Mark Palko asks: What are the worst examples of curriculum dead wood?
Here’s the background:
One of the first things that hit me [Palko] when I started teaching high school math was how much material there was to cover. . . . The most annoying part, though, was the number of topics that could easily have been cut, thus giving the students the time to master the important skills and concepts.
The example that really stuck with me was synthetic division, a more concise but less intuitive way of performing polynomial long division. Both of these topics are pretty much useless in daily life but polynomial long division does, at least, give the student some insight into the relationship between polynomials and familiar base-ten numbers. Synthetic division has no such value; it’s just a faster but less interesting way of doing something you’ll never have to do.
I started asking hardcore math people — mathematicians, statisticians, physicists, rocket scientists — if they.’d ever used synthetic division. By an overwhelming margin, the answer I got was “what’s synthetic division?” Not only did they not need it; it made so little impression that they forgot ever learning it. . . .
Since we need to pare down the curriculum, what you choose to cut? Specifically, what mathematical topics that you learned in school can future generations do without?
I’m too distant from the high school curriculum to have much to offer. I think the math in the local elementary school goes a bit too slow, but of course I’d think that. There has been some progress over the decades, though. The teachers used to get angry at my sisters when they’d read ahead in the workbook, but by the time I came around, they’d let me curl up with a math book and leave me alone. As long as I kept my mouth shut they had no problem with me.
More recently, we spent a year on sabbatical in France (as regular blog readers will recall), and one thing I liked about the public schools there was the uniform national curriculum. It was completely clear what was being covered each year, you could go to a bookstore and get a book such as “L’Année de CP” which had exercises covering the whole year (“CP” is first grade), you could see what was coming next, etc. I liked this transparency. Maybe it wouldn’t work so well in the higher grades but I liked it for elementary school.
Does anyone have thoughts on Mark’s original question about the high school curriculum?
At least on the surface, it isn’t and apples-to-apples comparison. I think he’s sliced it to make this case. A one mile drive is a very “short” drunk drive home, while a one mile “drunk walk” isn’t. There’s a lot more opportunity for mistakes on the one mile walk, with far greater consequences and it probably takes over eight times as long.
My impression of grade school education is that there is sometimes a fight between “practical” mathematics and “college” mathematics. It’s a useful debate because a large number of students need to be able to compute tips and estimate bills but never need to know what a polynomial or derivative is. I’d highlight that during my education there was often a disagreement about where to draw the line between these two curricula, though. I remember tutoring other students in how to take derivatives of polynomials for which they had memorized the mechanics of it only and had absolutely no understanding of it’s connection to approximation, maximization, slope, velocity, areas under curves, integration, differential equations— without understanding the mathematical object at least a little, it’s an utter waste of time in the “college” curriculum and I can’t see how “practical” taking mechanical derivatives is.
I had the great fortune of taking a lot of experimental grade school classes which included things like cleverly designed board games which taught properties essential to abstract algebra in 4th grade. I am biased, but actually expect that methods like that—ostensibly not mathematical, but more fun and synonymous to the real comprehension needed—would be a successful way to promote “college” mathematics without alienating students strongly on the “practical” side too much. They might even be convertible.
I took a quick look at:
http://www.mathcurriculumcenter.org/PDFS/HSreport.pdf
I’m too far away from this to say much, but if had my druthers, some probability and statistics would be required.
I might be tempted to cut back on some of the geometry and trig.
Calculus is required for some, but probability and statistics is likely to be way more useful for most people, if only for defense against their mis-use.
In Singapore a few years ago, I sat in on a class of 17-year-olds doing problems on Poisson processes and they were doing OK.
Perhaps a list of the topics he’s supposed to cover. A lot of us are pretty far from HS which gives us the advantage of knowing what is useful through practice, but we can’t remember all that useless stuff thankfully. ;-)
What I remember about 6th grade through HS math was that it was terribly repetitive in a boring useless way, and that except for a very few teachers, the teachers generally had no idea why they were teaching what they were teaching, except perhaps teaching towards a test like the AP calculus test.
