How I’m Learning to Step into Math Problems

The biggest inhibitor to my development as a math teacher is that I don’t teach or do math enough. That should make plenty of sense.

I’ve ramped up my teaching since fall with regular (okay, monthly) sessions at a local San Francisco high school. Opportunities to do math are bit easier to find and a bit easier to wedge into empty corners of my day than classroom teaching.

I was grateful, for example, that Jennifer Wilson built plenty of time for doing math into her workshop today at North Carolina’s state math education conference. She posed this problem (source unknown) and I experienced two insights into how I experience mathematical insights.

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First, I approximate an answer. I recognize that the diameter of the circle will be larger than the side of the square. That’s because I can draw the diameter in my imagination and compare the lengths, and also because I know that chords in a circle are never longer than the diameter. I’m guessing the diameter is around 25 units, not more than 30, and not less than 21.

Second, I try to figure out what makes the thing this thing rather than some other thing. I don’t have any details about how the square was constructed. The circle could be any circle, but what makes that square that square? I need to construct it myself. I start changing the square’s location and scale in my head, asking myself, “Is this square legal? What about this square?”

Here is what I see in my head:

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When the square becomes legal at the end, I hear an actual “ding” inside my brain. That’s when all the constraints make sense to me and I can start writing down variables and relationships.

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That last “20 – r” was only possible because of the exercise of mentally making different illegal and legal squares.

From there, I trotted over to Desmos with a Pythagorean relationship in my hand.

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Because I had approximated right and wrong answers earlier, I knew that 12.5 was too low. I realized that was the radius so I doubled it for the diameter.

I think these techniques are what Piggott and Woodham call “stepping into the problem.”

Here visualisations are used to help with understanding what the problem is about. The visualisation gives pupils the space to go deep into the situation to clarify and support their understanding before any generalisation can happen.

At least that’s the best term I can coax from the Internet. I don’t know if Polya’s work on problem solving speaks to that practice directly.

  • If you have another name for that process, let us know it.
  • If you’ve made mathematical problem solving a part of your development as a teacher, let us know how.
  • And if you have an interesting problem to share, let us know about that too.

I’ll leave you with this awesome little number from Brilliant. I promise you can solve it.

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Featured Comments

William Carey:

If you ask a literature teacher what book they read most recently for pleasure and they don’t have an answer that’d be really worrying. But I bet it’s pretty rare. If you ask a math teacher what math problem they most recently worked through for pleasure, I bet the results are much scarier.

Lori M:

For decades, there has been a focus in ELA classes around a push that teachers who read, know how to teach (and reach) readers. Let’s start a similar movement among math educators.

Dick Fuller:

With some over simplification, real problems are not about mathematics, certainly not about arithmetic. The problem is the formulation of the problem. To suggest a problem is particular to values of parameters points toward evaluation as the critical component of its solution. It is not. Evaluation is particularization of a general formulation. A bald assertion: this is at the root of the difficulty students have with “real” problems.

Simon Gregg offers his own solution and then links to a fascinating question about perimeter.

About 

I’m Dan and this is my blog. I’m a former high school math teacher and current head of teaching at Desmos. More here.

18 Comments

  1. Gah – I got that last one too fast to pay attention to my thought process. I’ve got Polya’s tactics for problem solving on the wall in my room – I’m pretty sure I went straight to “make use of symmetry” to solve it.

    I notice that my kids lack metacognition the same way I did on that problem – when we’re doing two step problems, they want to solve them in their head rather tha understanding the process, and when I ask them about process they just say that they know the answer. I assume that this means they do a rapid version of guess and check, without any awareness of what they’re doing.

    I’ve become more interested in *why* I have the insights I do, and how they are commonly (though not always) better than those of my students.

    • What specifically do you have from Polya on your wall? I would like something like that.

