Asking Good Questions In Math Class, Part 1

One glaring weakness in many American mathematics classrooms is the nature of questions teachers ask. This manifests itself in two pedagogical realms: first, the kinds of mathematical questions or tasks teachers tend to pose are often of a closed nature which call of nothing more than a calculation (or short series of calculations) and for which the answer is a single number that requires no further interpretation or thought on the part of the student. The second area of difficulty arises in the types of questions teachers ask when they want to help students who are having difficulty performing a given task (which generally consists of something like that mentioned previously, although this problem arises in more complex problem situations).

My focus here will be on the the first situation: what kinds of tasks might comprise "good questions" for math students in elementary school?

Mathematical Tasks

In their book MAKING SENSE: Teaching and Learning Mathematics With Understanding, Hiebert, et al., propose three features for appropriate mathematical tasks: 1) the tasks make the subject problematic for students; 2) the tasks connect with where the students are; and 3) the tasks engage students in thinking about important mathematics.

Problematizing mathematics

What do the authors mean by the word "problematic"? It is clear from their book that it does not mean to make the mathematics puzzling or difficult (though either or both of those may be collateral results). Rather, it means to ask questions that make students have to think, to make use of what they know, to move further along the path of their mathematical knowledge and abilities. Thus, a given problem may effectively problematize mathematics for one group of students that would not have that impact on other students. Asking a simple addition question to Kindergarten students who haven't learned what addition is has the potential to problematize what has heretofore been easily addressed by counting. Given the same problem, most third grade students would see the situation as a routine exercise. But for the third graders, a question about division could be problematic.

Meeting students where they are mathematically

That makes the second requirement for good problems crucial. And it also raises an important issue that is often missed in the Math Wars: teachers who have good understanding of where their students are mathematically are the ones best positioned to judge when a problem is appropriate for them. The better the teacher knows each student, the better able the teacher is to adjust tasks to individuals. A problem that is completely beyond the capabilities of all students in a class would likely not be a good one. But for far too many teachers, if a problem occurs in the textbook as part of a lesson, the teacher feels obligated to give that problem, as written, to all students. As a math coach, I've seen teachers follow blindly whatever is presented in their text, even when a given problem may take all or most students far off track with the main goal(s) of the lesson. When guest-teaching in their classrooms, I have frequently deleted or changed the sequence of the problems in the book when presenting them to the students. The teachers are generally very surprised by this, but when we discuss it afterwards, most are also unable to think of any clear reason I might choose to do this.

For example, in teaching a lesson on equivalent fractions to fourth graders from the EVERYDAY MATH book, I purposely omitted a problem that introduced mixed numbers, feeling from previous observation of the students that it would unnecessarily bog down the lesson for them in what was to my mind a secondary consideration. The notion that the text is a resource, not a bible, and that the teacher is the best judge of how to use that resource with a given student or classroom seemed radical to even the veteran teachers with whom I worked.

Mathematical residue

The third point the authors raise connects perfectly with the previous two while also raising another question about teacher content and pedagogical-content knowledge: many teachers do not seem to be able to readily identify the key mathematical idea(s) in a lesson or to distinguish between major and minor mathematical ideas in a unit or lesson. This makes successful decision making about what problems to use, omit, or revise very difficult if not impossible for them. Coupled with either relative ignorance of and/or lack of concern with where students actually are mathematically, confusion about how mathematical ideas are interrelated and what a reasonable hierarchy among them might be leaves teachers little choice but to follow what is in the text with complete passivity. While for some, this may be a desirable path of least resistance, it cannot be an acceptable teacher practice if we seriously expect to see significant improvement in mathematics teaching and learning. Thus, teachers must consider (and textbook authors must strive to make as clear as possible) what the important mathematical residue is for each lesson, chapter, and unit. By mathematical residue, I mean the key idea(s), concept(s), procedure(s), and connections that students are expected to take away from a task, lesson, or larger structure. The mathematical residue should also point clearly towards what students can actually DO with what they are expected to have learned from successfully engaging in the task.

What Next?

In Part 2, I will explore ideas from a book by Peter Sullivan and Pat Lilburn, : Good Questions for Math Teaching: Why Ask Them and What to Ask [K-6].
I will also look at some particular problems and how they might be good examples of what teachers can do in their classroom regardless of what curriculum they teach from.

In part 3, I will look at an expanded list of ideas about good math questions from a related book by Lainie Schuster and Nancy Canavan Anderson : Good Questions for Math Teaching: Why Ask Them And What to Ask, Grades 5-8

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.

Why Do Anti-Progressives Need To Lie?

The following appeared in a video at a Ridgewood School Board meeting about the INVESTIGATIONS IN NUMBER, DATA AND SPACE elementary mathematics program in use there:

"The literature of reform math has changed history, painting a dichotomy wherein its proponents state that all former math teaching required memorization of algorithms without understanding and that students had to “check their brain at the door” before math class.

I didn’t invent that expression. The official TERC Investigations web site reads, “otherwise intelligent and curious children who check their brain at the door as math time begins.” That’s how they characterize 'math before TERC.'"

Is this an accurate quotation? Yes, as far as it goes, and up to the word 'begins.' But the last sentence is an outright lie, as is her claim in her opening sentence. Look at the context of the quotation as it appears on the TERC site:

"Teachers see it all the time: otherwise intelligent and curious children who check their brain at the door as math time begins. "Just give me the rule" they demand, when asked to solve a problem. These children have learned, from previous school experience or from home, that mathematics is just following a bunch of rules that don't often make sense. Fortunately, teachers using "Investigations" are seeing fewer and fewer students with this behavior. Most students who have had "Investigations" since the early grades expect math to make sense and look for reasons behind steps and procedures they encounter."

Any mathematics teacher will tell you that this is accurate. Kids do demand to be given the rule, they do not want to be asked to think (it's not what they've been led to expect in math class; quite the opposite, in fact), and they resent being asked to articulate their thinking, even though it is only through showing how you do your work that anyone can evaluate what you've done: a raw number at the end could be a good guess, a wild speculation, copied from someone else, or the result of erroneous thinking that resulted in a lucky right answer for the wrong reasons. Every mathematics teacher knows this, and I suspect every thinking adult (and most children) know it, too.

But where is the blanket claim about "all math teaching before TERC" or "all math teaching" at all? It's not there. It's not implied. It's a total invention of an anti-reform activist parent Linda Moran, who dislikes the Investigations program and is determined to see it eliminated by any means necessary, including utter distortions of the truth. This isn't a fluke. This is the fundamental Big Lie approach that has become the stock in trade of anti-liberal, anti-progressive thinking, both regarding education and other aspects of our society, in the last 35 years or so. And in the years following 9/11, it appears that we're back to Barry Goldwater's old claim that "extremism in the defense of liberty is no vice." I'm not sure that the late Arizona Republican believed in out and out lying, however (which might be why he wasn't elected president). I think even Senator Goldwater would have thought that the misquoting, quoting out of context, distorting, exaggerating, selective use of statistics, and outright lying that have become ubiquitous since groups like Mathematically Correct and HOLD sprung up in California in the 1990s are patently unethical attempts to manipulate the public view on what comprises good mathematics teaching.

Does Linda Moran not realize she is distorting the truth? You can watch her video statement and you can read the transcript as posted to one of the anti-reform sites in Ridgewood, and judge for yourself.

On my view, this isn't an accidental distortion or a careless addition of a statement that no one at TERC and no reform advocate would make or take seriously. We all know that there have always been good mathematics teachers out there. Ideas that cropped up in the 1989 NCTM Standards volumes did not all emerge from nowhere at the sessions during which the book's many parts were drafted. I've seen veteran teachers who were clearly implementing many of the precepts and ideas from that volume in their practice and had been doing so long before the volume appeared, let alone its companion and successor volumes. But it is clear that there are too few such teachers and too little such teaching.

What NCTM, NCSM, AMTE, and other groups have tried to do is to promote better mathematics teacher education and better mathematics teaching for all. The anti-progressives appear deeply threatened by any attempt to broaden the outreach of mathematics to every child. They have called this "dumbing down" the curriculum (while at the same time complaining that programs like INVESTIGATIONS, EVERYDAY MATH, CONNECTED MATH, CORE-PLUS, IMP, and many others are "too difficult" for kids or "too difficult" for teachers). They call the books "fuzzy" and "Rainforest Algebra" and a host of other charming epithets, the result of which has been to inflame some parents, scare school boards and administrators, and ultimately engender counter-epithets such as "nostalgia math" and "parrot math." For good or for ill, however, the shrill voices of the anti-progressives have the ear of the media and have been able for the most part to drown out those who challenge their views.

It's time to stop the lies. It's time to confront ideologues who rely on distortions and disinformation. It's also time for those who advocate for reform to stop sticking their heads in the sand about what's going on out in the world. NCTM and other groups that claim to represent mathematics teaching and teacher education need to take a stronger stand to support the best teachers and programs against these unscrupulous and irresponsible attacks.

At the same time, the publishers and research projects that have helped to promote and develop innovative mathematics programs need to do a vastly better job of helping teachers understand what these new programs are about and to vigilantly eliminate the kinds of mindless teaching that might forbid students from using standard algorithms, insist that all children MUST use manipulatives (if a given child prefers other methods and is successful with them) or any other extreme view or practice that could be turned against reasonable reform methods and ideas. While I've never personally witnessed such teaching, the anti-reformers swear that this happens routinely. Given the dishonesty that seems to so completely inform their movement, I will believe these anecdotes when I see a teacher behave so unreasonably. However, it is the responsibility of NCTM, AMTE, NCSM, as well as all the NSF project leaders and the publishers of the progressive curricula to actively work against any such extreme and inflexible practice. Doing so will benefit everyone and make it that much more difficult for people like Linda Moran to get away with making irresponsible and unfounded claims.