By far the thing students have the most trouble with in calculus classes and on first year Physics test and things is doing their algebra correctly. This suggests that more emphasis on basic algebraic problem solving would be valuable.
My opinion is that the biggest offender is plane geometry. While geometric intuition is useful, many of the theorems students memorize (in order to invoke their usage in proofs) occur in very specific situations: triangles that happen to the similar; lines that happen to be parallel; shapes that happen to be perfect circles, square, etc. The plane geometry curriculum used today is more-or-less the one set down by Euclid in his Elements, but vectors and trigonometry are more general tools for solving the same types of problem. The main purpose geometry serves in the modern curriculum is to teach logic and how to prove things rigorously. However, I believe the vast majority of high school teachers do not realize this because so often they are concerned about getting the theorems out that they forget to teach the metacognitive issues of why the material is important or how to use the material. Indeed, my friend is currently training to be a high school math/science teacher, and next semester he will be assisting a geometry teacher. The geometry class he describes as “geometry appreciation.” Students will learn about how to find geometry in nature, geometry in art, and experiment with various geometry computer programs; proofs feature very little in this curriculum.
I believe this class is an elective, not a replacement for the required geometry credit, and this underscores an important point: geometry intuition is useful, and it shows students that math is not just a series of equations. However, many of Euclid’s theorems are not useful in themselves. I could imagine them being excised from the curriculum, geometric intuition being folded with subjects like vectors, linear algebra, and trigonometry, and mathematical proofs and rigor being emphasized in many other parts of the curriculum. The underclassmen engineering students I teach at Berkeley flinch when we say the word “proof,” probably because they relate it to archaic geometry classes, and many don’t know how to do a proof by contradiction/contrapositive. However, I hear no complaints about boolean logic, which they learn in their computer science classes. The conclusion I draw from this is that logic and proofs can be taught, but geometry is not the way to do it, yes every student spends a whole year devoted to it.
The topic of rationalizing the denominator is absolutely deadwood. The only reason we ever rationalized the denominator is because our sliderules couldn’t handle the division by anything other than an integer.
Fastforward to the year 2011, and we still require our learners to rationalize the denominator under the guise of “number sense”. Pure bs. We do it because we have “always” done it.
I almost agree. My exception is rationalizing complex denominators. Of course, that can be covered along with the rest of the complex number material.
Reading ahead raises an interesting point. A little common-sense flexibility can make a tremendous difference. By letting kids who have mastered a topic read ahead or do some enrichment activity (math puzzles and games, geometric constructions, etc.) you can make school a much better experience for bright kids. One of my concerns about some movement reformers is the tendency to take a somewhat Procrustean approach.
Topics I’d like to see in high school mathematics courses.
1. How to do “Fermi estimation” problems. Doing approximate calculations with little or no data using dimensional analysis, approximation, and informed guessing. The classic Fermi estimation problem asks: How many piano tuners are there in Chicago? You can do this in your head and get remarkably close to the real number. This is the kind of thing one does on the job all the time in engineering. During a meeting you can come up with an approximate answer quicker than anyone else. No calculators necessary. Clarance “Kelly” Johnson head of the Lockheed Skunk Works was a master at doing the Fermi estimation in aeronautical engineering. Ben Rich, who took over after Johnson retired, tells a story where Johnson came up with an estimate for a thermodynamics problem “on the fly.” Rich spent a week (or something like that) getting a refined answer only to see that Johnson had come withing 10%. Students need to learn that exact answers are usually unnecessary. Often they are meaningless anyway because the data are not exact. Some economists have yet to learn this. Christine Romer (Obama’s first economic advisor) used a spending multiplier of 1.52 in the first stimulus package. But we don’t really know if the multiplier is greater than one, let alone to three significant figures.
2. Teach the sum of a finite and infinite geometric series. Most elementary finance calculations are essentially cash flow problems. You can do mortgages, bond analysis, net present value etc with the geometric series as the starting point.
3. Do a better job with complex numbers. Calling the square root of minus one “imaginary” leads a lot of students to regard complex numbers as something not as “real” as real numbers.
