Category Archives: Problem Solving

Polynomial Guess

 

MOTIVATING COMPLEXITY             THROUGH PUZZLES

I found a really nice number guessing game several years ago, and I’ve used variations of this puzzle several times over the last few years in my Algebra class. Kids can’t help but want to know the answer to a logic riddle like this, and this year it occurred to me that I might be able to leverage this “want to know the puzzle answer” to motivate some more focus on understanding quadratics or higher degree polynomials.

The idea is that you choose a “secret” number, and give clues one at a time until students can narrow down the possibilities to a single answer.  I decided to try the same idea with more complicated expressions, so I created a couple of quadratics puzzles, and a polynomial version. I had these posted on the Sweet Math wall, and would add clues roughly one each day. I ran a simple number version, alongside the more advanced ones to allow entry for middle schoolers, and extension for high schoolers. I definitely noticed kids lingering at the clues as I added them. Some kids even asked me what time of day I would be adding another one. In an unexpected turn, it was a history teacher who submitted the correct guess for the first number puzzle. In your face Algebra students!

Although I haven’t done this yet, I like the idea of creating examples for sequences, and I think I’ll try this next year. Is the glory of being the first to guess correctly important enough to take the chance of guessing when there might be two possibilities for the common ratio of a geometric sequence? Do you team up with another student when you’ve narrowed it down to two possibilities so one of you is guaranteed to be victorious?

Here are a few of the puzzles I made in case you’d like to try them out. Please do let me know if you find them useful or if you think that I should sequence the clues differently or if you have other ideas for how to make them better.

 

Number Guess 1

Number Guess 2

Polynomial Guess

 

Put Your Marks Where Your Mouth Is

I was struck recently when reading Robert Kaplinsky’s post, Why Are You Using That Problem?.  In this piece, he articulates 3 different reasons that we might choose to use a particular problem: to introduce a concept, for productive struggle, or for problem completion – each of which has its own value.  Kaplinsky convincingly argues that we should be purposeful not just in which problems we choose, but in why we choose a particular problem.  I have been considering his thoughts in reference to productive struggle, and specifically in how to best to support students when we choose a problem with this as our goal.

I have always talked about encouraging and nurturing a classroom culture in which it is safe to take risks, and to experiment.  When I introduce a novel problem, I tell students that my expectation is that they try – that making an attempt is what I value.  My marking scheme always always includes credit for getting started, and for each step of a problem.  But in some ways, my marking scheme has been generally geared toward a sequence of steps leading toward a correct answer.  In awarding credit for specific answers, I have been communicating to students that I value those answers.  If I want students to put value on productive struggle, I need to demonstrate that that is what I value – and by extension, that is how they can earn marks.  (For now, I’ll leave the question of whether earning marks in general is productive at all…)

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I worked with Dr. Andreas(@DrAecon), my excellent colleague in the chemistry department, to create a set of guidelines and a rubric to support students in becoming independent and confident problem solvers.  In our guidelines, we tried to give students some explicit ideas to try when they are presented with a novel situation, and in our rubric, we tried to express criteria that values the process more than the answer.  We want students to make valid arguments, to justify their reasoning, to persevere in problem-solving,  and to feel that they can take risks, including taking a path that might not lead to a correct solution.  I hope that our guidelines and materials communicate these values.  I want to be explicit and transparent with students about this.  We’ll need to set aside time where we are not focused on learning specific content, but instead are focused on becoming strong and reflective problem solvers who know that they can earn respect and grades with this focus.  I want to put my marks where my mouth is.

Here are the rubric and problem-solving guidelines, based on Polya’s four steps.  I’ve included both Word and PDF versions, and a blank version, which is intended for students to use as an organizer for their work.  These are written with 10th grade students in mind, but I plan to create simplified versions of these for middle and elementary school students.  Please let me know if you can use these, and if you have any suggestions for improvement.

Problem_Solving_Rubric_Grade_10

PDF: Problem_Solving_Rubric_Grade_10

Problem Solving Protocol_Grade 10

PDF: Problem Solving Protocol_Grade 10

Problem Solving Protocol_Grade 10_Blank

PDF: Problem Solving Protocol_Grade 10_Blank