Out Of The Mouths of Babes

The following recently appeared on a blog of one of those fiercely opposed to reform mathematics instruction in Ridgewood, NJ. "GK" stands for "Gifted Kid"

GK prefers fractions and division to simple multiplication

A mom of a Gifted Kid gave him some multiplication drill. She keeps it to a minimum, but once in a while it's a good idea. However, it turns out she didn't give him a hard enough problem.

Here's the problem she gave her 9 year old and the expected partial products and final product:




4.2
X .24
----------
168
840
----------
1.008


But that's not how GK worked the problem. Instead, he did all this in his head:

21/5 X 6/25 = 126/125 = 1 1/125 = 1.008

Just like with reform math, Mom asked him to explain his reasoning, but not because of any ideology about language and math, but rather, because she had no idea what he had done until he spelled it out for her. S-L-O-W-L-Y.

Then she asked him why he didn't use the standard way, and he said because he liked his own strategy better.

Given harder problems, he does do them the standard way, which has now been dubbed "mom's way." Mom scratches her head a lot these days.

To which I replied: "Your son has made an excellent argument for so many of the things you argue against so bitterly, and he's done so articulately and with no ideology. Too bad you don't hear it."

Would not this child's teacher and classmates benefit from hearing about his strategy and discussing it? Are the other alternatives that this child would benefit from hearing that might emerge if the teacher were able to conduct a good class discussion about this problem, about his strategy, about how other kids think about this problem?

There is no "ideology about language and math" involved in asking children to explain their thinking. The ideology is in believing that it's a "waste of precious instructional time" that "should" be spent listening to a lecture by a teacher (who might well, like the "Clueless Mom," never think of this approach or might not understand it, either, on first blush) or doing drill and practice. As I already suggested, this post unwittingly cries out for precisely the things that this blogger and her allies have worked so hard to stamp out in their schools. The irony is beyond belief.

Math Anxiety: Where Does It Come From?

The following appeared on-line today:

"Math. Throughout my childhood, adolescence, and most of my adult life, the very thought of the subject struck genuine fear in me. I thought about fractions and decimals or even addition or subtraction and would actually feel pinpricks of sweat break out on the back of my neck and the palms of my hands."

I found myself wondering, as I read one person's glib comment that it "is unfortunate that Denise Noe did not have the same type of teachers that I had," if the terms "math anxiety" and "mathphobe" aren't misnomers to some extent.

That is to say, the implication seems to be that there's something about mathematics that is inherently anxiety-producing. While anything is possible, and unlike my esteemed colleagues on the other side of the Math Wars, I don't profess to know any great universal truths, I suspect that "math anxiety" is not something that would occur naturally in many people. I have a hard time imagining, if books like THE MATH GENE and THE NUMBER SENSE are well-founded, as I believe they are, that anyone would, with no external forces at work, find anything about childhood mathematics tasks inherently anxiety-provoking.

Thus, I am inclined to strongly suspect that like other kinds of academic anxieties, "math anxiety" or "math phobia" are learned (dare I say "constructed"?) responses that like most neurotic conditions represent the best coping strategy an individual can muster in the face of an unhealthy situation. (Only when the unhealthy stimuli no longer obtain, but the feelings associated with and the strategies no longer necessary are predominant does a "best coping strategy" become neurotic). What is the unhealthy situation? I would propose that it is the kind of abuse I have witnessed all-too-often while working with elementary school teachers: humiliation, impatience, tolerance of peer mockery, unreasonable application of pressure to perform, intolerance of error, demand for perfection, etc. Of course, some of these may originate not with the teacher, but with the student's home situation, but that is something I don't generally get the opportunity to observe. What I have observed suffices, however, to support the notion that bad teachers, be it through ignorance of the mathematics itself, rigidity in responding to alternate thinking or student speculation, or any of the previously-mentioned malpractice, are highly culpable in the creation of math anxiety, math phobia, or out and out fear and loathing of mathematics. Indeed, given some of the horrors I've seen (and consider that I wasn't spying: the teacher was operating with full knowledge that there s/he wasn't operating without other adult professionals in the room), it's a wonder anyone learns to enjoy mathematics at all.

Square Roots, part deux

There is a group of numbers for which the process previously described won’t work. For example, try to use it to find the square root of 100.

Grouping as before: 1 | 00

Subtracting 1 from 1 = 0.

Write 1 above the 1, bring down the next pair of digits, 00, and append to the 0.

Multiply 1 x 10 and add 11 = 21.

Can't subtract 21 from 0. Hmm. Although we know the answer is 10, to make things work, we can note the following, which is Rule #3:

If you want the square root of a whole number that ends in two or more zeros, write the number as a product of a number and an even power of ten.

So 100 = 1 x 10^2.

We get that the square root of 1 = 1, append one zero for every pair of zeroes in the original number, and Bob's your uncle. (Or something like that).

For example, to find the square root of 3,610,000, remove two pairs of zeroes from the original number, then apply the original procedure:

Group: 3 | 61.

Subtract 1 from 3 = 2

Can't subtract 3 from 2, so write 1 above the 3, bring down the next pair of digits and append them to the 2 => 261.

Multiply 1 x 10 and add 11 = 21.

Subtract 21 from 261 = 240.
Subtract 23 from 240 = 217
Subtract 25 from 217 = 192
Subtract 27 from 192 = 165
Subtract 29 from 165 = 136
Subtract 31 from 136 = 105
Subtract 33 from 105 = 72
Subtract 35 from 72 = 37
Subtract 37 from 37 = 0

So write a 9 above the 61. Append two zeroes to the 19, one for each pair removed.
Then the square root of 3,610,000 = 1900.

=====================

Finally, this process works for whole numbers that aren’t perfect squares and for decimals. It just won’t stop in those cases. For a decimal, also break the number into pairs of digits to the right of the decimal point.

For example, finding the square root of 3 to 3 decimal places.

Append pairs of zeroes for each decimal place you want in the answer, plus two more to be able to round to the given place.

So write 3 as 3 | 00 | 00 | 00 | 00

Subtract 1 from 3 = 2.

Write 1 above the 3. Bring down a pair of zeroes, append to the 2 => 200.

Multiply 1 x 10 and add 11 = 21.

Subtract 21 from 200 = 179
Subtract 23 from 179 = 156
Subtract 25 from 156 = 131
Subtract 27 from 131 = 104
Subtract 29 from 104 = 75
Subtract 31 from 75 = 44
Subtract 33 from 44 = 11.

Write 7 above the first pair of zeroes.

Bring down the next pair of zeroes and append to the 11 => 1100.

Multiply 33 x 10 and add 11 = 341.

Subtract 341 from 1100 = 759.
Subtract 343 from 759 = 416.
Subtract 345 from 416 = 71.

Write 3 above the second pair of zeroes.

Append the next pair of zeroes to the 71 => 7100.

Multiply 345 x 10 and add 11 = 3461.

Subtract 3461 from 7100 = 3639.
Subtract 3463 from 3639 = 176.

Write 2 above the third pair of zeroes.

Append the last pair of zeroes to the 176 => 17600

Multiply 3463 x 10 and add 11 = 346241.

We could continue, but it suffices to realize that the next digit will be 0 and so our answer is that the square root of 3 is 1.732 rounded to three decimal places.

A Different Square Root Algorithm

One algorithm for doing square roots by hand was recently mentioned (though not actually yet described in any detail) on math-teach@mathforum.org. It was averred that finding square roots required estimation. I posted part of an algorithm I taught to teacher-education students last year that does not rely on estimating, but instead uses subtraction of successive odd integers. What follows is part of the story using two examples that illustrate most of the situations that might arise. The third situation, as well as how to deal with the square roots of non-perfect squares will be posted later. If there are typos, let me know:



As usual, there are more ways to the woods than one. Here's one in which estimation is NOT necessary. As mentioned in a previous post, I used this with elementary education students in a math content class and we most definitely DID explore why it works. If anyone is interested, I can send the labs that were used to explore this approach, first in base five (Fen) and later in decimal. The algorithm is the same, and the logic behind it is, as well. Since I didn't learn the one Wayne refers to, perhaps he'll offer his algebraic explanation along with the algorithm itself.

The one I used is based on the fact that the nth perfect square is the sum of the first n odd integers. So this fact can be used to subtract successive odd integers from the number for which one wishes to find the square root. If the number isn't a perfect square, this method can be extended by adding pairs of zeroes to the original number and continuing the process for each additional decimal place one wishes to have in the answer.

It helps to look at a couple of examples to illustrate two "special cases" that arise with some numbers, requiring one or two additional "rules" or steps.

Using the example of 54,756:

Start by marking pairs of digits from the right-most digit: 5 | 47 | 56

Then subtract 1 from the leftmost digit or pair: 5 - 1 = 4.
Continue with the next odd integer: 4 - 3 = 1.

We can't subtract 5 from 1, so we count how many odd integers we've subtracted thus far (2) and mark that above the 5.

Bring down the next pair of digits and append it to the 1 yielding 147.

