There is a common math problem that is given to students studying geometry. It asks them to find the biggest area possible with a given length of fence. After exploring different shapes, the students generally come to the conclusion that a circular fence provides the biggest area. There is a good account of this activity in Jo Boaler’s What’s Math Got to Do with It. I know for a fact that all of my students have seen the problem at least once.
Fast forward to precalculus. My students are studying limits. I have given assessments where they are asked to compute limits at infinity and at a point, algebraically compute limits of rational functions, and explore discontinuities. But that’s not enough. I want them to be able to make arguments about limits. I want my students to be able to use limits to solve the paradoxes that stumped the Greeks and answer fundamental questions about the infinite and the infinitesimal.
I present to you, the farmer & the fence problem:
TL/DR version: A farmer notices that adding more sides to the fence results in a bigger area. He concludes that adding infinite sides will result in an infinitely large area.
I talked with the English teacher and got advice on how to structure an essay prompt. I had the format mirror that of the ACT argument essays. In short, I asked the students to use what they have learned about limits to disprove the farmer’s logic and show the faulty conclusion.
What I got blew me away…
Some students put the data points into a calculator and modeled the function to show that it had an asymptote. Some students showed that as you add more sides the shape begins to resemble a circle so the limit of the area would be the area of a circle (side note - did you know an infinite sided polygon is called an Apeirogon?). Some students calculated the area for more and more sided shapes and concluded that, while the area does increase, the rate at which the area increases, decreases. Students defined functions and used formal limit notation to illustrate their point. They included diagrams of graphs, sequences of polygons, and just about every other image you could imagine.
My favorite introduction was, “Just because a farmer can count sheep, does that make him a mathematician?”
But as I graded the essays I started to notice a trend. Many of the students who usually score low on procedural work wrote brilliant essays and got perfect scores (which was amazing). Many of the high academic status students wrote the most mediocre essays where they answered the question with no justification. They got a mediocre score which reflected their mediocre work (which was amazing in a different sense).
So here is my question: how can I say that any assessment I give is valid if the format in which I give it determines the academic success of my students. I’ve always known that assessment can only give me a vague idea of what students might know, but this essay has really shaken the foundation of how accurate I think my assessments are. Are my perceptions of my students’ academic capabilities merely a commentary on my preferred form of assessment?
In short, how do I ensure that I am assessing my student’s mathematical ability rather than my ability to make valid assessments?
I think you may be asking the wrong question when the more interesting question is, “How can I get my students to think like this more often?”
But if you want a good primer on writing higher order thinking test questions, you might find this of help:
http://docs.google.com/viewer?a=v&pid=sites&srcid=bWF0aGlzZnVuLm9yZ3xob21lfGd4OjJmY2E3MTFmNTgwODI5MQ
Maybe your test questions have been only at the application level and this was the first time you’ve given them a synthesis/evaluation level question. Writing good test questions is tough! Glad to see that your giving this the thought that it deserves.
Paul Hawking
Blog:
The Challenge of Teaching Math
Latest post:
Teaching word problems (systems of linear equations)
http://challenge-of-teaching-math.blogspot.com/2011/03/teaching-word-problems-systems-of.html
I think that the question, “How can I get my students to think like this more often?” implies that assessments which require other modes of thought are not as valid. I am slowly coming to the conclusion that a class which only assesses for higher-order thinking will be as much of a failure as a class which only assesses for rote memorization (of course one of those is quite common while the other is not).
I believe that I can justify this assessment only if I acknowledge that a wide variety of forms of assessment is one way to cancel out the noise caused by the forms of individual assessments. In other words, in order to develop a holistic picture of every single student’s ability level I need to use assessments which tailor to their individual strengths. Having the students explain their argument in an essay is only a good format for some of my students. As a result, it should not be the only assessment in my arsenal (of course the same could be said for quizzes, tests, portfolios, presentations, and any other form of assessment imaginable).
I wonder if a student who does well on the writing in your class, earning say a B for the class, would subsequently do worse next year in a class that only does non-writing assessments, compared to a student in your class who did better on your non-writing assessments, earning a B for the class. In other words, is helping your weaker students to do better on non-writing assessments going to help them more in the long run, such as in next year’s math class and university courses, than by giving them an alternative form of assessment to demonstrate proficiency.
I don’t have an answer for you, but it’s something to consider before you go full bore on incorporating written assessments into your classroom next year.
Essay writing and problem solving are completely different skills. Why would you expect ability at one to be very predictive of ability at the other? We can’t ever really assess what is going on in people’s heads, only what they produce. Most mathematics teachers are more interested in the students’ ability to solve mathematical problems than in their ability to write essays about math.
It is possible to have classes where writing essays about math is rated more important than being able to solve problems, or given comparable weight. These classes favor students who have developed different skills, and tend to be hard for the kids with language problems (especially not having English as a native language or dysgraphia).
As I grade college senior theses from engineering students, I certainly wish that more of them had learned to write about technical material—the college composition classes do not give them that training, and even tech writing class I created 23 years ago (and taught for 14 years) has gotten watered down to the point where they don’t really learn how to express technical content.
Despite that, I’d worry about putting my son in a math class that had more than a very occasional essay—writing takes a lot of time and effort, and most of that time would have to be taken away from learning the core mathematical skills.
In short, I’m not at all surprised that the good essay writers were a different group of students from the good problem solvers, occasional essay writing is valuable in any field (even math), and having too much essay writing is hard on the kids with language problems and takes away from time on core mathematical skills.
I completely agree that essay writing and problem solving are different skills. I just didn’t expect that asking students to submit their solution in the form of an essay would detract from some students to demonstrate their ability to solve the problem in the first place. Nor did I expect that students who usually struggle to answer a question would have so much to say about this problem and formulate such a good argument.
When I graded students I focused on the mathematical argument they made rather than the quality of their essay. What really surprised me was the abundance of good mathematical arguments from students who rarely demonstrate such a competent understanding of a topic.
I loved that problem. What a terrific idea. Whether is pseudocontextual or not, I’m not sure, but the way that you framed the assignment- getting the students to critically think about why the farmer was wrong- was brilliant. It’s like you disguised a math problem as a persuasive essay. Bravo. This is incredibly difficult to do.
I was a bit puzzled by your reflection, though. You wrote:
‘Many of the students who usually score low on procedural work wrote brilliant essays and got perfect scores (which was amazing). Many of the high academic status students wrote the most mediocre essays where they answered the question with no justification. They got a mediocre score which reflected their mediocre work (which was amazing in a different sense).’
This doesn’t seem possible, as students who have strong procedural foundations almost always can conceptually generalize at least as effectively as though who do not. It just fits a little too nicely, like a romantic comedy where the heroine realizes she’s been wrong all along about the dashing and kind man she thought was rude.