College Math Teaching

April 17, 2014

Rant: misconceptions we give to our students

Filed under: editorial, pedagogy — Tags: , , , — collegemathteaching @ 1:53 pm

I gave a take home test to my numerical analysis students. This was one of the problems (a “warm up”):

mth510gamma

I wanted them to choose a technique (say, a Lagrange polynomial or a cubic spline) to approximate the function at those values.

Most did fine. However one student’s answer bothered me: the student used a proper method but then rounded…to an integer each time.

Yes, in this special case, by pure chance..this turned out to be correct as this data comes from the gamma function.

But in general, that is terrible intuition. There is no reason a function should take integers to integers, even if that always happens with polynomials with integer coefficients.

Unfortunately, too many elementary calculus textbooks and too many math educators reinforce such bad intuition.

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April 30, 2013

Slate Post on Math Teaching

Filed under: mathematics education — Tags: , — oldgote @ 1:41 am

Ben Orlin in Slate:

As a math teacher, it’s easy to get frustrated with struggling students. They miss class. They procrastinate. When you take away their calculators, they moan like children who’ve lost their teddy bears. (Admittedly, a trauma.)
Even worse is what they don’t do. Ask questions. Take notes. Correct failing quizzes, even when promised that corrections will raise their scores. Don’t they care that they’re failing? Are they trying not to pass?
There are plenty of ways to diagnose such behavior. Chalk it up to sloth, disinterest, out-of-school distractions—surely those all play a role. But if you ask me, there’s a more powerful and underlying cause.

Math makes people feel stupid. It hurts to feel stupid.

Aw. So he goes on to relate his experience as an undergraduate at Yale, in a topology class:

So I did what most students do. I leaned on a friend who understood things better than I did. I bullied my poor girlfriend (also in the class) into explaining the homework problems to me. I never replicated her work outright, but I didn’t really learn it myself, either. I merely absorbed her explanations enough to write them up in my own words, a misty sort of comprehension that soon evaporated in the sun. (It was the Yale equivalent of my high school students’ worst vice, copying homework. If you’re reading this, guys: Don’t do it!)
I blamed others for my ordeal. Why had my girlfriend tricked me into taking this nightmare class? (She hadn’t.) Why did the professor just lurk in the back of the classroom, cackling at our incompetence, instead of teaching us? (He wasn’t cackling. Lurking, maybe, but not cackling.) Why did it need to be stupid topology, instead of something fun? (Topology is beautiful, the mathematics of lava lamps and pottery wheels.) And, when other excuses failed, that final line of defense: I hate this class! I hate topology!

You can read the rest.
Here is his conclusion:

Teachers have such power. He could have crushed me if he wanted.
He didn’t, of course. Once he recognized my infantile state, he spoon-fed me just enough ideas so that I could survive the lecture. I begged him not to ask me any tough questions during the presentation—in effect, asking him not to do his job—and with a sigh he agreed.
I made it through the lecture, graduated the next month, and buried the memory as quickly as I could.
Looking back, it’s amazing what a perfect specimen I was. I manifested every symptom that I now see in my own students:
Muddled half-comprehension.
Fear of asking questions.
Shyness about getting the teacher’s help.
Badgering a friend instead.
Copying homework.
Excuses; blaming others.
Procrastination.
Anxiety about public failure.
Terror of the teacher’s judgment.
Feeling incurably stupid.
Not wanting to admit any of it.
It’s surprisingly hard to write about this, even now. Mathematical failure—much like romantic failure—leaves us raw and vulnerable. It demands excuses.
I tell my story to illustrate that failure isn’t about a lack of “natural intelligence,” whatever that is. Instead, failure is born from a messy combination of bad circumstances: high anxiety, low motivation, gaps in background knowledge. Most of all, we fail because, when the moment comes to confront our shortcomings and open ourselves up to teachers and peers, we panic and deploy our defenses instead. For the same reason that I pushed away topology, struggling students push me away now.
Not understanding topology doesn’t make me stupid. It makes me bad at topology.

Ok.

First of all: it IS in part, about natural intelligence. The really smart math people, in general, don’t have trouble with undergraduate math classes, even those at Yale. I mean, of course, REALLY smart people (no, I am not one of those. 🙂 ).

Now he has an interesting observation about student “employing defenses”; at least some of them do.

But there are a host of other reasons too: some just don’t like the material, some ARE lazy (e. g., they won’t do what isn’t fun) and yes, some aren’t up to the task intellectually. Seriously: there are some subjects that many will never be able to master, even at an undergraduate level.

But one issue: someone on Facebook made the following comment:

Oh boohoo. If you’re bad at something you either make an attempt to improve at it or direct your attention to things you’re better at. Everyone is not good at everything and feeling stupid is not something people should be protected from. If you don’t get told you make mistakes or aren’t made to realize that some things take effort then you’re not improving. Learning disabilities aside, especially in higher math, the kids that are failing aren’t showing up to anything, aren’t doing the work, aren’t asking questions, aren’t studying and they don’t repeat to try and do better the next time. Children need to know what failure feels like (and math teachers were children and they do all likely know what failure feels like) so that they learn to try.

Emphasis mine.

My Opinion:
1. There are things that are too difficult for most of us to learn (e. g. quantum field theory).
2. It is useful to have a grasp of one’s intellectual limitations. All too often I see average people dismissing expert findings because those findings “don’t make sense to them.” People need MORE intellectual humility, not less of it.
3. If you haven’t failed at something, then you haven’t tried enough difficult things.

That said: this article is useful because it does give at least one angle of approach that might work with some students.

February 8, 2013

Issues in the News…

First of all, I’d like to make it clear that I am unqualified to talk about teaching mathematics at the junior high and high school level. I am qualified to make comments on what sorts of skills the students bring with them to college.

