The Rookery

Algebra — it's often a topic reserved for middle school and, for some it's a topic that produces a shudder.  Many remember it as the time when math "stopped making sense."  But times have changed.  We begin teaching algebraic thinking in kindergarten and we've just completed a unit on algebra in the 4/5s.

We grouped the students according to their familiarity with the concepts.  Many of the 5th graders had worked with these ideas before and were ready to apply their learning.  For the 4th graders, there were a lot of new concepts to learn.

My group had had the least experience with the ideas we were going to explore.  I began by sharing what I've found to be an excellent way to describe algebra, "You use what you know to figure out what you don't know."  Of course, it can become very difficult to figure out what you know but the phrase is a useful tool.  It sets up this strand of mathematics as one based on puzzles.  Puzzles aren't black and white — you don't get the answer the second you look at a good puzzle.  Instead you have to piece it together.

Too often in math, students look at a problem, don't see the answer imediately and declare that they don't know it.  Approaching problems as puzzles freed the students up to be O.K. with not getting it right away.

All of the groups began their explorations of algebraic thinking with a puzzle form called "pan balances."  The scale represents the equation and the student works to keep the sides of the scale balanced while figuring out what each of the shapes or symbols is equivalent to.  Those students who had worked with pan balance problems before learned how to write the equation they represented.  Students then went on to work with coordinate grids.  For some this was an introduction while others learned to graph functions on coordinate grids and use those lines and curves to predict answers to the function as the variables changed.

One of my favorite moments came when I introduced the coordinate grid.  Since I had the group who had the least experience, I had decided to have them work with the grid and get used to coordinate pairs such as (4, -2).  I explained that while this was related to algebra, I wasn't going to have them use the grids to do algebra and that they'd do that when they got to middle school.  As we plotted our first couple of points, one of the students stopped me.  "I think I've figured out what this has to do with algebra!"  He went on to explain that "when you put in a 3, it weighs 9!"  We had just plotted (3,9).  I squealed.  (Really, I did, it was kind of embarassing).  I plotted (1,3) next and then a few other students started to see.  "Oh!  It's like you have one on one side and three on the other so if you had three it woudld be nine!"  I squealed again then raised an eyebrow, "So how much would a 2 weigh?  Where should I put the dot that goes with 2 on the x-axis?"  "SIX!" responded a chorus of voices.  One voice said, "I can't believe we figured out something that middle schoolers do!"

That sense of discovery is crucial when kids do math.  Without it, math is only a series of unrelated things to memorize.  There is no joy and no deeper learning.  When one "discovers" something one has made new connections and has broken through the cognitive dissonance of learning.  It feels great.

By the way, the pan balance puzzle above can be solved – have some fun of your own.