Thursday, February 28, 2013

Math problems of the week: polynomial division in UCMP vs. Wentworth's Algebra

I. The only problem in the entire University of Chicago Math Project Algebra involving the division of a multinomial by a polynomial (p. 730):

There are no problems in University of Chicago Math Project Algebra that involve polynomial long division.

II. The first problem set in Wentworth's New School Algebra involving polynomial long division (pp. 70-71) [click to enlarge]:

My 6th grade (home-schooled) daughter's work on the 38th of these problems (she did this problem just yesterday, and, though she's no math genius, needed no help from me):

III. Extra Credit

Polynomial long division provides lots of practice with the distributive law: here, one is constantly distributing both monomials and the negative operator across polynomials. In addition, polynomial long division shows you place value at a more abstract level: here, one can discuss, for example, the c2s and the cs places. It also shows you the long division algorithm playing out at a more abstract level--which I recall finding quite enlightening back when I did polynomial long division (a few decades before Reform Algebra eliminated it).

1. Is there a good reason not to have today's algebra students do lots of polynomial long division?

2. Is there a good reason to wait until p. 730 of an algebra text before doing any binomial by monomial division?

Tuesday, February 26, 2013

Narrowing the Achievement Gap: how current trends backfire

Narrowing the achievement gap, as I noted earlier, has become a national obsession. But many of the efforts by today's educational institutions to narrow the gap actually end up widening it.

One such strategy, as I argue in my last post, is Reform Math. Its deficiencies have prompted the better educated, more resourceful, more financially able parents--the ones whose children are most likely to be on the upper side of the gap already--to seek outside enrichment or private education. Those whose entire K12 math education comes from classroom Reform Math, therefore, not only get a worse math education than children in most other developed countries, but also fall even further behind their more privileged classmates here in America.

A second strategy aimed at narrowing the gap is insisting that all kids do grade-level work, as defined earlier by the No Child Left Behind tests and now by the Common Core standards. This not only holds the top performers back and prompts their parents to seek private schools and/or extracurricular enrichment; it also makes it impossible to effectively remediate those on the other side of the gap. Many kids in this country either enter 1st grade with below age-level academic skills, and/or later fall (even further) behind because they're particularly vulnerable to deficient phonics instruction, deficient penmanship instruction, and all that's wrong with Reform Math. Perhaps they begin school with a weaker cognitive aptitude or mild learning disabilities, or perhaps it's simply that their parents aren't intervening with all that extracurricular help that their more privileged counterparts are getting.

In this blog I've focused mostly on how the one-size-fits-all approach hurts high-achievers. But I've also seen it hurting those on the other side of the gap. In the after school teaching program I'm involved with, for example, I see 5th graders who are still counting on their fingers and baffled by numerators and denominators struggling unsuccessfully through homework assignments that ask them to multiply numbers or add fractions. To make any progress at all, they need at this point to begin with math lessons several grade levels below what the Common Core dictates. Will the typical Common Core implementation allow this?

A third gap-narrowing strategy that ends up backfiring is the reliance on accommodation over instruction, which I blogged about earlier. Here I discussed a Philadelphia-area school that attempts to educate students with dyslexia and other "language-based differences" via an "arts-based approach that includes costumes, games, activities, and classrooms decorated as medieval castles and prehistoric caves." To catch up with their peers, these students need more practice with language and reading, not less.

A similar strategy aims to accommodate different "learning styles": providing multiple "entry points" for assignment completion. A science teacher at one model school, for example, assigns weekly homework with options that range from doing an experiment or solving some science problems, on the one hand, to tracking down a science article in the mainstream media and writing a summary.

But "learning styles" research is bunk. What the actual research suggests is that students differ not in their learning styles, but in their cognitive strengths and weaknesses. And what this means is that strategies intended to acommodate styles end up widening gaps. How likely is it that those with weak science skills will choose the science problems while those with weak language arts skills will choose the summary? Essentially what we have here is a voluntary form of tracking, vaguely reminiscent of that found in European high schools: one for the scientically inclined, another for those who prefer language arts.

But most of today's education professionals say they're totally opposed to tracking--precisely because it widens the gap. Narrowing the gap, however, means putting children ahead of ideology, and putting an end to the many new practices that are largely responsible for how wide the gap has gotten in the first place.

Sunday, February 24, 2013

Reform Math: is it just about the achievement gap?

Some people have proposed that what drives the Reform Math movement is a desire to lower the achievement gap. I agree-up to a point. The gap has become a national obsession. Hopeful gap-narrowers are everywhere. The better intentioned of them honestly believe that the collaborative, student-centered, drill-and-algorithm-eschewing, symbol-de-emphasizing, "no one right answer" approach to math resonates with those who (supposedly for cultural, gender-based, and/or "learning style" reasons) have languished under "traditional math." These people mean well, but are ignorant of what peer-reviewed cognitive science research has revealed about what students need to master mathematics.

Then there are those with a more cynical attitude towards the gap: narrow it mainly by lowering the top. Dumb everything down, water it down further with excess verbiage and non-mathemnatical material, and only give full credit to students who write out verbal explanations. That way, students who once would have excelled will become bored and disengaged, and those whose math abilities far exceed their verbal and penmanship skills will get bogged down in the non-mathematical stuff.

For some, hostility to traditional math and to those who most excelled in it may be personal. Perhaps they themselves did not excel and have long felt stupid about that and harbored grudges against those who did. To them, Reform math provides validation, vindication, and, possibly, a perverse form of revenge.

But would-be gap closing isn't the only agenda behind Reform Math. Big swathes of the general populace buy into the idea that traditional math is boring and baffling and that classroom math needs to keep up with modern times. Then there are three specific groups whose interest in Reform Math is professional: those in the education business, those in the educational publishing business, and those in the math business.

Within the former group, we have the true believers: those who truly believe, whether from idiosyncratic prejudice or from brainwashing by ed schools, that Reform Math teaches deeper understanding and better prepares students for living in the 21st century.

