From additive to multiplicative thinking

A few weeks ago our colleague and collaborator Pam Weber Harris led a really interesting numeracy workshop at Math in the City (City College). A former high school teacher and now teacher educator based in Austin, Texas, Pam has expertise in many areas: technology, assessment, K-12 mathematics, and more recently numeracy routines.  Her latest publication, Building Powerful Numeracy for Middle and High School Students, is a direct outgrowth of number strings that extends our work to college level mathematics.  Pam renames number strings as problem strings, but the essence of the routine remains the same.

During the workshop Pam led us through a variety of problem strings, moving along a continuum that she suggested was typical of the development of strategies for most kids:

Counting Strategies

Additive Thinking

Multiplicative Reasoning

Proportional Reasoning

Algebraic Reasoning

In one string we explored, the problems toggled back and forth between pure addition and related factored multiplication problems. This design then encouraged many of us to look for common factors and re-arrange those groups to make addition easier.  Here’s the string:

Screen Shot 2013-11-21 at 9.34.25 PM

Did you love the last problem?  We did, and thought an explicit connection to factoring and grouping like terms in algebra made good sense.  “That was really cool,” noted one participant to me on the break. “When we treat algebra as something new, unconnected to all of the numeracy work that came before it, it’s no wonder that kids are confused about what to do. That was kind of seamless. It’s making me rethink how I teach this stuff.”

Pam’s now back home in Texas, putting the finishing touches her next book Lessons and Activities for Building Powerful Numeracy, due spring 2014. Luckily for us, she continues to be a collaborator and promoter of our shared work. And, she’s agreed to be a judge in our number strings contest.

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