Who Invented "Lattice Multiplication"?

I can't seem to get the issue of lattice multiplication off my mind. The negative reactions to this perfectly sound algorithm are not grounded in any reasonable arguments, as far as I can see. But I can't help but suspect that many people, both those who are vehemently opposed to teaching alternative approaches such as lattice multiplication, as well as those who are open to or neutral on this and similar ideas, believe that somehow some hippie math ed reformers at the University of Chicago Mathematics Project (UCSMP) dreamed it up one night and stuck it into EVERYDAY MATHEMATICS (unless other hippie math ed reformers at TERC beat them to it with their INVESTIGATIONS IN NUMBER, DATA & SPACE curriculum). Alternate theories, some involving Satan, Osama Bin Laden, or Josef Stalin have not been verified as of this writing.

However, looking at Frank Swetz's CAPITALISM & ARITHMETIC, I came upon the following earlier today. I hope this doesn't cause riots among some of the anti-reformers:

A ready variant of this method is the following gelosia or graticola technique, so named for its similarity to the contemporary lattice grillwork used over the windows of the high-born Italian ladies to protect them from public view. Following Byzantine custom, the wives and daughters of Venetian nobles were usually kept sequestered. Spying unobserved from such vantage point, these ladies often saw scenes that disturbed them or made them "jealous."

The computational technique employed is basically the same as that used in per quadrilatero; however, the cells are partitioned by diagonal lines so that when the product of a row entry and a column entry result in two digits the units digit is written below the diagonal line and the ten's digit above. In this manner, carrying becomes an integral part of the algorithm. When all partial products are obtained, the entries within diagonal columns are added as before and the resulting total product written along a side and base of the array.

This method is quite old. It probably originated in India, was known to be popular in Arab and Persian works, and was finally accepted into European arithmetics in the fourteenth century. Due to its organizational efficiency and the ease it provides in multipling any two multidigit numbers, it was quite popular as a computational scheme; however, it was difficult to print and read and thus fell out of favor. It is from the gelosia grid and principle that the computational device known as Napier's Bones (1617) evolved.

Horrible to discover that this "fuzzy" approach has been around for many, many centuries, was popular and fell into disfavor only because of the limitations of 15th century printing methods (Swetz's book is focused on arithmetic in 15th century Europe). But of course, if it has organizational efficiency and ease of use, it just might be an okay thing to teach to kids as ONE way to do multiplication.

I read only a couple of days ago on an anti-reform math list that "[s]ince your child is only in first grade one thing you really need to be aware of going forward is that it is unlikely you child will be taught the standard techniques (now elevated to "algorithms" ) for doing basic arithmetic operations in EM." This was quite a surprise to me, since Everyday Math most definitely presents the "standard techniques or algorithms" for addition, subtraction, and multiplication. Long division, a point of much contention in the Math Wars, is not presented in the EM curriculum, though during my work in Pontiac, MI, the teachers at all five of the elementary schools where I coached introduced it anyway. I suspect they are not isolated cases. But aside from division, there is no doubt that the claim by this parent about EM is completely false. Sounds great, though, when you want to scare the bejeezus out of folks, especially those inclined to believe anything they read or hear that puts unfamiliar or "new" mathematics teaching tools in a negative light.

In any event, it appears that we can't lynch the authors at UCSMP or TERC for lattice multiplication. I have no doubt, however, that some charge or other can be made to stick and that sooner or later those who dare to try to help kids think about and do mathematics more deeply and effectively will be punished for their temerity.

Looking Further at Multiplication

For some reason, the lattice method of multiplication really annoys some people. I wish I could understand what appears to be an irrational rejection of a perfectly sensible approach that is just as mathematically sound as the "traditional" way most Americans were supposed to learn to do multi-digit multiplications.

I recently had another fruitless "dialogue" (read: smash-your-head-against-a-brick-wall, you'll-get-further argument) with an entrenched foe of anything and everything viewed as tainted by the evil worm of fuzzy, reform thinking. I will try to edit it into something that some readers may find useful. What started this was the posting by another notorious reform opponent of a link to a YouTube video that shows a way to do multiplication by drawing a series of crossing lines. Of course, the underlying mathematics is the same as what makes the lattice method (and pretty much all methods I've seen taught) work. But the video appears to be presented more as a "neat trick" than as something to be taken seriously, and I know of no one who is teaching it to kids. It wouldn't be disastrous if someone were, but it seems a bit too much like a bit of showmanship than something anybody wants kids to learn.

The person who posted this to math-teach sarcastically called it "the lattice method made easy" or something like that. I commented that it WASN'T the lattice method, but that this video hardly constituted a reasoned critique of lattice multiplication or any other alternative model or algorithm. And once again into the breach stepped an old antagonist to insist that the Vedic multiplication really WAS the lattice method. What follows is comprised of some of the things I wrote in response to this anti-reform critic.

Why Not Teach Multiple Methods?

It's rather hard to get away from the fact that many students like and prefer the lattice method to the "traditional" one. So foes need to either "get over it," as the saying goes, or deal with the real issue: this and any method that makes mathematical sense should be taught "transparently": that is, the sense as well as the procedure needs to be investigated by teachers and students. Otherwise, it's no more or LESS of a trick than is ANY algorithm. If you deny that, you'd best have some logical argument to support why it's different to do what you learned in school, that the "traditional" method is "real" math, while the lattice method is not, etc. I can't help but insist that such a viewpoint has no merit, as I've yet to see any logical argument to persuade anyone who isn't simply entrenched in the Mathematically Correct/HOLD dogma

Who cares whether something is "standard" as long as it works and makes sense to the person(s) using it?

Of course, we don't expect teachers to teach EVERY single algorithm they've ever heard of or seen. If a KID were to present the algorithm from the video, it would be nice for the teacher to be able to recognize the underlying structure. But why identify it specifically with the lattice method? What you really seem not to get (though maybe you do and just can't acknowledge for political or religious reasons) is that ALL the algorithms one sees in EVERYDAY MATH and other books, reform or not, are based on the same idea: find a fast way to do repeated addition based on the fact that we have a place-value system.

Four Models Worth Exploring

Area Model

1) Note: the example illustrated above uses smaller numbers than the example I have explicated in the text, but the basic procedure and interpretation are the same.

Take a piece of decimal graph paper and find the product of 32 and 27 on it in the manner I will describe: write along the top of the page "32" and draw a line along the top boundary line of the graph (not of the paper itself) that runs across three blocks of ten and then two single squares in the next adjacent block of ten. Then write 27 down the left margin and draw a line down the left boundary line of the graph that runs down two blocks of then and then seven unit squares in the next adjacent block of ten.