Within the second group we have the stakeholders of textbook and software companies, as well as those in the professional development business. They benefit financially from the constant "reforming" of K12 math. The more you reform things away from what's curently traditional, the more easily you can convince schools to keep buying new material and paying for new training sessions.

Within the third group we have the mathematicians (or those who claim to be). Many of them have no sympathy for the drills of traditional math, which have little to do with what they like about math--and with their work as professional mathematicians. Perhaps, as young math whizzes, they didn't depend on such drills as much as ordinary humans do. It may never occur to them that most human beings can't simply play around with numbers and be ready for BC Calculus by 12th grade.

Others do concede that ordinary people are different, but go too far in separating the ordinary from the elected few. They think that the solution is to teach math to most kids entirely through concrete examples and real-life application--forget proofs and symbolic reasoning.

Finally, many college math professors hear Reform Math's buzz words--"higher-level thinking," "conceptual understanding"--and assume these will ameliorate, rather than exacerbate, the growing deficiencies they see in their undergraduate students, many of whom can't seem either to understand conceptually or do math except at a mindless, recipe-following level. When the more concerned of these math professors venture into workshops led by "math education experts," they assume that these experts mean the same they do by "deep, conceptual understanding." They take what they hear about Reform Math vs. traditional math on faith, and it doesn't occur to most of them (unless they have kids in school) that the so-called experts--many of whom seem like smart, well-intentioned people--may not be trustworthy. And that they themselves need to take a close look at the new curricula and at what sorts of problems are assigned at which grade levels.

One mathematician friend of mine who attended one of these "math education expert"-led workshops a few years ago was initially taken in by some of what he heard from the apparently smart, well-intentioned presenter. Below is the accompanying handout, with his notes at the top of the first page [click to enlarge]:

All this looks so much more conceptual than a mindless execution of the standard algorithms, doesn't it? But what this handout doesn't show is (1) the grade level of the student in question, and (2) all that the curriculum leaves out in the name of conceptual understanding and explaining one's answers in words, numbers, or pictures.

It was only after I showed my professor friend page by page, problem set by problem set, grade level by grade level, the actual curriculum that this workshop presenter was peddling that he completely changed his mind.

I've said it before and I'll say it again: no one--whether they are an "math education expert" or a math curriculum developer or a math professor or simply someone who believes in "21st century skills"--can claim to support Reform Math until they look closely at the curriculum and then agree to have their children get their entire K12 math education from it. No "but my child is different," and no Kumon or after-school Singapore Math or extra-curricular test prep allowed!

Friday, February 22, 2013

Math problems of the week: Reform vs. traditional algebra

Introducing the Quadratic Formula I. In Integrated Math 2, Unit 4, "Quadratic Formula and Graphs," p. 215 [click to enlarge]:

II. In The University of Chicago Mathematics Project Algebra, Chapter 9, "Quadratic Equations and Square Roots," p. 574 [click to enlarge]:

III. In Wentworth's New School Algebra, Chapter XIX, "Quadratic Equations," p. 278 [click to enlarge]:

III. Extra Credit:

The Reform Math texts don't show students where the quadratic formula comes from; the second text simply tells students that it's "famous" and to "memorize it today." Wentworth provides a complete derivation.  All three texts show students how to apply the formula to quadratic equations, but only Wentworth assumes that students already know how to turn a quadratic equation into an equation of the form ax2 + bx + c.

Given this, consider one of the good points made by Keith Devlin in the piece I blogged about earlier this week:
If mention of the word algebra automatically conjures up memorizing the use of the formula for solving a quadratic equation, chances are you had this kind of deficient school math education. For one thing, that’s not algebra but arithmetic; for another, it’s not at all representative of what algebra is, namely, thinking and reasoning about entire classes of numbers, using logic rather than arithmetic.
Question 1:

Should Devlin be focusing his efforts on creating computer games that somehow teach math non-symbolically, and, as reported in the following excerpt from an article in Inside Higher Education, defending Jo Boaler's highly flawed study of Reform vs. traditional high school math programs:

Keith Devlin, director of the Human Sciences and Technologies Advanced Research Institute at Stanford, said that he has "enormous respect" for Boaler, although he characterized himself as someone who doesn't know her well, but has read her work and is sympathetic to it. He said that he shares her views, but that he does so "based on my own experience and from reading the work of others," not from his own research. So he said that while he has also faced "unprofessional" attacks when he has expressed those views, he hasn't attracted the same level of criticism as has Boaler.
Of her critics, Devlin said that "I suspect they fear her because she brings hard data that threatens their view of how children should be taught mathematics." He said that the criticisms of Boaler reach "the point of character assassination."
Or should he be critiquing today's Reform Math texts for teaching algebra largely as the mindless arithmetic application of formulas rather than as the logical sequences of algebraic manipulations taught in traditional texts? And, as Barry Garelick discusses in EdNews, for teaching geometry as "the application of theorems rather than the proving of propositions"?

Question 2:

Should Devlin happen upon this blog post and/or my previous one about him, will he characterize what I write as an "unprofessional" attack and/or as a "character assassination" that ignores "hard data that threatens my view of how children should be taught mathematics"?

Wednesday, February 20, 2013

Devlin's Lament: the "symbol barrier"

In an article in the most recent issue of American Scientist entitled "The Music of Math Games," Keith Devlin (head of the Human-Sciences and Technologies Advanced Research Institute at Stanford University and NPR's "math guy") says that learning math should be like learning to play the piano. In doing so, he recalls (but does not credit) Paul Lockhart's Lament ("A piano student's lament: how music lessons cheat us out of our second most fascinating and imaginative art form"), which I blogged about here.

Though Devlin is no literary virtuoso, not all of what he writes here is mushy metaphor. He begins with a discussion of educational software, and here his points are clear and consistent with my own experience. Most "math games" and "math education" software programs I've seen don't make mathematics an organic part of the games or activities. Instead, math problems--mostly arithmetic problems of the "mere calculation" variety--are shoe-horned into non-mathematical situations. Here they serve simply as tasks you must complete before moving through the current non-mathematical activity or on to the next non-mathematical activity.

