Invented Math Strategies

Upon working with a student in my math group this past week, I discovered a very interesting strategy of solving computation problems that I have not yet seen. I would say that this strategy is a mix between using manipulatives and drawing out a visual representation. Therefore i think that it is somewhere in the mix of the direct model strategy and the counting strategy.

When I first observed the student solving a problem such as 34-19, I thought that he was just drawing 14 tick marks and then crossing off 9. Although this is similar to his method, his strategy proved to be a little more complex. He would draw out three large tick marks to represent the three in the tens place in 34, and he would draw out four small tick marks to represent the the 4 in the ones place in 34. He would do the same thing for the number 19 - draw out one big tick mark to represent the one, and nine small tick marks to represent the nine in the ones place. Then he would look to see that he did not have enough tick marks in the #34 tick mark group to take away nine, and so he would X out one of the large tick marks and turn it into ten small tick marks. Then he would X out nine small tick marks, and count up the number of small tick marks he had left. He would write this number down. Next he would take away 1big tick mark from the #34 tick mark group, and then also write down the number of big tick marks he had left right in front of the number of smaller tick marks he had left. He would fill in the answer right under neath the algorithm of the problem that he had previously drawn out.

It took me quite a while to understand this student's method of solving the problem, because frequently I had a hard time distinguishing between which was a big tick mark, which was a little tick mark, and which tick marks had already been crossed out. However, this is a strategy that this student has used for a while now, and as long and I was able to have him explain it to me so that I finally understood what was happening, and also so that he was able to properly let someone else know how to do the strategy, then I am alright with letting him use it to solve problems.

If I had to think of two other strategies that this student might use, I think I would see if he could fade the tick marks into just drawing out 34 tick marks ( all the same size) and then just crossing out 19 tick marks to see how many are left. However, I do realize that this method might be a little more remedial, because when the student gets to the higher numbers - like 68, it is not realistic to have them draw out 68 tick marks.
Another method might be to use that same tick mark strategy, but only write one tick mark for every five numbers. Therefore the student would get practice counting by fives and he would also not have to use as many tick marks. However this may also become confusing to transition between counting by fives and ones if you did not provide the student with multiples of five.

big tick marks :                 smaller tick marks:
I I I       ( 3)                                          i i i i ( 4)

I     ( 1)                                             i i i i i i i i i  (9)

                  after borrowing :

big tick marks:                 smaller tick marks:
11    ( 2)                                      i i i i i i i i i i i i i i  (14)      
 I       (1)                                    i i i i i i i i i (9)                          



=15