Next, extend a line from to the left at the end of the line you just drew equal to the length of the line you drew at the top. Extend a line from the right end of the new line up to meet the end of the first line drawn.

The rest, which is easier to show than write, involves picking four colors and with markers, crayons, or colored pencils, color the 10 x 10 squares one color, the 10 x 7 rectangles a second color, the 2 x 10 rectangles a third color, and the 2 x 7 rectangle a fourth color. Counting up, you have six 10 x 10 squares + three 10 x 7 rectangles + two 2 x 10 rectangles + fourteen unit squares.

This gives us 6 x 100 + 3 x 70 + 2 x 20 + 14 = 600 + 210 + 40 + 14 = 864

Expanded notation

2) Compare this with the "expanded notation" model:

32 x 27 = 30 x 20 + 2 x 20 + 30 x 7 + 2 x 7 = 600 + 40 + 210 + 14 = 864.

The Lattice Method

3) Note: the example illustrated at the beginning of this entry uses different numbers. I am staying with the same two numbers throughout my explanations, but again, the directions and analysis remains the same.

Compare this with the "lattice method." You draw a 2 x 2 box and draw a diagonal in each box from the lower left to upper right corners.

You write 3 and 2 over the top boxes, respectively.

You write 2 and 7 down the right side next to the top right and bottom right boxes, respectively.

You multiply 2 x 7 and write 1 and 4 in the upper and lower compartments of the bottom right-hand box, respectively. You multiply 2 x 2 and write 0 and 4 in the upper and lower compartments of the top right-hand box, respectively. You multiply 3 x 2 and write 0 and 6 in the upper and lower compartments of the top left-hand box, respectively. And then you multiply 3 x 7 and write 2 and 1 in the upper and lower compartments of the lower left-hand box, respectively.

Now starting in the lower right hand corner, you do diagonal addition from "top to bottom," moving from right to left and writing one digit under each box, carrying any extra digit, if it occurs, to the next diagonal. You get in this example, from right to left, 4, 6, 8, and reading this from left to right as we normally do, we have the correct answer, 864.

Because of the numbers chosen for this example, there are no carries. I've found that they are the only stumbling block likely to emerge that isn't related to errors in the individual multiplications or final additions due to carelessness or lack of knowledge of one-digit math facts. After practice, most kids handle this carry issue with ease. The model extends to more digits and to decimal numbers as well.

Is this algorithm "better" than others? Not for me, personally, but then, I didn't learn it until a few years ago. Lots of students like it. It seems like a "neat" (in more senses than one) way to do the partial products.

What's clear is that it's really NOT significantly more time consuming, no matter what the nay-sayers insist upon arguing, and certainly not to the extent that they would have us believe. It's a compact algorithm, like all of them. The only one discussed so far that I do not present to teachers or kids as a way I actually would like to see them do their work for more than the purposes of THINKING about (and in this case VISUALIZING) the partial product <=> area similarities is, of course, the first one, because we really don't need to do a drawn out set of rectangles and coloring (which clearly DOES take a relatively long time by comparison to all the others) to get the answers. We only want to see that there is some underlying relationship between area and multiplication. For very visual kids, however, sometimes the coloring, which I present fourth, not first, really helps tie everything together, and for some it's a key breakthrough model, so it's definitely worth spending some time on it, at most part of a period, and then re-examining ALL the models one uses to see how they relate to one another.

Keep in mind, too, that we could use other tools here, including blocks, tiles, or other hands-on models. But I think we have enough for now to make the important points.

The "Standard" Algorithm

4) Compare this with the "standard algorithm"

32 x 27 = "2 x 7 = 14. Write down the 4, 'carry' the 1'; 3 x 7 = 21. Add that 1 you 'carried' and write down 22. Now move down to the next line underneath and write a 0 under the 4 (alternatively, "mentally shift one place to the left before writing down the next digit); 2 x 2 = 4. Write down the 4 (under the middle digit above). 3 x 2 = 6. Write down the 6 to the left of the 4. Now draw a line underneath what you just wrote and do the resulting column addition. 4 + 0 = 4; 2 + 4 = 6; 2 + 6 = 8. Read your answer from left to right: 864.

(Yes, I purposely mechanized the description of the last algorithm, but not unfairly so: that's what kids ARE taught to do, after all, and it's not all that hard to see some of the places that they can and often do go wrong. No algorithm is fool-proof, and any algorithm is subject to errors in the sub-calculations as well as in where one writes the partial results (here, the partial products). What's GLARINGLY missing from this particular algorithm is any conscious consideration of place-value. Except for that shifting, which is generally taught as a mindless step to be religiously observed, there's no acknowledgment that with the exception of the first multiplication, everything you say to yourself is a lie. You never multiply 3 x 7, but actually 30 times 7, and so forth. The compression process gains speed but loses information.)

Now, if every kid who was carrying out this or any other algorithm had a reasonably good grasp of what s/he was being asked to do, none of that "lying" would matter. But for far too many kids, that standard algorithm makes as much sense as voodoo (perhaps less). Is that the fault of the algorithm itself? Not really. It's a fact, however, that our standard algorithms for both multiplication and division are all about speed, and so we sacrifice some information (what the REAL partial products are in the case of multiplication; what we're subtracting in pieces from the dividend at each step of the division algorithm) without, we hope, loss of accuracy, compressing the repeated additions or repeated subtractions, respectively, because place value lets us do this.

But you REALLY have to think like a kid: not a kid who either doesn't give a rat's behind about comprehension, but is aces at following instructions and knows the basic number facts well enough to do these two algorithms with accuracy and minimal screw-ups, nor like a kid who REALLY gets what's going on, practices, and then can do the steps automatically, but rather a kid who has holes in his knowledge of the facts and/or is not adept at following a set of steps that make little or no sense to him. This kid is going to make repeated mistakes, either in the sub-calculations or in where he writes things, or in the additions or subtractions, or, likely, all over the place. And this kid will be so bloody confused about where he's going wrong that he likely will sink into deeper confusion, quite possibly convinced that either multiplication or long division (or both) is way beyond him, or that these are some pretty messed-up algorithms.

