What makes "Singapore Math" so effective?
1. Focus on the use of the Concrete - Pictorial - Abstract Cycle. And not as just a remediation tool. As students move through these stages of understanding they build important strategy connections and move to generalization of concepts. CPA is a cycle not a linear progression.
2. Focus on Visualization and the Use of Heuristics. Visualization comes from the process of building and drawing. It is that magic moment when the concrete or pictorial have suddenly moved to an image in your mind. You no longer have to build or draw it. For me, this happens in multi- digit multiplication now. I finally have lost the visual of stacking for multiplication in my mind. Breaking apart the numbers is what I see now. For instance, I see 314 x 7 as (300 x 7) + (10 x 7) + (4 x 7). No regrouping for me.
Heuristics are structures that students create to make meaning for themselves. One heuristic I like is: What do you know, what do you need to know, what can you draw, what can you do. Another is bar models. Neither have a "right" way to do them. It is like writing a paper in high school, if it makes sense to me and to my reader, it makes sense.
3. Focus on the Use of Mathematical and Perceptual Variation. Every problem has a purpose in developing student understanding. Problem strings vary by mathematical complexity or by the visual models used to represent them. Learning occurs in the bridges between the problems.
4. Focus on Problem Solving as the Heart of Mathematics. The goal of mathematics is to solve real world problems. Notice I didn't say WORD problems but real-world problems. There is a difference. No longer is the role of the mathematician calculating endless strings. Mathematical modeling and application of mathematics to solve problems in the world around us is the goal. Doing this in a way that is collaborative, flexible, able to be communicated and justified is not a bonus but the whole purpose.
All of this occurs with the support of the readiness, engagement and mastery cycles and a three part lesson structure that helps students move through exploration, guided practice and independent practice.
I have had the wonderful opportunity to work with interventionist and Special Educators all over the world. Each time I get asked, is Singapore math for "my" kids.
Good intervention needs to be systematic and not just focused on fact fluency.
There are four main principles of good intervention:
1. Use of the Concrete to Pictorial to Abstract approach
2. A Systematic approach to number strings
3. A recognition to production to extension approach.
4. Contextual Relevance
Back in graduate school at Northwestern, when the field of learning disabilities was still in its infancy, my mentor, Dr. Doris Johnson used an analogy of a black box to describe learning disabilities.
She said, we need to be thinking about the input (how we are giving students the information); cognitive load and processing inside the box (or the student's mind) and the output (how they will communicate their understanding)
To me intervention lies in the bridges. How can I systematically increase the cognitive load while creating bridges between the Concrete, Pictorial, and Abstract? How can I help students to generalize rather than learn discrete tasks.
For instance: Grade 3 standard of subtracting across zeros, when given a subtraction story:
1. Does a student recognize when regrouping is necessary? Using tens frames, in the concrete, with numbers less than 20? If I show them 15 -7 can they tell me that they will have to break into the full ten?
2. Can they produce the regrouping themselves? Still in Concrete with number less than 20?
3. Can they recognize when regrouping is necessary with base ten materials with numbers less than 100? In problems that only require regrouping in the ones place?
4. Can they produce a regrouping in the concrete with regrouping in the ones place.?
As soon as I see an understanding in this trajectory, I quickly move to the next. And by systematically moving through steps, I can help a student to grow not only their understanding of content by their mindset of success. Ohio has produced some amazing support for breaking down the standards. See them on the Great Teaching Resources pages.
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...