12 2/3 + 7 3/5 = huh

The other day I was watching my intern teach a math lesson on using the algorhythm method to divide fractions. I was working with a young person who had no real concept of what a fraction was a month prior, indeed she did not understand what multiplication actually represented, much less her multiplication tables. We had been doing conceptual work such as open ended fraction questions (a la Dan Meyer) and manipulatives and diagrams so that she could visualize fractional numbers and problems.

She had made good progress in the weeks prior and now we were teaching the algorhythm method to divide fractions (invert and multiply, divide the top, then the bottom, etc.). As I worked with her, it became more obvious than ever that teaching the algorhythm alone is a tremendous obstacle to real learning in math. With this student, in this instant, I decided that if she could estimate the answer of a dividing fraction question, that would be enough.

On Friday, a few teachers had a discussion in my room as to why kids do not do well in math. I made the statement that much of it is due to the poor teaching methods of teachers. Alfie Kohn has posted a great article on Math instruction! He says:

"...that traditional forms of teaching, and an emphasis on the basics, contributed significantly to the low standing of older American students....instruction in this country still seems – compared to instruction in some other countries – more centered on students as passive absorbers of knowledge rather than as active participants who construct, transform, and integrate knowledge.”

He cites studies from Japan where math instruction may be different:

"...three out of five U.S. teachers said they were chiefly concerned with “skill building.” Only one out of four Japanese teachers responded that way: the overwhelming majority said they wanted their students to understand a particular math concept. That goal led those teachers to include deductive reasoning in their instruction, which played a role in 62 percent of Japanese lessons and 0 percent of U.S. lessons. Japanese teachers also explored the intricacies of specific mathematical concepts with their students rather than just naming those concepts, American style. In Japanese classrooms, fewer math problems were considered in more depth, and students participated actively in suggesting different ways of solving those problems. Also, interestingly, homework was rarely assigned."

Teaching the algorhythm alone seems to me to be the most unproductive way to teach math, or conversely, the most productive way to produce disengaged, confused math students! Much of what I read these days backs this up. In a recent post, Conrad Wolfram claims that we spend 90% of teaching time in math on computation and not enough on the conceptual and real life application. Kohn goes on to say:

"the research conducted on such programs has been concentrated in the primary grades, and it points to a result that can be summarized in six words: better reasoning without sacrificing computational skills...in one study, forty first-grade teachers in Wisconsin were given special training in how to make problem solving the organizing focus of teaching arithmetic. When achievement tests taken by their students were later compared to those of traditionally taught children, the results showed a modest, though consistent, edge for the former group. “A focus on problem solving does not necessarily result in a decline in performance in computational skills..."

I love teaching math and the challenges it brings. There are a world of educational opportunities out there for math teachers. 21st century learning applies to math in a big way.