There really are lots of places to mess up this algorithm (and the division algorithm). So rather than scream that these are perfectly good algorithms that EVERY kids MUST learn and that all other models and/or algorithms are stupid, inefficient, impractical, dumbed-down, not "real" math, etc., why not accept the fact they are all grounded in perfectly solid mathematics, that some models will connect first for some kids, while others will connect first for other kids, and eventually a competent teacher will do her best to see to it that all kids have the chance to make the underlying connections among these models and algorithms. Then, it's a tad easier, in all likelihood, to make a truly convincing case for the standard algorithm as the fastest (because you don't have to write down as much), but it will STILL be up to the student to choose.

If your goal is understanding and competence, I see no other choice. If it's a fanatical and irrational opposition to alternatives, well, be my guest, but you'll never get me or thousands of math teachers who work with real kids to accept such bizarrely rigid thinking.

The actual lattice method makes sense if you take the time to think about it. I find it objectionable, however, that it, like the "standard" algorithm and most of the rest of grade school math, is generally taught with no eye towards why and how it works, only as a black box to be followed mindlessly.

My antagonist wrote: "Of course the actual lattice method makes sense if you think about it. No one questions its validity, rather the questioning is whether it has any real instructive value, or lasting value."

How are the terms "real instructive value" or "lasting value" meaningful? They're undefined and unexplicated terms here that are, I suspect, offered for rhetorical rather than logical impact.

What I find objectionable to the anti-reform approach to "analysis" is that it seems to care only about any straying from the "traditional" black boxes.

I'll simply state that any black box is undeniably a black box. If you look inside the box and show or learn how it works, then for you it's no longer a black box. I've repeatedly said that I'm incensed by any teacher who teaches ANY method as a "black box." The fact that this one gets taught that way infuriates me. As for efficiency and volume of paper, I think those are clearly red herrings, especially the paper issue, though the time one isn't much more relevant in practice. A kid who knows this method can whip through it with facility. Ditto all the above models EXCEPT for the area model. There, both time AND paper are definitely issues. But it's a model to be explored only for understanding, not as a long-term strategy.

I was asked, "In the classrooms you prefer, does a lattice method get taught in addition to a standard algorithm, or instead of the standard algorithm?"

I have NEVER seen anyone teach the lattice method by itself. Never. What I prefer here is irrelevant to what is actually done, but to be perfectly clear: I want at least the four methods I outlined above taught, for reasons I hope I've made clear. For my money, the partial-products method is really at the heart of ALL other methods, so I would like to see it taught as the first "efficient, compact" algorithm. However, I believe that students should first do some one digit times one digit problems as repeated addition and then at least one two digit times one digit problems (e.g., 13 x 7 and 7 x 13) in "both directions so that the idea that repeated addition is far too time-consuming and space-consuming to be either efficient or (as the numbers get bigger) terribly accurate. How easy is it to write too many or too few 7's or to miscalculate when doing just the example given?

The idea is to have students appreciate what is gained by compression, but also to see what could be lost in term of information and what we "say" to ourselves as we do these other algorithms.

My opponent wrote:

For example, from the California Grade 4 Standards:

" 3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multi digit number by a two-digit number and for dividing a multi digit number by a one-digit number; use relationships between them to simplify computations and to check results."

How is "demonstrate an understanding of" operationalized in any assessment you or the MC/HOLD crowd would accept? I know that "the ability to use" is operationalized on EVERY test I've ever seen.

He next wrote:

"There is still a use for teachers, Mike. And does your beloved lattice method makes it easier to detect 'understanding'?"

There's nothing more beloved about the lattice method, for me, than any of the others. They each have strengths and weaknesses, advantages and disadvantages. Any can be taught mindlessly. None SHOULD be taught that way.

"Demonstrating an understanding of ..." is not the same as 'mindlessly apply a method learned by pure rote drill and kill".

I agree

Then we're getting somewhere.

No, we're not. Because you still won't explain (and deliberately omitted my questions about) how you would assess the former in a way that you and I and Wayne and most reasonable, knowledgeable, intelligent people would accept as meaningful.

And you're still trying to denigrate all methods but one. I have no intention of doing anything of the kind, as I've explicitly outlined here and previously. I'm not trying to "sell" anything. I'm trying to get just one anti-reform person to finally admit that all these methods are grounded in real, valid mathematics, and that all can serve useful PEDAGOGICAL purposes. I don't really care which method kids choose to use, though I, too, would likely try to sell them on #2 and #4 over #1 (for sure) and #3 (though I have not objection to anyone who uses it if s/he uses it "well."

If your retort to all this is going to be more empty claims about time, paper, etc., leave me out of it. If you've got something non-rhetorical to offer, indicating that you've decided to drop your usual stance and engage in meaningful conversation, and if you want to answer that assessment issue and to operationalize the terms you used in your previous post on this topic, I'm all ears.

Conclusion

I wish I could report that the reply I received was useful. Unfortunately, as the reader likely has gleaned from just the dialogue in this entry, my opponent remains just that: a dedicated foe who has no interest in conceding anything positive about any "non-standard" approach to thinking about, teaching, or doing multiplication (or long division, or anything else in mathematics). Naturally, since I have a long history with him, I was not surprised. I no longer write on public lists in hopes of convincing those who are incensed about every and any reform idea or method with which they themselves were not taught, be it in mathematics, science, literacy, or any other educational area. I write to help clarify my own ideas and for those readers who are at least moderate or neutral or simply looking for varied viewpoints. When I mentioned that I would not continue with this fellow on these issues, but that I expected that what I presented about these models was likely useful to some readers, he retorted, "I doubt it." And of course, he really does.

Reasons to be Cheerful, Part -3

Reasons to be cheerful (part three)

Summer, Buddy Holly, the working folly
Good golly Miss Molly and boats
Hammersmith Palais, the Bolshoi Ballet
Jump back in the alley and nanny goats

18-wheeler Scammels, Domenecker camels
All other mammals plus equal votes
Seeing Piccadilly, Fanny Smith and Willy
Being rather silly, and porridge oats

A bit of grin and bear it, a bit of come and share it
You're welcome, we can spare it - yellow socks
Too short to be haughty, too nutty to be naughty
Going on 40 - no electric shocks

The juice of the carrot, the smile of the parrot
A little drop of claret - anything that rocks
Elvis and Scotty, days when I ain't spotty,
Sitting on the potty - curing smallpox

Reasons to be cheerful part 3

Health service glasses
Gigolos and brasses
round or skinny bottoms

Take your mum to paris
lighting up the chalice
wee willy harris

Bantu Stephen Biko, listening to Rico
Harpo, Groucho, Chico

Cheddar cheese and pickle, the Vincent motorsickle
Slap and tickle
Woody Allen, Dali, Dimitri and Pasquale
balabalabala and Volare

Something nice to study, phoning up a buddy
Being in my nuddy
Saying hokey-dokey, singalonga Smokey
Coming out of chokey

John Coltrane's soprano, Adi Celentano
Bonar Colleano

Reasons to be cheerful part 3

If the above rings no bells, you're probably too young or too old to recall the sprightly Ian Drury and the Blockheads. You owe it to yourself to download the above song immediately and listen to it several times before proceeding. This way, there be monsters.

