Math insanity in Washington state

I've taught calculus at the college level and, now that my daughter is taking algebra in middle school, I see why students have so much trouble with college math: they aren't taught math in middle school. Let me clarify this statement by first illustrating what I mean when I say that too many students have trouble with calculus and why you should care about that. Calculus is generally considered to be the first college math course. It's nice if students have taken calculus already, and it's even possible that their high school calculus class was good enough to allow them to skip the first quarter of college calculus, but it's not absolutely necessary. Students generally take calculus from the beginning in college, they get credit for it, and degree curricula are designed with the assumption that they'll take calculus. On the other hand, few pre-calculus or earlier classes count for college credit -- they are offered on a supportive basis to make up for background that students should have already had.

Moreover, calculus has become a gatekeeper course for many degrees in the sciences, engineering, and even business. For many degrees, calculus is an absolute prerequisite for later core classes. In other degree programs, calculus may only be a prerequisite for certain electives, but the faculty have made the decision that they don't want students moving through the program who are unable to take these electives. So, a student unable to succeed in calculus (notice I didn't write "pass"; students need to become facile enough with the subject matter to apply it naturally and without effort in classes that build upon calculus) will not become a science, engineering, or business major. I cannot emphasize this point enough: if a student has not been prepared to succeed in calculus by high school graduation time, it is much less likely that he or she will be able to have some of the most rewarding careers.

So, why are students having trouble in calculus? While there are many reasons, the biggest is a lack of facility with algebra, and by this I mean the ability to solve algebraic equations by symbolic manipulation. A simple example of this might be to solve the equation 3x + 12 = 45. A symbolic approach would be:

3x = 33     (subtract 12 from both sides)
x = 11      (divide both sides by 3)

I'll explain why this is so important for calculus in a moment; let me first describe what children are taught instead.

Basically, they're taught something more like math appreciation than math. Let's use as an example my daughter's textbook, which is Contemporary Mathematics in Context: A Unified Approach, part of the "core-plus mathematics project" from McGraw-Hill. First off, much of this book is statistics, rather than algebra, even though this is supposed to be an algebra I (eighth grade honors, or ninth grade regular) class. The kids spend enormous amounts of time in groups, measuring each other's fingers, counting which thumb is on top, etc., so that they can gather some of their own data. They spend even more time writing paragraphs describing histograms. Then they move on to linear equations, motivated by linear fits to data -- linear regression -- rather than just algebra. Part of the reason for this is that they have yet to see an equation in the book!

Finally, they get to linear equations, though first through a little detour in which the words "NOW" and "NEXT" are used, instead of letters, for variables. Why they do this is mystifying, since all of these kids have seen problems in elementary school in which shapes were used instead of variables. Along the way, they learn about slopes, even of slopes of nonlinear functions, which is a calculus problem. Of course, they don't need to actually do any math in that last case (thought they're given the quadratic equation involved), just write about what the slope of a curved graph might mean, how rate of change relates to slope, etc. By this point, they've note seen more than three equations on any page of the book, and most pages have none at all. No techniques have been presented for solving linear problems; students are asked to solve such problems by inspection of graphs or tables of (x, y) pairs. Finally (precisely 2/3 of the way through the book), a symbolic approach is mentioned (in a chapter entitled "Quick Solutions"). Here's what the book says:

...it is often possible to solve problems that involve linear equations without the use of tables and graphs.

This is absolutely untrue! It is always possible to solve such problems this way (i.e., symbolically). In fact, the fraction of problems that can be solved with tables or graphs -- the only method used so far in the book -- is vanishingly small.

Let me disabuse you of the idea that the book will now introduce methods such as adding/multiplying both sides of the equation by the same value. Instead, what the book does is say that different people might reason about solving such equations different ways. A couple examples are given, and student groups are asked to try to figure out what the reasoning process is in each case. Then, kids are invited to solve equations in whatever way makes sense to them. That's it!

So, to sum up, symbolic manipulation is erroneously introduced as a specialized shortcut that can work in some cases: a sort of mental trick with no fixed methods that you just have to figure out for yourself. This is in a five-page section, after more than 200 pages that are taught with graphs, tables, and calculators in the context of statistics.

So, why is this a problem? Because calculus requires facility with the symbol manipulation approach. Unlike simple algebraic problems, which can (again, in simple cases with integer solutions or the like) be approached graphically, calculus is the mathematics of change. Graphical aids in calculus involve computer programs in which you interactively move things around and watch how solutions change. While solutions are numbers in algebra, they are equations in calculus. If the symbol manipulations of algebra aren't as easy as breathing for a student, he or she will have a tough time in calculus. I cannot see how a student who has used the Core-Plus curriculum can pass calculus, unless his or her math education has been supplemented outside of school. Thus, in Washington state we are preventing most of our children from getting the degrees that lead to the highest-paying, fastest growing career paths that exist: science, technology, and business.