As Devlin writes:

To build an engaging game that also supports good mathematics learning requires... understanding, at a deep level, what mathematics is, how and why people learn and do mathematics, how to get and keep them engaged in their learning, and how to represent the mathematics on the platform on which the game will be played.
The same is true of language learning. Most linguistic software taps only superficial aspects of language, and, as I know from personal experience, it takes great effort to build a program that does more than that.

Where I begin to part ways with Mr. Devlin is in his discussion of traditional math and what he thinks is an excessive emphasis on symbols:
Many people have come to believe mathematics is the memorization of, and mastery at using, various formulas and symbolic procedures to solve encapsulated and essentially artificial problems. Such people typically have that impression of math because they have never been shown anything else...
By and large, the public identifies doing math with writing symbols, often obscure symbols. Why do they make that automatic identification? A large part of the explanation is that much of the time they spent in the school mathematics classroom was devoted to the development of correct symbolic manipulation skills, and symbol-filled books are the standard way to store and distribute mathematical knowledge. So we have gotten used to the fact that mathematics is presented to us by way of symbolic expressions.
This approach to math, Devlin suggests, is at odds with the resolutions of a "blue-ribbon panel of experts" serving on the National Research Council’s Mathematics Learning Study Committee ("Adding it Up: Helping Children Learn Mathematics," National Academies Press, 2001). In Devlin's words: these resolutions hold that math proficiency consists of:
the aggregate of mathematical knowledge, skills, developed abilities, habits of mind and attitudes that are essential ingredients for life in the 21st century. They break this aggregate down to what they describe as “five tightly interwoven” threads. The first is conceptual understanding, the comprehension of mathematical concepts, operations and relations. The second is procedural fluency, defined as skill in carrying out arithmetical procedures accurately, efficiently, flexibly and appropriately. Third is strategic competence, or the ability to formulate, represent and solve mathematical problems arising in real-world situations. Fourth is adaptive reasoning—the capacity for logical thought, reflection, explanation and justification. Finally there’s productive disposition, a habitual inclination to see mathematics as sensible, useful and worthwhile, combined with a confidence in one’s own ability to master the material.
Ah, "21st century skills," "habits of mind," "conceptual understanding," "real-world situations," "explanation," "disposition"...--all this makes me wonder about the ratio of mathematicians to math eduation "experts" on this blue-ribbon panel. (It should be noted that Devlin himself is not, strictly speaking, a mathematician; he holds a Ph.D. in logic from the University of Bristol, and, while affiliated with Stanford, is not a member of the Stanford math department.)

Standing in the way of these lofty goals is what Devlin calls the "symbol barrier":
For the entire history of organized mathematics instruction, where we had no alternative to using static, symbolic expressions on flat surfaces to store and distribute mathematical knowledge, that barrier has prevented millions of people from becoming proficient in a cognitive skill set of evident major importance in today’s world, on a par with the ability to read and write.
To the rescue comes... Devlin's math education software program:
With video games, we can circumvent the barrier. Because video games are dynamic, interactive and controlled by the user yet designed by the developer, they are the perfect medium for representing everyday mathematics, allowing direct access to the mathematics (bypassing the symbols) in the same direct way that a piano provides direct access to the music.
Devlin's notion that a well-designed math video game can help students meet the National Academy's goals for math education rests on two assumptions. One is that students can achieve a sufficient level of mastery in mathematics without symbols. The other is that playing such video games is to math what playing the piano is to music.

To address the first claim, Devlin elaborates the analogy to music:
Just how essential are those symbols? After all, until the invention of various kinds of recording devices, symbolic musical notation was the only way to store and distribute music, yet no one ever confuses music with a musical score.
Just as music is created and enjoyed within the mind, so too is mathematics created and carried out (and by many of us enjoyed) in the mind. At its heart, mathematics is a mental activity—a way of thinking—one that over several millennia of human history has proved to be highly beneficial to life and society.
But there's an important difference between math and music--and a reason why no one confuses music with a musical score. Music has a privileged place in subjective experience. Along with sensations like color, taste, and smell, it produces in us a characteristic, irreduceable, qualitative impression--an instance of what philosophers call "qualia." Just as there's no way to capture the subjective impression of "redness" with a graph of its electromagnetic frequency, or of "chocolate" with a 3-D model of its molecular structure, so, too, with the subjective feeling of a tonic-dominant-submediant-mediant-subdominant-tonic-subdominant-dominant chord progression. Embedded in what makes music what it is to us is the qualia of its chords and melodies.

Like most other, more abstract concepts ("heliocentric," "temporary"), mathematic concepts don't generally evoke this qualia sensation. What makes math beautiful are things like eloquence, patterns, and power. Unlike a Bach fugue translated homomorphically into, say, a collage of shapes, mathematical concepts can be be translated into different representational systems without losing their essence and beauty.

Devlin argues that while we might write down symbols in the course of doing real-life math, it is primarily a "thinking process," and that "at its heart, mathematics is a mental activity—a way of thinking." I agree. Indeed, math is much more appropriately compared with thoughts than with music. But this makes math symbols the mathematical equivalent of linguistic symbols. While thoughts, like math, can be expressed in a number of different symbol systems, you need some sort of symbol system in order to represent your own thoughts and to understand the thoughts of others.

This is especially true of abstract thoughts--and of abstract math. As Devlin himself admits, "the advanced mathematics used by scientists and engineers is intrinsically symbolic. "What isn't intrinsically symbolic, Devlin claims, is "everyday mathematics":
The kind of math important to ordinary people in their lives... is not, and it can be done in your head. Roughly speaking, everyday mathematics comprises counting, arithmetic, proportional reasoning, numerical estimation, elementary geometry and trigonometry, elementary algebra, basic probability and statistics, logical thinking, algorithm use, problem formation (modeling), problem solving, and sound calculator use. (Yes, even elementary algebra belongs in that list. The symbols are not essential.)
OK, but what does this mean for education? Are we going to decide before the end of middle school which students are going to become scientists, engineers, and mathematicians, and only help those students scale the "symbol barrier"? For a barrier it certainly is, as Devlin himself notes: "people can become highly skilled at doing mental math and yet be hopeless at its symbolic representations."