Bad Day At School

There's nothing quite like having a classroom full (and I do mean FULL: as in 36 of 42 allegedly enrolled students in a room with desks and books for 30) of "precalculus" students with too much heat in the room and too much ire in their souls as you try to take them through a review of something that on the one hand you know they SHOULD know, and on the other you suspect with good reason is going to be news to most of them, despite having allegedly passed two years of algebra and one of geometry. The review topic in question: the point-slope form of a linear equation. The first task, having ascertained that the majority of students at least believed they knew what "slope" meant and could find it given the coordinates of two points, was to find the equation of the line that passed through them. Little did I realize just what sort of tiger trap I was about to fall into.

Picking two points from one of the textbook problems, I began guiding the students through the process of using the point-slope form, with my usual plan of showing that regardless of the point one chose to use in the equation, it would result in the same graph, the same line, and, upon applying a little algebraic manipulation into the slope-intercept form, stark evidence that the equations arrived at with either point were equivalent, as of course should be the case if they have the same slope and pass through the same two points.

Unfortunately, I never got that far. So much time had been spent dealing with logistical issues regarding the shortage of seats and books, as well as dealing with various disciplinary issues, trying to take attendance on-line (PowerSchool, where is thy sting?) given that I was using my laptop, not the promised but as yet invisible Pentium-based desktop Mac that would have made the process much less time-consuming (for reasons not worth going into here), I knew that if things continued to go less than smoothly, it would be difficult to finish what was supposed to be the only major mathematical point I thought I might be able to look at with them before class ended. However, as I started to proceed with the equation with one of the two points, I was rudely and persistently told by a vociferous subset of the class who had been making things difficult (when not making them impossible) for much of the two weeks we'd been meeting that I was clearly wrong.

At first, I thought they meant that I'd made some sort of calculation error, hardly an impossibility under the best of circumstances, and these were anything but the best. I began to recheck my work, but the sound and fury from their part of the classroom made rational thought or even simple calculation a doubtful process at best. Finally, I realized what was amiss: they had taken PART of my lesson as gospel, but refused to attend to the crucial caveat. Since I labeled the points as "point 1" and "point 2" respectively, and since the coordinates were labeled with the appropriate subscripts, clearly I was in error when I chose "point 2" as the one to plug into the equation. After all, hadn't I written the point-slope form with subscripts x1 and y1?

How could I possibly be stupid enough to now be telling them that the point that had 2s in the subscripts were kosher (okay, no one said "kosher" and in fact, none of what was said approached in any way the slightest degree of civility, but this is a family blog, after all)? Obviously, I'd made a huge mistake and anything I did from this juncture on was going to be horridly wrong.

Now, this was hardly the first time I'd taught this topic, either to high school or community college students. Indeed, having taught a lot of algebra courses over the years, I was used to there being skepticism that it didn't matter which point was used and that it would be possible to show that whichever was selected would be fine, and that the resulting equations, though initially appearing to represent different lines, would ultimately turn out to be the same, without any doubt at all.

However, nothing in my experience with non-alternative education students had prepared me for a group of students who were supposedly ready for precalculus and who not only were so confused by a notation issue, but more importantly were NOT going to be patient or trusting enough to see if just possibly their teacher had a clue about what was going on and could, in short order, demonstrate that fact and part the clouds.

Be Careful What You Wish For

I hoped, of course, for a teachable moment. I figured that either I'd get to finish the problem with input from the students, and, having tried both points, they'd start to see (or in some cases REMEMBER from previous experience) that all was indeed well, or that if I were extraordinarily fortunate, the light would go on for most of them before I even got to that point, and, mirabile dictu! they would look at me with new-found respect, paving the way for a successful and productive year.

Instead, things bogged down as no one in the class would agree to see what happened if they finished the problem with those coordinates with the OTHER subscripts and I finished it with the point I'd selected: the hue and cry became such that more time was lost trying to maintain some order, and when the bell rang, the problem remained unfinished, several students loudly agreeing with my chief antagonist in the class when he proclaimed that I "didn't know what I was doing."

While of course there have been moments in my teaching career when I really DIDN'T know what I was doing, either mathematically, pedagogically, or a combination thereof, this was most decidedly not one of the times I was unclear about the mathematics. I was very confused, however, about what kinds of experiences would lead a class of seniors and juniors to be so invested in proving that the teacher couldn't possibly right, and not by making a convincing mathematical argument, but merely by making it effectively impossible for the teacher to show that he might actually be (dare I say it?) right.

What the fudge?

I'll stop at this point, leaving readers to consider their own experiences in similar circumstances, if any, giving everyone ample opportunity to contemplate what might have saved the day in this or similar situations. I'm sure that my "solution," such as it was, will not be terribly satisfying, so there's no hurry on my part to offer it. By all means, I'm sure others would have done much better in my shoes than did I, and I'm interested to learn about the alternatives that I might have employed but did not.

If only I'd had a Vincent Black Lightning 1952 and a red-headed girl waiting outside, instead of a Japanese sedan and a horrid, traffic-snarled 80 minute slog home.

Pardon The Interruption

If you've been wondering about new entries on this blog, wonder no longer. After a month's hiatus, I'm back, blogging from a new perspective. I have taken a job at a public charter high school on the west side of Detroit and reentered the classroom full time after four years of doing professional development work, mostly with elementary and middle school math teachers. As you might imagine, if feels both strange and strangely familiar to be back. Unlike my two previous full-time positions in K-12, where I taught at an alternative high school with students whose average math and literacy level was between 4th and 5th grade, and at a middle college where I basically spent one or more semesters with students trying to get themselves in a position to pass an intermediate algebra course at the community college that sponsored our charter and physically hosted our school, this time around I've got one group of sophomores for Algebra 1, three classes of seniors for Algebra II, and one class of seniors for precalculus.

