If We Actually Taught Math…
…we would get rid of course designations like Algebra I, Algebra II, Geometry, Calculus, etc.. Mostly because they convince kids that they can be finished with material when the course ends.
…we would stop teaching a linear progression of math “courses” as if every student was planning to major in math at university. That is demonstrably false.
…we would worry more about those times outside of math class when mathematical thinking would have solved a social issue. Like when my students struggle with racism and prejudice and have no context because they live in rural Iowa. Statistics, and being able to parse them, would have helped a lot there, but alas, they are not used.
…we wouldn’t be so quick to give quizzes over material that was just covered. Cramming is a pandemic issue in the learning of math.
…we would ask students to generate the “word problems” they hate so much.
…we would only give our highest marks to the students who can answer “when will we ever use this?” on their own. Hell, we’d only give them to the students who start there.
…we would ask an effing question that is actually interesting. Like, how fast do boats sink? Does the size of the gash in the hull matter? Or, How many people have there been, total? Or, is there a relationship between latitude of a country and its GDP? Or, what’s with the Hawaiian island hot spot? Or, or, or, or …
…we would teach kids to program computers as early as possible (~10 yrs). We don’t now because it would make most of the high school math curriculum redundant and esoteric, and many teachers lack the training. Example redundancy: finding areas with integrals using anything but numerical methods, or you could keep teaching the myriad “rules” about odd powers of cosine multiplied by even powers of sine…
…we would see those “art” kids taking math electives.
…we would see students connecting math skills to the limitations of those skills instead of the chapter or sub-chapter number. Like, “Oh, vertexes, that was some parabola thing from chapter 2.3.4,” when instead “Oh, a vertex of a parabola is useful when you want to reflect stuff in a coherent way” would be music to my ears.
…we would see math teachers spending as much out-of-pocket cash as science teachers.
… and finally, we would never have to have our aunts on Christmas Eve tell us that “math was just way over their heads” in front of their own children. That’s not a very good present.
This list goes on forever.
Why?
Because I once had a student ask me when she would ever have to take the derivative of:
I said maybe some crazy periodic system like the eddies surround the input of a flute’s mouthpiece, but what are the chances she’d ever study that?
Really, I’m only making them do that because some professor later will make them do that for points, and I don’t set my students up to do badly in college. Honestly, finding the derivative of sin(3x+90) is probably capable enough, but enough or capable isn’t the constraint we’ve put on math class, now is it?
Colin Beveridge:
September 13th, 2012 at 5:30 pm
Bravo, sir.
We’d have students working together to figure out how to fix the snarl-ups around the cafeteria at 11.25am.
We’d have them consulting for local businesses on tricky decisions.
We’d have them communicating their results engagingly and clearly.
Kelly Holman:
September 14th, 2012 at 12:22 am
… We would teach about all the amazing and beautiful things people never see unless they become math geeks, like knots and the fourth dimension and game theory and optical illusions and if you rotate a parabola slowly, when does it cease to be a function?
But am I the only one who finds sublime beauty in doing math by hand and seeing how things Fit. Together. So. Perfectly? I wouldn’t see that if the computer did it all. Still, if you can program a computer to do something, that’s pretty good proof you know how to do it.
Theron Hitchman:
September 14th, 2012 at 1:22 am
So, if part of the issue is downward pressure of expectations where I teach (a certain university not too far away), what kind of changes in signal would help?
What changes do you want to see in the University mathematics curriculum, and in expectations from admissions that will remove that one obstacle?
Ryan Buck:
September 14th, 2012 at 8:26 am
Shawn-
Re: The programming early, is there anywhere around you that participates in the First LEGO or LEGO Robotics competitions? John Deere has grant money available to start those up, which they have in the QC.
Jim P:
September 14th, 2012 at 1:13 pm
I agree with most of your list, and I also agree with the comment below:
“But am I the only one who finds sublime beauty in doing math by hand and seeing how things Fit. Together. So. Perfectly? I wouldn’t see that if the computer did it all. Still, if you can program a computer to do something, that’s pretty good proof you know how to do it.”
There is some beauty that does find its way into our curriculum (surprisingly enough), and some students do catch on to that. It is difficult most of the time for that to shine through over all the “redundant and esoteric” stuff.
I think a large part of your list we can already do within our restrictions. For example things like “ask[ing] students to generate the “word problems” they hate so much,” or “ask [ing] an effing question that is actually interesting. Like, how fast do boats sink? Does the size of the gash in the hull matter?”