To get the next odd integer to subtract, multiply the last odd integer subtracted by 10 and add 11 (this is Rule #1) to the product. Here, we have 10 x 3 + 11 = 41. Proceed as previously, subtracting 41 from 147 = 106.
Subtract the next odd integer, 43 from 106 = 63.
Subtract the next odd integer, 45 from 63 = 18.

Again, we can't subtract 47 from 18; counting, we have done 3 subtractions and place 3 above the pair 47. Multiply 45 x 10 and add 11 = 461.

Bring down the next pair of digits, 56, and append them to the 18, yielding 1856.

Subtract 461 from 1856 = 1395.
Subtract the next odd integer, 463 from 1395 = 932.
Subtract the next odd integer, 465 from 932 = 467.
Subtract the next odd integer, 467 from 467 = 0.
Stop.

Counting how many subtractions, we see it is 4 and we write 4 above the 56.

Our answer is that 234 is the square root of 54,756. (alternately, instead of keeping a running total of the subtractions and placing the digits above successive pairs of digits from the left, take the last number subtracted, 467, add 1, and divide the result, 468, by 2 = 234, same as what we determined the other way.

==========================

A second example introduces another rule not previously required: find the square root of 4,121,062,016 using the subtraction of successive odd integers.

Begin as above by making pairs of digits from the right-most digit: 4 | 12 | 10 | 62 | 40 | 16

Subtract 1 from 4 = 3.
Subtract 3 from 3 = 0.
Write down 2 for the two subtractions above the 4.
Bring down the next pair of digits, 12.
Multiply 3 x 10 and add 11 = 41.
Note that 41 is too big to subtract from 12.
Write 0 above the 12, since we did 0 subtractions.

Bring down the next pair of digits, 10, and append to the 12 => 1210.

Insert a 0 to the left of the last digit in 41 => 401. (This is Rule #2)
Subtract 401 from 1210 = 809.
Subtract the next odd integer, 403, from 809 = 406.
Subtract the next odd integer, 405 from 406 = 1.

For the three subtractions, write 3 above the 10.

Bring down the next pair of integers, 62 and append to the 1 => 162
Multiply 405 by 10 and add 11 = 4061.
We need to apply Rule #2 again. Write 0 above the 62, bring down the next pair of digits, 40, and append to the 162 => 16240.
Insert 0 to the left of the last digit of 4061 => 40601.

Note that this is still too big to subtract from 16240.
Apply Rule #2 again (and it may have to be applied more than twice in particular cases).
Write 0 above the 40, bring down the 16 and append to the 16240 => 1624016.
Insert a 0 to the left of the last digit of 40601 => 406001.

Subtract 406001 from 1624016 = 1218015.
Subtract the next odd integer , 406003 from 1218015 = 812012.
Subtract the next odd integer, 406005 from 812012 = 406007.
Subtract the next odd integer, 406007 from 406007 = 0.
Write a 4 above the last pair of digits, 16.

The square root of 41210624016 = 203,004.

Again, alternately, the answer = (406007+1) / 2 = 203,004.

I will post the situation that calls for Rule #3 as well as examples with non-perfect squares a little later.

When Is A Topic No Longer Vital For The Curriculum?

Recently, on math-teach@mathforum.org, the issue of whether changes in the standard K-12 curriculum are a natural, reasonable reflection of changes in both society and education, or whether they are yet another harbinger of the coming of the Apocalypse. The debate centered on an opinion piece by Stuart Wachowicz in which he begins,

“Pride in craftsmanship obligates the mathemati-
cians of one generation to dispose of the unfinished
business of their predecessors.”
-E.T. Bell, The Last Problem
The above statement most accurately describes the
legacy of one generation of mathematicians to the next.
However, on might be tempted to ponder whether this
will continue to be possible in North America. The dis-
cipline of mathematics, as we have known it, is clearly
under threat. The threat is a consequence of allowing cur-
riculum writers to change the centuries-old definition of
mathematics and what needs to be learned based on utili-
tarianism, combined with the current practice of allowing
unproven fads to influence pedagogy.

For the full text of Mr. Wachowicz's piece, see "Perils of Modern Math Education"

This led a number of participants to posit that there were certainly things that could be dropped and in fact have been dropped from the curriculum in the past century without any clear-cut harm resulting, but it was difficult to get those who were apparently in agreement with Mr. Wachowicz to stipulate what they feel it has been okay not to teach. Then, in an attempt to turn the tables, Wayne Bishop wrote:

Would you give us a list of a few topics (beyond table look-up and associated interpolation) that would have been appropriate high school preparation for a math-based college major decades ago that are no longer appropriate? Obviously, not every high school student - not every college-bound high school student - has aspirations for a math-based major but we certainly don't want to screw up those who do or who have not yet committed themselves against that long-range goal.

My response, slightly edited for clarity, follows:

Wayne Bishop's theory appears to be, eliminate nothing that, if eliminated, might "screw up" potential math-based majors. Using this logic, we have publishers producing those tree-killing doorstops that both Dom Rosa and Kirby Urner, perhaps for very different reasons, decry, and which seem consonant with the complaint that Wayne and other MC/HOLD flacks echo (when it's convenient) about the mile-wide, inch deep American curriculum, a sin they lay at the feet of those potentially blown-up Schools of Education, especially the reform-oriented progressives who've taken them over.

It's also this logic that has kept topics in SAT, ACT, GRE, GMAT, and LSAT prep books by some publishers that haven't actually appeared on the respective test for years, if not decades. Rather than cut, they simply rewrite and append. After all, it can't HURT to leave in those extra pages, right? The old item types might make a comeback, and fatter book => higher price with not all that much additional cost in production. More trees die, but that's really collateral damage. And anyway, the OTHER guy's books still cover that topic.

There are other problems connected with this philosophy that are more serious for teachers and students.

States suffer from many of the same problems from which individuals and publishers suffer. They are loath to drop a topic if "the other guy" still has it in their curriculum framework. Couple this with the very legitimate push to put some important topics and strands that previously weren't treated until much later (and often in a very cursory manner, if at all, in the high school curriculum, e.g., probability and statistics, transformational geometry, discrete math), and you have a real quandary: in fact, you have state frameworks that, while attempting to go in a reform direction, wind up as . . . yep, a mile wide and an inch deep.

Out there in the real world of struggling inner-city and poverty-stricken districts, high stakes tests and NCLB pressure teachers to get their kids up to snuff not only on the OLD elementary school curriculum with its focus on arithmetic through the rational numbers and maybe integers (in a K-6 school, both would likely be addressed by "graduation" to junior high school), but all this new stuff. But since NOTHING ever gets dropped and new stuff is getting added in, and the approach being used in any given district may not be getting enough kids up to snuff on a host of topics (probably different ones for different kids), the results are predictable: math students enter high school, when and if they do, like a bunch of Swiss cheeses - full of holes in their knowledge and each one with holes in different places.

The teachers are caught between the proverbial Scylla and Charybdis: teach every topic to mastery and understanding for the vast majority of kids and you don't get to a bunch of required topics and risk being fired for insubordination; teach at the usual district or textbook pacing guides and you leave a lot of kids in the dust and you get fired anyway (or your principal does, or the school fails to make AYP and eventually is closed, after it loses funds through various provisions of NCLB). Bummer, dude.

The obvious solution? Fire all the teachers, close the public schools, give vouchers so that the wealthy kids can even more readily afford really nice clothes while attending the ritzy private schools that just happen to be in the neighborhoods where they live (this is, of course, an ENORMOUS coincidence) or off at some exclusive prep school in scenic New England that unfortunately doesn't have much in the way of provisions for the kids in those inner city or rural poverty pockets in their underperforming schools and districts. This is regrettable, of course). The poor kids can attend a local Catholic school, of course, as they are more likely to crop up in ghettos and far-flung areas in rural Idaho.

What we most certainly SHOULDN'T do, of course, is support efforts to repeal or greatly revise a punitive, ineffective law like NCLB, change our assessment focus from using testing to embarrass and discriminate unfairly towards using testing for getting useful information that actually is used intelligently to make things better for those most in need. Nor should we seriously look at any sort of curriculum that actually attempts to address issues for both elementary teachers and students of profound understanding of fundamental mathematics; or that uses any approach or tools other than paper-and-pencil algorithms, generally taught either initially or in the end out of desperation, as "black box" procedures with no understanding whatsoever. The goal becomes passing a test rather than learning and understanding what's actually going on. Student thinking is kept to a minimum to serve the great gods of multiple choice testing.

(For the irony-impaired, proceed with caution following the words "The obvious solution?")

Double Standards and the Anti-Reformers

The following appeared recently on the Eclectic Educator blog (6/14/07) and is an excellent example of some of the countless contradictions that seem to inform so many of the opinions one reads about mathematics education from the anti-reformers:

" I think the methodology of Singapore is easy to learn. What's been difficult for American elementary school teachers, from my research so far, seems to be their lack of understanding of mathematics at any deep level.

It's interesting to me that with Singapore math being in some ways similar to reform math (although in my opinion much better) it still requires lots of drill beyond the school day.

It seems that when reform math was implemented here, that was another blind spot. It didn't occur to anybody that kids would need a whole lot of drill beyond the school day.

And so parents, not having been given warning, are tutoring as a band-aid instead as a way to complete the program. They're doing so unevenly, and some parents don't even realize its necessity.

