Casting Out Nines

BusinessWeek’s TechBeat blog has this article about the federal panel report on K-8 mathematics instruction that I blogged about here. It’s good to see this report getting attention in the blogosphere and MSM. It needs more. One thing from the BusinessWeek article that needs a slight bit of correction, though — it says:

The sad thing about the report that despite the unanimity on a panel that represents a broad spectrum of the mathematics and math education communities, it will take a decade or more for its recommendations to be implemented. It simply takes that long for curriculum guidelines to be recast, textbooks to be rewritten, and teachers to be trained or retrained. And in that time, a lot more damage can be done.

That may be true of traditional public schools, where red tape and opposing political forces must be overcome at every turn, but it does not have to be true of private or charter schools where reaction times to changing pedagogical climates can be much faster. I think this report, and the dire consequences of ignoring it, create a prime situation for charter schools and private schools to lead the way to better education for our kids. If governments would open up schools to market forces to a greater degree, we might even get the traditional public schools on board once parents choose to send their kids to places that are getting on the ball faster.

A federal panel examining K-8 mathematics education in the USA has made some forthright recommendations, according to this article in the NYT today. Unlike many federal panels, this one has an uncommon amount of common sense in its conclusions. For example, this finding that is striking in the way it refrains from choosing sides in the math wars:

Parents and teachers in school districts across the country have fought passionately over the relative merits of traditional, or teacher-directed, instruction, in which students are told how to solve problems and then are drilled on them, as opposed to reform or child-centered instruction, which emphasizes student exploration and conceptual understanding. The panel said both methods have a role.

“There is no basis in research for favoring teacher-based or student-centered instruction,” said Dr. Larry R. Faulkner, the chairman of the panel, at a briefing for reporters on Wednesday. “People may retain their strongly held philosophical inclinations, but the research does not show that either is better than the other.” [...]

“To prepare students for algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency and problem-solving skills,” the report said. “Debates regarding the relative importance of these aspects of mathematical knowledge are misguided. These capabilities are mutually supportive.”

Say what? An appeal to actual research rather than anecdotes and personal biases when thinking about effective math teaching? Amazing. And this shocking discovery:

[T]he panel found that it is important for students to master their basic math facts by heart.

“For all content areas, practice allows students to achieve automaticity of basic skills — the fast, accurate, and effortless processing of content information — which frees up working memory for more complex aspects of problem solving,” the report said.

Dr. Faulkner, a former president of the University of Texas at Austin, said the panel “buys the notion from cognitive science that kids have to know the facts.”

“In the language of cognitive science, working memory needs to be predominately dedicated to new material in order to have a learning progression, and previously addressed material needs to be in long-term memory,” he said.

Why, it’s almost as if they think that mastery precedes creativity or something. And finally:

The report makes a plea for shorter and more accurate math textbooks. Given the shortage of elementary teachers with a solid grounding in math, the report recommends further research on the use of math specialists to teach several different elementary grades, as is done in many top-performing nations.

The article goes on to give some of the panel’s recommended benchmarks for mathematical skills in grades 3-7. There’s also a link to the panel’s report.

All I can say is that I hope math educators, prospective teachers (especially prospective elementary school teachers), curriculum designers, ed schools, school boards, and everybody else who is a stakeholder with some influence in this process are listening. We’ve got 2 years until our oldest starts kindergarten and she needs teachers and curricula who get math right.

[h/t God Plays Dice]

Following up on his three posts on classical education yesterday, Gene Veith weighs in on mathematics instruction: 

I admit that classical education may be lagging in the math department. The new classical schools are doing little with the Quadrivium, the other four liberal arts (arithmetic, geometry, astronomy, and music). The Trivium, which is being implemented to great effect (grammar, logic, and rhetoric), has to do with mastering language and what you can do with it. The Quadrivium has to do with mathematics (yes, even in the way music was taught).

