## Place Value Representation

**Tagged**base ten blocks, concept, concrete, concrete representational abstract, CRA, hands-on, mathy

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This research has major implications for math for students with special needs…but some of us already knew this!

Here’s a common word problem used for linear functions and equations (y=mx+b):

There are 6 inches of snow on the ground. Snow is falling at a rate of 2 inches per hour. Write a linear equation showing total snow as a function of time (in hours). The equation would be y=2x + 6.

Often the word problems like this are presented on a sheet of paper in isolation as an attempt to make the math relevant and to develop conceptual understanding. For students who have trouble with conceptual understanding, words on paper are likely too abstract or symbolic to allow applications like the one above to be meaningful.

The real life application is useful if presented more effectively. Here’s an approach to use the same scenario but in a more relevant and meaningful presentation. The photo above shows the current amount of snow – call it 6 inches. Students can be shown the photo to allow for a discussion about accumulation and for their estimates of the amount of snow shown. The photo below shows an excerpt from a storm warning. Showing this warning and a snow fall video can allow for a discussion about rate of snow fall and the purpose for storm warnings. Combined, this approach can lead into the above word problem.

Once the application is presented students can be asked to compute snow levels after 1 hour, 2 hours etc. Then they can be asked to determine how long it would take for the accumulation to reach 18 inches (the prediction for the day this post was composed). After computing the answers WITHOUT the equation the students can be shown how to use the equation – the “mathy way.”

The photo above shows 3 levels of task demands for children based on Vygotsky’s levels of development.

- On the left is a level in which the student can readily perform the task independently, i.e. he is doing something he already knows how to do.
- On the right is a level that is too challenging for the student to accomplish independently. It is something he cannot do and does not know how to do.
- In the middle is a sweet spot. The level involves tasks that are accessible to the student but with support – scaffolding.

In reading this is known as the “instructional level” – see photo below. Reading material is evaluated by determining how challenging it is for a student. Material that the student can read independently allows for some growth in reading ability. Material that the student finds too challenging would not allow for substantive growth. In the middle is the sweet spot – the Zone of Proximal Development.

We can do the same with math using scaffolding. In the photo below is work performed by a former 7th grade student of mine with Asperger’s who tested at a 1st grade math level. I used colored pencils and 2 sided tokens to support his work with integers (red for negative and yellow for positive) in a CRA approach. The color coding and tokens were like the swimmies in the photo above of the child in the ZPD. Eventually these supports were faded. Throughout this process I was constantly pressing him to do more with a little less assistance.

I want to emphasize 2 major points regarding this.

- Substantive learning occurs when a student has to step beyond his or her current ability level – the ZPD.
- Often in schools educators avoid this, especially for students with special needs, because we want students to be engaged and successful (in the short term). We often confuse being active with learning. The guy on the tricycle in the top photo was performing a task but was he learning? (Note: this is not a student with special needs but a guy having some fun.)

Here’s are a video that fleshes out this idea.

Factoring, as used in elementary school, is the act of changing a number into numbers that multiply to produce the original number. For example, 12 can be factored to 3×4. 3 and 4 are called **factors** of 12.

In the photo above, a “factor tree” is used to help identify the factors. I have often seen the factor tree used as the initial approach to teaching factors. I’ve also seen it used as the primary means of providing intervention for students struggling with factoring. Think about that. Students who didn’t understand the initial instruction that likely involved the mathy approach shown above were provided the SAME approach.

Math topics can be presented with a more concrete introduction which can allow for more in-depth understanding. The photo below shows a CRA approach to factoring. This approach can be used as part of UDL or as an instructional strategy for intervention.

Found this (above) cool example of corresponding angles (see photo below for explanation). This window photo could be a nice introduction to this type of problem by printing it out on paper and having students match angles as the teacher shows the photo on the Smart Board or screen.

Slope is one of the the most important topics in algebra and is often understood by students at a superficial level. I suggest introducing slope first by drawing upon prior knowledge and making the concept relevant (see photo above). This includes presenting the topic using multiple representations: the original real life situation, rates (see photo above) and tables, visuals, and hands on cutouts (see photos below).

A key aspect of slope is that it represents a relationship between 2 variables. Color coding (red for hours, green for pay) can be used to highlight the 2 variables and how they interact – see photo above and below.

The photo below can be used either in initial instruction, especially for co-taught classes, or as an intervention for students who needs a more concrete representation of a rate (CRA). The clocks (representing hours) and bills can be left in the table for or cut out.

Factors are “things being multiplied”, e.g. 2 and 3 are factors of 6.

Factoring polynomials like 5×2 + 15 above is one of the most challenging algebra topics, especially for students with special needs. To make these problems more accessible a more concrete approach is possible. The photo above shows how finding a greatest common factor (GCF) is possible using a hands on, visual approach. Click on this link to a folder with a document explaining this approach and with a document that is a master for these card cutouts.

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