And who is to blame for this mess? Why, the Superintendent of Public Instruction, Terry Bergeson. She's the one who has pressed this kind of instruction for a decade now. She's the one who has instituted a special test for Washington state that prevents our children's performance from being compared to children in other states and countries -- a test that costs much more than commonly-used alternatives.

That's why I'm advocating in this post for Randy Dorn for Superintendent of Public Instruction. If you're a Washington state voter, click on the title of this post to go to his web site. Read about him. Compare what he plans to do to the current abysmal state of education. It's time for a change in this Washington, too.

Actual sexism

I recently received an email for fundraising from a friend's child's school. It includes a link to a web site that's selling magazines. The usual deal is that the school gets some cut of the subscription price and the kids get crappy "prizes" from Oriental Trading. So, I clicked on the "Children" section link and was almost immediately struck by two magazines offered:

• Boys' Life, with a cover on future manned missions to Mars, and
• Discovery Girls, with a cover touting stories like, "Jealousy ruined my friendship" and "Mix & match fashion".

Is it any wonder that so few girls make it through middle school with their natural interest in science and mathematics intact?

Math & science: it's good for the economy

An interesting article in that well-known socialist rag, The Wall Street Journal. Math and science do matter, not just for the individuals involved, but for the US economy as a whole. And, despite decades-ago calls to improve science and math education, the US still lags other industrialized nations.

Needing "school math" without using it

I was thinking about my previous posts about the UW College of Education's (CoE's) recent political polemic about so-called reform math. One of their major points is that engineers don't use what they call "school math": they just use computers. Please allow me to outline my own work, which is highly compute intensive and rarely involves what they would call school math, but which nevertheless I could not do without a healthy dose of school math -- not just in my education, but also in my work.

My research is in the area of computational neuroscience, in which I build mathematical models of individual nerve cells (neuron)s and groups of neurons, develop simulation software for single computers and clusters of computers, and analyze data from simulations and also from experiments on actual living tissue. This sort of work is very much like that done by anyone simulating physical systems, be they biological, chemical, mechanical, electrical, etc.

Like the engineers described in the CoE's publication, my work is heavily computational, as it isn't feasible to do this work with pencil and paper, as in "school math". The basics of the mathematical models involve a number of differential equations: equations that describe how some part of the system changes in response to other parts of the system. Now, it turns out that differential equations is covered by a pretty much standard college sophomore mathematics course. So, why isn't the stuff I do "school math"? It's simple:

1. We only cover the mast basic type of differential equations in that class, linear equations. These are actually quite good for describing simple systems: electrical circuits made up of resistors and capacitors, mechanical systems with springs, etc. The advantage of these equations is that we can solve them on paper and they're easy to learn. The disadvantage is that they aren't very good descriptions of complicated systems like neurons (and many other, nonlinear systems). Once we move to nonlinear systems, we almost certainly need to use computers to do numerical simulations.
2. We mostly just solve single equations in that class (there are other classes where we learn to solve groups of differential equations, later on in the curriculum). The systems I'm interested in can have hundreds or thousands of differential equations, and so I have no choice but use computer simulation.

If you were to watch me work, you would see the following (between the long periods of time in meetings, in class, preparing for said things): I decide on a question I'd like to answer, such as how the behavior of a network of neurons changes as some parameter (think: "tuning knob") is changed. I set up the parameters for a simulation or maybe bunch of simulations and, anytime from a few minutes to a few days later, I have some results. I load those results into MATLAB (numerical mathematics software) and plot the results. I then either exclaim, "Wow!", and hurriedly start writing a summary and thinking of what else I need to do to finish telling the story for a publication (rare), or I say, "Nuts!" and think again why the system either displays uninteresting behavior (Who knows; maybe its lack of interest is in itself something noteworthy? Or is that just wishful thinking?) or doesn't behave like the living nervous system. So, the observer sees that I don't "do" "school math". End of story?