But some are definitely difficult like “get[ting] rid of course designations” or “teach[ing] kids to program computers as early as possible.”
I think a lot of us are out here on the internet to search for ways to innovate within the confines of whatever curricula we’re working under. I would say that ultimately to break new grounds and to achieve new heights, we’d definitely need to shatter the current structure of mathematical education.
Bah I ramble too much.
Trial By Blogging:
September 14th, 2012 at 10:01 pm
As one of the college math teachers that make their students know how to find derivatives for problems that they will (admittedly) most likely never encounter in life, I’ve begun doing the following:
1. I teach the basic rules (and prove them all even though I know most students don’t care about the proofs there are a few math majors in the class so it’s important for them to see “basic” proofs).
2. We practice a few relatively simple derivatives in class.
3. We go back and practice simplifying exponents (again, from experience!)
4. I assign a review worksheet that I write that includes a few different derivatives (all of which are relatively similar to what we do in class) and one that looks something like: cuberoot(sqrt(sin(cos(4x^2 – 7)^-2))+(7x-4)/(3x^5-x^-2)).
5. Students inevitable get frightened by the previous problem, most attempt it (only a few get it correct).
6. We cover it in class, they see the progression of all the rules (plus the exponent practice).
7. They ask for a similar problem again to try on their own (ok, not all the students ask but the majority want a new one).
8. They get another one during a later homework assignment.
9. The majority of the class gets it correct.
10. They have a great sense of accomplishment AND they know they are ready to master the exam without having to cram at the end of the chapter.
Admittedly, there’s not a lot of direct application to the terrible functions I assign, but in my view that’s ok. Part of the charm of math is the puzzle aspect – and as long as it’s done in small doses, assigning problems that serve as nothing more than a challenge or puzzle isn’t such a bad idea, or at least I don’t think it is.
On the other hand, I have absolutely no qualms with trying a brand new approach (it’s what led me to the edu-blogs after all).
Shawn:
September 16th, 2012 at 8:50 pm
Theron:
Awesome question. This all started when a few students handled the ALEKS system poorly and ended up getting discouraged away from math. These kids weren’t geniuises, but they were creative in math.
Shawn:
September 16th, 2012 at 8:54 pm
Jim: Agreed. It’s been hitting me pretty hard lately, that all the digital ink I’ve spilled over the years is really dedicated to finding the right shade of lipstick for the pig we call the standard curriculum.
Nick:
September 18th, 2012 at 1:22 pm
I have similar frustrations with some of the science curriculum that we force every student to learn in Ohio. Maybe it is because I am from an area with a higher poverty level, but some of our state standards content will probably never make a reappearance in the lives of my students. I think there is a lot of stuff that could be taught that would do more good for my students than what we currently teach.
Theron Hitchman:
September 21st, 2012 at 8:48 pm
Yeah, standardized tests are often a hurdle.
Well, the ALEKS placement test/remedial module thing is not going away any time soon. In fact, it is just starting. All three Iowa Regents Universities are only beginning to use them.
ALEKS is here for a bit because it is supposed to meet a need that we have more than enough evidence for: DFW rates in lower level math courses are terrible. (esp Calc I, where that rate is about 50% at UNI.) Simply put, we get a large crop of new students each fall, and we need some way to sort them into the proper courses. Ideally, this could be achieved with advising…but that wasn’t working, and ALEKS is Something We Can Buy!
By the way, I taught a section or two of Calc a bit ago and things weren’t much better. The students weren’t ready (basic algebra, oh my). I failed to get them to change their approach and attitude. Many of them failed the course.
Of course, there are many ways to address the DFW problem. Say, changing instructional practices, rethinking _what_ we teach, etc… but every change requires more time and effort from an already strained tenure-stream faculty. So now it is a bit about time and money, and entrenched power structure.
I am a bit of a radical, and I am open to rethinking the standard college curriculum. But large scale change of curriculum at the college level is a long way off. I once tried to make it so that pre-service teachers could take an experimental class on the structure of the real numbers in place of Multivariable Calculus–by the reaction I got, you would have thought that I had insulted a large group of people because of their ethnicity. I can (and have) changed how and what I teach
This leaves me with my original question. What other ways might things be changed?
You know, I feel the frustration, too. Real mathematics is as you describe it: open ended questioning and problem solving. But college mathematics, all of that happens way up high on the “ladder of abstraction.” How should things be arranged to prepare the general population without neglecting the fact that we might still need to find a next generation of mathematicians?
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