But Devlin is too busy appreciating the (well-studied) math skills of Brazilian street vendors, who do complex arithmetic calculations in their heads with 98% accuracy, and supposedly without the help of symbols (even mental ones?), to realize the educational implications of the fact that "when faced with what are (from a mathematical perspective) the very same problems, but presented in the traditional symbols, their performance drops to a mere 35 to 40 percent accuracy." No, not everyone is going to become an engineer. But not all non-engineers are going to become Brazilian street vendors.

It's ironic how deeply Devlin appreciates the difficulty that "ordinary people" have with the symbol barrier without appreciating what this says about their educational needs:
It simply is not the case that ordinary people cannot do everyday math. Rather, they cannot do symbolic everyday math. In fact, for most people, it’s not accurate to say that the problems they are presented in paper-and-pencil format are “the same as” the ones they solve fluently in a real life setting. When you read the transcripts of the ways they solve the problems in the two settings, you realize that they are doing completely different things. Only someone who has mastery of symbolic mathematics can recognize the problems encountered in the two contexts as being “the same.”
Instead of seeing this as a reason for exposing children to mathematical symbols early and often, Devlin sees this as reason to create computer games that somehow teach math non-symbolically.

He calls this "adaptive technology," a term that should raise red flags. In a recent blog post, I wrote about how assistive technology often becomes yet another excuse not to teach basic skills. Kids with dyslexia struggle mightily with the symbol system of written language; should they instead learn everything through text-to-speech and speech-to-text devices, and never learn how to read and write?

Devlin makes a few other strained comparisons to the piano:
The piano metaphor can be pursued further. There’s a widespread belief that you first have to master the basic skills to progress in mathematics. That’s total nonsense. It’s like saying you have to master musical notation and the performance of musical scales before you can start to try to play an instrument—a surefire way to put someone off music if ever there was one.
No it's not; it's like saying you have to master simple scales and exercises before you move on to Rachmaninoff.
The one difference between music and math is that whereas a single piano can be used to play almost any tune, a video game designed to play, say, addition of fractions, probably won’t be able to play multiplication of fractions. This means that the task facing the game designer is not to design one instrument but an entire orchestra.
Can one create a video game that functions "as an instrument on which a person can 'play' mathematics?"
Can this be done? Yes. I know this fact to be true because I spent almost five years working with talented and experienced game developers on a stealth project at a large video game company, trying to build such an orchestra.
What does Devlin's software do? The last two paragraphs of this article function as an extended but not very informative infomercial. Here's the most informative excerpt:
Available in early March, Wuzzit Trouble is a game where players must free the Wuzzits from the traps they’ve inadvertently wandered into inside a castle. Players must use puzzle-solving skills to gather keys that open the gearlike combination locks on the cages, while avoiding hazards.
Puzzle solving? As I argue in my last post on math games, existing games already offer some version of this, and it isn't math. This, indeed, is one of the other problems with so-called math education software.

Devlin suggests his software is different:
Unlike the majority of other casual games, it is built on top of sound mathematical principles, which means that anyone who plays it will be learning and practicing good mathematical thinking—much like a person playing a musical instrument for pleasure will at the same time learn about music.

Wuzzit Trouble might look and play like a simple arithmetic game, and indeed that is the point. But looks can be deceiving. The puzzles carry star ratings, and I have yet to achieve the maximum number of stars on some of the puzzles! (I never mastered Rachmaninov on the piano either.) The game is not designed to teach. The intention is to provide an “instrument” that, in addition to being fun to play, not only provides implicit learning but may also be used as a basis for formal learning in a scholastic setting.
If you say so. But I wonder how much it will cost schools (and society) to find out whether this latest incarnation of "math education" software helps prepare students to become mathematicians, scientists, engineers--or Brazilian street vendors.

Monday, February 18, 2013

Peer conferences and peer editing: more outlets for classroom bullying

A reader of this blog forwarded to me an exchange she had with two teachers regarding a peer conferencing/peer editing incident involving her son (Thomas) and a classmate (Anastasia). (All names have been changed to protect privacy). In peer editing/conferencing, also known as “Writer’s Workshop,” and the second stage of a comprehensive writing process that begins with “self editing,” students are supposed to contribute “constructive feedback” and “new perspectives” in response to one another’s writing.

Thomas’ mother begins her own writing process with the following:

