Once upon a time, there was a man named Dr. Yeap Ban Har who came to Houston to save us all from our misconceptions about learning math. Dr. Ban Har is incredibly knowledgeable in his field, so he was a confident presenter. He is also a working teacher, which gives him current and vivid experiences to share with his audience.
I was initially overwhelmed with all of the new information, but I was able to go home after the first day, do a little studying, complete some practice problems, and enter day two with more confidence. Thank you to these authors and titles for assisting in my daily growing knowledge:
|Check it out on Amazon|
|Check it out on Amazon|
The Singapore Model
Probably the most important idea behind the Singapore model is in the way that the learning is presented. I see it as being a continuum, moving from the concrete to the pictorial to the abstract. They call it the "CPA" approach.
Students begin by problem-solving using concrete manipulatives, things they can move around and hold in their hands. I was of course picturing counting cubes and plastic bears, which were the things I recalled using in my early mathematics years. Manipulatives can be anything, though, as I quickly learned. However, the consideration that should be given when choosing these tools was surprising to many of us. It is important to vary the manipulatives being used, so students don't attach a concept to a certain item. It is also important to make sure that all of the objects being used in a lesson are the same noun. Meaning, you don't just want to put a smorgasbord of items for students to use. Younger students need even more consistency with their manipulatives, so even keeping their counting cubes the same color when introducing a new concept is helpful. It all comes back to the way young brains work and the way we learn our worlds. The brain is fascinating!
This first step, the problem-solving, is crucial. The kids really need to be allowed to work out as many possible solutions as they can think of, and these solutions need to be questioned and valued by the teacher. They need to be able to explain their thinking processes. The key is not to learn the rules, but instead to learn WHY these rules work. They are not presented with the rules in advance, but simply asked to solve a problem. I will say, this approach would have definitely sparked interest in my young mind. I liked the challenge of solving problems; I could learn rules, but there was little reward in getting a right answer that I had been told how to get. Figuring out something on my own gives me a since of pride, and I think this approach can give the kids that same since of pride as they are learning key math concepts.
Once students gain more confidence in a concept (like addition or subtraction) they are ready to move to the pictorial phase, where they will instead draw their own models on paper. Model drawing was a brand new concept for me, but I LOVE IT!!! I don't think I can explain it and give it justice here, but the Step-by-Step Model Drawing by Char Forsten presents it in a very easy way. I just started at the very beginning of the book and worked through each of the lessons. Then, I did some practice problems in my notebook.
This isn't the only way students work through the pictorial phase. They can also draw pictures of the same objects they used in the concrete phase. The important thing is that they are using pictures to assist them in problem-solving.
Finally, students are able to move into the abstract phase, which is where they can work problems on paper and understand the rules that govern these processes. This can't happen until they understand the WHY behind the various mathematical concepts. They have to have AMPLE time to test out their theories and problem solve without the fear of being "wrong." This isn't about memorization, which is what I did in school--I still have flashbacks about the multiplication flashcards. This is learning to think and learning to explain that thinking. It is fantastic!
A Single Lesson
There is so much that I am not going to be able to include here, but I feel that there is one more important element to add before I end this post. After the CPA approach, which takes place across the years as students are introduced to and continue to expand their knowledge on mathematical concepts, a valuable thing I took away from the workshop was the 3-part lesson format. Each lesson follows generally the same format.
1. Anchor task (about 20 minutes): Students are asked to solve some sort of problem using manipulatives. More advanced students may instead use pictures, especially if this is a review of a certain concept. This is the phase where the teacher provides instruction and assistance in the task. She doesn't tell them the answers or how to do it, but instead gives them assistance and asks questions about their discoveries. She is also very aware of her language choices, making sure to use mathematical vocabulary consistently.
2. Guided practice (about 20 minutes): Students work in small groups to continue problem solving. They may use manipulatives and come up with possible equations. Again, this will depend on the concept and the stage of the learners. The teacher observes and offers support. Students may get assistance from peers, who will also be using their own manipulatives or models to solve the problem.
3. Individual practice (10-20 minutes): Now that students are understanding the WHY and the HOW, they practice while the teacher observes and assesses. Some students may need to continue using manipulatives to assist them in this process, while others may be moving into pictorial or even abstract methods. Again, it depends on the learner's stage of development. The beauty is, this method allows for easy differentiation.
I feel like there is so much I could not add in a single post. Hopefully I was able to give you an overview, though. If you would like more information, feel free to contact me and I'll gladly share my knowledge with you. :)
I will leave you with a sample word problem that Dr. Ban Har gave us. If you figure out an answer, post a response below. (HINT: READ CAREFULLY)
Four sisters, Alice, Betty, Charmaine and Dolly, had a total of 260 sweets at first. Mother gave Alice 20 sweets. Betty ate 10 sweets. Charmaine bought some sweets and her share doubled. Dolly have half of her sweets away. As a result, they each had the same number of sweets left.
How many more sweets did Dolly have than Charmaine at first?