Well, not quite. Because the observer doesn't see what's going on "behind the scenes" (i.e., in my mind). First of all, I would have no hope of even being able to start understanding what simulations I need to run without a very firm and extensive "school math" background. For instance, I work with a number of bright undergraduate students in my research. Some of them have math backgrounds that include differential equations and beyond and some don't. This has nothing to do with how smart they are; math beyond calculus isn't required for computer science and so only those students who come to us via a "nonstandard" pathway (e.g., changed major, previous degree/career) will have the more advanced math. Though all of these students can help out in my research, only those with more advanced "school math" are able to understand the underlying mathematical model well enough to mess with that aspect of the project (unless I teach a student some of the required "school math"). After all, unless you want to resort to randomly poking something just to see what it does, you really have to understand what's going on inside it; that's the only way you can intelligently select what kind of "poking" is likely to tell you something interesting. In fact, it's the only way you can begin to ask questions about the system, let alone start formulating experiments to answer those questions.

Even after the simulations are over, I still need to interpret the results, and this requires yet more "school math" running around in my head. What kind of result did I get? What relationship does it have to previous results I've gotten, or for that matter, results others may have gotten? What does this result mean in the overall context of the system in question and the thinks I'd like to know about it/do with it? And so on.

In other words, it is emphatically not the case that the computer has relieved me of the need to know math. All the computer has done is take over the grunt work: it has become an additional tool in my mathematical arsenal. But the computer can't think, and that thinking is where all the "school math" is. It just isn't apparent to the observer because I know it well enough that it happens in my brain automatically. This is no different than the automaticity with grammar that we use in everyday life. Just because we don't carefully label each of our utterances with "subject", "verb", "object" doesn't mean that grammar isn't necessary.

Finally, does this apply to "everyday" engineers, or just people doing research? Of course it applies to engineers (at least those who haven't "moved up" to management)! That's why businesses hire engineers: they need people who can think about solutions to problems and have the depth of background to understand the interrelationships among parts of solutions from "first principles" on up to final product. Some tasks may become routine and thus almost automatic or thoughtless, but its important to have someone who can look at a problem (or a solution proposed by some software) and say, "Wait a minute; something's fishy here." And, in the final analysis, that's the most important contribution of "school math": it is the language of creativity.

More on "modern" math

I'm continuing my reading of the UW College of Education's little treatise on mathematics education. The author(s) are writing about multiple-choice testing on page 25, such as many standardized tests:

Taylor’s research shows that, although males tend to perform well on multiple-choice questions, females do not. The test questions that are most effective for non-Asian minorities and females are conceptual math problems... show how they arrived at their solutions.

These “performance-based” questions offer partial credit for partial understanding...

“The kind of algorithmic math traditionally taught in middle and high school might make sense — with no further explanation — to future theoretical mathematicians, but it seems a fairly elitist thing to push algorithmic math as mathematics instruction for all students.”

“Most kids are not going to become mathematicians, but they are still going to need to use mathematical ideas. What happens is that the largely abstract mathematics instruction becomes a turn-off for many students so they drop out of mathematics. We are one of the very few nations in the world where it’s acceptable to say, ‘I don’t do math.’”

Allow me to deconstruct:

• Again, another false dichotomy. The type of tests that one uses is mostly independent of the way one teaches, and one could use either "show your work" or "choose the correct answer" testing (or even other testing approaches) with either "traditional" or "modern" math.
• Just another straw-man argument meant to equate "traditional" math with what the author(s) consider poor math practices, whether or not such a connection actually exists.
• I'm not sure what to make about the statement that women and minorities don't do as well on multiple-choice tests. Even assuming this is true, this is not necessarily an argument for modifying testing (assuming there are good reasons for such tests). Instead, it would seem to me to be motivation for getting at the underlying reasons and addressing those. And what are these students supposed to do when they hit a point in their education that requires them to take a standardized, multiple-choice test? Bitch and moan about how unfair things are? Well, yes, the world is unfair. Complaining rarely helps. This is a recipe for setting these children up for failure later in life.
• Algorithmic thinking is just for theoretical mathematicians?!? Algorithmic math is elitist?!? What about chefs; they write and follow algorithms all the time. Are they elite mathematicians? The process of computation (which is what we're talking about at the lower grades) is algorithmic; there is no other kind of computation. This is simply incomprehensible, and smacks of someone who doesn't understand what an algorithm is (yes, this is quite a nerve to hit for me as a computer scientist).
• Very few students will become mathematicians? Strictly speaking, this is true; allow me to neglect to discuss what fraction may actually need math beyond those who become mathematicians. The problem is, which ones will go on to need the math? Not so easy to answer. What if the alternative approach to instruction rules out mathematically intensive careers for a good chunk of students? I submit that that's what has been happening: the absolutely brightest students, with parents who have the resources to help them go beyond "modern" math instruction, will do OK, students who will actually never need math may not be harmed one way or another, and a big group in the middle who would struggle under "traditional" math but gain sufficient mastery to continue onward in their studies will be shut out of a wide range of careers.
• The overall tone of the quote implies a mind-set (like much of this document) that "math is hard", "math is irrelevant to everyday life", "math is for mutant theoretical mathematicians", "math is elitist". A wise colleague of mine once said, "Mathematics is a social justice issue." I certainly agree. We need to stop treating math, including algorithms, like something so complex that only Star Trek-like disembodied brains can understand and start treating it as a common human birthright and the only truly international and intercultural language.
• I especially like the last sentence in the quote. After giving a bunch of reasons why algorithms aren't central to mathematics, why certain groups of children need alternative approaches to testing, and why most children won't need "abstract" or "algorithmic" mathematics ("school math" earlier in the brochure), Taylor bemoans the acceptability of people saying that they don't do math! Wait a minute... OK, I've banged my head against the wall, and that still seems like a contradiction of the thesis of the earlier material ("certain types of math are too hard for most children").