Thomas apparently made what he thought were constructive suggestions to which Anastasia took offense. She did not appreciate his suggestions, so re-edited his and ripped the s*** out of it.
What are they supposed to be getting out of this? Aren't the results of these evaluations going to depend entirely on whether or not the evaluators are friends? What do you expect to happen when 10-year-olds are told to grade each other’s work?
One of Thomas’ teachers (not present at the time) responds with the following perspectives on the virtues of peer editing and the problems Thomas has doing it properly:
I have to say that it [peer editing] is quite helpful for the students. They learn now to kindly and effectively communicate constructive criticism. Students learn to be thoughtful listeners and enjoy hearing feedback from their peers. This is usually quite a supportive process.
Having just observed Thomas working in his PBL [Problem-Based Learning] group, my guess is he was less sensitive than he could have been about his constructive criticism. He has a tendency to represent his opinion as fact rather than his personal feelings or beliefs about something. While working with his group he would become very insistent that his idea was followed even though other members had different ideas... It is wonderful that he is so sure of himself, however, he does need to make room for other opinions and possibilities.
Similar perspectives are contributed by Thomas’ other teacher (the one who was present):
I specifically paired Thomas and Anastasia together for peer conferencing as they are both talented writers. I thought that they would provide each other with valuable new perspectives. I strongly feel that peer conferencing is a useful strategy for getting students to apply skills learned in class, notice common mistakes in writing that they often overlook in their own writing, and help a peer in a constructive way. It is a widely-used and effective teaching tool, based on the understanding that kids learn best through application of skills and collaboration.
Yesterday morning, I sat down with Thomas and Anastasia to discuss what was going on with their peer conferencing, because Thomas had expressed some frustrations. Although Thomas is commended for his desire to complete the conferencing process, he had demanded a conference with Anastasia during the morning gym time, which was not an appropriate time given the fact that we would be working on conferences during regular class time. Anastasia did not respond to Thomas’s request in the most respectful way, so we discussed what could have been done by both parties that could have had a more positive outcome. We discussed using respectful language even when expressing frustrations with each other and finding an appropriate time and place to address our concerns. I also explained that they were both excellent writers who could offer each other some very helpful suggestions. By the end of the conversation, Thomas and Anastasia both seemed to understand each other’s feelings and were in a good position to continue their conference.
Once students are done with their peer conference, the next step is to have a one-on-one teacher conference. This is an opportunity for me to either confirm some of the suggestions for revision (done in both self and peer stages), make additional suggestions, and to make corrections to misinterpretations. While peer conferencing is a valuable step, it is by no means the final step. My evaluations of individual student work is based on the work of that student, not on the peer’s comments. I require students to go through the process of peer conferencing, but again this is only one step of the entire process.
Thomas is a talented writer and I respect how dedicated he is to his work. Being open to different perspectives and styles of writing can only make his writing even stronger. Tomorrow I will review ways of tailoring peer comments in a respectful way with the class again. I hope that this helps to bring more clarity to the structure of Writing Workshop and my intention to provide Thomas with the best possible instruction. If you would like to discuss this further, [the other teacher] and I would be happy to schedule a meeting with you.
Hopeful of their openness to novel perspectives and constructive criticism, Thomas’ mother replies with the following:
Writing is a very personal form of communication, because it can be picked over and critiqued after it has been written. The purpose of writing is to express thoughts and ideas. If a child is in fear of peer criticism and will be laid bare by teachers in front of a random child who may not like him very much, how is he going to feel free to express himself in writing? It does not seem that Anastasia was pleased with her assignment to work with Thomas and I suspect that Thomas was aware of that.
Thomas’ writing skills improved dramatically last year, largely because his teacher gave a lot of very constructive criticism. Thomas trusted her and respected her opinion, and would spend hours editing and changing his writing. He learned about style and writing for interest and he tried to copy the writing styles of his favorite authors. He could see that he was improving and was so pleased with the results that he was saying that he wanted to be a novelist someday.
It is hard to see how a ten year old child can inspire and direct the way that a teacher can. There is enormous peer pressure at this age to conform and fit into the social group. Thomas is a bit of a social wallflower and does not comfortably fit in. For him, the thought of humiliation in front of peers is the hardest part of school, and with his prior experiences at this school it is easy to see why he feels this way. If he is obligated to get the approval of Anastasia or any other child in the class before a constructive interaction with the teacher it is going to significantly impact what he is willing to put on the paper.
Peers at this age are not naturally kind. You may be able to get them to act kindly to one another when you are looking, although it seems that even that requires some work, but there is a lot more to their interpersonal relationships. When asked to evaluate each other, all of this will come into play. Children who are friends and who are popular will want to work together and are likely to evaluate each other positively. Children who are less popular and want to make friends will want to work with popular kids and will likely evaluate popular kids positively, because it may improve their social standing. But kids who are popular will not want to work with kids who are not popular, and will likely evaluate them more harshly and find subtle(and not so subtle) ways to make it clear that they do not want to be paired with unpopular children. Children like Thomas are very aware of their social standing and this kind of peer evaluation puts them in a no-win situation. It results in teachers essentially codifying their social position.
Thomas is not enjoying the writing process at all this year and no longer puts much effort into it. His writing has become formulaic and dull. He answers the questions succinctly and with as little personal information as possible. When asked to write a personal journal, he made up a fictitious child and an imaginary life, because he did not want to write about himself. I believe peer editing is a big part of the reason why.
I suspect that Thomas is not the only kid who feels this way. I would have felt the same at his age, and I think peer editing is difficult for many. The boys especially are very competitive now, and they can be very harsh with each other. Social hierarchy and bullying should be kept out of the classroom as much as possible and I think that this kind of classroom activity invites it in. Writing should be about freedom of expression, not fear of peer criticism.
In reaction to these novel perspectives, the second of Thomas’ teachers expresses appreciation:
Thank you for your message. I appreciate your perspective and will continue to support Thomas as a writer in ways that both support his academic and social growth.
Since this is all that she writes, it would appear that she doesn’t have any constructive criticisms or additional perspectives to add.

The perspective that is particularly novel to me in all this is Thomas’ mother’s perspective on peer bullying. This blog has discussed various recent classroom practices that create opportunities for bullying--particularly group activities, and differentiated instruction--but never before had I considered the opportunities opened up by Writer’s Workshop.

Saturday, February 16, 2013

The Distributive Law and the Special Rules Revisited: what instant polynomial pattern recognition can do for you

Here are two problems from a diagnostic quiz given to a Calculus 1 for engineers class this semester at University of Maryland. Thanks to Leigh Lieberman for sending me these!

1. 1.000002 × 1.000003

2. (√3 − √2)^4 x (√3 + √2)^4

Most of the students were unable to calculate the correct answers. Can you?

Challenge problems:

3. The most typical answer given by these students to the second problem was 65. Explain what they did wrong.

4. Relate it to lack of emphasis on the Distributive Law and the Special Rules of polynomial multiplication discussed in this week's Problems of the Week.

