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2015 August 22

Don Cohen, Calculus by and for Young People: 1930–2015

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I just found out that the author of Calculus by and for Young People died this year:

Donald “The Mathman” Cohen was a great educator and influenced thousands of children and adults throughout his life. He began teaching in the 1950s and continued his work through The Math Program which started in 1976 in Champaign/Urbana, IL with partner Jerry Glenn. Don continued teaching right up until he passed away in 2015 at age 85. Don’s books and other materials continue to be treasures for all who are interested in math or math education.[Don Cohen – “The Mathman” – 1930-2015]

His family is making his books available as free PDF downloads, as a memorial: Don Cohen – “The Mathman” – 1930-2015

We didn’t use his books (by the time we found out about them, our son had already acquired enough calculus not to benefit from the books), but lots of kids did get an early start on calculus using them.  He aimed is material at middle school students, rather than college students. His approach to teaching math seems to have been highly regarded both by math teachers and in home-school circles. More info about his math program at his web page.

2014 April 11

Arthur Benjamin: Teach statistics before calculus!

I rarely have the patience to sit through a video of a TED talk—like advertisements, I rarely find them worth the time they consume. I can read a transcript of the talk in 1/4 the time, and not be distracted by the facial tics and awkward gestures of the speaker. I was pointed to one TED talk (with about 1.3 million views since Feb 2009) recently that has a message I agree with: Arthur Benjamin: Teach statistics before calculus!

The message is a simple one, though it takes him 3 minutes to make:calculus is the wrong summit for k–12 math to be aiming at.

Calculus is a great subject for scientists, engineers, and economists—one of the most fundamental branches of mathematics—but most people never use it. It would be far more valuable to have universal literacy in probability and statistics, and leave calculus to the 20% of the population who might actually use it someday.  I agree with Arthur Benjamin completely—and this is spoken as someone who was a math major and who learned calculus about 30 years before learning statistics.

Of course, to do probability and statistics well at an advanced level, one does need integral calculus, even measure theory, but the basics of probability and statistics can be taught with counting and summing in discrete spaces, and that is the level at which statistics should be taught in high schools.  (Arthur Benjamin alludes to this continuous vs. discrete math distinction in his talk, but he misleadingly implies that probability and statistics is a branch of discrete math, rather than that it can be learned in either discrete or continuous contexts.)

If I could overhaul math education at the high school level, I would make it go something like

  1. algebra
  2. logic, proofs, and combinatorics (as in applied discrete math)
  3. statistics
  4. geometry, trigonometry, and complex numbers
  5. calculus

The STEM students would get all 5 subjects, at least by the freshman year of college, and the non-STEM students would top with statistics or trigonometry, depending on their level of interest in math.  I could even see an argument for putting statistics before logic and proof, though I think it is easier to reason about uncertainty after you have a firm foundation in reasoning without uncertainty.

I made a comment along these lines in response to the blog post by Jason Dyer that pointed me to the TED talk. In response, Robert Hansen suggested a different, more conventional order:

  1. algebra
  2. combinatorics and statistics
  3. logic, proofs and geometry
  4. advanced algebra, trigonometry
  5. calculus

It is common to put combinatorics and statistics together, but that results in confusion on students’ part, because too many of the probability examples are then uniform distribution counting problems. It is useful to have some combinatorics before statistics (so that counting problems are possible examples), but mixing the two makes it less likely that non-uniform probability (which is what the real world mainly has) will be properly developed. We don’t need more people thinking that if there are only two possibilities that they must be equally likely!

I’ve also always felt that putting proofs together with geometry does damage to both. Analytic geometry is much more useful nowadays than Euclidean-style proofs, so I’d rather put geometry with trigonometry and complex numbers, and leave proof techniques and logic to an algebraic domain.

2013 December 23

Different levels of the “same” course

Filed under: Uncategorized — gasstationwithoutpumps @ 12:46
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In thinking about the redesign of the bioengineering curriculum, I’ve had to pay a lot of attention to what level of courses the engineers would be required to take.  Our campus offers physics at three different levels (one algebra-based, the other two calculus-based) and calculus at 4 levels (honors for math majors, for physicists and engineers, for life scientists, and for economists). Do I allow the students to take either of the calculus-based physics courses? Do I allow any of the three calculus classes (excluding the one for economics majors)?  I’ve wrestled with this problem for a while (see for example, my post Physics for life-sciences majors from last June).

In favor of allowing the lower level courses:

Usually there is the most scheduling flexibility for the second-lowest level—the level aimed at biology majors—because that is where the largest numbers of students are, so the courses get offered repeatedly during the year, while the more advanced courses get offered only once.  So from a scheduling standpoint, it would be best if students were able to take those courses.