4. Minimize the use of hand calculators. I would delay their introduction at least to the Junior year. Start the kids off in elementary school with the abacus. Then the slide rule. Yes the slide rule where you have to keep track of the decimal point and exponent. I’m not sure I would have the students use computers at all until college.
If we have to copy another country’s math curriculum then choose Russia, not France.
Maybe we should take a cue from Apple and call complex numbers “iMath” ;)
I agree with James Fung about geometry. Worth covering, but not for very long.
yes to 2.
Generally yes to 1., though I think that shouldn’t be limited to math. Also, you’re wrong about Romer: If you’re actually looking at what Romer and Bernstein write, they emphasize “rule of thumb,” “considerable uncertainty” etc. all over the place. The reason they use those point estimates is partly simplicity (they’re using other people’s estimators so no one can accuse them of doctoring their own) and communication. I the report themselves they actually do provide a ballpark estimate for jobs created – 3.3- 4.1 million. (and the woman’s name is Christina). http://otrans.3cdn.net/45593e8ecbd339d074_l3m6bt1te.pdf
3. No. I don’t believe the psychology of this – I don’t think people would understand Pi better if it was called Pie either. And there are costs to using names that no one in actual math/engineering/physics uses.
4. That sounds cranky and old fashioned – “When I was young…”. Do we have any empirical evidence that calculators do damage to the ability to think mathematically? Without such evidence I believe it’s generally a bad idea to have kids – especially younger kids – do something in an unnecessarily complicated way.
Yes to 2, 3, and 4.
But not 1. Well, to be more precise, 1 is an extremely valuable life skill. But it ain’t mathematics. A large part of Fermi estimates requires intuitive guesses towards the order of magnitudes of certain key numbers. And that cannot possibly be developed in a classroom setting.
For example, if I were to ask, in the abstract, an alien who only knows that an elephant is “the largest extant land mammal on Earth” to give an estimate of the average mass of an elephant, the alien may reason starting from the knowledge that Earthbound mammals are carbon based life-forms with a calcium based skeletal system and mostly water. Then estimate the gravitational acceleration at the surface of the earth based on its size and the generic density estimates for the rocky inner planets of this solar system. Then using knowledge of the tensile strength of carbon derive an estimate on the maximum sustainable size of a land animal…. and he will end up somewhere in the region of one of the larger dinosaurs.
The problem of the current instructions in mathematics goes much deeper than just the choice of curricula. If you just impose Fermi estimates as one of the things that is taught, after one generation of standardized testing, you would end up with “Word problems” that supposedly tests the “Problem solving skills of the children” of the “Fermi estimate variety” that reads something like
“We know there are 250 thousand families in the Chicago area. A living wage is one that earns around 50 thousand dollars per year. Each piano tuning session takes about 1.5 hours, and skilled professionals are often paid 100 dollars per hour. With about 1 in five families owning a piano, estimate how many piano tuners live in Chicago.”
which, as you see, is totally besides the point.
Posted this at Mark’s blog but I’ll cross-post it here with additional content:
Of the six trigonometric functions, only three are necessary: sine, cosine, and tangent. Cotangent, Secant, and Cosecant are useless. (“But wait!” the conservatives will cry. “Without secant, how will we differentiate tangent?” Easy, the derivative of tangent is 1/cos^2.)
Here are two additional pieces of evidence that sec,cot,csc should be dropped: 1. they do not appear in calculators, 2. they do not appear in “sohcahtoa”. These aren’t rigorous litmus tests, but when the real world has seemingly decided to ditch something, we should at least *consider* removing it from the curriculum.
Sure, it’s not a big cut, but it’s a start.
Well, my C in real analysis changed me from a math major to an economics major, so I make a retroactively self-serving plea for real analysis. To those who note, probably correctly, that a C in real analysis is actually a fair assessment of my mathematics acumen, I note that a lot of mathematical training works that way — used to separate rather than educate. Thus, the only things that are deadwood are those things that are neither built on nor used as separating devices. Both count for something. Lots and lots of math is never used even by mathematicians, much less those of us in mathematics-related professions. A practical math curriculum would de-emphasize lots of things that are more easily simulated today. Why learn calculus when you can simulate optimization? While I think there are good reasons, you have to answer the question: for whom?