5. Use your answers to 3 and 4 to predict what is likely to have been the most typical answer to problem 1.

Thursday, February 14, 2013

Math problems of the week: 1900s algebra vs. Interactive Math Program

I. From the beginning of the "Special Rules" chapter (Chapter VI) of Wentworth's New School Algebra (published in 1898) [click to enlarge]:

II. The two problem sets within the entire Year 4 Interactive Mathematics Program that come closest to providing practice applying the Distribute Law to algebraic expressions [click to enlarge]:

(Oops, the first two of these can be simplified without applying the Distributive Law. The other two, by the way, among just a handle of problems throughout the entire book that require factoring)

(Oops, these problems, like the majority in the IMP book, don't involve any algebraic manipulation at all; just plugging in numbers.).

III. Extra Credit:

There are no problems in the Year 4 Interactive Mathematics Program that involve demonstration or application of "Special Rules" in polynomial multiplication--Square of Sums, Square of Differences, Product of Sum and Difference. Is application of these "Special Rules" a 21st Century Skill?

Tuesday, February 12, 2013

What kinds of peer groups and parenting styles are most problematic for "peer orientation"?

In some really insightful comments on my last post, people raised the question of what kinds of peer groups are most problematic in terms of the issues raised in Hold on To Your Kids.

Anonymous 2/10 points out that the groups of unsupervised neighborhood kids of yore were generally a mixed age group, and that it is among same-aged peers that things get difficult.

C_T hypothesizes that what's particularly problematic are "peers-as-displayed-by-modern-media," in which there are fewer of the "parent-figure adults" who in older TV shows were "wise mentors who generally ran things and helped solve their kids' problems." Today's shows teach kids that "parents aren't very necessary" and giving them "fictional teen celebrities as role models."

Anonymous 2/11 points out that, if a kid goes out by himself in this day and age and starts playing with whichever kids he meets, "they won't necessarily be the kids you'd like him to hang out with."

All this seems right to me. Growing up I experienced both unsupervised play with neighborhood kids brought together by choice, and schoolyard (and classroom, and cafeteria) dynamics with large numbers of same-aged peers brought together by circumstances. Only in the latter cases do I recall nastiness, and there was plenty of it to go around. As for the former, though I live in once again a neighborhood full of families and yards and alleys, only rarely do I see anything like what I remember of the neighborhood play of my childhood.

What about the fact that most of today's kids are indoors and in structured activities, relentlessly supervised and micromanaged by parents and other adults? In this age of helicopter parenting, Auntie Ann astutely asks, do we really need to hold on to our kids even more? She proposes that the real problem is different: the low expectations we have of kids, especially for their capacity for responsibility. "When children are responsible for caring for siblings, feeding and milking the livestock, gathering firewood, fetching water, etc.," Autie Ann writes, "they grow into their role in the community."

I'd add that they also had less time back then to hang out with peers, let alone watch child-centered shows on TV that offer up teenaged role models.

The kinds of "holding on" that Gabor and Mate recommend, it seems to me, are in fact in decline: family dinners; interactive family activities and outings. No matter how much time today's kids spend at home rather than out in the neighborhood, and no matter how much time they spend in cars with parents while driven from activity to activity, they're kept out of reach by iphones and ipods and computer screens. No matter how much time kids spend in playdates that are scheduled and (nominally) supervised by their parents, most of the interactions are entirely peer-based, with kids playing with kids and adults socializing with adults, often at least one room away. What Gabor and Mate say we need more of is parents playing with, talking to, and otherwise bonding with children: rather than micromanaging them, building emotional attachments.

Friday, February 8, 2013

Math problems of the week: introducing algebra in 1900s vs. Singapore Math

I. From the end of the 5th chapter (out of 27 chapters) of Wentworth's New School Algebra (published in 1898), pp. 70-71 [click to enlarge]:

II. From the end of the 5th chapter (out of approx. 16 chapters) of the Singapore Math Discovering Mathematics 7A (Common Core), p. 29) [click to enlarge]:

III. Commentary:

My daughter finished up the Singapore Math Challenging Word Problems for 6th grade back in October and, after checking out the two Singapore Math sequels, New Elementary Mathematics and Discovering Mathematics, I opted instead to start her on Wentworth's New School Algebra. She's no math genius, but, after all that solid 1st-6th grade Singapore Math, complete with the Challenging World Problems series, I felt she was ready to move beyond arithmetic straight into algebra--something that neither the Singapore Math sequels do. There is some algebra in New Elementary Mathematics and Discovering Mathematics, but there's still a lot of arithmetic mixed in.

Wentworth's 1898 New School Algebra, on the other hand, moves straight into algebra, in the most straight forward, systematic way I've seen in any algebra book. Chapter I: Definitions and Notation; Chapter 2: Simple Equations; Chapter Three: Positive and Negative Numbers [in algebraic expressions]; Chapter Four: Addition and Subtraction [of polynomials]; Chapter Five: Multiplication and Division [of polynomials]. So straight forward and systematic is Wentworth's curriculum that my 6th grade daughter, after doing every single problem leading up to this, is now working on the above problem set with minimal assistance from her mother.

It makes me wonder how many other 6th graders could get this far with algebra, if only they were given a Singaporean foundation in arithmetic followed by a Wentworthian systematicity in introductory algebra.

Wednesday, February 6, 2013

Problems with child-centered "sociability," II: digesting those excerpts

This past weekend I finished Gordon Neufeld and Gabor Mate's Hold on to Your Kids, and had just enough time to excerpt some key passages before lending the book to one of my many friends who urgently needs to hold on to his kid. And today I have just enough time to digest what seem to me to be the most important points within these excerpts.

But first, a bit of background.

Neufeld and Mate's focus is on the rise of peer-orientation, or children who orient to their peers rather than their parents--and, as a result, are increasingly raised by these peers, educated by these peers, and look to these (highly fickle) peers as their primary source of love, acceptance, and emotional support.

Neufeld and Mate date the rise of peer-orientation to the 1950s, but argue that it's worse than ever. While they don't flesh out entirely why this is so, it seems to me that the following factors are at play:

(1) more and more kids, from infancy on, spending their weekdays in day care centers, and, once they reach school age, in after-school centers.

(2) the unprecedently large fraction of the school day that students spend in groups--and facing one another in desk "pods" rather than facing the teacher.