In bioengineering, we also get a lot of students who start out in biology, but who later realize that other majors are more interesting (freshman year everyone thinks they want to go to med school—most have given no thought at all to engineering).  Because the biology majors are advised to take the calculus and physics courses intended for biologists, the students have taken only those and not the higher level calculus and physics courses intended for engineers.  So a change of major is easier if students are allowed to take the biology-level calculus and physics.

One thing I’m trying hard to avoid in the bioengineering curriculum redesign is “creeping prerequitism”—the tendency for most courses to gradually increase the prerequisites in order to have better prepared students in the course.  In many cases the prerequisites are irrelevant to the material of the course (like multi-variable calculus for a data structures course or genetics for a cell biology course), but are just filter prereqs, to make sure the students have more “maturity” by having passed a gantlet of other course.  Because of these prerequisites (both real ones and filter ones) being added independently by each of the 8 or 9 departments that teach courses required for bioengineers, we end up with a program grossly overloaded with lower-division “preparation” courses, and not enough upper-division “application” courses.

Against allowing the lower-level courses:

In exit interviews with seniors last spring and this fall, we asked them about their experiences in calculus and physics.  Those who had taken the lower level of calculus-based physics course felt that it had been a waster of their time—neither their classmates nor their professors seemed to care much about whether the material was learned, and everything was covered rather superficially.  (We didn’t get the same info about calculus, because most had been forced to retake the higher-level calculus class if they had only taken the biology-level one.)  So from a pedagogic standpoint the students get a better course if they take the higher level with students who expect to use the material and with professors who expect their own majors to be taking the course.

Some upper-division courses do rely on math and physics skills of the more advanced courses.  For example, the upper-division probability and statistical inference classes do rely on students being adept at integration, the statics and dynamics course relies on students knowing Newtonian mechanics well and being able to handle differential equations, and the electronics courses require some skill with calculus and differential equations.

Concluding thoughts

I read an blog post today by a high-school physics teacher addressing a similar question at the high-school level: Jacobs Physics: How do you tell the difference between AP and “regular” physics?.  He doesn’t have to face what courses students are required to take, but only which ones they should be advised to take, but the underlying questions are the same. In the post, Greg Jacobs writes

If an AP and a Regular course cover the same “standards,” how are the two classes different?

Don’t use standards to define courses; use tests and exams, preferably as written by someone external to the course, to define courses.  Once you’re clear on the level, topics, and depth of question that your students will be expected to answer, then you can make up a concordance with any state standards you need to.

The AP Physics 1 exam covers much of the same material as regular/Regents. The major difference is the depth of that coverage, as evidenced in the test questions.

A regular question can generally be categorized in a single topic area, and can be answered in one step, or two brief steps, or a one-two sentence explanation with reference to a single fact of physics.

An AP question generally requires cross-categorization across two or three topic areas. Most require multi-step reasoning, or a two-three sentence explanation with reference to more than just one fact of physics. AP questions, for the most part, require students to make connections across skills and topics.

As an additional comparison, you might consider a conceptual class. Conceptual Physics can cover many of the same topics as “regular” physics, but without using a calculator.  …  A conceptual approach provides a greater contrast between AP and non-AP physics.

The key idea here is that the difference between levels is not in what subjects are covered, but in the expected skills of the students after taking the course. That holds true at the college level as well—I can’t decide based just on catalog copy what level of course students need, because the catalog copy only lists topics, not the complexity of the problems that students who pass will be able to solve.

In the interest of minimizing filter prereqs, but making sure that all genuine prereqs are met, I’m suggesting requiring the higher level for the bioelectronics and assistive technology: motor tracks, but allowing the lower calculus-based physics for the biomolecular and assistive technology: cognitive/perceptual tracks.  I am suggesting requiring the physicist/engineer track for calculus in all tracks, since it is needed for a higher-level course in all of them. It’s not the same course in each track, but electronics, statics and dynamics, and statistical inference all require greater facility with calculus than the calculus-for-biologists track provides.

2012 May 26

Free calculus book

Filed under: home school — gasstationwithoutpumps @ 22:02
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There is a free calculus book (by Gilbert Strang) available on line though the MIT OpenCourseWare.  The book covers all of single-variable calculus and most of multi-variable calculus, with over three times as many pages for the single-variable stuff as for the multi-variable stuff.