There are myriad reasons, including what I've explained above, that reform math is failing in our country.

At least in Singapore it's built into the culture that kids need tutoring for drill outside of school. Not to mention that in Singapore they don't expect the kids to discover all this deep understanding on their own."

What is problematic here is that the Eclectic Educator ( in fact Ridgewood, NJ blogger and anti-reform activist, Linda Moran, someone who has been central to the recent storm over the use of the INVESTIGATIONS IN NUMBER, DATA, AND SPACE curriculum and the withdrawal of acceptance by Martin Brooks of the superintendency for the Ridgewood Public School district), seems more than willing to accept reality when so doing explains and excuses difficulties in implementing a favored program, Singapore Math, for which she openly advocates, but will not accept this same limitation as a reasonable explanation for difficulties associated with INVESTIGATIONS, EVERYDAY MATH, or other programs which she dislikes.

Most mathematics educators in the US acknowledge that the vast majority of elementary school teachers here lack what Liping Ma has called "profound understanding of fundamental mathematics" (PUFM). I would agree on this assessment, based on all my experiences as a student, teacher, parent, teacher-educator, researcher, and field-supervisor for student-teachers. Very few of our teachers or would-be teachers can, for example, give an appropriate word-problem to illustrate division by a fraction. Ma, drawing on research by Deborah Ball, now the dean of the University of Michigan School of Education and professor of mathematics education, showed that some of our teachers cannot even do the indicated operation when asked to solve and create a context for 1 3/4 divided by 1/2. Not surprisingly, many wind up with problems that would be appropriate for dividing the number IN half (it's important to note when using examples like this whether the problem is given orally, because it's not clear in such cases if the solver has misheard him/herself being asked the wrong thing. If the problem is written down, that explanation doesn't obtain).

Of course, one issue that arises in trying to understand why K-12 students, teacher education students, or in-service teachers struggle with fraction division is the mechanical way arithmetic has been TRADITIONALLY taught in this country, going back at least as far as the 1950s, when I was in school (but in fact, from most evidence, much further than that). We are told early on, for example, that multiplication "makes things bigger" and that division "makes things smaller." This perspective becomes very deeply ingrained in the constructs we all carry with us about arithmetic. So while it is sensible to say that for positive dividends and divisors greater than one, the quotient will be smaller, but things get more complex when one allows for rational and negative numbers, and the various combinations and general rules for what happens to the quotients actually require some thought. If one divides -3 by 2, the resulting quotient, -3/2 is actually greater than the initial dividend. If one divides 3 by 1/2, the resulting quotient, 6, is also greater than the dividend. But 3 divided by -1/2? The quotient is -6 which is of course smaller than 3. And so on. The various possibilities can be very confusing for adults, let alone for young children. However, it is imperative that teachers understand what's going on and why, and that they have sharpened number sense so that none of these outcomes strikes them as at all surprising if they are going to be able to explain, model, and teach these concepts to kids.

Unfortunately, as Ball's and Ma's research makes clear, such is far from the case. But this isn't new (though it may be news to some), and it didn't emerge suddenly the day the 1989 NCTM Standards was published, or in the 1990s when the first NSF-funded curriculum projects were being developed and piloted. Far from it.

Why won't the anti-reform extremists admit this? Because to do so would be to admit that the entire approach that has been used in the United States to teach the vast majority of our students is woefully mechanistic, grounded in blindly-taught and blindly-followed (often poorly in each case) procedures, rather than deeply thought-about and investigated concepts in order to master strategies and algorithms. A huge part of the Math Wars is over this issue, but sadly one gets mostly heat and virtually no light if one reads only what the anti-progressive haters say on the subject.

To read most of them is to be taken to an imaginary land and time in which all American children were taught by highly-knowledgeable, patient, insightful teachers who knew the math and understood children (all children, of course) and how they learned (and it goes without saying that all children learned the same way, at the same pace, and at the same age). If you're reading this and think I'm exaggerating, I can readily point you to thousands of posts to various math discussion lists that are based on assumptions like this. Further, there seems to be an enormous emotional investment on the part of some people to ignore the fact that millions of Americans were ill-served by the mathematics education they were offered in the post-war era (and, likely, before that). It's now a cliche, but nonetheless true, to point out that mathematics teachers are told all the time when they state what they do for a living, "Oh, I was always awful at math. I always hated math. I never could do math," etc., something English teachers are almost never told openly by adults regarding reading and writing. Illiteracy is a mark of shame in this country, but innumeracy is nearly a badge of honor: "Yes, I've been brutalized by math, just like you, and I'm STILL bad at it. Ain't it awful? (Nudge, nudge, wink, wink)."

Thus, it is an odd aspect of the Math Wars that not only some mathematicians and advanced end-users of mathematics (engineers, physicists, etc.) think that the way they were taught was great because clearly it worked for them (and hence must be the best, only valid way for math to be taught to everyone), but also those who themselves suffered and withered in math classes will decry so-called fuzzy methods because "I know math, and that isn't math." Of course, it IS math, but it may not be the same approach to it that they didn't connect with when they were in school.

Not everyone is blinded by either nostalgia or a kind of "frat boy" or "no pain, no gain" mentality ("I suffered in math class and my kid will too, if s/he knows what's good for her/him!"). Some very successful mathematicians and math users recognize the limitations of the methods they saw as students and have generally used themselves as teachers. Some who failed to thrive in math realize now that it was the limited methods and limited teachers who failed them, not their innate ability to do math. Which brings me back to Ms. Moran and her comments on Singapore Math and her blind, unrelenting hatred of INVESTIGATIONS and other progressive math programs.

I will not attempt to interrogate her motives in my currrent entry, but I will point out that everything she says about the weaknesses of American elementary school teachers in mathematics is directly relevant to understanding the difficulties we face in moving forward towards more effective mathematics teaching and learning in the United States. Hypocritical selectivity in to whom one gives a pass (e.g., Singapore or Saxon Math) and whom one takes over the coals (e.g., INVESTIGATIONS, EVERYDAY MATH, TRAILBLAZERS, and a host of middle and high school programs) serves our children badly, while serving to make any reasonable examination of what works, for whom, and possibly why it does just about impossible.

I knew before I began this blog that: a) I would come under personal attack from people both anonymous and on occasion honorable enough to give their real names or identities; b) that I would be accused of being a flack for one or more programs or for NCTM, or something of the kind (I've never worked for any of the projects or publishers, though I have worked for and with school districts that use progressive curricula, and I have taught a couple of high school programs that Ms. Moran would no doubt criticize as fuzzy and the like); c) I would be criticized for not being "fair" in my entries or my responses to those whose comments I find myself in disagreement with: while I never promised anyone a rose garden OR fairness here, I have yet to edit or block a single response from anyone. I've made the existence of this blog as public as possible, and expect that sooner or later it will draw much harsher criticisms and comments than it has thus far (at which point I suppose I'll face some interesting decisions). But the blog IS, after all, mine, and I will respond to critics as I see fit, just as I expect they would do were I to post in spaces they control. As stated, that's not been an issue for me, so far, but if I'm supposed to "roll over" "play dead," and generally be nice because someone thinks I'm being a bit too slanted, I will only point out the obvious fact that this is a blog, and my slant is precisely the one I'm planning to offer.

That said, I think that EM, INVESTIGATIONS, and many other reform programs are valuable first- and second-generation attempts to carve out new directions for mathematics education. (And I see good things in Singapore Math we can use, and even some useful things, though far fewer of them, in Saxon Math). Reform programs have been around long enough to be criticized responsibly and to have undergone revisions (for which, ironically, they are ALSO attacked by the extremist critics, as if other textbooks emerge from the heads of their authors and publishers perfect and inviolable and never need corrections, rewriting, or wholesale changes).

They are not, however, the final word on where to go with mathematics teaching and learning in K-12. Nor will anything ever be. Math grows and changes. So do our uses for it. So does our culture, our technology, and the tools we have available. The very nature of school is changing, though often painfully, almost glacially slowly.

There are interesting things going on that don't involve textbooks at all, something that should be utterly unsurprising if you're reading this on line. There are projects that see computer science and programing and applications of programming as central to learning mathematics effectively. Buckminster Fuller long ago predicted approaches to learning that anticipated the availability of video lectures "on demand" to people who would not need travel to a central location like a university to gain access to a college education, and it is not unreasonable to think that Mr. Fuller would have found math education via books alone, viewed during lectures in traditional K-12 classrooms unbelievably archaic and anachronistic were he alive today.

The bottom line isn't INVESTIGATIONS vs. Saxon or Singapore Math vs. EM, but rather an overall understanding of how kids learn and how we can best help them learn and decide what they want and need to learn. The Math Wars, regardless of what the anti-progressives tell you, have never been about the math content, but rather about education and what it means in a democracy. We have so many good choices, but also so many bad ones, and the ones that have for the most part failed us in the past are not the way we should be pointing now. If schools and math teachers aren't to go the way of the evening newspaper and even network nightly news and the reporters and anchors whose livelihoods depend upon them, they need to look towards the future, not the past alone. The lack of PUFM among our teachers, the low value we place upon K-12 teaching and especially K-5 teaching, and our cultural attitudes and beliefs about mathematics teaching and learning are far more central to whether we will raise the level of numeracy in this country than are the choice of math texts. This isn't to say that such choices don't suggest a lot about the vision of those making the choices, but I am sure that with or without a text, or with a text one doesn't completely favor, a competent, motivated, imaginative teacher can bring mathematics to life, and a student willing to engage that teacher fairly will have every fair chance to reach his or her maximum potential in mathematics and beyond.