The first day was deceptive, to say the least. Fewer than half the students showed up for school, if attendance in my classes was typical, and things went remarkably smoothly. I think it was the first time I ended the first day of classes feeling calm, cool, and ready for whatever the year had to offer (as opposed to wanting to crawl into bed before my head exploded). I thought perhaps the real students were being cleverly hidden away while a group imported from Cass Tech or some other magnet school had been brought over to heighten my false sense of security. Well, maybe not Cass Tech: in going over an SAT-type problem about combined rates of work, I discovered that many if not all of my students, or at least all of those who were willing to venture an opinion, believed that 1 1/3 was between 1 1/2 and 2. Perhaps not shocking from freshmen, but from 10th and 12th graders? From 12th graders enrolled in precalculus? What was I getting into? But since behavior was decent, I was optimistic that much could be accomplished.

Day 2 made clear that the honeymoon, if not over, was clearly not fated to last through Monday. Indeed, with two of my classes, it was pretty much time to seek couples counseling, individual therapy, and a good divorce lawyer. While I had been told that the maximum enrollment in my classes would be 30 (which frankly is more than I've EVER had in a class I taught in K12 alone), by today there were at least two classes that had 30+. With most of those enrolled showing up today, it was hot (no windows in my room), humid, and too often annoyingly loud. And me without my earplugs, my whip, or my teddy bear. :(

So my challenge here will likely be for much of the ensuing year to explore my teaching experiences with you, time and brain allowing. We'll see how it goes for as long as it seems productive to do so. Feel free to come along for any or all parts of the ride.

An Interesting Conversation

While not quite the Lincoln-Douglas debates, over the last several days, an old argument has been resurrected on math-teach@mathforum.org. Those not recently conversant with that long-standing free-for-all list likely don't know that after a decade or so as the wild west of mathematics education lists, where anything went and often did, in spades, the list owners have stepped in to bring peace, harmony, and bliss to this particular battleground in the Math Wars. Thus, the lion now lies down with the lamb, all is civility, no epithets are hurled, no ad hominem attacks launched, and decorum is the order of the day.

And yet, intriguing undertones, sometime imported from more venerable and genteel locations such as NOTICES of the AMS, 54(2007), pp 822-3

There, one may find an exchange between Jerry Rosen, a founding member of Mathematically Correct and a Cal State Northridge mathematician and Sol Garfunkel of COMAP regarding old questions about who is to blame for the alleged contemporary crisis in US K-12 mathematics education. Professor Rosen wrote:

NSF-Sponsored Educational Programs

In his Opinion article “Because Math Matters”, Solomon Garfunkel outlines true problems but draws wrong conclusions. His statement “We are not doing a very good job. U.S. students are falling behind students in most industrialized countries” is true, but his claim that the NSF [National Science Foundation] has “led the effort for innovation in mathematics education since the 1950s” is loaded as well as overtly political (something he claims we should avoid). I am not sure what the NSF was doing in the 1950s and 1960s with respect to math education, but it has been quite involved since the 1980s and in mostly bad ways. Look at some of the results of NSF programs’ de-emphasis on arithmetic calculation and algebraic manipulation and the (NCTM-encouraged [National Council of Teachers of Mathematics]) substitution of calculator usage for long division. Implicit in his analysis and many such writings is the opinion that teaching children strong computational skills is bad. But the example given is the poor performance of U.S. students. However, this poor performance has taken place during the NSF’s most innovative period, i.e, from the 1980s to the present. This is why the backlash against fuzzy math on the part of parents has been so extreme; during this time U.S. children have plummeted in math. A good example to study would be the New York City public school system and the CUNY [City University of New York] college system. It is acknowledged that in the 1940s–1960s the CUNY colleges produced more scientists (including mathematicians) and doctors (and various professionals) than any other college system. Furthermore, many of these people came from the working and middle classes. Doesn’t this need to be analyzed? I went through the NYC public school system and CUNY in the 1960s and 1970s, and many of my classmates ended up in highly successful professions. However, there were bad things, one of which was tracking minorities into programs where they were denied traditional approaches to education. One of the dubious achievements of reform movements of the 1980s and 1990s was that basic arithmetic competence was denied to minorities as well as to the white working and middle class children. Let me summarize: The U.S. became a world leader in mathematics between the 1940s and the 1960s. In fact, many of the countries that are beating us now sent thousands of their future scientists (and still do) to be educated in the U.S. Since the 1980s, concomitant with the rise in “innovative” teaching techniques, the U.S. has declined considerably in international comparisons. Garfunkel is wrong in saying this isn’t political; it is most definitely political. In fact, the NSF needs to immediately stop funding all education initiatives and start subjecting future education grant proposals to the same rigorous standards they use for mathematical/scientific research.

Sol Garfunkel replied:

Reply to Rosen

Rosen misstates the actual time frame for reform funding and as a consequence misplaces the blame for poor performance. The NCTM Standards were published in 1989. They were universally endorsed by all of the major mathematical professional societies. The Standards were in fact undertaken because of apervasive sense that we were doing an inadequate job of educating students in mathematics at the K–12 level. NSF funding of reform efforts began in the early 1990s, and the major reform curricula did not appear until the mid- to late-1990s. And at their height (there has been some drawback of high school programs since the math wars) the elementary, middle, and high school curricula achieved no more market share than 25%, 20%, and 5% respectively. It is clearly inappropriate to blame poor performance in the 1980s and 1990s on these innovative curricula. The inconvenient truth is that there were no good old days, just a lot of hard work left to be done.

Math-Teach Gets Involved

Lou Talman, a mathematician from Metropolitan State University in Denver posted a link to the above to math-teach under the subject "More Nonsense From Jerry Rosen, and Solomon Garfunkel's Reply"

After that, Wayne Bishop, another Cal State (LA) mathematician wrote:

"Jerry's letter strikes me as being spot on.. Could you be more specific as to what you consider to be nonsense?"

Lou Talman replied: "Sol Garfunkel explained it pretty well, Wayne. Do you disagree with Garfunkel?"

I posted, independently:

As Sol Garfunkel accurately states, Jerry Rosen's dates are badly
amiss, for starters.

As for the rest of the claims Jerry makes, they appear to be a very
unscientific attack on something he personally dislikes, grounded on
his n = 1 experiences as a student, and then adding 2 + 2 to get 5
regarding who and what is primarily to blame for underperforming
inner city students in the 1980s and 1990s. Any argument that pins
the blame on NSF/NCTM ideas and programs is badly flawed given the
reality of when the programs appeared, as well as their penetration
into the market. Moreover, their penetration in NYC in particular was
not very great until about 2004, and then only at the elementary
level. Jerry knows this. So does anyone paying attention.

