math inquiry

In our last math class, my "aha moment" came when we learned how to use algebra tiles to uncover algebraic concepts. It took a couple of example problems, but once I saw how the algebra tiles worked I realized their potential as a learning tool - especially as a visual representation of rules like "FOIL" - rules that we were simply given and expected to memorize back in the day.

A question I have about using inquiry based techniques like algebra tiles for teaching math concepts is how to introduce them to a classroom with a scripted, direct instruction based, curriculum. My gut feeling is that these techniques would increase comprehension in my main placement classroom. However, I can see an inquiry based approach like algebra taking up more teaching time, therefore, the idea would meet with resistance.

In my main placement classroom, as stated, my feeling is that using manipulatives like algebra tiles would help some of our struggling kids "get it." Simply re-iterating the rule isn't working for them. We're currently using a FOIL technique called multiplication wrestling to teach double digit multiplication. The kids take each double digit number in the problem, separate it into 10s and 1s, and then multiply all the numbers together using FOIL. For instance, the kids would write out 32*49 as (30+2)*(40+9). Then the 30 and the 2 need to wrestle the 40 and the 9 (30*40, 30*9, 2*40, 2*9). The results are added together to get an answer. Many kids get this, but many do not, and going over the rules again isn't helping things sink in. I'd be interested to see whether these kids are helped by using a manipulative technique.

patterns and rules

This past week, I learned how pattern recognition can help students find a rule and develop algebraic thinking. In honesty, while I've certainly detected patterns to solve problems before, using patterns to define rules is not something I've spent much time doing. Growing up, rules were given to you. You memorized them. It did not matter why or how they worked. Using inquiry methods to derive a mathematical rule from a perceived pattern - and expressing that rule algebraically - isn't something I ever remember doing in school. While I developed a real appreciation for inquiry methods in last quarter's math methods class, for some reason last week's activity "Crossing the River" (in which we determined how many trips, subject to certain rules, it would take to ferry a given number of adults and children across a river) really reinforced their value.

One question I have about using patterns to develop algebraic concepts is how to differentiate instruction for varying ability levels. For example, some students may not be able to detect a pattern when solving a problem like "Crossing the River" and members of their learning group may not be able to successfully explain it. Perhaps there is another entry point for this exercise, but I am not sure what it would be.

In the classroom, while we may choose to do an exercise such as "Crossing the River" and be resolved to allow as much time as is needed for every student to achieve comprehension, the reality is that we may not be able to. Running out of time seems to be a common occurrence, in my classroom experience thus far. On the other hand, if we're able to successfully differentiate an exercise like "Crossing the River," small group or math station work lends itself to differentiation quite nicely. This strategy might better enable every student to succeed.