Timely Tech News

It's amazing the depths some folks will plumb to smear, defame, and intimidate decent people, while justifying their anonymity because of the repercussions they maintain will fall on their heads from teachers and administrators. In this regard, it seemed timely that Yahoo! reported yesterday that police departments in Boston have set up a way for citizens to use text messaging anonymously to offer tips about crimes. Not that I oppose responsible citizens having a safe way to report real criminal activity, but this seems like the perfect way for our Ridgewood friends and many like them to further attempt to assassinate their perceived enemies with impunity.

http://news.yahoo.com/s/nm/20070615/us_nm/usa_crime_telephones_dc

Correction regarding "Paranoia?"

I see that I've erred in stating that it was Greg Goodknight rather than the more carefully "shielded" 'Haim Pipik' who accused me yesterday of paranoia. My apologies both to Greg for mistakenly accusing him of both calling me that and for playing at amateur psychologist in this regard. In fact, Greg accused me of "projection" yesterday, in his actual failed attempt to be a, as he put it, "shrink"; and of course I apologize deeply to "Haim Pipik" as well, for not properly giving him credit for this particular bit of idiocy.

It's tragic that when being attacked by two such mental giants as was the case for me yesterday that I relied on my short-term memory rather than simply checking back through the flood of hate e-mails I got from these two charming individuals.

Paranoia?

Yesterday, I was falsely accused of paranoia on the math-teach@mathforum.org list by amateur psychologist and professional know-nothing "Haim Pipik," yet another anonymous foe of progressive politics and education. While I and others believe we know precisely who uses that nom de net, efforts to get him to drop his charade and simply admit who he is have been in vain. Of course, in the wake of the travesty of democracy we have witnessed in Ridgewood, NJ this week, thanks to the clandestine efforts of a tiny group of activists, some of whom don't even live in New Jersey, let alone in Ridgewood, "paranoia" becomes a very relative term.

By making anonymous calls and sending anonymous e-mails, this little group of brown shirts undid the legitimate hiring of a top-notch superintendent because he'd written favorably and intelligently about constructivism and was perceived to support the legitimately selected K-5 math program in Ridgewood, INVESTIGATIONS IN NUMBER, DATA, and SPACE. From what I've read, the personal attacks on his integrity and that of his wife were the usual vicious "parents with pitchforks" lies that have been the stock in trade of followers of the leadership of Mathematically Correct and NYC-HOLD. I'm sure those who hold with the beliefs of those groups are congratulating themselves for their "math warrior" victory. The rest of Ridgewood is wondering how fewer than 100 people could cost the town the $20,000 already spent on a job search to land Martin Brooks and the additional money that must now be spent to find a new superintendent.

I had a cordial e-mail from Dr. Brooks this morning and I'm confident that had these unprincipled assassins not been allowed to operate with impunity in Ridgewood, the town would have gotten an outstanding leader for its public schools. The effect of what this band of jackals has done is likely to be very chilling as the town tries to find a person of equal ability in the wake of the unjustified personal attacks on Dr. Brooks and his wife.

This is the Fox News, Rush Limbaugh, Newt Gingrich, Karl Rove, and George W. Bush legacy. The good news is that I believe this incident will gain enough local and national attention that perhaps "paranoids" like me will be less easily ignored by people in a position to do something before it's completely impossible for anyone calling her or himself a progressive educator or liberal to seek work in public schools. More importantly, those who have yet to take these hate groups seriously will perhaps now be galvanized into principled and serious opposition to this sort of hijacking of our schools and our democracy.

What's Really Going on With Multiplication in Everyday Math?

As promised elsewhere, I'm going to look carefully at what Everyday Math offers about multi-digit multiplication, how it advises teachers to teach this topic, and what I've seen in practice (and how I attempted to rectify problems with how it was being taught in some classrooms).

First, from p. 140 of the first volume of the EM 4th grade teacher's manual, the very first thing it says about multiplication. If you've never seen this before, it's no doubt because no one who attacks EM would dare quote it to you.

"The authors believe that most students should learn the basic facts for addition and multiplication to the point of instant recall. Mastery of the basic facts will give students surprising power in making quick estimates and operating with larger numbers."

I hope you were sitting down when you read that. Surprising, isn't it, if you've been getting your propaganda feed from various hate blogs or the Mathematically Correct/HOLD websites and those they link to?

Don't you wonder why the anti-reformers don't mention that quotation? It's not hidden somewhere deep in the unit. It's the very first paragraph teachers are expected to read.

Well, let's look a little further:

p. 144 "Calculators are used in this unit for games . . . In BEAT THE CALCULATOR, students quickly realize that their brains are much more efficient than their calculators when finding a product like 7 * 3. . . .

"However, the no-calculator icon (see margin) does appear on many journal pages, including those on solving number stories and Math Boxes pages in which the intention is to encourage practice with algorithms for adding and subtracting numbers. Alert students for the icon."

Hmm. Gosh, that doesn't sound much like the usual version of the EM philosophy I read about all over the internet. Sounds dramatically different, in fact. Could it be that the anti-reform bloggers and hate-mongers just overlooked those glaringly obvious examples that undermine their propaganda about EM? Are they all a little myopic? Yes, they're VERY myopic when it comes to having to see evidence that works against their fanatical dislike of programs like EM and INVESTIGATIONS.

Okay, I'm going to skip the rest of Unit 3, but you should not. If you have access to a copy of the 4th grade student book or the teachers' guide, do read over those parts of the unit that deal with multiplication facts (fact families, etc.) and how to think about and apply multiplication in context.

NEXT: A look at Lesson 5.5 in the unit entitled Big Numbers, Estimation, and Computation. Here we get the first formal algorithm for approaching multidigit multiplication, partial products.

Open Letter to Dr. Martin Brooks regarding Ridgewood, NJ

Dear Dr. Brooks:

I'm very saddened to learn that you've chosen not to take the
Ridgewood, NJ position. I grew up next door, in Fair Lawn, and
graduated high school in 1968. I'm now a mathematics educator based
in Ann Arbor, MI. I am an active supporter of good, high-quality
mathematics teaching and follow the "math wars" closely and with
great interest. I became aware of Linda Moran and her band of
activists earlier this year, and then learned through reading her
lists and blogs that you were taking the job there, much to my
pleasure. I've read your book on constructivism with pleasure and had
looked forward to seeing you keep things going in the right direction
in Ridgewood.

Unfortunately, your withdrawal, which I can understand must have been
an unpleasant personal decision, has given a great deal of
satisfaction to the anti-progressives in Ridgewood. Surely you
understand that they represent a small minority. Had you come, you
would have found great support for your ideas among both parents and
the teachers. Several have already written me privately to express
their dismay at your change of heart, and there is evidence on-line
that these activists are not beloved in their community, except
amongst themselves and like-minded people in other communities.
Indeed, some of the most frequent posters to Linda Moran's anti-TERC
list are from other cities, most notably Irvington, NY
(coincidentally, my high school girlfriend grew up there, so I know
the town better than I probably should).

I know you are unlikely to change your mind, but I want to urge you
to reconsider. The people of Ridgewood spent $20,000 to find you. It
will be a feather in the cap of some very awful people if you allow
that money to be wasted. And you could do the job there, I am sure,
and would now probably find even greater support that you might have
before. We cannot let these small, vocal, anti-democratic groups to
run roughshod over our schools and our kids.

Please see my blog and a web-
site I helped create . I will be
addressing many of the substantive issues about reform mathematics,
what it really calls for and how it actually works, as well as trying
to expose the lies and distortions of people like Linda Moran. But
again, to let her and those like her win this battle is a disaster.
It has already emboldened them, as is evident on her list, and I
shudder to think what will come next: veto power over what books may
be read in Ridgewood? Control over what political ideas may be
examined? Creation "science" and "intelligent
design" in biology class? Where does this stop? You truly do have to
consider the very deep implications of your withdrawal beyond just
your personal situation. These people routinely insult me on-line,
have done so for 15 years, have tried to cost me my academic standing
and several jobs, and yet I persevere. I think you could do so if you
reconsidered and went to Ridgewood. Don't let them win without a real
fight.

Sincerely,

Michael Paul Goldenberg



"The new life needs to be inspired with the realization that the new
advantages were gained through great gropes in the dark by unknown,
unsung intellectual explorers."
R. Buckminster Fuller

What Does Liping Ma REALLY say?

I saw Liping Ma speak at Rutgers University on July 19th, 2000. I
took extensive notes during her talk, and in thinking about the many
misrepresentations I have read of her actual thinking about
mathematics education, both here and in Mainland China, I decided it
was time to re-read KNOWING AND TEACHING ELEMENTARY MATHEMATICS, the
book that shot her into the limelight in the midst of the on-going
fight over what mathematics teaching and learning is about. It has
disturbed me to no end to see here balanced and reasonable ideas
distorted and "reinvented" and misused to justify what is usually
referred to as "traditional" math teaching in the United States, both
by activists in national groups like Mathematically Correct and NYC-
HOLD, and on local anti-reform blogs and discussion lists such as
Linda Moran's beyond-terc and many others. One would think that
Professor Ma had at some point signed a secret accord with such
folks, but my best guess is that she remains now as she appeared to
be in that summer of 2000: doing her own thing based on her
understanding of what matters, outside the usual politicized fray.
Given that she is one of the three people on the current National
Math Panel who is both qualified to speak intelligently about both
mathematics AND mathematic teaching and learning in elementary
schools (the other two being Deborah Ball and current NCTM president,
Francis "Skip" Fennell, I really wanted to refresh my understanding
of her views.

