Differentiated Coaching for Educators

Frequently, I lead mathematics teachers through a problem from the wonderful site, www.nrich.maths.org, designed to help students develop mathematical conversations skills and value cooperative work. The problem asked participants to solve equations such as 19*24; 227 + 198; 57.6/2; 101*16*4, and so on, and then work as a group to agree on the most efficient method for each problem. For the first one, teachers often use the standard algorithm for multiplication. However, some will multiply 20*24 and subtract 24. Much faster. THEN the rest of the teachers immerse themselves in finding elegant ways to do the rest of the problems. They begin playing with the numbers. My favorite method is changing the last problem to 101 * (2 to the 6th power). The teacher who came up with it said, "It's really not a fast way but I can't recall the last time I was so engaged with an arithmetic problem!"

The ensuing discussions highlight many dilemmas teachers face, but the fact that they all have the common experience of delighting in problems that are usually given to students as rote, procedural tasks creates an atmosphere of openness. Instead of defending practice, the teachers begin to seek tools and solutions. A few discussion points:

--"But sometimes the algorithm is faster. We need to teach them." Yes we do. But groups also agreed that students make careless mistakes because they don't think while calculating. We talk about how a string of two-digit multiplication problems, with some lending themselves to different pathways and others to the algorithm, would help students think about strategies before starting a problem.

--"This would confuse special education students." The task would work for these students if the first set of equations they received involved 2-digit addition, all of which benefited from thinking about the numbers rather than blindly applying the algorithm (such as 18+22). "Mathematicians are lazy" could be a fun frame for helping these students begin to enjoy arithmetic.

--"Some students will just copy from classmates." We talk about the power of asking for justification, since copying won't help in explaining the method used. Since students often lack experience in cooperative learning or rich all-class discussions, we discuss how they could begin with these problems as an all-class activity, reinforce getting students to attempt the problems before beginning the discussion, and using teacher moves that hold all students accountable for participating in the discussion.

My real goal is teachers to see that by turning more of the work of thinking over to students, not only would students have more fun with math, but they as teachers would hopefully spend less time dragging fact-based answers out of students and more time enjoying discussions. One teacher who had already tried an all-class discussion we designed together said, "It didn't go perfectly--in fact I had to backtrack and have them set rules for discussion. But when we tried again, every student participated. And I was far less tired at the end of the period than on other days with that particular class. It worked--and we did far more math, too!!"

Ready for fun, anyone?