Teaching students to add appears to be a very linear, skill driven endeavor. Hidden in this is the concept of what it means to add and how to assess this conceptual understanding. Here is an approach to address and assess the concept of adding.
In the photo above a student is prompted to pull both groups into 1 pile (see photo below). The word, add, is not addressed. The symbol is absolutely not introduced yet.
Once the student has demonstrated a consistent performance of pulling the groups into 1 pile (addition) two other tasks are introduced, taking away and sorting. The student is presented each of these individually (field of 1).
After showing consistent performance in demonstration of these skills, the skills are then presented using a generalized mat (see below).
Then two skills as pairs. First “pulling together” and “taking away” are randomly prompted individually, e.g. “pull into a pile” using the generalized mat above. Then combine “pull together” and “sort” then “sort” and “take away.” Finally all 3 are randomly chosen (field of 3).
Word problems are used in an attempt to address conceptual understanding. Often the problems are diminished to following a mere algorithm. For example, The cookie and tadpole problems are reduced to identify key words such as “total” and “left” (see photo below). I suggest a concept first approach. Show or provide hands on or visuals (you can cut out the cookies and tadpoles for hands on) and engage the student in the concept of adding and subtracting before introducing word problems. Develop this conceptual, intuitive understanding so the words and situations in the word problem are more concrete to them.
Educators teaching math typically start with the “mathy” stuff first. For example, for finding the sales price teachers may start with showing students the steps to calculate (photo below).
I start with the concept, either with a pictorial representation or actual objects to represent the underlying concept. In the photo above, I show an object (related to the student’s interest – this student is into weight training) on sale. The $50 circled in yellow represent the original price. I explain the concept of being on sale and discount and show that 20% is $10 to take away (marked out). This leaves $40 (in green) which is the sales price. This allows for conceptual understanding before showing him the “mathy” way of doing the problem.
All too often math topics are introduced first with the skills and steps. This is backwards. The photo above shows how I introduced solving equations a high school student with autism using the concept as an entry point.
We discussed what was involved in buying a car, including payments (no interest) then I posed the problem seen at the top. I asked him to figure out the monthly payment. He worked out the problem, overlooking the down payment. With a minimal prompt he self corrected. I followed this by “showing him the mathy way of doing the problem.” (Seen in the bottom half of the photo). He conceptually understood why the -1,000 was the first step and x had meaning.
This is a version of CRA.
Math is an esoteric subject for most people. Good instruction makes information meaningful. One method for making information meaningful is to connect new information to prior experience.
In this situation the new information involves determining whether shapes are similar (see photo below). One example of student prior experience with this topic would be shrinking people down. In the photo above I use Mini Me and Dr. Evil and their respective (and fabricated) weights and shoe sizes as measures that will eventually give way to measures of sides of a polygon (below). When working on the problem below the students can be prompted by recalling the analogy of Mini Me and Dr. Evil.
The photos below are used to introduce length and area as part of a CRA approach. First a student is asked to build a Lego garage. He first builds the bottom row of a wall and the teacher asks for length in terms of how many Legos are lined up. After building a wall the teacher asks for area in terms of how many Legos are used in the wall. Then the student is given a handout with the following photos. Following this handout the student finds length and area of tiled floor and walls made up of cinder blocks, if available. Eventually a ruler is introduced and multiplying to find area is presented at the end.
For the photo above the student is asked to count Legos to compute length. In the photo below the student is asked which is longer and to explain.
In the photo above the student is first asked to determine the area of the red wall in terms of number of Lego squares. Then the student is asked which wall has more area. This is followed by the photo below. This allows a different perspective of area.
This is a photo of a handout from a special ed conference shared by Dr. Shaunita Strozier (firstname.lastname@example.org) of Valdosta State University. It shows the use of algebra tiles to provide a concrete level of the concept of an equation and solving the equation. The photos on the left show the R or representational level of the concept. Her approach is called SUMLOWS which is an acronym explained in this handout.
This is a follow-up to a previous post about using comics to engage a student with autism. In the photo above the comics are on sale for $2 each which is a situation modelled by a linear function. The student finished this assignment independently, quickly and accurately (aside from the bars on the graph). (Note that he did not need the table at the top that shows the different number of comics.) In another previous post I explained the different levels of representation (CRA) with the equation being the most abstract. In this handout the equation was the last item addressed and was computed using the graphing calculator (Linreg function).
The following photo is from a warm up given prior to the assignment above. It addressed prerequisite skills for the intro to linear functions assignment. This student confused the x and y axes (you can see some of the points from his initial effort) and I used color coding to clear this up.The following photo is the student’s effort on a follow-up problem, which he completed correctly (and again did not need the table at the top).