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What is the Concrete →Pictorial →Abstract Approach to Teaching Math?

Part ONE: Kindergarten – 3rd Grade 

Singapore Math is known for being based on decades of research about how children learn mathematics. How does it work? Why has it been so successful in international tests? Let’s take a look at some classrooms:

“Concrete” experiences in math class – the use of physical objects

Kindergarten:

Kindergarten is of course the grade level using  the most concrete activities. Here’s a recent example:

Students drop a handful of  blocks onto a sheet of paper. They predict whether more blocks are on the left or right side of the middle line. They then “check” their prediction by lining up the blocks on the ten frame as pictured at right.

What are the benefits of an activity like this?

a) The students get excited and involved in the dropping process, and the predicting of a “winning side”. They are pleased when they prove their predictions right, and rightfully proud. This produces a feeling of joy in he classroom that would not exist in a lesson based on worksheets, pencils and memorization.

b) The students are creating a visually memorable organization of what used to be a chaotic mess of blocks. They are seeing the patterned grouping of fives that is so fundamental to our base-10 number system. Six is obviously one more than 5, nine is easy to see as one less than 10. We can ask them “What would happen if they moved a block from the 6 to the 9 group?” The child moves the block, and sees that the old total (15) is exactly the same as the new total. If they say 10 and 5 are easier to count than 9 and 6, we nod and agree. “Math is good for getting things organized,” we comment.

 

1st Grade

Early in October, Keys 1st graders spent a good half hour inventorying the hundreds of math manipulatives in their classroom: Blocks, dice, counters, shapes, cubes, etc.  They realized that counting them one-by-one would not work well – there were just too many of them! So they decided to bag them in groups of 10 (“Because we always do our counting to 10, and then we start over.”)

Working in groups, they managed to bag over 1000 items, and proudly lay them in groups on the carpet. Each group totaled their own row (by counting by 10s). There was a palpable “buzz” in the air – everyone loved the project.

What are the benefits of an activity like this?

a) The concept of “one hundred” is difficult for young children. They can count that high, but usually do not yet have strong visualization of how much 100 actually is. Or that 82, for example, is 8 tens and 2 ones.

b) The relationship between numbers is just emerging. For example, verifying that 17 and 22 are just as far apart as 57 and 62 would still require counting for most young students. Our goal is an internalization of this concept.

c) The act of bagging by tens reinforces the fact that tens do not equal ones. They realize why they must count four whole bags and 2 small blocks as 10, 20, 30, 40, 41, 42 – not 10, 20, 30, 40, 50, 60.

2nd Grade

Students played “Cubes under Plates” by hiding base-10 blocks while a partner turns their back.  If Student #1 hid 30 cubes in total, how many cubes are still hiding under the 3rd plate?  Since this is played without pencils, it requires children to “see” objects they cannot really see – the definition of visualization.

Students will solve this differently, depending on their current development level.

  • One student might count the 14 blocks on the right, then continue counting the 12 on the left (15, 16, …) to get to 26, then count up to 30 in their head and say “4”.
  • Another student might count 12 on the left, “see” the 14 (ten+4) on the right and add them together mentally (“12 + 10  +  4 =  26”) and know that 26 is 4 away from 30.
  • A third child might be ready to play with much larger 2-digit numbers, like 57 and 44, given that 110 blocks were hiding. This requires them to group the tens mentally (90), then hold that in their memory. Then they group the ones (11) and add it to the 90. Often they break up the 11 into 10 and 1, so that they can count mentally, “90…. 100 …. 101”. If we started with 110, and already uncovered 101, there must be 9 hiding under the last plate.

What are the benefits of an activity like this?

(a) Number sense. The use of activities without pencils forces children to truly observe base-ten blocks and internalize their relative values. The sentence 57 + 44  is an abstract symbolic representation of the actual quantities shown in the blocks. It is the final goal of our curriculum, but is inaccessible to many students if they have never had access to concrete materials.

(b) Long-term Achievement. The Concrete>Pictorial>Abstract (C>P>A) method succeeds for a much greater percentage of students than the traditional memorization approach. [1] [2] [3]  Math anxiety affects almost 50% of the US adult population, countries that teach using C>P>A score higher on international tests, and our colleges are increasingly having to teach remedial mathematics to their incoming students. All of this in an era where graduating with a firm mastery of mathematical concepts is increasingly important to success in the job market, to a lifetime as an informed citizen, and to thoughtful, logical decision-making in one’s private life. This is a waste of ability and human potential that we can no longer afford.

(c)  Equity. The speed at which a student acquires number sense is not as much a sign of intelligence as it is a sign of individual learning style and preferences. In our experience, about a third of all students can learn math abstractly, without its concrete foundations – the traditional “math people”. But this can also be deceptive:  some pupils who learn to calculate rapidly may not have as fully developed an understanding as we might expect. They may not be flexible in their thinking, they may not be able to apply their learning to tricky word problems, and they may struggle in algebra and beyond, when faced with the sheer volume of memorization needed.

On the other hand, about a third of students need to spend a lot more time at the concrete level. (with the remaining third somewhere in the middle.) We call it the Gift of Time – given time, all students  can master grade-level concepts and end up finishing 8th grade with a solid preparation for advanced high school math and beyond. We’ve seen it over and over in our graduates.

3rd  Grade

Try this question from a 3rd grade practice sheet. In 2nd  grade, we rearrange actual blocks that are breaking the base-ten “fire department rules”. (At the right, there are too many blocks in the Ones and Tens columns, for example – we call these “silly numbers”).

In 3rd grade, we use drawings of base 10 blocks. (This is the “Pictorial” phase.)Questions (e) and (f) afford us an example of how blocks can explain math. The “silly number” originally used 45 blocks (1 thousand, 7 hundreds, 15 tens, and 22 ones).

When we regroup, we have a normal base-10 number of  1,872.  (which uses 18 blocks)

Hmmm….45 is a multiple of 9, but interestingly, so is 18. Why is this?
In order to explain it, you almost have to hold the blocks and exchange them!  For example: you exchange 10 ones for one ten. You have a net loss of nine blocks (lose 10, gain 1) in every exchange. You end up always removing 9’s.

Voilà  – the mystery is explained!TO BE CONTINUED… Blog #2 will look at C→P→A in the upper grades.

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