Thus, I was thrilled to discover that I had saved those lecture notes
from 2000. As a service to those who actually want to know what she
was thinking at that time, quite shortly after her book appeared, I
am going to reproduce those notes here. Of course, it is unlikely
that my doing so will sway those with entrenched beliefs that require
them to filter out anything that might cause cognitive dissonance.
But getting this on-line into publicly-accessible form will be a
benefit for many others who find it.

The talk began at 1 PM on July 19th, 2000 and was held at Rutgers
University in New Brunswick, NJ. It was a public lecture. If memory
serves, there were probably about 100 people or so in attendance.
Among them were teachers from School #2 in Paterson, NJ, who have
been participating in a long research project involving the use of
Japanese-style lesson study as a means to improve the quality of
mathematics teaching and learning in a multi-racial urban district.
This caught my attention at the time because my late uncle, who held
a Master's degree in mathematics from Columbia University, taught
industrial arts at School #2 until he retired in the mid-1970s. (He
was never allowed to teach mathematics in New York City, where he
originally sought work, because of his heavy Russian accent. In the
1930s and '40s, an oral examination was required of prospective
teachers, and it appears that accents could be used to keep otherwise
qualified teachers from working in their areas of expertise, an irony
given Professor Ma's accent as well as those of many people now
teaching throughout the country). Also attending were several
professors of mathematics education from Rutgers, as well as Fran
Curcio from Queens College in NYC, and Patsy Wang-Iverson, a Senior
Scientist with Research for Better Schools. Her primary
responsibilities, under the Mid-Atlantic Eisenhower Consortium for
Mathematics and Science Education, include overall management of the
Third International Mathematics and Science Study (TIMSS)
dissemination throughout the Mid-Atlantic region. Dr. Wang-Iverson
was responsible for the Paterson teachers being at this public talk.

Here are my notes, slightly edited for clarity or to insert questions
where I'm not sure what I have written down in terms of whether Prof.
Ma made the point or I was jotting down parenthetical questions. For
the most part, however, these are her thoughts and ideas as presented
during that hour lecture.

=================

Introducing herself, Liping Ma told us that she taught elementary
mathematics in China for seven years and was later a principal
there. She came to the US to get her doctorate in mathematics
education, first at Michigan State where she worked with Deborah
Ball, Maggie Lampert and others, and later at Stanford, to which she
transferred because here husband didn't enjoy Michigan weather. She
finished her Ph.D at Stanford with Lee Shulman.

Ma states: We can't change US culture. We can't change the language
[does she mean the language Americans use to think and speak about
math education?] We CAN change teachers' content knowledge.

US has more resources to change teachers' knowledge (than does
China). In China, to give student a cup of knowledge, the teacher
needs a bucketful of knowledge. Ma didn't find information here on
what constitutes "good understanding," e.g., what is supposed to be
in each teacher's bucket?

Her research was with teachers from five Chinese schools. They
included some "best" Chinese teachers. Three of the schools were in
Shanghai and included one school considered to be a "best" school.
Also, one school from a small town, and one rurual school (her own
former school). She had 72 teachers in her sample.

The US data was drawn from research done at MSU by Deborah Ball. Ma
conducted no interviews or original research on US teachers. One of
the groups in the US sample was at the end of their first year of
teaching, earning Master's degrees at MSU at the same time. Others
were "above-average" and experienced teachers [no information offered
on what criteria went into determining their standing as "above
average"].

The terms "decomposing a higher value unit" and "composing a higher
value unit" in Chinese come from abacus terminology [technology rears
its head in unexpected ways]

Just two syllables: "jin yi"

One reason Chinese teachers have richer base of forms of regrouping
is because they let the kids give their ways and then discuss those
ways, including where it makes sense to use those various ways. And
the profound grounding "within 10" and "within 20" makes the kids
have lots of ideas, so there is an interactive process between "old"
teaching classroom practice (discussions), and what teachers have
available to draw upon for teaching other related topics.

N.B. : Ma only addresses subtraction and multiplication given the one
hour public lecture format. With about 15 minutes left, she chooses
to look at the multiplication issue rather than the division of
fractions or geometry: these are briefly mentioned only during the
introduction about the Ball questionnaire.

Chinese teachers see similar errors to those in the US, but address
them at the 2 digit times 2 digit level and try to get at the
misconceptions even at the 1 digit x 1 digit level by asking about
what units are being dealt with (e.g., 5 x 2 = 10 what? 10 'ones')

[Not 100 % sure if this following is her comment or mine, but I think
she said it: "It's wrong to say that US teachers 'only' want kids to
understand what the teachers understand themselves, which, generally,
is strictly procedural; they are no different from Chinese teachers
in what they DESIRE; they only differ in the DEPTH of their own
knowledge. How can someone want something for kids that s/he doesn't
realize even EXISTS?]

Ma says she has her students "invent/discover" how to do three-digit
multiplication and has those who succeed teach the other students how
to do it.

The seeds of advanced math are embedded in "elementary" mathematics;
the more solidly students and future teachers know arithmetic, the
easier higher math will be for them. But teachers of elementary math
don't need advanced math to teach elementary math effectively.
Stigler and Stevenson claim that US teachers teach "too fast." Ma
claims that Chinese teachers in reality only go slowly on key topics,
however.

Why do Chinese teachers grow in knowledge from their teaching
experience, while US teachers don't? Chinese teachers don't assume
you (new teachers) know the math [?] when you start, but study and
grow and learn from colleagues and books.

Following this was a Q & A session:

Ma: In China, they've learned from some US/Western practices. For
example, kids sit two to a desk for pair work, and two groups of two
(two desks) for small group work. Also do larger group work.

First grade students engage in classroom discussion, always given
time limits for tasks; e.g., use 8 sticks to make figures and write
number sentences ot describe your figure. For example, 1 + 3 + 3 + 1
= 8.

Kids tell stories about the number 8 (for example) based on what they
discover from these activities.

As a result, they really know composing and decomposing small
numbers. By the end of first grade kids know all the multiples up to
5. In China, especially in the cities, teachers are specialists and
teach only math in elementary grades (Ma's caveat: in Taiwan,
Singapore, Japan, the elementary teachers are NOT specialists and
still do a very good job on their math teaching). Kids spend about
45 minutes a day on math. Textbooks are small (shows the Japanese
textbooks for six-year-olds: about 3 inches wide in TOTAL. Each book
is about 1/2 to 3/4 inches thick.) In Japan, kids have their own sets
of manipulatives, unlike in China, and the quality of them is better
than the ones teachers have available in China. Chinese kids share w/
classmates. Teachers who have PUFM (profound understanding of
fundamental mathematics) believe they can teach effectively within
the 45 minute structure except for the lowest-level kids. In China,
drill is much better designed than in the US. There isn't "drill for
drill's sake," but rather based on understanding. In California, Ma's
son gets drilled on the same material 2 or 3 times a week. She thinks
badly of this practice. In China, drill "moves up" [not sure exactly
what she means by this].

In China, teachers often concentrate on the students' textbook and
feel that the teachers' book is mostly useless. The teachers prefer
to interpret the student book themselves. They ask each other why an
example is given by the author and they evaluate it as useful and
connected, or not. They meet once a week (usually by grade level, not
subject) to discuss their teaching. Most teachers know at least a
"small loop" of topics: e.g., 1st & 2nd grade topics or 1st, 2nd, and
3rd grade topics. Some (those who have PUFM) know grade 1-5 topics
well, and they get to teach all five grades.

Teacher guides don't generally include anticipated student errors &
typical responses, but teachers share this information with each
other from their practices. Ma was given the chance to spend a month
in some good classrooms when she started out, to observe and learn
from good teachers. But the same principal who allowed this trained
another teacher by noting and analyzing EVERY student error made in
that teacher's classroom. So there are varied approaches for
different teachers possible.

The term "knowledge package" is Ma's, not something generally used in
China. In China, the principal decides, mostly based on his/her
immediate needs, what new teachers will teach. If the principal
believes a teacher is "promising," that teacher gets to loop more
among the different grades and becomes what is termed a "backbone"
teacher for that school.

When asked how she "corrects" her son's education in California: I
teach him the 'most important concepts.' You can learn from "what you
already know. Simple things." Her son isn't tops in speed, but she
says she feels that is fine. Someone else asks a question that
expresses negative views on calculator use in elementary math
classrooms: isn't lack of PUFM 'masked' by calculator use? Ma says
her son learned negative numbers not from his older sister but from
playing with a calculator. Ma tested him and found his understanding
of negative numbers to be good. She thinks it was rooted in his
number sense. She believes we need much more research before saying
"yes" or "no" to calculators. In China, there is no official national
policy on them.

Asked why America has such resistance to cooperative and
collaborative work among TEACHERS, Ma gives a very diplomatic answer:
"I'm a foreigner and don't have a why."

Asked, "what are the Big Ideas of elementary mathematics, and where
does one find this information? Ma says she's working on this as her
next project but warns that she's not read that much of the literature.

