"Master Comes from Variation not Repetition" -- Yeap Ban Har

I have been watching the increased focus on using good tasks in math. Thankfully, Singapore Math users have know about the importance of Anchor Tasks for a long time.  But is using a good task enough? 

Sure great tasks help students connect to learning better, retain information longer, and make math relevant but do they help build generalizable mathematical patterns that help students to retain mathematical concepts and procedures longer? 

Years ago, I taught from a curriculum full of great tasks. But my students, in the long run, still didn't remember the patterns that lead to the the rules they needed to solve  novel problems. 

So how do you teach generalization as well as concept development? The key lies in not only having purposeful tasks, but in also using deliberately crafted number trajectories.

Consider the following:   
242-128 in which students discover that regrouping in the ones place might be necessary
278 - 39 in which students discover that the rule still applies even if the hundreds in not represented
763-207 in which students learn how to deal with a zero place holder
317 - 172 in which students discover that regrouping might also apply to the tens place
317 - 25 in which students discover that the rule still applies even if the hundreds in not represented
432-178 in which students discover that the regrouping might be necessary in both the tens and ones place
500 -45 in which hopefully students use mental math but could regroup and encounter zero placeholders.

This deliberate attention to the number trajectories and the order in which students experience what Piaget called Disequilibrium and then Accommodation (the process of encountering information that doesn't fit your schema and then expanding your schema to accommodate the new information) builds generalization of mathematical principles. This is a key hallmark of the Singapore Math. 

Developing number trajectories is hard work. In writing Singapore Math Fact Fluency, it took days to develop even a half page of problems and then several emails with my Singaporean colleagues to discuss the build of the problems and rationalize their choice. 

"Mastery comes from variation not repetition" is one of my favorite quotes from Ban Har. (There are many after almost 10 years of working together!)  Variation means number trajectories but also variation in physical and visual models and in finding multiple methods or multiple solutions. Topics for another time... 

For now, I continue to marvel at the beauty and complexity of these carefully designed strings...