(3) the fact that parents themselves,  ever since the 1950s or so, have themselves been increasingly peer oriented--such that we care more and more about our kids fitting in, and spend more and more nights out with our adult friends rather than home with our children.

(4) the rise of social media--making kids and parents alike more peer-oriented

Neufeld and Mate blame a host of problems on the rise of peer orientation and peer culture:

Societal: the rise of bullying and criminal behavior

Psychological: pediatric and adolescent psychological disorders (especially oppositional defiance, social pathology, anxiety, and depression),

Academic: the decline in teachability, academic knowledge, vocabulary, reading, and culture.

Turning now to Neufeld and Mate's key points:

First, what seems so obvious that nearly everyone in our society acts on it simply isn't true:

The belief is that socializing--children spending time with one another--begets socialization: the capacity for skillful and mature relating to other human beings. There is no evidence to support such an assumption, despite its popularity.
In fact, what data there is supports the opposite conclusion:
One of the largest studies ever done on [the subject of daycare and sociability] followed more than a thousand children from birth to kindergarten The more time a child had spent in day care, the more likely she was to manifest aggression and disobedience, both at home and in kindergarten.
In other words:
The more children spend time with one another, the less likely they are to get along and the less likely they are to fit into civil society.
And--a second key point--the less likely they are to become creative, imaginative and curious:
Because of the strong emphasis on peer socialization, emergent play--play arising from the child's creativity, imagination, and curiosity about the world--has become endangered.
Third, anti-bullying measures are backfiring:
In response to the intensifying cruelty of children to one another, schools all over this continent are rushing to design programs to inculcate social responsibility in youngsters.
In particular, self-styled anti-bullying "experts" think the way to prevent bullying is to make students do cooperative group activities; but, as I've argued, this actually creates more opportunities for bullying, and, by intensifying peer orientation, may cause more bullying in the long term.
Fourth, teachers are making students less teachable:
Given that peer orientation is devastating our educational system, one would think that we would be alarmed, seeking ways to reverse the trend or at least slow it down. On the contrary, we as educators and parents are actually aiding and abetting this phenomenon.
Peer teaching--all that heterogeneous grouping and "differentiated instruction." Peer editing--all that "Writer's Workshop". More generally, all that "cooperative" group work that also facilitates bullying. One thing I've noticed in my after school tutoring ventures is that we're having to undo not just the effects of Everyday Math, but of Constructivist teaching. Students don't expect to sit facing the blackboard or to listen at length to their teachers.

Fifth, and perhaps the most poignant point of all:
Peer interaction is routinely prescribed for yet another purpose: to take the rough edges off children who may be a bit too eccentric for our liking. We seem to have an obsession in North America with being "normal" and fitting in. Perhaps we as adults have become so peer-oriented ourselves that instead of seeking to express our own true individuality, we take our cue for how to be and how to act from one another...

Monday, February 4, 2013

Letter from Huck: My Triumphant Return: Trying to Find a Cure for the Flu and the Common Core

Out in Left Field proudly presents the eleventh in a series of letters by an aspiring math teacher formerly known as "John Dewey." All personal and place names, with the exception of Miss Katharine's, have been changed to protect privacy.

I have decided to take up my correspondence once more with Miss Katharine about my experiences rafting along the ideological, political, and cultural river known as math education. Some of you may dimly recall that about a year ago I wrote some letters using the name Huck Finn, describing my experiences as a student teacher. I am not yet teaching full time, but I am subbing, so you’ll have to be satisfied with that for now. I’ve also abandoned the raft for a canoe for easier maneuvering.

 I chose the name Huck Finn for two reasons: 1) I am looking to get hired as a teacher and because my opinions may not be popular amongst those on the other side of the river, my real name is best not told; and 2) the sheer poetry of it.   My experiences have for the most part been very positive and instructional.  I missed my old class and my teacher from my student teaching days and thought that no one would ever take their place in my heart.  For the most part it’s true, but I do recall on one of my early subbing jobs at a middle school, the first student entered boldly into the classroom, overjoyed at the prospect of having a sub instead of the real teacher and cheerily announced: “Hi! My name is Lupita, but you can call me ‘sexy’. ” I replied “Hi! My name is Mr. Finn, and I will call you Lupita.”  I was amazed at the reflexive and automatic nature of this response.

 It is the heart of flu season and my greetings from students have gone from “call me ‘sexy’” to “Can I go to the nurse’s office; I feel like I’m going to throw up.” In the course of my work, things don’t get any more complicated than that.  I usually don’t find myself in the thick of political or ideological battles concerning math education with anyone. But the other day, something did happen to make me aware once again of my trip down the ideological river.  I was subbing in a high school geometry class, and preparing to put up a quiz on the projector when I noticed on the table at the front of the class some extras of handouts. One handout was titled “Standards for Mathematical Practice" which consisted of the eight standards that relate to how math should be taught, per the Common Core Math Standards, which have been adopted in 46 states.  I picked it up, and realized it had been handed out to students as a list of things for them to do in the course of “doing math”. 

With the handout in my hand, I was suddenly surprised by the sound of my own voice. "I really hate to see stuff like this,” I said.  The students looked up.  “It means nothing to you, I know, but it comes from something called Common Core which will go into effect next year. It is going to result in math being taught in certain ways.  In fact, it’s where your teacher is today, at a conference on how to teach the Common Core standards, learning the new ways she has to teach you math.”  Fortunately, the piqued interest of the students had now turned into blank stares.  Their years of acquired expertise had taught them how to tune out teacher soliloquies. 

The Standards of Mathematical Practice that got me so excited are the following:
1. Make sense of problem solving and persevere in solving them 
2. Reason abstractly and quantitatively 
3. Construct viable arguments and critique the reasoning of others 
4. Model with mathematics 5. Use appropriate tools strategically 
6. Attend to precision 
7. Look for and make use of structure 
8. Look for and express regularity in repeated reasoning
Taken at face value, they are as seemingly benign as when the company you’re working for says that the latest reorganization will not change anything and your life will continue as it always has. “Make sense of problem solving and persevere in solving them” seems like good advice, and the way I interpret it is to help students tackle problems by giving them the tools, instruction, guidance and practice to do so.  One of the many ways it is being interpreted, however, is to require students to find multiple ways to solve problems.  (I say this because I noticed in another classroom a poster bearing the logo of “Common Core” at the top, showing multiple ways of doing particular types of algebra problems.)

