Comments on: The Difference Between Pure And Applied Math

By: Emily

Emily — Wed, 18 May 2016 18:59:01 +0000

This is a fantastic dilemma, but the dilemma arises from the practical task of assigning points to the student’s answer. I imagine those same teachers would be largely in agreement about what the students’ work demonstrates about their mathematical thinking, or what instructional approach might be appropriate to take next (although that’s a moot point to graders of AP exams). It’s the relative importance of the pure versus the applied, and how to assign point values to these, that creates the debate. In everyday teaching, how we adjust our instruction when students make mistakes like these is more important than how many points we take off. Let the AP graders get hung up on points.

By: Dan Meyer

Dan Meyer — Tue, 17 May 2016 00:06:44 +0000

Nice. Thanks, Benjamin. I've added those links to the body of the post.

By: Benjamin D.

Benjamin D. — Mon, 16 May 2016 02:40:21 +0000

FWIW: The MSRI piece by H.O. Pollak is from the COMAP Mathematical Modeling Handbook.

Full copies are available for free online; here is a brief sample with the article (begins on PDF 8/25):

: http://www.comap.com/modelingHB/Modeling_HB_Sample.pdf

But here is a longer excerpt from the Handbook (though with an abbreviated version of Pollak’s intro):

: http://www.comap.com/modelingHB/CCSSModelingHB.pdf

One reason to point this out is that the lessons are each followed with a section entitled “Teacher’s Guide — Extending the Model.” Although not attributed in the document, these were (essentially or entirely) written by Pollak.

So, e.g., in the linked MSRI version Pollak writes:

1. “An introduction to the modeling of epidemics can be associated with Viral Marketing.”

2. “…Sunken Treasure has discrete, continuous, and even experimental aspects.”

“Sunken Treasure, besides using a variety of forms of mathematical reasoning, even suggests using physics in order to do mathematics!”

3. “For example, in connection with several modules involving probability and statistics, the notion of optimal stopping occurs more than once. It is the central idea in Picking a Painting…”

I wrote (but did not edit…) the three lessons numbered above, and report with surety that Pollak’s extensions are much better than my contributions! (E.g., the “association” of modeling epidemics with Viral Marketing is totally due to Pollak.)

By: Katenerdypoo

Katenerdypoo — Sat, 14 May 2016 06:30:48 +0000

In a situation where the answer is so clearly wrong that it borders on the ridiculous, I typically take a point for the initial mistake and a point for not recognizing that the answer is wrong. If a student writes something like a question mark afterwards or “this doesn’t seem right by I don’t know what I did wrong,” I take that as recognizing the unreasonableness of their answer and take only the point for the error.

Of course a student without anything written may also have realized it was wrong, but they should keep in mind that their job on the test is to fully communicate all of their understanding to me.

By: Chester Draws

Chester Draws — Thu, 12 May 2016 05:21:55 +0000

Carl, to clarify.

Students should notice that they have made an error when the answer is clearly wrong in the context.

My point is that going back and looking for the where error occurred is not necessarily practical in an exam.

I often know that I have made an error in a worked answer I want to give students. In extreme cases it has taken me hours to track it down. The thing is, once I made the error I kept making it again because I was incapable of seeing it cleanly. That requires taking time off and coming back fresh to it.

By: Carl malartre

Carl malartre — Thu, 12 May 2016 04:02:22 +0000

Maybe what is real has implication on catching obvious errors. From my understanding of the book Thinking, fast and slow, if it was real for the student, brain system 1 would have easily catched that error. Maybe because this is unreal for the students, lazy system 2 let it slide.

By: Chester Draws

Chester Draws — Wed, 11 May 2016 09:42:05 +0000

I think we need to be careful about accusing our students of “not checking”.

I know that I am not alone in my inability to see mistakes in what I have just typed. To see mistakes I need to take a long break and come back and re-read it fresh. This is an adult, who knows the sorts of mistakes he makes, mind you, so should spot them. Not a kid who has no ideas what he can’t do.

Students rechecking their work generally make the same mistake again. So much so that I no longer ask students to check their work, just to learn to be more careful in the first place.

In fact if my students know that their answer is wrong I tell them to do it again from the start. That is pretty much the only way they will do it right (and even then they’ll still often make the same mistake).

In an exam situation students simply do not have the time and calmness to check properly. It is literally a waste of time to ask them to do so, when they could be doing more questions (or the same ones slower).

By: C Calculus

C Calculus — Tue, 10 May 2016 22:55:01 +0000

So you’re telling me that the AP graders didn’t have a rubric or any other type of systematic approach for grading open-ended problems? I hope they do now.

By: Howard Phillips

Howard Phillips — Tue, 10 May 2016 20:49:58 +0000

To David Garcia. I used to tell this to my students. It worked. They saw the fairness.

By: Scott Swita

Scott Swita — Tue, 10 May 2016 20:06:35 +0000

This reminds me of a problem I gave my students about the cost of car tires one time. A student worked out the problem and said each tire had to cost something like $5000. The other students said that didn’t make sense. The student replied “it doesn’t have to make sense, it is just a math problem”.