I hope that this little brochure isn't indicative of the overall level of scholarship at the UW College of Education...

Even engineers apparently don't need math

From the UW College of Education comes the very attractive publication linked from the title above. It addresses the "math wars" between "traditionalists" and advocates of "modern math education". Of course, like many political debates, they use titles such as those to pre-dispose their readers to see things their way. This is a sign right off the bat that this is not a scholarly work, but a political argument. Other signs are straw-man examples that are supposed to show how "traditional" math is misguided, such as this (p. 9):

A toy is hidden in one of two cakes. One cake is a circle, cut into fourths. The other is a rectangle, cut into sixths. Students must choose the cake that gives them the best chance of finding the toy.

Some choose the rectangle. Why? Because “most toys come in square boxes.”

Of course this is a poor question for students who would answer that way, because it is varying two things at a time (cake shape -- and therefore slice shape -- and fractions of total area). Presumably, this question is trying to get at more abstract thinking; that the shape of the cake and its slices doesn't matter, all that matters is fraction of total area. But the publication doesn't say anything about this, all it does is use this as a straw man to set up the argument that we shouldn't tell students how to do things (like optimal methods for mathematical calculations). All methods are equally valid:

One student may add 28 + 34 with traditional column carryover. Another adds 2 to 28 and subtracts 2 from 34 before adding the two results. A third student adds 8 and 4 to make 12, then 12 and 30 to make 42, and 20 more to make 62. In an effective classroom, all those solutions are studied, the links between them established, and the connection made to larger mathematical concepts (such as place value, the properties of addition, and developing generalized strategies).

This sounds very nice until you consider, "How did these students all arrive at different methods for addition?" The answer is that they weren't taught how to add; they were expected to "discover" it themselves. Go read a history of mathematics book sometime and consider how long humanity has worked to discover what we know about mathematics; how many geniuses have been involved. Does it make sense to systematically (not as an occasional teaching device) expect children to re-create any fraction of this? And is the only way to teach about place value, etc. to compare multiple methods?

Oh, and the implication is that "traditionalists" teach by giving out problems and just marking them right or wrong and "modernists" look at student mistakes and seek to understand why they make them. Nice false dichotomy.

I find this anecdote on page 13 especially interesting. One of the UW Education faculty has spent time observing engineers, scientists, and architects working, and here are his conclusions:

The architects, he discovered, worked problems out with visuals, not textbook algorithms. Engineers use mathematics, but much of that is embedded in their computational tools, and they too use forms of quantitative reasoning that looked very different from the activities of school math. It turned out that school math was a fairly rare species of activity outside of school.

“If you spend a month with architects, you’ll never once see them write an equation,” says Stevens.

The story was the same when he studied roadway engineers. “All the calculations were done on the computer,” says Stevens.

As the brochure continues, the distinction is between "school math" and the math that people actually use in the real world. Well, except for mathematicians, who are like poets, viewing the world in a different way than most people. Apparently, engineers don't need to know math; it's already in the computers (how it got there is unanswered). It's unfortunate that engineering schools and the accreditation folks require math through differential equations, multivariate calculus, etc. They must not know what engineers do as well as UW Education folks.

Letter to the editor

Follow the link above to my letter published in The Seattle Times. It's in response to the Washington State Superintendent of Public Instruction trying desperately to keep WA using a failed math curriculum.

Math wars

The title links to a Seattle Times article about the conflict between advocates of "reformed math" education and those of us who think our children need to develop solid computation competency (yes, I have a clear bias). Here's one teacher's summary of "reformed math":

"It makes higher math more accessible to them," said Zandria Hopper, a fifth-grade teacher at Elizabeth Blackwell Elementary School in Sammamish. "They are pressed to justify and reason from kindergarten on."

