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.