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.