The teacher day in Chinese elementary schools is typically starts at
7:30, with classes for the kids ending around 2:30 or 3. Teachers
teach six periods, but many run math clubs and groups for slower
kids, so teachers often leave at 5 pm or later. Teachers are expected
to grade every problem and the kids are expected to correct every
error. The school year is 42 weeks

Very few kids are held back; the teachers design their own tests and
re-use them.

Blind spots in American mathematics education: ordinality is taught,
but not cardinality (or not sufficiently so), and that is one of the
important Big Ideas. American teachers over-stress counting-on and
counting back on the number line, as opposed to seeing numbers as
sets [uh, oh!].

Ma is worried about using concept maps and writing to assess students
specifically because the quality of the teaching of writing here
isn't strong enough.

Q: Should students be interviewed on their knowledge (like in the
Ball data, and Ma's use of TELT (?) questions)? Ma: hard to get the
right people as interviewers. She thinks you can know without
interviews what the kids know [she doesn't elaborate on this].

Q: How important is teacher content knowledge?

Ma: You cannot separate teacher content knowledge from the teachers'
knowledge of math pedagogy and and their knowledge of how kids learn
math. They must know what something in math is, how to do it, and
what it means.

Q: About division: when is this taught in China and why isn't it so
hard for Chinese students (as opposed to Americans)?

Ma: This is rooted in linking division to multiplication as its
inverse. e.g., 5 times 3 means 3 groups of 5 elements each. it is
important to stress the partitive and quotitive models early, as is
done in China by late 3rd grade or early 4th grade). Skip over the
"circle" idea (?) quickly and go to fractions as another model for
division.

Q. About limits on what numbers are explored, e.g., numbers to 9,
within 20, etc.: what about more open-ended exploration?
Ma: I've been thinking a lot about that. Many Chinese teachers do
lots of discussion. Teachers put various methods of, say, regrouping
on the board and ask kids why we have them and why they and
conventional ways are used.

Q. What is the relationship between advanced math classes for
teachers and their ability to teach mathematics?
Ma: Better to do deep exploration in classrooms of elementary
mathematics.

"Decomposing a higher value unit" is Ma's term. In Chinese, it's more
like "stepping back" on an abacus.

I asked: Could you compare and contrast the Ball and Lampert
classrooms, based on the available videos and other data from their
1989-90 teaching of grades 3 and 5, respectively, at Spartan Village
Elementary School in East Lansing, with their discussions over
several days on a central problem of the day which kids explore and
work on, with what you've seen in Chinese classrooms?

Ma: I looked at the [very well-known] Shea numbers video and showed
this to teachers in China. The teachers there with good PUFM really
liked it, but felt that given constraints there, they could only "go
wild" like that a few times each year.

Ma: A good textbook is the most important thing we can offer teachers
to improve their knowledge of mathematics. But 90% of teachers here
believe they already know enough elementary math. So a good textbook
would need to be able to show them that they are more ignorant than
they believe and then help to reduce that ignorance.

She feels that as far as student materials go, we need to develop our
own textbooks for American students.

Patsy Wang-Iverson: In the US, kids try to figure out what the
teacher is thinking (and then do that). In Japan, the teachers try to
figure out what the students are thinking so that they can help the
students make more sense of the mathematics.

In China, there is no tracking or ability grouping until 10th grade.

Ma: "China (and EVERY country) has serious problems in its public
education."

From Susan Ohanian

I urge everyone to read the petition and seriously consider signing it. Forward it, link to it, get the word out.

Michael

From: Susan Ohanian
Date: June 10, 2007
Subject: Re: [eddra] Announcing the Rational Mathematics Education blog

Educator Roundtable invites all of you to join the grassroots movement.
Sign the petition to end NCLB.
http://www.educatorroundtable.org/petition.html

Buy 100 booklets explaining, in dispassionate, factual fashion, just what is IN NCLB. The booklet, written by a teacher/researcher, Elizabeth Jaeger, is written in the hopes that activists will distribute information to the community. We can start the revolution if each of you hold a conversation with 100 people.

We are selling the booklet at cost: 100 for $50.
Our object is not to sell booklets but to start those conversations.

You can send me your snail address, and I will send you the booklets.

Susan Ohanian

Little Ironies of the Math Wars

I found this quotation on an anti-Everyday Math site this morning:

"To have a great idea, have a lot of them." Thomas A. Edison

Could anything be more ironic? This site has links to the NYC-HOLD "Math Myths" piece I'm currently interrogating, and the blogger makes clear that those "multiple methods" for doing arithmetic that are developed in the Everyday Math books (and in other curricula such as INVESTIGATIONS IN NUMBER, DATA AND SPACE (aka, "TERC" by its detractors, who apparently find it hard to say or type "INVESTIGATIONS") are anathema to its predictably anonymous author. I imagine the wrath of the entire state of Connecticut and the mythical "Education Mafia" will come down on the author's head and the heads of the author's child(ren) for criticizing Everyday Math. :)

In any event, Edison calls for lots of ideas in order to have a great one. The reform haters want ONE idea about EVERYTHING in and out of mathematics and education, want ONE way to do EVERY calculation (gods forefend there could be more than one!), and want that idea to come NOT from the thought processes of children but from a Zeus on Math Olympus: the teacher, who of course must only teach the One True Way. Note well that if tomorrow some child in some classroom somewhere were to come up with a perfectly good, perfectly useful, perfectly sensible, perfectly logical way to do addition, subtraction, multiplication, division, or other elementary school mathematical task, and if - the notion is of course unthinkable! - her classmates agreed that this was a very good, clear, understandable way to do the task at hand - Edison and lots of other bright folks would cheer, but our hate-filled Connecticut parent and others of similar narrow mind would run to the Courant Institute or Cal State-LA for assurances from the Math Gods that there really is only one right way to do that task.

I'm not sure what is more outrageous: that parents who have a reasonable degree of education can be so blind to how limiting and ineffective this approach to mathematics teaching and learning is for most kids, or that professional mathematicians who should have a clear sense of the thought processes that go into doing mathematics can be so ignorant of how kids learn and think. Of course, mathematics education and teaching are NOT the purview of most professional mathematicians, and while some - Hyman Bass, Alexander Zvonkin, and others - have recognized their ignorance (not a bad thing) in this regard as they've moved from the complexities of doing original math to investing the complexities of teaching and learning and training teachers for school mathematics.

I suppose the point here might simply be: don't quote people who are far smarter than you unless you have thought deeply about how what they've said reflects on your worldview. You might just wish you'd stuck to quoting ideologues and political pundits.

Let's Debunk Some NYC-HOLD Nonsense (Part 1)

As readers familiar with the Math Wars know, one of the most active groups who attack reform and progressive ideas in mathematics education (and other educational areas) calls itself HOLD, which ironically stands for "Honest, Open, Logical Debate." That is a grand slam of linguistic abuse, given that there is nothing honest or open about this group of angry professionals and parents, they do NOT welcome debate, and while there is sometimes logic in their arguments, just as there is a kind of logic in MOST opinions, that doesn't make what they assert true. If you start with false assumptions, you can draw some lovely logical conclusions that are utterly false, even if they are logically valid.

While HOLD began in California as part of the assault on whole language and reform math, it leapt across the country to form a second (and as far as I can tell, the only active branch), NYC-HOLD. What has made this offshoot effective is that many of its members are professional mathematicians affiliated with NYU's prestigious Courant Institute (though not all the members are so affiliated, not all from NYU are mathematicians, and the ostensible head of the group, Elizabeth Carson, is a self-described "actress." And yes, she certainly CAN act, as her testimony before various public hearings and at school board meetings, etc. attests.

Like any number of other leaders and members of "parents with pitchforks" groups dedicated to overthrowing any and all instances of what they perceive of as progressive education (if you, like I, grew up thinking that "progressive" is a positive word in this regard, rest assured they would vehemently disagree), including "whole language" literacy instruction and reform ideas in mathematics education that they dismiss as "fuzzy" (one of the less offensive epithets they and their forerunners at Mathematically Correct have used to attack ideas they dislike), Ms. Carson can whip up all the drama needed to gain attention of the media and scare parents and other educational stakeholders about the evils they associate with progressive reform.

While it would be impossible to debunk the entirety of the disinformation that has appeared over more than a decade on the NYC-HOLD web site, I have chosen one of their web pages as an appropriate place to focus my first substantive blog entry following yesterday's announcement and introduction of the Rational Mathematics Education blog. I'm going to look at their "Ten Myths About Math Education And Why You Shouldn't Believe Them" by Karen Budd, Elizabeth Carson, Barry Garelick, David Klein, R. James Milgram, Ralph A. Raimi, Martha Schwartz, Sandra Stotsky, Vern Williams, and W. Stephen Wilson. This appeared on May 4, 2005 at .

While you may go to their page for the original, I will include here only the ten "myths," the "reality" the NYC-HOLD folks offer up, and my own responses, taking them in order.

Under the misleading heading of "NCTM (Fuzzy) Myth we get:

Myth #1

"Only what students discover for themselves is truly learned."

And the alleged "Reality" :

"Students learn in a variety of ways. Basing most learning on student discovery is time-consuming, does not insure that students end up learning the right concepts, and can delay or prevent progression to the next level. Successful programs use discovery for only a few very carefully selected topics, never all topics."

My take: This is one of the innumerable assaults on constructivism in mathematics education. Nearly every such attack distorts or completely misrepresents both what constructivism is (a theory of human learning) and what constructivists believe are the theory's implications for mathematics teaching and learning.

