in an interview, Zig Engelmann said that he doesn't believe in using manipulatives for teaching math. I thought it was universally accepted that kids start with counting blocks, fingers, etc

Here is what Engelmann and Carnine have written on the topic or the need for making mathematical connections:

*Connecting Math Concepts*presents skills in applications in a way that permits children to connect concepts.

...

If the program did an effective job of articulating connections between various concepts, the child who completes the first level would have a strong conceptual foundation on which to build...

How does

*Connecting Math Concepts*forge these basic connections?

The principal vehicle is the number line. The number line is potentially dangerous for teaching number facts because it provides children with a method of figuring out the answer without attending to relationship. To figure out 4 plus 3, the children simply go to 4 on the number line and then move three spaces further. Although children learn the right answer, they don't easily learn the relationship between 4+2 and 4+3 because they have never been required to do anything that focuses on the relationship. The just follow the number line as a "rote" procedure.

A similar problem exists with any method that permits children to count and figure out the answer to problems like 4 plus 2. Finger operations are logical. (Hold up 4 fingers; hold up 2 more; count all.) However, these operations don't prompt systematic learning of relationships because children are not required to attend to relationships. Line-making operations suffer from the same problem. They provide children with a reliable method for figuring out answers to problems; however, they do not induce fact knowledge or knowledge of relationships. Furthermore, they often militate against systematic learning of facts because so long as the child is able to use finger-counting operation or a number line, what motivation is there for learning facts?

*Connecting Math Concepts*, Level A, Teacher's Guide,

*pp*. 6-7.

## 8 comments:

I don't understand the quotation. CMC has plenty of work using number lines, and Zig's early work had kids using fingers to count. Does the quote mean they don't believe in it?

I used both methods with my 6-year old, and found they didn't work, because it just slowed his eventual progress. I found that memorization of math "triangles" (i.e., 5 can be broken into 0/5, 1/4, 2/3) seems to be the best long-term method, as it promotes learning of both addition and subtraction and is quick and easy once learned.

CMC has plenty of work using number lines with covered-up numbers. That is the difference that Zig was getting at. When numbers are covered up, students must attend to the underlying relationships to predict answers. The number line is used merely as a confirmatory device, not as a method for figuring out answers.

Two years ago I did a systematic literature search on this topic (use of manipulatives) and was surprised to find there was virtually nothing to support their use. There were qualitative studies galore (showing improved affect, student engagement, whatever), but with one or two exceptions, confined to using manipulatives to introduce a topic, there was no evidence that their use improved achievement or understanding.

Recently I was on a committee with some high-powered university folks (on another topic). Over lunch, I mentioned this manipulatives issue to one of the research experts present, who replied, "The reason you couldn't find anything is because there isn't anything to find." Everybody loves manipulatives, but they have not been shown to make a difference.

I have a copy of Power Math K from Singapore Math. There are no exercises with manipulatives but there are many pictures of things like 3 giraffes, 2 are crossed out and that represents 3-2=1.

The closest thing to a number line are examples on only one page (178). It looks more like a road than a line and the numbers are not covered.

I have found some examples from CMC

http://www.specialconnections.ku.edu/~specconn/page/instruction/di/pdf/math_feature_a.pdf

http://www.specialconnections.ku.edu/~specconn/page/instruction/di/pdf/math_sample_lesson_b.pdf

Is that what you mean by a math triangle, jh?

ari-free

ari-free

I couldn't see your link.

Please use tinyurl: www.tinyurl.com

It will shrink your web address.

jh

http://tinyurl.com/6r8t9j

and the supposed math traingles

http://tinyurl.com/6n29k6

ari-free

ari-free,

same idea.

I write a triangle with the big number on the top, and two numbers that make up the big number on each side of the base

10

/ \

6 - 4

Students learn to add 6+4 = 10, and can also learn that 10-4=6 and 10-6=4. I'm not sure it's DI-approved, but I think it might work for my son.

EM does the triangles.

DI does something similar; they call them number families.

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