Finally, it's unclear at best that anyone is "beating us"; the
educational sky has been falling in the United States, according to
one or more critics, since about 1845. During the ensuing century and
3/4, we went from a backwater to the most powerful nation in the
history of the world. I'm sure that lots of nations wish their skies
were falling in the same ways ours have been. While no one seriously
disputes that there's room for improvement in the quality of
mathematics education in this country, there's still a lot of debate
about where and how that improvement needs to be.

So what don't you consider to be nonsense about Jerry's letter?

Jerry Rosen replies to my post

I cc'd the above to Jerry Rosen, since he had not be a participant on math-teach for many years. He responded as follows:

Let me see if I can explain this as simply as possible:

The CUNY system had produced more doctors, lawyers and scientists than any other system in the country from 1940's -1970's and this is a system with a mostly middle and working class student body. Anyone intent on doing educational research, not pushing some agenda, would want to understand what was good about the the NY public school system and what wasn't. The rebellions of the 60's lead to open admissions in CUNY in the 70's (I went to Brooklyn College during the start of open admissions - anyone with a HS diploma could go). The CUNY colleges became moer integrated (obviously a good thing), but many of of the inner city minority graduates were not properly educated. I was involved with tutoring in the CUNY system and many of these
open admissions students couldn't do basic math and basic English. So there was a failure, a large failure, on the part of the NY publiuc school system. This was a failure due to racism. But, what is the answer??

Starting in the 80's, we have people starting say, with absolutely no research behind it, that we need to improve education of inner city students by reducing computation and whole language learning. But the problems that the inner city students were having was exactly that their basic skills were so poor. Those of you who support holistic, feel good, "let's keep the poor minorities feeling good about themselves'" have bought into very racist and amazingly harmful policies. You did no research and many obtained grants and restarted dormant educational careers on the backs on inner city minority students - it is racism,
mike.

My reply to Jerry Rosen

As of this entry, what follows is the most recent addition to the conversation, having appeared on Saturday, August 4, 2007. Interested readers should continue to check the math-forum website to see if things go further.

I posted:

First, the idea that current reforms could have caused past
shortfalls cannot remain unchallenged. You refuse to acknowledge that
the reason there were so many students who couldn't do basic math and
English must be PRE-reform. There's no logical basis to lay that at
the feet of what you call "whole language" and whatever name you
currently care to give to the reform math efforts (which are
manifold, not monolithic) in the 1990s and beyond. I'll call these
efforts, collectively, progressive reform of math education.

Since that initial premise is false, any conclusions you draw from it
are suspect. And as I've asked for the past 15 years, where is the
research that supports that the methods with which you were taught
were successful for the vast majority of students? Your own account
below indicates quite clearly that they weren't helping many inner
city kids. Some of us would argue that no one is truly well-served by
the kind of instruction that typified the mathematics teaching that
was not only typical in the 1960s, but remains extremely typical
today, especially at the high school level. But that's a debate not
likely to be settled readily. What is clear, however, is that there
were huge numbers of students of all ethnicities and economic
backgrounds who didn't learn much math in this country well before
any reform effort you care to name. Richard Rothstein's THE WAY WE
WERE? clearly documents beyond any doubt that educational experts and critics were finding deep problems with American literacy, math, and other aspects of public education as far back as Horace Mann in 1845 or so. Whether any of these critiques were well-founded isn't the issue so much as the notion that we've always had people in the US saying that public education isn't working. And we also all have good
reason to believe that while the sky isn't falling and never has been, we could improve the quality of mathematics teaching (as well
as teaching in other subjects).

But where is the research to support the idea that things were
actually working well for most kids, white, black, purple, or blue,
under the system in which you learned? Many individuals who thrived,
relatively, in that era, appear blissfully unaware that many, many of
their contemporaries were floundering in math (and other subjects).
But you can't really be one of them, given your social awareness, can
you? This isn't ONLY a matter of "racism": It's a matter of class,
and it's a matter of pedagogical ignorance. The idea that we knew
MORE about how people learn, either in general or in mathematics in
particular, 100 years ago than we do now is hard to swallow. The
methods we consider "traditional" actually aren't as timeless as we
may think, but they've been around for long enough to show that they
are, at best inadequate.

The attack on progressive reform as "experimental" ideas being
foisted upon innocent children by mad scientists or fuzzy ideologues
is a myth that has been spread as a result of the cultural and
educational wars of the past couple of decades. We can debate the
quality of the research and try to come to some consensus (if a
miracle occurs) about what comprises good, valid, and reliable
research methodology and refrain from changing the rules to suit a
given data set that favors or runs counter to our ideologies, but no
one can fairly claim that progressive reform in math education didn't
ground itself in research.

I won't muddy the waters here by going too deeply into the literacy
wars other than to say that having read a lot more about so-called
'whole-language' vs. phonics-first or phonics-only reading
instruction than I had when those issues started surfacing on this
list in the mid-to-late 1990s, I'm not convinced that the advocates
of the latter aren't guilty of exactly going off on a political,
ideologically-based tear to push an ineffective program of literacy
for motives that, if nothing else, appear to be very much about
personal profit and, as you say below, "pushing an agenda." The
recent scandals surrounding the US Dept of Education, Reading First,
and "reading czar" Reid Lyons and friends attest to some very
questionable practices. And this from the fellow who recommends
"blowing up the schools of education," with no reference to getting
the people out of the buildings first, I might add, suggesting a call
for terrorist tactics that would have landed any progressive
educators who suggested something similar behind bars under the
Patriot Act. One useful reference that documents the anti-progressive
movement's strategies and tactics in the Reading Wars is Denny
Taylor's well-researched BEGINNING TO READ AND THE SPIN DOCTORS OF SCIENCE: The Political Campaign to Change America's Mind about How Children Learn to Read. I would also recommend Frank Smith's work.

As for your last couple of sentences, they seem to be accusing those
who promoted the progressive reforms in mathematics education as: a)
wanting to "reduce computation"; b) of doing no research; and c) of
being racists.

As to the first, while there is undeniably a call for less emphasis
on mathematics AS computation for K-12 curricula in the reform
efforts that began with the 1989 NCTM Standards for Curriculum and
Assessment, very few people, if any, have ever called for eliminating
computation from the curriculum. It's simply not honest to claim that
teaching alternative algorithms and strategies, many of the first
leading students and teachers to more deeply examine why basic
algorithms for arithmetic work, and most or all of the latter being
perfectly legitimate problem solving methods used by mathematicians
and scientists, are attempts to water down curriculum. Quite the
contrary. The reform efforts of the late '80s and early '90s were not
put forth as the final word, but rather as frameworks for further
exploration and refinement. Had the responses from some people who
didn't like these efforts been less rhetorical and more fair-minded
and honest, I suspect we'd be much further along towards finding even
more effective materials and methods. Unfortunately, much energy and
momentum has been lost as a result of the ceaseless and unfair
propaganda campaigns to undermine any and all reforms.