      I have developed a rubric based on Polya’s problem solving techniques but I don’t use it with my students any more because I find it isn’t helpful enough. The idea is that students who are stuck can continue to do productive work (showing perseverance in doing it) by exploring their interpretation of the problem or looking explicitly at the plan they have developed for solving it, etc. I want a student who is demonstrating true perseverance but failing to solve the problem to be able to earn just as high of a score as a student who quickly solves the problem. I don’t know why the rubric hasn’t worked well. Perhaps this is because I’m not asking good enough questions? Or maybe it is because the grading burdent it generates is too high and so I tell myself it doesn’t work when it really is helping my students (yet another argument for truly good AI-*assisted* grading). Here is my rubric:
      https://docs.google.com/document/d/1Q8TVKt_3_wUU–pbj0TNU9_8PXtRTU0UTYEc_YyzJE4/

  2. I suspect that question came from Paul Lockhart’s *Measurement*, which is a text full of great questions like that (including that one). Polya also has a short book, whose title escapes me, where he describes exactly the process you work through here: working from constraints towards the “ding”.

    It’s always fascinated me that there’s a culture of adults, say, getting together to read literature but there’s not really a culture of adults getting together to do math.

    If you ask a literature teacher what book they read most recently for pleasure and they don’t have an answer that’d be really worrying. But I bet it’s pretty rare. If you ask a math teacher what math problem they most recently worked through for pleasure, I bet the results are much scarier.

    • I can’t pass up the opportunity to comment on your statement: “If you ask a math teacher what math problem they most recently worked through for pleasure, I bet the results are much scarier.” As math teachers, we can model the attraction to math by engaging both independently and with our students in rich, fun problem solving (like these problems offered by Dan). This may be the draw for local math circles and robotics clubs. Modeling that culture of collaborative math may pay off in encouraging our students to be problems solvers and independent mathematicians. For decades, there has been a focus in ELA classes around a push that teachers who read, know how to teach (and reach) readers. Let’s start a similar movement among math educators.

    • 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!)

  3. Thanks for sharing your actual experience with problem solving. In the interest of full exposure: I did physics and engineering. I was a consumer of mathematics. In retirement I decided to spend some time trying to figure out why it took me so long to comfortably do college freshman physics problems. With some over simplification, real problems are not about mathematics, certainly not about arithmetic. The problem is the formulation of the problem. To suggest a problem is particular to values of parameters points toward evaluation as the critical component of its solution. It is not. Evaluation is particularization of a general formulation. A bald assertion: this is at the root of the difficulty students have with “real” problems. Numbers make for easy assessment; in practice this leads to student understanding problem solving as retrieval of formulae to plug into and evaluate. Excuse the rant, but entertain the possibility that the current approach is causing pain to those who have to learn how to solve real problems.

    • > Modeling that culture of collaborative math may pay off in encouraging our students to be problems solvers and independent mathematicians.

      I don’t think there’s any “may” there. I think it definitely pays off. We’ve started a monthly all school math question (usually something similar to Dan’s original post. I’m shamelessly raiding *Measurement*). Teachers and students alike have a month to give me an answer in writing. The prize is your name up on the board at the end of the month when I post the solution.

      The students sort of expect the math teachers to answer it. They’re always surprised when non-math teachers answer it. A larger and larger group of students are working the problems too, and that’s shaping how they approach things in the classroom.

    • 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.

  4. I think if I had been handed a worksheet with that diagram on it, I would have had a much tougher time trying to solve this problem. The reason why I was able to make sense of this particular problem was because, like Dan, I wanted to figure out how to reproduce the problem on paper before trying to solve it. This is significant to my teaching, because I usually think I’m helping students by giving them the worksheet so they don’t have to worry about their drawing skills and can start the “math” more easily. I see that reproducing the figure is half the math. Thanks for sharing this, Dan.

    Also, I strongly recommend teachers spend some time on the Brilliant website. It really does make a difference to spend time engaging with interesting problems.

  5. 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.

  6. 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.

    • 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.

  7. 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.

  8. 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.

  9. 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.

  10. Malcolm Roberts

    October 30, 2016 - 1:40 pm -

    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.