However, check the article's sidebar, which compares two fifth-grade problems -- one "reformed math" and one traditional. The reformed math problem is 14x9=? The traditional math problem is 492x98=? Which students do you think will get into top-tier universities?

This is math?

Follow the link to a Kirkland Courier Reporter article on a math class in a suburban school district located near where I live. In a typical "reform math" class, fifth graders spend time inventing their own ways to solve the "problem of the day": 55 x 20. Their teacher waxes poetic about the superiority of this approach over "memorization of formulas". No mention of the fact that solving 55 x 20 is not important enough to spend that much time on. And that this time might be better spent on inherently more interesting things, like why multiplication works, the structure of number systems, etc. And that this is, at best, a fourth grade math problem.

Meanwhile, the students either lose the possibility of future careers in science or engineering, or their parents tutor them, or their parents spend money on math tutors for them (a booming business in Washington state). And then their parents join Where's the Math? to pressure school district officials to inject sanity and rigor back into K-12 math education.

Math Education: A University View

Following up on a previous post, there's a new video about "reformed math" by Clifford Mass, a UW Seattle professor of Atmospheric Science. He doesn't say anything that most other university faculty would say: our K-12 math education system is failing our students, as they discover when they reach college. Unfortunately, that's a bit late.

Topics: mathematics,education.

My Mathematics Genealogy

Follow the link above; unfortunately, my advisors and myself are not close enough to being mathematicians to produce a very full genealogy.

Math Education: An Inconvenient Truth

A little over two years ago, in a posting comparing schools here in the Seattle suburbs with those in the Gainesville, Florida area, I wrote, "... Northshore uses the Everyday Math textbooks, while Alachua uses Harcourt Math. Both series are, in our opinion, excellent." That was from my wife's and my points of view as parents of a second-grader, with whom we work outside of school on supplementary math and other subjects. As the YouTube video linked from the subject line above indicates, our assessment probably isn't reflective of the full K-8 experience of most families.

First off, go right now and watch the video; it's excellent. Seriously, watch it.

My opinion of Everyday Math was based on two of its aspects:

1. Because of its spiral approach, it introduces algebraic concepts very early and at a level that seemed appropriate for those children.
2. It has some very interesting ways of teaching concepts such a addition and subtraction that seem, to me, to make their relationship as inverse operations intuitively obvious. Again, a more advanced concept introduced in what seemed an age-appropriate manner.

It is important to keep in mind that this was coming from a parent of a child who was far ahead of her grade level in math (not to brag, but as a fourth-grader, she is doing seventh-grade math) and who did extra work at home. The YouTube video points out some matters that have become much clearer as time has passed and my children have worked with math textbooks in higher grades.

• There's not much practice in the textbook. This is offset by extra materials used by teachers in our school here (such as Mad Minute), and by our own use of the Singapore Math books.
• There's all sorts of irrelevant materials in the textbook. A chapter on patterns has a page on Native American crafts, but that page says nothing about the patterns in those crafts. It's basically a generic, too-brief overview of such crafts. Certainly, there is a strong connection between weaving and patterns, but nothing is said about that there. We fix this ourselves by talking about such issues.
• Exercises at the end of sections are a mish-mash of all sorts of problems. On the one hand, review is good. But why have the first problems in a section of a late elementary or early middle school textbook be a series of very simple addition or subtraction problems? It distracts the child from the topic just learned, and makes her wonder if maybe there isn't some sort of trick involved and those problems aren't as easy they seem. It could have made her less secure about her math ability! (Actually, by this point, our older daughter is quite cynical about these "baby problems" and isn't bothered by them.)
• It has all sorts of bizarre approaches, such as those detailed in the YouTube video (did I mention that you should really watch it?). Stuff which leaves me wondering if the authors were on peyote when they wrote that section. For our daughters, these are just random and interesting things, and they rightly dismiss them as inefficient and confusing. They ask their teachers if they have to do it that way, and do so for the few problems where it's required, reverting back the much more efficient, standard algorithms they are familiar with.

As a Computer Science professor, my observation is that I can see the effect of this approach to education. At the end of my third week teaching freshman calculus (I plan to blog about that shortly), I have noted that the de-emphasis of topics such as division of fractions impacts college-level math. Unfortunately, we're on the quarter system here, and so there's not much leeway to address these issues in class. However, I am making a mental note for myself to discuss this with our Quantitative Skills Center folks; maybe a review of K-12 textbook content would provide some ideas for remediation. In the final analysis, that is what these textbooks have done: pushed topics they think too time-consuming into university education time.

Additional Links: Reviews of UCSMP Everyday Mathematics, Where's the Math?, Donald Simanek's documents and links on education.

Topics: mathematics, education.