The second point about lack of research simply does not stand up to
the slightest scrutiny. Again, you may wish to debate here the nature
of what counts as research, but to say that there's no research and
that people went off without doing any first or doing on-going
research after is simply untrue.

It is the last point, however, that I find most disturbing. I have
long complained that the charges of "liberal racism" that seem to
underpin and resound throughout the rhetoric of anti-reformers is one
of the most heinous and unfounded tactics I've ever witnessed as a
professional educator. I'm happy, at least, that you don't pussyfoot
around and actually use that word several times in your post. But I
think your accusations are not well-founded and not at all well-
directed. I will leave it to you to answer how you can find yourself
politically allied with so many openly right-wingers, conservatives,
neo-conservatives, and reactionaries, and yet still believe that they
all have a fundamentally similar social agenda to you and the other
anti-reform spokespeople who label themselves as liberals,
progressives, or socialists. I, for one, have rarely seen a more
utterly ironic and incredible alliance. Fortunately, I don't have to
rationalize my work in this regard. I don't find myself daily in
agreement with people about mathematics education issues with whom I would otherwise be horrified to agree with about much else. But for
you to hint that the people I've worked with and come to respect as
deeply dedicated professionals in mathematics/mathematics education
are "racists" or practicing "racism" is both utterly unfair and
utterly untrue. By the standards by which this list is now supposed
to be operating, I'd say "dem's fighting words." But out of deference
to the list owners, I will not respond in kind. I'll simply state my
previously-offered position that blanket accusations or insinuations
of "racism" against either members of this list or professionals in
mathematics and mathematics education have no place in this forum or
in this debate. Absent clear documentation that establishes beyond a
reasonable doubt that reformers or anyone else are acting out of a
"racist" agenda, that word and related forms should not be used.

A Book You Need To Read If You Care About Education

In 1998, as California was in its first year of a counter- revolutionary mathematics curriculum framework thanks to efforts led by anti- progressive mathematics education groups like Mathematically Correct and HOLD, Richard Rothstein's THE WAY WE WERE? The Myths and Realities of American's Student Achievement was published as a Century Foundation Report. The first chapter should be required reading for anyone who thinks s/he knows about the history of American public education. It explores and debunks many widely-believed myths about "falling" achievement in literacy, general academic knowledge, and the whole notion that our public schools are failing us, kids today are far less competent in basic skills than were those of previous generations, and that our lousy schools are paving the road to hell for our kids and our nation. In particular, Rothstein examines two currently hot-button topics - social promotion and bi-lingual education - and reveals many surprising realities about the history of these two educational practices that will likely shock many readers, whichever side of these issues they might support.

In particular, the first chapter of the book is an eye-opener. It looks at the views held by critics of contemporary American public education during past eras from the 1960s and '70s all the way back to before the American Civil War. What he concludes about nostalgic portrayals of a bygone "Golden Age" of American schooling is something I and many other advocates have long said about mathematics education in particular: if there was such an age, it goes back more than 175 years. In fact, there is no evidence to support the notion that anyone in this country lived in an era in which contemporary critics were not bewailing the decline and fall of our schools and the abject ignorance of the next generation's young people.

Although this is not a long book (all told, it's only 140 pages) and even new it costs only about $12.00, it's possible to pick up a used copy for under $2.00 on-line. But for those who want to get a direct sense of what Rothstein says and don't want to take my word for it, Richard C. Leone's Foreword, as well as Chapter 1: Our Failing Public Schools and Chapter 2: It Just Doesn't Add Up, are available to read on-line free of charge on the Century Foundation's web site.

I urge everyone to read at least the first chapter. It will likely whet your appetite for the whole book. And it will give you both food for thought as well as some well-documented ammunition the next time someone starts telling you about how wonderful things were in American schools (at least compared to today) and how brilliant all the students were back in some previous decade. To quote Rothstein's clever paraphrase of Will Rogers: "The schools ain't what they used to be and probably never were."

Asking Good Questions In Math Class, Part 2

Last time, I explored some ideas about what comprises good mathematical questions in mathematics classrooms taken from MAKING SENSE: Teaching and Learning Mathematics With Understanding, by Hiebert, et al. In this entry, I want to introduce ideas from a book by Peter Sullivan and Pat Lilburn: Good Questions for Math Teaching: Why Ask Them and What to Ask [K-6]

Sullivan and Lilburn's Criteria

In the introduction to their book, Sullivan and Lilburn list three main criteria for good questions: a) They require more than remembering a fact or reproducing a skill;
b) Students can learn by answering the questions, and the teacher learns about each student from the attempt; and c) There may be several acceptable answers.

The first of these features, while not something everyone sees as important in all mathematics classrooms, is essential to non-routine questions. It takes no teacher skill to come up with questions that only ask for students to show that they have memorized a fact (e.g., "What do we call the answer in a multiplication problem?" or "What is the sum of the interior angles in any triangle?") or reproduce a skill (e.g., "Find the product of 27 x 36" or "What is the slope of the line that passes through the points (3, 2) and (0, 6)?") Teachers can easily make up such problems or find them in textbooks, ancillary materials, supplementary workbooks, on-line, or generated by software. There isn't much to debate here: no one is suggesting that there is no place for such problems, and as they are in fact ubiquitous in American classrooms, I have no interest in debating or analyzing their use. As practice, as routine exercises, as reinforcement, as "drill and skill" or "drill and kill," depending on your viewpoint, they simply are part of the landscape. What is of more interest to me, and perhaps of more controversy for some, emerges from the other two criteria.

The second property these authors want to see in good questions are those that are both somehow instructive in their own right to students and which carry added information that the teacher learns about each student as s/he attempts to solve the problem. This suggests that good problems themselves add something to student knowledge and understanding that might not be readily obtained from, say, listening to direct instruction or doing routine exercises.

At the same time, the authors value problems that tell the teacher something about the student that, I assume, helps better assess student thinking and learning, allows for better feedback and instruction, and which, once again, would not be evident from having students complete more routine exercises.