Rational root theorem, while at it, most of the stuff about higher degree polynomials, factoring “nice” quadratic functions without the quadratic formula (if one gets to the point of Vieta’s formulas, they’ll be pleased to know that some quadratic functions can be factored in 5 rather than in 10 seconds). A lot of plane geometry, as said above. Many of the topics (vertical angles, parallel lines cut by a transversal, tangents/secants of circles, etc.) boil down to solving equations: if one vertical angle is 2x + 5 and the other is 3x – 10, what is x? Lots of trig – agree about secant, co-secant, co-tangent. Venn diagrams. Stem-and-leaf plots, box-and-whisker plots. 90 if not 99% of time spent on the idea of function as one input – one output.
What is your objection to Venn diagrams? I find them pedagogically very helpful when introducing basic set operations.
It’s mostly their use in elementary/middle school, where again, it’s often just another way of setting up equations. Also, for the most part, it’s Euler diagrams, the universe if frequently ignored. I think it makes sense to use them for teaching set theory/logic, though not sure if it should be a big independent topic showing up on high stakes tests.
Given that it’s probably the only easily presentable in school math topic created by an American, I think, they are not going anywhere.
Hmm … I’m pretty sure Venn was British. So, maybe we can get rid of his diagrams after all. ;)
Oops, time to stop relying on memory. On the bright side, the diagrams may be restricted to their proper place, and even Euler may get recognized one day.
With regard to use of computers vs human computation, as usual Asimov had something to say: The Feeling of Power or the actual story here.
I’m also to far from these to say much usefull, al least at this point. But maybe wahts needed is not small changes, but a full revolution: http://www.sudval.org/
If I knew about this school ten years ago, I made something like that for my kids.
First and foremost, this: http://www.maa.org/devlin/LockhartsLament.pdf
Math is such a rich subject in and of itself, and moreover can be tied into history, philosophy, art, and on and on. But most teachers are math illiterate and so we are stuck in a culture of math illiteracy.
A big part of this is that people don’t understand math. Not specific subjects within math, but math itself. They don’t understand the idea of working with numerical, geometric and logical intuition, translating these intuitions into symbols and rules for their manipulation, and seeing the order that emerges.
So our kids go through school forced to push symbols around with no discussion of WHAT they are doing, WHY it works or HOW it fits into the great sweep of human knowledge and culture.
For example, I spent years learning what I thought to be useless rules about polynomials, with NO mention of the existence of Taylor series! I assumed all this stuff with polynomials was just symbol pushing, that “real” math used all sorts of strange and exotic functions that represented the real world but were far far beyond what we were doing; turns out I was only half right. Yet no mention of Taylor series, approximation, or even the long and distinguished history of polynomials as subjects of mathematical study! NO rationale, no connection, just symbol pushing.
So math education needs to seriously revise its focus. Practice and forcing kids to learn is of course crucial, but the reasons must be provided! And in giving those reasons, we necessarily create students with a deeper appreciation and understanding of math.
Working out an appropriate way to do this at the earlier ages is a job for educators and developmental psychologists, but I do wish we’d strongly de-emphasize computation for computation’s sake, and focus more on having students understand WHAT they are doing when they multiply, WHY it works, and so on. Frankly, I still have trouble remembering whether 8*7 is 54 or 56, but recently made it through a pretty rigorous Advanced Calculus sequence where I successfully learned to really do proofs, and got a great understanding of the real numbers and convergence out of it. It’s the difference between doing math and pushing symbols. They’re related, but we need to remember which takes precedence.
Similarly, we can create many interesting puzzles for kids, and for middle-schoolers, tie math in with MUSIC (!!!!!!!!!!!!!!!) and history, introducing QUESTIONS and guiding students through the logical search for answers. We need to teach students that asking questions and rationally seeking answers is part of our civilization.
As for highschoolers, I can strongly recommend de-emphasizing a lot of the geometry, which I think is only taught for use on the SAT, and other rococo subjects mentioned, and replacing it with linear algebra, which is incredibly useful and bizarrely neglected.