This, I think, is the new frontier for classical educators. Yes, there is Saxon math, but it seems traditional (which is better than the contemporary), rather than classical, as such.

Prof. Veith ends with a call for ideas about how mathematics instruction would look like in a classical education setting. I left this comment:

I think a “classical” approach to teaching math would, going along with the spirit of the other classical education posts yesterday, teach the hypostatic union of content and process — the facts and the methods, yes (and without cutesy gimmicks), but also the processes of logical deduction, analytic problem-solving heuristics, and argumentation. Geometry is a very good place to start and an essential to include in any such approach. But I’d also throw in more esoteric topics as number theory and discrete math (counting and graph theory) — in whatever dosage and level is age-appropriate.

At the university level, and maybe at the high school level for kids with a good basic arithmetic background, I’d love to be able to use the book “Essential College Mathematics” by Zwier and Nyhoff as a standard one-year course in mathematics (and in place of the usual year of calculus most such students take). It’s out of print, but the chapters are on sets; cardinal numbers; the integers; logic; axiomatic systems and the mathematical method; groups; rational numbers, real numbers, and fields; analytic geometry of the line and plane; and finally functions, derivatives, and applications. You have to see how the text is written to see why it does a good job with both content and process.

(I took out the mini-rant against the gosh-awful Saxon method.)

Any thoughts from the audience here?

One of my linear algebra students is an education major doing student teaching. Today he showed me this method of simplifying radicals which he learned from his supervising teacher. Apparently it’s called the “Illini method”. Googling this term returns nothing math-related, so I think that term was probably invented by his supervisor, who went to college in Illinois.

The procedure goes as follows. Start with a radical to simplify, say . Look under the radical and find a prime that divides it, say 5. Then form a two-column array with the original radical in the top-left, the divisor prime in the adjacent row in the right column, and the result you get from dividing the radicand by that prime number in the left column below the radical. In this case, it’s:

Now look for a prime that divides the lower-left term, say another 5. Again, put the dividing prime across from the dividend, and the quotient below the dividend. With our example, the array at this stage looks like:

In general, continue this process of dividing prime numbers into the lower-left entry in the array, writing the prime across from that entry, and writing the quotient beneath that entry, until you end up with a 1 in the lower-left entry. So the final state of our example would be:

Now, look at the left-hand column of the array. Group off any pairs of numbers you see. Multiply together all numbers which are representative of a pair. In our case, there is only one such pair, a pair of 5’s. Any numbers that occur singly are placed under a radical and multiplied. In our case, that’s the single 2. Then multiply the product of numbers which are in pairs times the radical which contains the singleton numbers. So we end up in our example with .

Here’s another example with a larger number, :

There are three groups of 2’s, so outside the final radical we’ll put . And the 3 and 11 are by themselves, so under the radical we put 33. Hence .

Pretty clearly, all this method is doing is presenting a different way to do the bookkeeping for doing the prime factorization of the number under the radical. The final step of grouping off the prime pairs and leaving the un-paired primes under the radical is analogous to finding all the squared primes in the prime factorization.

This method is nice and systematic, and I can see why students (and student-teachers) might like it. But it seems to be obscuring some important concepts that students ought to know. With the method of factoring, looking for squared primes, and then removing them from the square root, at least you are dealing directly with the inverse relationship between squares and square roots. The Illini method, on the other hand, uses an approach of “put this here and then put that over there” with minimal contact with actual math. It does work, and it does keep things in order. But do students really understand why it works?

Your thoughts?  What does this method make clearer, and what does it obscure? Should high school algebra teachers be teaching it?

Here’s a problem I have with the way most calculus textbooks are written, and therefore by default the way most calculus courses end up being taught. Tell me if I am crazy or missing something.

We teach calculus from a depth-first viewpoint. That means that whenever we encounter a concept, we go as deeply as possible in that concept before moving on to the next one. There are some subjects where this makes sense, but in calculus we have a small number of main ideas that are made out of several concepts, and if we stop to attain maximal depth on every single thing, there’s a good chance that we never arrive at the main idea with any degree of understanding.

