The Power of Patterns

My Number Sense (MATH119) students are taking their midterm for our course today.  A pattern problem was the very first question I asked them to think about.  I am a big fan of pattern work at all grade levels.  There is so much richness that can come out of pattern work.  Here's what I anticipate seeing from my students with this pattern.

There will be at least three different kinds of response.

• One group of students will identify a consistent pattern of addition and simply follow that pattern of sums until they get to the 50th stage.

• Another group of students will find a variable expression that connects the stage number to the number of blocks at a given stage.  These students will use their expression to find the number of blocks at the 50th stage.

• Yet another group will come up with a(n) approach(es) that I would never, ever, not in a
  hundred years had ever thought of on my own, causing me to smile, say, "That's cool!," and help me to appreciate the flexibility and beauty of math.

Pattern work allows the students to see the problem in a way that makes sense to them.  Problems like this encourage our students to see the math as more than a set of procedures to be followed, but rather as a puzzle to be solved.  Jo Boaler calls problems like this ,"low floor, high ceiling."  I really like this idea.  While there are varying levels of sophistication within the solution strategies of my students, it allows me to continue the conversation with my students about the math after the assessment occurs.  This particular pattern is quadratic in its nature.  This grounding of quadratics within a pattern allows me to have my learners connect and use representation of the mathematics in a variety of forms.  When my learners can see and make those connections on their own, that is where the real power of mathematical thinking emerges.