In general, I support introducing advanced concepts in a relaxed and loose fashion throughout the educational process (along with proofs masquerading as puzzles), and tying them up at the end of highschool (or during community college for slower kids) in linear algebra (where the objects of study readily lend themselves to all sorts of fascinating interpretations and conclusions) by asking students to, god forbid, reason on their own.
We also need to make sure kids understand that all these exalted math geniuses had their limits, things they couldn’t prove and/or fudged, and that such limitations are part of being human, and part of the joy of math is exploring these limitations.
In conclusion, four words: Donald in MathMagic Land.
With regard to the value of geometry, I would agree that this is not the best environment to introduce proofs. The problem is that we’re asking people to do proofs within a branch of mathematics to which they have just been introduced, so they’re struggling with integrating new concepts and on top of that they’re asked to do this even newer thing, proofs. It would make much more sense to introduce proofs in an algebraic context, probably in either their 2nd algebra class or pre-calculus. That being said, there are some basic concepts needed from geometry to go on to trigonometry and calculus, like 30-60-90 and 45-45-90 right triangle concepts; these could be taught as a prelude to trigonometry. When I was in high school I skipped geometry and had no difficulty with trigonometry — but taking college geometry a few years later was quite a shock.
The biggest stumbling block that I saw when teaching (I taught high school math for 6 years, including geometry and calculus), was a generally poor understanding among students of what to do with negative numbers, and of course fractions. Admittedly the school where I taught was academically weak (the important thing in that community was that the football team go to State), but this particular weakness persisted even into calculus.
Geometry and trig beyond the very basics are useless for most students, even most college-bound ones. My DH was an engineering major and he used them for that, but I haven’t used them at all since high school. Arithmetic, basic algebra, statistics, and occasionally calculus are the math I use IRL. The problem high schools have is that in 8th to 10th grade, when students traditionally take a geometry course, we don’t know which college-bound students will actually be going into fields that use it and which won’t. So we make everyone take it.
Absolutely agree on more Fermi-type problems. And I think this is linked to less use of calculators, maybe even moving to the abacus. ‘Alex’s Adventures in Numberland’ was pretty persuasive that the abacus helps build intuition about calculation. And I’ve had way too many people come and work for me who would pull out the calculator without even stopping to think about whether an order-of-magnitude estimation would answer the question.
I like jamdox’s idea of showing where things are going earlier on, as well as your invocation of Donald. I seriously disagree about de-emphasizing computation though. Maybe this gets at a difference between teaching math for general use and teaching math for future mathematicians, but most people need a solid foundation of numeracy. (And no, I’m not sure how I square this with my appeal to estimation problems.)
I’m more pro-geometry than most people here. I think the logic is good. I remember doing a ridiculous amount of abstract algebra. Not sure if that was part of the general curriculum or not, but I still can’t see the point. I’d happily see that dropped in favor of more probability and statistics. I might drop the rigor around limits and proofs of why the integral is the anti-derivative in favor of more problem solving…. but I guess I’m an engineer at heart rather than a mathematician.
One thing which could be done in high school is simple design of Experiments, like a simplified version of Box,Hunter&Hunter. And yes, that uses group theory! Simple factorial experiments, actually designed & done in class!
I liked geometry both as a student and a teacher. Kids who are struggling with the basic concept of manipulating abstract entities get to work with very concrete visual objects. Proofs are awkward on all subjects, and I agree that the style from plane geometry does not carry over well to intermediate topics. It’s a nice stand-alone topic and might be more appropriate as an elective / summer enrichment activity.
I also found the historical nature of it neat. Although how we teach geometry is very different from Elements, this learning activity has been common to nearly all educated Westerners for centuries if not millenia. Saying a mathematical people should not learn plane geometry is like saying Christians should not read the Bible, after all, how often do they need to use all those obscure bits?
While I understand that there are many useful applications of matrices (very few of which, if any, are elucidated in the curriculum), why do we make students solve systems of equations in linear algebra using Cramer’s rule by hand? In practice, it amounts to a test of procedural memory. I am willing to be convinced of its merit if someone can offer a compelling argument.