My son just finished single-variable calculus a month ago through the Art of Problem-Solving online course and took the AP Calculus BC class, and so I’m thinking about what math he should do next year.  Doing the MV calculus from a book like this would be one reasonable choice.

Although multi-variable calculus is the most common course to follow single-variable calculus, I don’t think I’ll push him into it next year (though he could take it if he chooses to).  I think that it would be more valuable for him to take Applied Discrete Math (combinatorics, proof by induction, Boolean algebra), probability, and statistical inference courses before going further in calculus.

2012 April 18

Distance learning for gifted kids

Suki Wessling, a local writer who is home-schooling her kids, recently wrote an article about distance-learning oppoturnites for gifted kids: Boutique distance learning offers variety for gifted kids – National gifted children | We have not used any of the “boutique” services she mentioned, nor, for that matter the large services like Johns Hopkins University’s Center for Talented Youth or Stanford’s Education Program for Gifted Youth.

There are several reasons we’ve been reluctant to use many on-line courses:

  • Many are quite expensive. EPGY courses are around $500 to $750, plus $50 registration and shipping fees, JHU-CTY courses are $500–$1280.  I’d want to know that the course would be a very good fit and of higher quality than a corresponding community college class (about $300) before committing to an online course.
  • Too much screen time.  My son already spends more time in front of a screen than is healthy (as do I, so I can’t chide him too much). At least with community college classes he gets the exercise of bicycling to the class (in fact, this provides so much exercise that it counts as his PE class: about 4 hours a week).
  • Difficulty in finding courses that fit his educational needs and interests.  There are undoubtedly a number of courses that would be an excellent fit for him, but it is very difficult to distinguish them from other courses that have similar descriptions but would be at the wrong pace, wrong level, or have too much busy work.

So far we have only used one on-line course provider: Art of Problem Solving.  A year ago, I posted about our experience with with their precalculus course: Good online math classes.  My son did their calculus class this year with the same instructor, and we had similarly good results.  The AoPS calculus classes are not cheap ($500 with books), but they were an excellent fit for my son. If I could be assured of as good a fit in other online courses, I would be more willing to use online providers.

This year my son has been keeping time logs for his consultant teacher in the home-school umbrella.  For the AoPS calculus class that just ended, he did almost all the weekly and challenge problems, but not quite all. We added up the total hours (class and homework) for February and March, and got 56 hours—just under 7 hours a week.  His total workload for all courses (including the cycling that counts as PE) averaged 40.75 hours a week in February, which I regard as about the right amount of time for a high school student to be spending on school.  It is certainly much larger than the 2–3 hours a day that some home schoolers regard as adequate.  The main advantage for us of home schooling is not a reduction in workload, but a spending the time on appropriate work, rather than busy work or dead time.

I think that the calculus class was a good deal higher workload than the Precalculus class last year, but we did not keep time logs then, so I may be mistaken.  My son did not take any of their lower-level classes, so I can’t comment on the workload of any of them (though we did use the intro algebra and intro geometry books some earlier, and were happy with them, which is why I was willing to give AoPS online courses a chance).

My understanding is that by the end of the AoPS calculus course well over half the students had dropped, possibly because they could not keep up with the pace or the workload.  You only get your money refunded if you drop in the first 3 weeks, so a lot of families ended up wasting the tuition money.  I’m afraid of a similar thing happening if we pick an online course that is not a good fit for our son.

He will probably do one AP practice test before taking the AP Calculus BC test next month, but that should only take about 3.5 hours.  The AP test should be a good review of the essential material of the course, but so far as I can tell, the AoPS Calculus class covers more material in greater depth than the usual AP calculus BC course or the usual first-year college calculus class.  It is definitely a calculus-for-mathematicians course, with a lot of emphasis on problem solving and rigorous foundations (like using Darboux integrals, a somewhat cleaner equivalent to Riemann integrals).  Some of the differential and integral equations they had in the last challenge set seemed difficult even for me (though I must admit that ODE was never my favorite subject, and it has been over 30 years since I last did any differential equation other than a trivial exponential decay).

The AoPS courses also cover complex numbers fairly well, something that is not always done in other precalculus and calculus classes. Another gifted high school student I know has taken calculus through multi-variable calculus at the local community college.  I was amazed to find out that he’d had almost nothing about complex numbers: not even such fundamental things as Euler’s formula: e^{i\theta} = \cos \theta + i \sin \theta.  This lack came to light during physics class, when I was deriving acceleration for something moving in a circle by taking the second derivative of R e^{i \omega t} with respect to t.  It is so much easier to work with exponential functions than trig functions that it didn’t occur to me that the community college calculus classes would not have covered it.

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