The final property they see as part of good problems is the possibility of more than one acceptable answer. Some parents and critics of reform teaching find this sort of thing highly troubling. Unfortunately, coupled with an increasing emphasis on process over product, critics and parents (and some pro-reform teachers), appear to believe that the idea here is that the "right answer" doesn't matter. From my perspective, the "process vs. product" debate is fruitless if people on either side insist on either supporting one extreme or the other, or upon claiming that people on the other side only value either process or product. For what may be a small minority of pro-reform teachers, I can only say that believing that the answer doesn't matter at all leaves out a piece of the learning puzzle that they can't afford to ignore. But the other extreme view, that all that matters is "getting the right answer" and everything else is a waste of time seems to me to be destructive for kids if taken too far (kids can do an excellent job, make a small mistake that could stem from misunderstanding, carelessness, or something as trivial as reversing digits in copying down the answer (or mis-marking a multiple choice answer sheet). It's not that it's destructive for kids to think there's a single right answer when in fact there is; it's not that it's destructive for them to know they made an error, if in fact they did; what is destructive is to denigrate excellent and clear thinking on the part of students if in fact that have done this and given evidence of it. And that is, of course, when of the problems with multiple-choice problems or any assessment that fails to credit students for demonstrating what they know (or rewarding them for a lucky guess): not only are students taught that the answer is everything, but also that it's better to be lucky than good (or at least just as good to be lucky), and that gaming the system suffices. It also teaches them that if you get a problem right (even through luck or a series of mistakes that just get you to the right place blindly), there's no point in thinking about or having to explain your answer (and I suspect this is behind SOME of the resistance on the part of kids to being asked to talk or write about their mathematical thinking: doing so can unmask their luck, their underlying confusion, or even cheating).

In any event, the point here that must be noted carefully: the authors say that there MAY be multiple acceptable answers to good problems. So they allow that problems with only one acceptable answer could also be good. But what does it mean for a problem to have multiple acceptable answers? Before I look at one example from their book, I want to suggest that a problem that has multiple solutions that are correct within the terms of the problem are quite possible. This fact can arise in a case where there is more than one right way to accomplish a task that has a mathematical component but which is not asking for just a number. Or perhaps more than one number meets the criteria of a problem. Even in cases where there is a sole correct numerical solution, there may be more than one good way to arrive at that answer.

Unfortunately, these reasonable possibilities do not strike everyone as "okay" for kids. This believe may arise from a narrow understanding of what mathematics is and how it develops historically. Unfortunately, the history of mathematics is not much stressed even for mathematics majors at the college level. Few university mathematics departments offer a course in math history any more, not even for students who plan to teach at the secondary level and who could benefit greatly getting a more honest historical view of how the field has developed and grown. Such an understanding can help make the subject more human for students and teachers alike, and can also increase its accessibility. The fact that Newton and Leibniz independently developed the calculus, one of many such examples of parallel invention in science and mathematics, helps validate the notion that not ever powerful idea must wait for a single, unique person and set of circumstances to bring it to fruition.

Finally, the idea of open-ended problems is not radical or unique to American mathematics education reform. Jerry Becker and Shigeru Shimada edited an excellent volume entitled, The Open-Ended Approach: A New Proposal for Teaching Mathematics. It explores many examples of quite challenging and engaging problems from Japanese K-12 math classrooms that illustrate what is meant by problems with more than one reasonable or valid solution. And there is clear evidence from the TIMSS videos taken in Japan that such problems are used regularly in ordinary middle school lessons. It seems almost obvious that a non-routine problem, even one drawn from quite typical mathematics content, can fit this criterion.

For a look at open-ended questions drawn from American elementary classrooms, along with ways to find, create, and evaluate student responses to them, see OPEN-ENDED QUESTIONS in ELEMENTARY MATHEMATICS by Mary Kay Dyer and Christine Moynihan.

An Example from Elementary Mathematics

Sullivan and Lilburn offer a host of examples from different content bands and grade levels. The following is in their chapter on number in a section of fractions problems for grades 3-4. On page 27, they pose:

"A friend of mine put these fraction into two groups: 3/4, 2/5, 1/3, 6/10, 1/10. What might the two groups be?"

Looking at this in terms of the authors' three criteria for good problems, we can note that it fits the first one: the problem calls for something other than the recall of a fact or demonstrating a procedure. Second, since students may not have been asked to do anything like this problem, they may learn from the process of trying to solve it. Further, the teacher can gain information about what students know about fractions, possible areas of confusion, and willingness to engage in and communicate about mathematics through how each chooses to answer the question. Finally, it should be obvious that while there is no clear-cut right answer to this problem, there are many acceptable answers.

Before going further, try to decide how you would sort these fractions into exactly two groups. Can you think of more than one way to do so? What did you focus on? Do you believe that there is a single best way to sort them? What alternatives might students you teach come up with? Try not to read further until you've come up with at least three ways to sort the fractions that make mathematical sense to you or that reflect things you think students would think of.

Okay, I'll assume you've tried to sort the fractions a few different ways. Here are some possibilities:

A) 3/4 and 6/10 B) 2/5, 1/3, 1/10.
A) 1/3 and 1/10 B) 3/4, 2/5, 6/10
A) 3/4, 2/5, 1/3 B) 6/10, 1/10
A) 2/5 and 1/3 B) 3/4, 6/10, 1/10
A) 6/10 B) 3/4, 2/5, 1/3, 1/10

Can you see at least one reasonable mathematical basis for each sorting above? Are the multiple valid mathematical reasons for any of the sortings? Are there sortings you found not listed above? Do they involve mathematical ideas not reflected in any of those I've offered? Can you think of any that students might come up with that would indicate confusion? How would you know?

I hope it is clear that tasks like the one above are most effective for learning when: 1) students are required to explain their thinking; 2) students share their explanations with the teacher and each other; 3) students offer and accept feedback constructively and freely from peers and the teacher; 4) a classroom culture has been established to facilitate all of the above, students have had opportunities to model or see modeled all relevant aspects of the expected behaviors, and students feel safe to share all their ideas and to offer and ask for feedback; and 5) this classroom culture establishes that it is important to understand other people's thinking and to make one's own thinking understandable.

What's Next?

In Part 3, I will look at a companion volume to the Sullivan and Lilburn book: GOOD QUESTIONS FOR MATH TEACHING: WHY ASK THEM AND WHAT TO ASK Grades 5-8 by Lainie Schuster and Nancy Canavan Anderson.

In part 4, I will investigate the other issue I raised at the beginning of this entry: what kinds of questions can teachers ask to help get students "unstuck." What kind of classroom discourse practices do teachers need to promote to effectively make students more independent, able problem solvers and learners of mathematics.