The big ideas of calculus — the rate of change (derivative) and accumulated change (integral) — are actually really simple if you consider them simply for what they are and what they were invented to do. Derivatives, for instance: You have a function, and it is changing in all kinds of ill-behaved ways. The object is to find out exactly how quickly it is changing at a given point. We quantify that rate of change by sticking a tangent line on the graph of the function at that point and measuring its slope. Really, that’s it. Slopes of lines. The rest are technical details on how to calculate this slope with some degree of accuracy, and those details range from graphical estimation to interpolation tricks to algebraic techniques.

But in Stewart’s Calculus book, the coin of the realm of calculus texts, here’s what students have to study before the derivative is defined: an entire chapter of precalculus review (a mind-numbing section 1.1 on functions and notation, mathematical models, families of functions, exponential functions, inverse functions and logarithms), then a chapter on limits in which students have to master finding limits from graphs, calculating limits using the Limit Laws, the epsilon-delta definition of a limit (mostly untaught these days), continuity, and limits at infinity.

Then there’s a section on “Tangents, Velocities, and Other Rates of Change” followed by two sections on the Derivative.*

This approach plays directly in to the greatest weakness of the average calculus student, which is algebra/precalculus content mastery and the ability to master technical details of calculations and theory. How likely is it, for the student who struggles to read mathematics or use algebra correctly, that this student will be in any shape to learn what a derivative is, and what one is for, by the time they get there?

You want students to master those technical calculations and theory, of course. But you also want those to be mastered in context, not just as mathematical tricks to be learned as parlor games. The few students who survive the onslaught of detail mastery and are still psychologically around to learn what a derivative is, often find it extremely hard to know what f’(3) = 2 actually means. All they know is that you bring the power down and subtract one, and maybe the Product Rule.

I’d prefer some kind of approach to calculus that is not depth-first but more like breadth-first, where students get a good grounding in the overall ideas of calculus and do some basic work before mining into the really deep details. Not all students really need those deep details, after all.

* OK, there is a section (2.1) where the ideas of tangent lines and velocities are briefly introduced. And then summarily ignored until the end of that chapter. The students typically ignore that material right along with the book.

Here’s a promotional video for a new math curriculum from Pearson called enVisionMATH. (It must be a sign of the times that grade school math curricula have promotional videos.) Watch carefully.

Four questions about this:

1. Should it be a requirement of parenthood that you must remember enough 5th grade math to teach it halfway decently to your kids?
2. Does the smartboard come included with the textbooks?
3. Did anybody else have the overwhelming urge to yell “Bingo!” after about 2 minutes in?
4. When will textbook companies stop drawing the conclusion that because kids today like to play video games, talk on cell phones, and listen to MP3 players, that they are therefore learning in a fundamentally different way than anybody else in history?

The last question is all about the research-free digital nativist assumption that is the source of many lucrative curriculum deals these days. Data, please?

[ht Teaching College Math Technology Blog]

The Everyday Math curriculum has been rejected in the state of Texas. I’ve blogged about Everyday Math and how it attempts to teach multiplication before. But I didn’t know that it had activities like this:

A. If math were a color, it would be –, because –.

B. If it were a food, it would be –, because –.

C. If it were weather, it would be –, because –.

I’m not sure exactly what the point of an exercise like this is — perhaps the curriculum is just trying very studiously not to get too deep into mathematics itself, thereby teaching math without the social stigma of being very enthusiastic about it. Or maybe the idea is to get kids to see math from a different point of view, as a sort of oblique path through math anxiety.

Either way, it’s the wrong approach. The only way to come to terms with math, conquer math anxiety, and appreciate (and learn) the subject is… to get good at it. And that only comes by doing, lots and lots of doing. You replace practice with long division for this stuff, you’re not doing what you ought to be doing. To paraphrase what somebody said a couple of thousand years ago to a similarly math-disaffected person, there is no royal road to understanding arithmetic or algebra, no cutesy affective end-arounds to get out of the hard work of learning.