One classic distortion is to write as if "constructivism" and "discovery learning" were the same thing, and that there is only one flavor of "discovery learning" (and it's horribly ineffective, time-wasting, etc. Opponents of progressive education are very fond hollering about precious instructional time, and this leads some of them to simultaneously lobby for longer school days and a longer school year, as well as a reduction or elimination of "extra" things like recess, music, art, etc. The result for kids and teachers has been quite horrific, and one need only spend a day in most elementary schools where recess has been cut back to 15 minutes once a day for fewer than five days a week (if not entirely done away with) to see the deleterious effects of such a short-sighted policy. Not only do kids absolutely need time to move their bodies and play, but teachers, too, need time to rest their brains a little, move, and do a little reflecting on what's happened thus far and how that might lead them to modify their plans for the rest of the day. While I will likely be critical at times in this blog of the lack of professional reflection on the part of most teachers of mathematics in K12 that I've observed, the blame should not fall solely on the shoulders of the teachers, especially not those who genuinely WANT to reflect and revise practice even in the course of a single day, but who don't have a moment to breathe or even go to the bathroom, let alone reflect on lessons and craft changes based on the day's experience (for the teacher and the students). Making teachers "teach" more and kids "learn" more is, of course, part of the big, fraudulent push for more "accountability" in public education: someone things teachers are sitting on their thumbs getting "big bucks" for goofing off. Do some teachers in fact goof off? Of course. There's nothing unique to education about lazy, cynical employees. But for the vast majority of teachers who care about their students and are trying to do a nearly impossible job at pay levels that generally do not reflect either the amount of education they've had to qualify them for their work or the number of hours they work on their craft and attend to their duties outside of their classroom and building.

That said, no one suggests seriously what is asserted to be the position of NCTM. That is, no one asserts that unless a student discovers something for her/himself, s/he doesn't learn it or doesn't "truly" learn it. The problem here to begin with is two-fold: we have an NYC-HOLD straw man version of what NCTM (and by implication other reform-inclined mathematics educators) supposedly believes, and no one at NCTM or anywhere else believes any such thing. But on top of that, we have this already-mentioned conflation of constructivism (which DOES speak to how we all learn) and discovery learning, which is one option among many.

The effect of this quite deliberate conflating is to lead unsophisticated readers to believe that when they hear someone advocate for a constructivist view of knowledge and learning, they are hearing a call for the exclusive use of discovery learning. However, constructivism as a theory of learning does not call for or lead logically to any particular teaching approach. What it asserts is that learning takes place in a cycle: no one starts as a tabula rasa, every person has an individual perceptual make-up that varies, and upon birth, every experience is processed in an ever-adapting, ever-changing, ever-modifying and being-modified framework. What we encounter is perceived and processed by us and at the same time changes us in ways that impact how we perceive and process each subsequent experience. This cycle of mental and physical experiences interacting with our already-existing but in flux constructs and mental and physical processes is what we call "learning." No two people perceive the same basic events identically; each brings unique constructs and perceptual frameworks to the experience and stands in at least a slightly different perspective to the "same" experience.

If the constructivist view of learning is correct, and I have yet to read one that makes more sense or undoes the fundamental tenets of the theory, then what, if anything, are the implications for teaching and learning mathematics? Does this viewpoint in fact lead to the conclusion being claimed by the NYC-HOLD authors to represent NCTM, et al?

Not in the least. There is nothing in constructivism that would say that we only learn what we discover, if by "discover" one means, as I think the HOLD folks intend us to think, "invent from scratch with no direction from teachers or peers (or texts or other sources." But that is simply nonsense. Constructivism doesn't assert, for example, that we don't learn when we read or hear a lecture or do a math worksheet or engage in other sorts of learning activities that the HOLD authors would find acceptable. What theory of learning could offer up something so obviously and empirically wrong and hope to win adherents among educated people?

["Students learn in a variety of ways. Basing most learning on student discovery is time-consuming, does not insure that students end up learning the right concepts, and can delay or prevent progression to the next level. Successful programs use discovery for only a few very carefully selected topics, never all topics."]

Thus, when we are offered the "reality" by HOLD, we first get a tautology that is supported by every competent educator on the planet: of course students do not all learn the same way. Indeed, this is an implication of constructivism. It has been more conservative and reactionary educational writers who have called for "back to basics" and "direct instruction" in mathematics, and for phonics-only literacy education. Yet here we have HOLD writers playing at being reasonable by offering a truism that more fittingly is associated with progressive educational theory and practice (this tactic, by the way, is rampant in the Math Wars and other battlefields in education).

Next, we get the "time-consuming" complaint about discovery learning, as if it were time-efficient to simply spoon-feed information to students who don't get it, don't engage with it, and who must be taught, re-taught, and often remediated for months, years, or even decades on the mathematics they never truly think about effectively when given "efficient" lectures.

Next, we're told that discovery methods (note that we're far from talking about constructivism) do not 'insure" [sic] students end up learning the "right" concepts. This notion plays on fears that, left to their own devices, students will waste time, go irredeemably off-track, become stuck, lost, frustrated, etc. But "discovery" learning comes in a variety of styles and flavors. One of the most common, especially in the textbooks that emerged from various NSF-funded projects, as well as some of their predecessors and successors, is more accurately described as guided discovery. Lessons are crafted to lead students along a known series of questions that lead inevitably to certain conclusions. Students aren't "inventing" new mathematics, but simply being allowed to find something out that is already known. The argument here is that it would be faster to just tell them, and that's true, but if they don't get it when told, it's necessary to spend more and more time reviewing and re-teaching, as mentioned above. And there simply is no historical evidence to support the idea that the vast majority of Americans have reached a high degree of proficiency in mathematics and been able to go on to mathematics-intensive careers through this traditional direct approach. It seems reasonable to invest class time to let students make connections for themselves before a competent teacher leads classroom discussion in such a way that all students are convinced of the truth of what has been "discovered" by themselves and some or all of their peers.

The red herring of "efficiency" is then ramped up with the notion that students won't be able to progress to "the next level" if they haven't "discovered the right concepts." While this might make sense if progressive mathematics education actually left students completely to their own devices, if the textbooks and teachers' guides didn't try to lead students to important mathematical ideas, and if the teacher was completely incompetent regarding both what mathematical ideas are at the heart of the lessons s/he's teaching and the implications of the ideas the students generate themselves. Again, this is simply another HOLD straw man, easy to knock down, but not in fact what is being promoted.

Finally, we're told that successful programs use discovery on a limited basis. This may well be true, but if there is a program that uses discovery learning of ANY type that is approved and recommended by HOLD and its allies in the Math Wars, it's news to me, and likely news to anyone who has been following the debate over the past 15 or more years. The programs for which they advocate tend to eschew any sort of exploration, discovery (guided or otherwise), investigation of student thinking, discussion of the strengths and weaknesses of alternative approaches, etc.

Again, this has nothing to do with what most reform advocates believe or what constructivist learning theory states or implies. And none of this goes towards the initial claim that reformers believe that we only learn what we discover.

There is no doubt that students learn from EVERY lesson. The relevant question is, however, WHAT do they learn? They may be in the classroom while the teacher offers an explanation of fraction addition, multiplication of integers, the Pythagorean Theorem, or a host of other procedures and/or concepts. However, what the students may learn is that mathematics is hard, boring, irrelevant, useless, confusing, etc. Or they may learn that if they can parrot what teacher does, they will get a good grade on the next quiz or test, regardless of whether they have a clue WHAT is being done or why one might do it, let alone why it actually MAKES SENSE. Far too often, what is learned is that mathematics does NOT make sense (sometimes not even to the teacher).

Can "discovery" learning also result in student confusion, error, and other undesirable effects? Of course. No method or teacher or book or lesson is foolproof (regardless of what the late John Saxon or some people who advocate for Singapore Math or worksheets from Kumon may believe). But teachers who are well grounded in constructivism are far more likely to teach lessons that repeatedly encourage and develop student thinking and communication in class. As a good teacher helps her/his students grow from a room of isolated individuals into a learning community, it becomes imperative that students and teacher learn to listen to and respond to one another's "constructions" of mathematics, and this community learns to refine and improve their understanding, both collectively and individually. Instead of letting students sink or swim based on what they can make of lectures and attempts at solving problems at their desks and trying poorly-understood homework exercises away from school, teachers are helping the class reach consensus about what methods and explanations are most sensible and effective. The class is unlikely to agree on one single metaphor or approach, but having more than one way to do or think about mathematics is a strength, not a weakness. Whatever 'extra" time is used to arrive at a more collective understanding that emerges from student thinking (guided by a good lesson and the teacher) is easily justified by the quality and depth of the learning.

This, then, is a more accurate representation of what NCTM and other progressive reformers mean by constructivism and "discovery" learning. I close by pointing out that there is also room on occasion for both looser discovery in which students are encouraged to take more risks and follow paths that may not be so closely planned by authors and teachers, as well as for every traditional or direct approach. It is only when such approaches dominate that we see historically the utter stagnation of mathematics for the majority of American students. This has been the case for decade upon decade, and it is disingenuous in the extreme for groups like NYC-HOLD to suggest that a majority of our citizens were at some golden point in the past deeply mathematically literate by means of lecture and drill-and-practice instruction. It simply isn't so, and never was.