Formative Per Day #1

Today was the first day of regular classes with the 11th and 12th grades at VPA.

Description of Today’s Formative Assessment
Today’s formative assessment went like this: students were direct to a page on the class web site where they read this passage from the beginning of The Art and Craft of Problem Solving, by Paul Zeitz:

An exercise is a question that tests the student’s mastery of a narrowly focused technique, usually one that was recently “covered.” Exercises may be hard or easy, but they are never puzzling, for it is always immediately clear how to proceed. Getting the solution may involve hairy technical work, but the path towards solution is always apparent. In contrast, a problem is a question that cannot be answered immediately. Problems are often open-ended, paradoxical, and sometimes unsolvable, and require investigation before one can come close to a solution. Problems and problem solving are at the heart of mathematics. Research mathematicians do nothing but open-ended problem solving. In industry, being able to solve a poorly defined problem is much more important to an employer than being able to, say, invert a matrix. A computer can do the latter, but not the former.

A good problem solver is not just more employable. Someone who learns how to solve mathematical problems enters the mainstream culture of mathematics; he or she develops great confidence and can inspire others. Best of all, problem solvers have fun; the adept problem solver knows how to play with mathematics, and understands and appreciates beautiful mathematics.

An analogy: The average (non-problem-solver) math student is like someone who goes to a gym three times a week to do lots of repetitions with low weights on various exercise machines. In contrast, the problem solver goes on a long, hard backpacking trip. Both people get stronger. The problem solver gets hot, cold, wet, tired, and hungry. The problem solver gets lost, and has to find his or her way. The problem solver gets blisters. The problem solver climbs to the top of mountains, sees hitherto undreamed of vistas. The problem solver arrives at places of amazing beauty, and experiences ecstasy that is amplified by the effort expended to get there. When the problem solver returns home, he or she is energized by the adventure, and cannot stop gushing about the wonderful experience. Meanwhile, the gym rat has gotten steadily stronger, but has not had much fun, and has little to share with others.

They were then prompted to leave a comment about what they’d just read.

How did it go? / What did I learn?
Better than expected!  I almost felt like it was a cop-out to call this a formative assessment, but it absolutely was.  From reading the (mostly short) student responses, I learned that a health 1/3 of students think math class really is more about practice exercises than problem solving, which is an opinion I’ll seek to change this semester.  I was able to see, in a few words, who brings with them a passion for mathematics, and who is a little less stoked.

Some favorite responses

I love where this student goes with the metaphor – though know we’ll want to work on writing mechanics:

I think the purpose of mathematics is to do exercise, which would help you solve problems. If you don’t practice; do exercises , you wont understand or be accurate with your answer.  when in solving problems i feel focused , sometimes confused.  This reading applies to my life outside of school because sometimes when im doing work or solving a problem , The problem solver gets hot, cold, wet, tired, and hungry. The problem solver gets lost, and has to find his or her way. The problem solver gets blisters – all those things happen to me also , or worst !

Math class as competition:

When solving a problem given to me, I feel as if I am being entered into a race and like any race you have to get to the finish line before anyone else to get that feeling of victory. It’s not enough to just be first, being able to explain how you got your answer and why you took certain steps, in my book, it adds style points to the finish. At the end of the day you get to walk away as the guy/girl who could solve that problem before anyone.


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