I think there could be an enormous market in coming years for “alternative” at-home math curricula to counteract the sloppy mess of “modern”, usually NSF-funded, math curricula — and those “alternatives” would look awfully similar to the math texts of the 50’s and early 60’s.

Editorial: This is installment #6 in this week’s retrospective series where I’m reposting some classic articles with updated comments. For me, there’s a deep interplay between pedagogy and culture that we don’t make nearly enough of. I’ve posted many articles trying to make the point that many of the problems in education, particularly math education, that we try to treat with curricula or technology or pedagogy are doomed to fail because they are not problems with teaching — they are problems with the culture. In this article, I make that point and go one step further, bringing in a theme from my Calvinist/Lutheran religious beliefs. 

The uncrossable line of math

Originally posted: April 16, 2006

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Darren at Right on the Left Coast has spot-on commentary on a spot-onpiece in the Seattle Times about math instruction. The part I want to highlight for now is this quote from the Times article:

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Editorial: This is installment #5 in retrospective week. When I announced retrospective week, I said that some of the articles I would be highlighting may not have gotten many comments but started larger conversations — and this is certainly one of them, although the conversation went totally to places I didn’t want it to go. 

Read the article for yourself, and you’ll see that it is a reflection on what makes a successful student in an upper-level math course, what education programs often cite as characteristics of successful teachers, how those two sets are often portrayed as mutually exclusive, and why math education majors have to work to possess both sets of characteristics as an integrated whole in order to become great teachers. 

But that’s not how a lot of readers took it. In particular, some of the education majors at my college read this article and took it to be a public put-down of their intelligence. There were even some people, for all I know not connected to my college at all, who anonymously sent messages to the dean and president of my college to complain about me and my negativity towards education majors. Without revealing details, I’ll just say that it all culminated in my quitting the blog for five weeks. 

Despite the controversy this article caused, and may cause again by re-posting it, I stand by my statements. Students have got to learn to be tough-minded, self-confident, and detail-oriented to succeed in upper level math courses; and they have to work at integrating this with being caring, compassionate people in order to be excellent teachers. Kids in school — my kids — deserve nothing less than excellent mathematicians who are excellent teachers. And for goodness’ sake, if you don’t agree with this, LEAVE A STINKING COMMENT rather than go running to the dean.  

Characteristics of upper-level math success

Originally posted: November 1, 2006

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I’ve been grading a wheelbarrow-load of papers from my upper-level geometry class this morning. It’s been making me think about the jump from taking calculus to courses beyond calculus. A lot of very good calculus students simply hit the wall when they move on to an “upper-level” course, like linear algebra or geometry. The jump is difficult, I think, because there are certain personality traits that have to be in place for a student to succeed past calculus:

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Midterms are coming up in a couple of weeks, and while most of the students in my precalculus class are doing reasonably well, some aren’t. Here are some questions I’ve struggled with every time I teach a freshman class, and maybe some of you out there have suggestions. If so, leave them in the comments.

1. How do you impress upon students (freshmen) the importance of coming to office hours? I don’t think I’ve had more than six distinct students visit office hours for help all semester long, and I’d consider this an active semester in terms of office hours. The rest go to the Math Study Center, study tables for football or fraternities, etc. but it does no evident good for a lot of them. I think it would do them good to come see me; but how to convince them of this?
2. How do you convince a student that their purpose for being here, their job, is to be a student? Some of the students don’t come to office hours because they haven’t touched the exercises all semester long, and that’s because they are involved in several different campus activities which are promoted in the name of “getting involved”. The cost to their time budget is unsustainable. How to get them to prioritize time properly?
3. How do you get students to transition from the typical high-school-math mode of “get the answer in the shortest possible time frame” to the college mode of “work hard over an extended period to really understand what you are doing”?

Thoughts?
[Photo by kodama.]