Quite right about Cramer’s rule– that’s beyond dead weight, it’s more like a cancer. While we’re at it, a good half or three-quarters of what is usually called “Calculus II” or “Integral Calculus” could be scrapped: that is, “a billion and one techniques for finding antiderivatives”.
Yes, Cramer’s rule is actually a cancer. And back in the day, I managed to skip calc II due to a really good AP test score, and I don’t feel like I missed much of anything. A few lessons in infinite series convergence could be rolled into calc I.
But the Calc II stuff is college level so it doesn’t help with the HS level stuff. I am still hoping for a list of topics so we can slice and dice them. I haven’t seen that list here, or on the original blog.
I’m going to make the opposite argument. I don’t think there’s very much that can be skipped.
You can skip synthetic division (but not polynomial long division), you can skipp DesCartes rule of signs (but not the rational root theorem). Don’t skip trig, and don’t give students formula sheets for trig (that’s a great way to guarantee that they won’t remember any of it when their Calc teacher in college asks them to do problems where you have to solve trig equations). Cramers rule can go (but not Gaussian elimination). Don’t get rid of factoring and solving trig functions by hand, just because your calculator can do it for you–if your calculator does it, you’ll never learn the basics you need to get anywhere mathematically in college. Granted you may never need to do any of these things by hand, but learning how to do them by hand strengthens the mental pathways that let you learn the next stuff up the mathematical ladder. Geometry is good stuff too–it’s true we kind of drop it all in a big lump in high school, where countries with a national curriculum can break things up so it’s more manageable, but you’re going to miss a lot if you gut the course.
Now I think that we shouldn’t require everyone to take the courses where you do lots of trig or lots of matrix work, or the binomial theorem or whatever, but those topics should be in the sequence that prepares students who are planning to major in math/physics/engineering in college. Don’t sell those kids short by cutting things from the curriculum that their college teachers will expect them to know.
I think the best choice to make the curriculum more manageable is to split off those topics that aren’t well integrated–probability, statistics, maybe matrices and series. Make them a separate course on an independent track–everyone takes the same classes up to algebra 1 (or geometry, or algebra 2 or whatever you want as a prereq for this new class). Require this new class (it’s what most people thing is important for non-science majors) as the final course in the sequence. Let the calculus bound students take this course simultaneously with precalc (or whatever they would be taking next), so they can stay on the same route to calculus. Not everyone needs calculus, but those that need calculus need almost all of those “skipable” topics.
–A college math prof.
Some computational aspects of deadwood perhaps worth considering:
1. Polynomial long division is an algorithm. It is simple to describe, yet has important consequences for understanding the structure of polynomial rings and serves as an introduction to modern computational algebra (see Cox et al., Ideals, Varieties, and Algorithms, 1992). More importantly for the present discussion, though students may never perform polynomial long division in daily life, they are daily users of algorithms and an understanding of what these are may be aided by understanding what polynomial long division is.
2. Synthetic division is a recursive formula for efficient polynomial evaluation. It is also known as Horner’s rule (Dahlquist and Bjoerk, Numerical Methods, 1974) or the chain method (Hamming, Numerical Methods for Scientists and Engineers, Dover, 1973). It is a far more efficient method of polynomial evaluation than direct substitution. Students can easily investigate these relative efficiencies by counting flops (floating point operations) and performing computations. Application of synthetic division to polynomial evaluation can elicit both surprise and pleasure, and offers a relatively accessible, yet authentic glimpse of the interconnections between seemingly disparate mathematical concepts and techniques. In short, synthetic division is an elegant algorithm that demonstrates the application of school algebra to a computational problem of real significance.
3. Rationalizing formulas to avoid severe cancellation is a common trick in numerical analysis (again, see Hamming or Dahlquist and Bjoerk). It is not difficult to devise simple numerical examples where rounding errors lead to absurd results and rationalization restores stability. Rationalization, like synthetic and long division, is a relatively simple algebraic technique having real-world computational importance.