You’d think as a 10th year teacher this revelation that math is so much more than computation would have come to me sooner, but it has taken a lot of experience and pondering over other’s work for the ideas that follow to congeal.  More specifically, I’ve come to realize the extent to which I undervalue, under validate, and under assess the structures and understandings outside the realm of calculating in my classroom.

It was Conrad Wolfram’s TED Talk, Teaching Kids Real Math with Computers, that first had me pondering with one of his opening questions; What is math?  Conrad WolframHe explains that to him, math encompasses the four areas pictured here and we should focus student’s time on the three areas that stress thinking, while partnering computers in the computation process.  As with most published work, I do not completely agree with all of his points the same as most would not agree with all of mine.  However, his viewpoint has had a tremendous impact on me, and I have returned to watch this talk many times.

I personally like the classic framework written in 1957 by George Polya in How to Solve It: A New Aspect of Mathematical Method of understanding the problem, devising a plan, executing the plan, and verification.  Polya 4 Step Problem SolvingThis classic four stage approach does miss the idea of posing the right questions in the first place, but provides a valuable look into where to validate student work in critical thinking and problem solving.  I do, however, think this iterative process can be effective on more than just math applications where an answer is calculated.

 

Towards the end of last school year I began to evaluate the exams we were giving students more closely due to the realization of how much weight we were putting on them in terms of their overall grade.  For instance, I’ve taken a screenshot of the firstpage of an area assessment that we give to our geometry students that accounts for 80% of their grade in that unit (along with other similar assessments).  Area TestNow questions certainly do get progressively harder, I just didn’t want to share the entire test as most teachers are still using the problems.  What I found was eye-opening.  Of the 25 total points, 23 of them were aimed directly at computation.  That’s 92%.  And the question I had to ask myself is, “Is that ok”?  Is the calculation of the area even what’s most important to me as the teacher in this unit?  The answer for me is no.  So I began to ponder what was the most important aspect of this unit from my viewpoint and why?  What I realized after this reflection was that I valued student understanding of the structure of area beginning with the rectangle and deriving all area formulas out from it.

“Interesting,” I thought to myself.  I actually value the structure and progression of the formula derivations the most.  As I was scrolling through Ed Southall’s (@Solvemymaths and author of Yes, but Why) Twitter posts earlier this month I came across this threadabout people’s most unexpected wow moments in math, which is definitely worth your time to read.  Within it I found that educator Howie Hua found this area progression to be his biggest epiphany moment in his math experiences to date!  Area Deriving RectangleWhat if I better structured the unit so students were able to experience that progression, while also valuing and validating it in the end of unit assessment 🙂

Last year I asked students to write a Scratch program that explained one of the formula derivations.  Unfortunately, I only allotted one 48 minute class period to do so, but was impressed by what many were able to accomplish in that amount of time.  One studentfocused in on the trapezoid, and described what it meant quite eloquently.  I only awarded him five practice points… #undervalued:(trapezoid

 

This year I have the ability to change that.  The ability to validate this creative thought on a much larger scale.  I can have the students express each derivation in a way that’s unique to them and let the full progression unfold and be part of their performance grade.  I can add value to analysis, planning, and reflection in a way that I haven’t yet done.

What about you?  What percentage of your grades are assessment based?  What percentage of those assessments are computation based?  Are the grades you’re awarding a true representation of student’s “mathematical” knowledge?

BOOM!

Mike

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