Comments on: How I’m Learning to Step into Math Problems

By: Hana Silverstein

Hana Silverstein — Wed, 02 Nov 2016 21:14:20 +0000

Hi William – there is a culture of adults getting together to do math! It’s called Math Teachers’ Circles (www.mathteacherscircle.org/). I organize a circle in San Jose, CA, but there are groups all around the country. We meet once a month to have dinner and work on math problems. It’s how we’ve made mathematical problem solving a part of our development as teachers. (PS. to Dan — we are starting a Math Teachers’ Circle this spring in San Francisco, to be located at the Proof School. We’d love if you could join us sometime!)

By: William Carey

William Carey — Wed, 02 Nov 2016 18:36:14 +0000

In reply to Dick Fuller. I really like what Dick says about the real problem being the formulation of the problem. This year I've been flipping my Algebra II classroom. Instead of giving the students a general formula and having them work out many particular evaluations of it, I have the students start by working out lots of particular examples, and ask them to work out the general formulation. The switch from general first to particular first has been great for the vitality of the class.

By: Malcolm Roberts

Malcolm Roberts — Sun, 30 Oct 2016 21:40:30 +0000

I have found the book by Mason, Stacey and Burton called Thinking Mathematically

https://www.amazon.com/Thinking-Mathematically-2nd-J-Mason/dp/0273728911

very helpful in providing practical advice on mathematical problem solving. I think that Polya’s discussion on problem solving is better suited to reflecting back on the process.

Just out of interest, I have used Mason et al’s ideas with high school students, undergraduates and teachers and all of them seem to agree that the ideas are helpful.

By: Dick Fuller

Dick Fuller — Sun, 30 Oct 2016 15:15:13 +0000

Let me use the square-circle problem to show where I am coming from. Again this is the perspective of a consumer of mathematics. Two interpretations suggest themselves: (1) this is problem about the relation of two geometric figures, it’s a geometry problem; or (2) this a problem about the layout of an area for, say, a flower garden. Presenting a figure that was, in effect, produced by a straight edge and a compass points to (1), but the value 20 suggests the garden, except there are there are no units. If I were suggesting a garden problem to a student I would use a sketch to suggest a precise answer is not the point, use a graphical approach if you want. On the other hand I wouldn’t put that gratuitous 20 on geometry problem. With it the presentation gives itself up as an academic problem whose answer has no intrinsic interest to the anyone besides the teacher. Why not suggest taking the side as 1, then I know it is a geometry problem whose solution can be scaled up if I have a particular problem. In any case the squared term drops out of the quadratic equation, and I can solve the resulting linear equation by hand to get a precise result that supports a geometric conclusion. In particular a student can take the 3-4-5 result to the bank. Others point out it has intrinsic interest, it has value to the student. I know teaching is a contact sport, but what you teach is too important, too significant, to warp it into academic exercise.

By: Karen

Karen — Sat, 29 Oct 2016 19:21:04 +0000

I think this problem would be a good “headache” for the equation of a circle. I solved it without scrolling down to your solution. I did immediately see to put this on Cartesian coordinates with the bottom-centre point on the origin and (10,20) as a point of interest. I then saw that the centre of the circle would be (0,r). The thought of solving this withOUT x^2 + (y-r)^2 = r^2, well, it gives me a headache, even knowing some stuff about transformations and that the equation can be derived from the Pythagorean Theorem and seeing a bunch of right -angled triangles in my diagram.

By: Dan Meyer

Dan Meyer — Sat, 29 Oct 2016 15:12:02 +0000

Thanks for the comments, everybody. I highly recommend the back-and-forth between Dick Fuller, Lori M, and William Carey on the state of problem solving in K12 mathematics curriculum. I pulled that exchange into the main post.

By: Simon Gregg

Simon Gregg — Sat, 29 Oct 2016 12:24:53 +0000

That circle-square question is one I tried too:
http://seekecho.blogspot.fr/2012/09/some-answers.html

By: Benjamin Leis

Benjamin Leis — Fri, 28 Oct 2016 20:29:51 +0000

In reply to Scott Farrand. Its actually a great followup question to ask why the 3-4-5 occurred. Hint: 3-4-5 triangle are built out of 1:2 triangles. So anytime you start with a square and create a triangle by adding its median, (as occurred here) you're half way towards finding a 3-4-5. This turns out to be all over the place if you're looking for it.

By: Scott Farrand

Scott Farrand — Fri, 28 Oct 2016 17:59:34 +0000

There’s another good reason to appreciate this problem — it provides a “natural” occurrence of a 3-4-5 triangle. Those seem to be somewhat rare, in the wild.

By: Robert

Robert — Fri, 28 Oct 2016 17:52:57 +0000

This is a good example illustrating how problem solving works in real-time. If the circle fully circumscribes the square, then the question would have been trivial. But in this case, the correct reasoning relies on knowledge about bisecting a side of a square, dual representation of the lengths and spotting a right angle, thereby turning a visual into an algebraic problem of solving an quadratic equation. It’s pretty cool to see how these different components come together.