Not only is the language in which the calculation is imbedded more dense than one might expect, but the language of division "73 divided by 98" is in the opposite order from the order in which students are taught to divide, leading many to then try to divide 98 by 73. And then, having made that error, they find their mistake reinforced by the fact that the calculator wants them to enter things exactly in the order they appear in print (at least in THIS example). Our way of expressing division in language is at least partly to blame for a lot of the confusion in even setting up a correct calculation, let alone carrying it out successfully.

There are other difficulties with division surrounding the conceptual basis for the algorithm, but here we see how students can be led astray regardless of whether they know how to do long division by hand on an exercise that simply offered 98, the side-ways "L" we put to its immediate right, and 34 placed above it. This student did the wrong problem correctly by hand. He then made precisely the error the test-makers set him up to make by selecting 34%. I would have been interested to see how he would have done with the calculator alone, or with the calculator first, at any rate, as far as the order of the numbers was concerned.

To conclude, I would hesitate to say that this student didn't know how to do long division. He could successfully carry out the algorithm. Where his problems lay had to do with issues that may have been more linguistic than anything else. How he said the problem to himself or heard me expressing it was not reliably leading him to set up the right calculation. And the answers were, of course, tempting him to make some unsound and unjustified moves to retrofit his results to the answer choice that seemed to fit. This occured both when he did the problem by hand AND with a calculator. The mistake had nothing to do with his relying on the calculator, in fact, since it was his choice initially to do the problem without it. He did the division he set up properly. Yet his answer was dead wrong, regardless of the technology or its absence. I don't claim this proves anything, but it is thought-provoking. And this incident does help emphasize the potential power of estimation and mental math.