There’s nothing wrong with finding multiple ways of solving problems.  But in early grades, students find it more than a little frustrating to be told to find three ways of adding 17 + 69.   Putting students in the position of not satisfying the teacher by producing a correct answer and showing how they got it unless they find multiple ways of doing it is a recipe for 1) disaster and 2) rote learning, the bugaboo of the purveyors of “find more than one way to solve it”.

Now it’s true that during my recent assignment, I went over a proof of a problem they had in their homework and, immediately after going through the proof the teacher had provided, asked “There’s another way to do this proof; can anyone see how?” I’m in a habit of doing that in high school math classes, and I suppose that’s good, because if it comes to the point that I actually teach full time and a strict principal decides to make sure I’m following the Standards for Math Practice, he or she would likely conclude that I’m doing it. 

What I will not be doing is requiring students to do things in multiple ways. I’m a bit stubborn on this point, but I tend to believe if a student can do a proof, or solve a problem and do so correctly by applying prior knowledge, then he or she doesn’t also have to do twenty five fingertip pushups.

All in all, teaching high school is a bit more straightforward than the lower grades.  By that, I mean that the span of content to be covered is such that there isn’t room for discovery- and inquiry-based group work and “read my mind—what answer am I looking for” type of catechisms. I hope that Common Core doesn’t bring that about.  If it does, you can look for me along with flu-stricken students in the nurse’s office, trying to keep the nausea at bay and figuring out what my career options might be.

Saturday, February 2, 2013

Holding on to your kids: problems with child-centered "sociability"

I've just finished reading Hold On To Your Kids: Why Parents Need to Matter More than Peers, with no time left today to digest it. Here, instead, are some memorable passages that I want to return to later on:

In response to the intensifying cruelty of children to one another, schools all over this continent are rushing to design programs to inculcate social responsibility in youngsters. We are barking up the wrong tree when we try to make children responsible for other children. In my view it is completely unrealistic to believe we can in this way eradicate peer exclusion and rejection and insulting communication. We should, instead, be working to take the sting out of such natural manifestations of immaturity by reestablishing the power of adults to protect children from themselves and from one another. (p. 103)

Given that peer orientation is devastating our educational system, one would think that we would be alarmed, seeking ways to reverse the trend or at least slow it down. On the contrary, we as educators and parents are actually aiding and abetting this phenomenon. Our "enlightened" child-centered approach to education has us studying children and confusing what is with what should be, their desires with their needs. A dangerous educational myth has arisen that children learn best from their peers. They do, partially because peers are easier to emulate than adults but mostly because children have become so peer oriented. What they learn, however, is not the value of thinking, the importance of individuality, the mysteries of nature, the secrets of science... Nor do they learn what is distinctly human, how to become humane, why we have laws, or what it means to be noble. What children learn from their peers is how to talk like their peers, walk like their peers, dress like their peers, act like their peers, look like their peers. In short, what they learn is how to conform and imitate. (p. 174).

Compared with adult oriented kids, peer oriented children come across as less needy and more mature... [A]tleast initially, peer-oriented children also tend to be more schoolable. The cost of that mistaken impression [is] the loss of teachability. (p. 236)

We usually think of shyness as a negative quality, something we would want children to overcome. Yet developmentally, even this apparent handicap has a useful function.  Shyness is an attachment force, designed to shut the child down socially, discouraging many interactions with those outside her nexus of safe connections.
Adult-oriented children are much slower to lose their shyness around their peers. What should eventually temper this shyness is not peer orientation, but the psychological maturity that engenders a strong sense of self... (pp. 238-9)

One of the largest studies ever done on [the subject of daycare and sociability] followed more than a thousand children from birth to kindergarten The more time a child had spent in day care, the more likely she was to manifest aggression and disobedience, both at home and in kindergarten. (p. 239)

The belief is that socializing--children spending time with one another--begets socialization: the capacity for skillful and mature relating to other human beings. There is no evidence to support such an assumption, despite its popularity. If socializing with peers led to getting along and to becoming responsible members of society, the more time a child spent with her peers, the better the relating would tend to be. In actual fact, the more children spend time with ane another, the less likely they are to get along and the less likely they are to fit into civil society. (p. 241-242)

Peer interaction is routinely prescribed for yet another purpose: to take the rough edges off children who may be a bit too eccentric for our liking. We seem to have an obsession in North America with being "normal" and fitting in. Perhaps we as adults have become so peer-oriented ourselves that instead of seeking to express our own true individuality, we take our cue for how to be and how to act from one another...
The more a child depends on accepting adults, the more room there is for uniqueness and individuality to unfold and the greater the insulation against the intolerance of peers. (pp. 248-249)

Another pervasive--and pernicious--myth is that peer interactions enhance a child's self-esteem... The ultimate issue in self-esteem is not how good one feels about oneself, but the independence of self-evaluations from the judgments of others.  (p. 249)

But don't children need to play with one another? We have to see the difference here between what children want and what they need. The play that children need for healthy development is emergent play, not social play. Emergent play (or creative solitute) does not involve interacting with others... If playmates are involved, they stem from the child's imagination... The parent is always the best bet for this kind of play, serving as an attachment anchor--although even the parent must not overdo it, lest the emergent play deteriorate into social play, which is far less beneficial. Children are not able to serve the function of an attachment anchor with one another, so their emergent play is almost always preempted by social interaction. Because of the strong emphasis on peer socialization, emergent play--play arising from the child's creativity, imagination, and curiosity about the world--has become endangered. (p. 252)

In the mean time, I welcome your reactions to these!