But I am interested in issues affecting mathematics education and so will mention a couple of them.

1. California is moving away from having all 8’th graders take “algebra 1”. Note: I was in 8’th grade from 1972-1973. Our school was undergoing an experiment to see if 8’th graders could learn algebra 1. Being new to the school, I was put into the regular math class, but was quickly switched into the lone section of algebra 1. The point: it wasn’t considered “standard for everyone.”

My “off the cuff” remarks: I know that students mature at different rates and wonder if most are ready for the challenge by the 8’th grade. I also wonder about “regression to the mean” effects of having everyone take algebra 1; does that force the teacher to water down the course?

2. What would college students want to tell their high school teachers?

By Drew Appleby

I read Epstein School head Stan Beiner’s guest column on what kids really need to know for college with great interest because one of the main goals of my 40-years as a college professor was to help my students make a successful transition from high school to college.

I taught thousands of freshmen in Introductory Psychology classes and Freshman Learning Communities, and I was constantly amazed by how many of them suffered from a severe case of “culture shock” when they moved from high school to college.

I used one of my assignments to identify these cultural differences by asking my students to create suggestions they would like to give their former high school teachers to help them better prepare their students for college. A content analysis of the results produced the following six suggestion summaries.

The underlying theme in all these suggestions is that my students firmly believed they would have been better prepared for college if their high school teachers had provided them with more opportunities to behave in the responsible ways that are required for success in higher education […]

You can surf to the article to read the suggestions. They are not surprising; they boil down to “be harder on us and hold us accountable.” (duh). But what is more interesting, to me, is some of the comments left by the high school teachers:

“I have tried to hold students accountable, give them an assignment with a due date and expect it turned in. When I gave them failing grades, I was told my teaching was flawed and needed professional development. The idea that the students were the problem is/was anathema to the administration.”

“hahahaha!! Hold the kids responsible and you will get into trouble! I worked at one school where we had to submit a written “game plan” of what WE were going to do to help failing students. Most teachers just passed them…it was easier. See what SGA teacher wrote earlier….that is the reality of most high school teachers.”

“Pressure on taechers from parents and administrators to “cut the kid a break” is intense! Go along to get along. That’s the philosophy of public education in Georgia.”

“It was the same when I was in college during the 80’s. Hindsight makes you wished you would have pushed yourself harder. Students and parents need to look at themselves for making excuses while in high school. One thing you forget. College is a choice, high school is not. the College mindset is do what is asked or find yourself another career path. High school, do it or not, there is a seat in the class for you tomorrow. It is harder to commit to anything, student or adult, if the rewards or consequences are superficial. Making you attend school has it advantages for society and it disadvantages.”

My two cents: it appears to me that too many of the high schools are adopting “the customer is always right” attitude with the student and their parents being “the customer”. I think that is the wrong approach. The “customer” is society, as a whole. After all, public schools are funded by everyone’s tax dollars, and not just the tax dollars of those who have kids attending the school. Sometimes, educating the student means telling them things that they don’t want to hear, making them do things that they don’t want to do, and standing up to the helicopter parents. But, who will stand up for the teachers when they do this?

then-and-now-comic-460x270

thennoweducation

Note: if you google “education then and now” (search for images) you’ll find the above cartoons translated into different languages. Evidently, the US isn’t alone.

Statistics Education
Attaining statistical literacy can be hard work. But this is work that has a large pay off.
Here is an editorial by David Brooks about how statistics can help you “unlearn” the stuff that “you know is true”, but isn’t.

This New England Journal of Medicine article takes a look at well known “factoids” about obesity, and how many of them don’t stand up to statistical scrutiny. (note: the article is behind a paywall, but if you are university faculty, you probably have access to the article via your library.

And of course, there was the 2012 general election. The pundits just “knew” that the election was going to be close; those who were statistically literate knew otherwise.

December 1, 2012

One challenge of teaching “brief calculus” (“business calculus”, “applied calculus”, etc.)

Today’s exam covered elementary integrals and partial derivatives; in our course we usually mention two variable functions and show how to calculate some “easy” partial derivatives.

So today’s exam saw a D/F student show up late (as usual); keep in mind this is an 8 am class (no class prior to it). He, as usual, got little or nothing correct. Of course we had the usual \int \frac{1}{x^2} dx = ln(x^x) + C, \int^1_0 3e^{5x}dx = (15e^5 -15) + C , etc.

But there was this too: note that we had barely discussed partial derivatives and how to calculate them “by the formula”. But I did give the following bonus question: “is it possible to have a function f(x,y) where f_x = x^3 + y^3 and f_y = 3xy ? Yes, this is a common question in multivariable calculus (e. g., “is this vector field conservative?”) but remember this is a “brief calculus” course.

A few students took the challenge; some computed \int(x^3 + y^3)dx = \frac{x^4}{4}+ xy^3 + C, \int (3xy^2)dy = \frac{3}{2}xy^2+C and noted that the two functions cannot be made to match (I didn’t expect them to recognize that functions of one variable alone represents constants of integration). Some took the second partials and noted f_{xy} = 3y^2, f_{yx} = 3y and that these don’t match. Again, this was NOT a problem that we practiced.

Another instance: given the ideal gas law PV = nRT I challenged them to show \frac{\partial P}{\partial V}\frac{\partial V}{\partial T}\frac{\partial T}{\partial P} = -1 and someone got it!

Bottom line: in one course, we have some bright, interested students who enjoy thinking and we have some who either don’t or can’t. This makes teaching difficult; if one tries to “teach to the mean” one is teaching to the empty set. It is almost: either bore half the class, or blow away half the class.

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