The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years.—Common Core State Standards for Mathematics, p. 8
CSS.MP1 Make sense of problems and persevere in solving them.
Christian was a first grader we worked with during the Spring of 2017. We were interested in investigating his mathematical thinking so we presented him with addition tasks, but did not have him solve the problems with paper and pencil. The task we gave him consisted of two numbers printed on a card, his first numbers were 20 and 13, and he was asked how much altogether. For this problem, he quickly responded 33. The second step of the task was to turn the cards over; on the backs of the cards were the corresponding number of worms printed for him to count. He counted the worms and his count confirmed his answer of 33.
We then gave him the numbers 19 and 13. He stared at the cards for 15 seconds and then said, “Oh, 23,” leaned back he appeared satisfied with his results. He was asked to count and upon counting he ended up at 32, looked confused, and said, “No, I got it wrong.” When asked how he got his first answer, he said that he thought of the three but forgot the one (pointing to the 1 in the tens digit of 13). “Oh, it should have been 33.” His confusion began when his count was not the same answer that he first determined. He tried to use his mathematical thinking to resolve this discrepancy but without success.
Our belief was that he had used a prescribed mathematical method that consisted of using place value to combine digits. As many teachers know, young children can be successful with this method. The difficulties occur when students are asked to cross a decade using this system and need to regroup. To confirm our belief, we gave Christian the numbers 14 and 13. He quickly answered 27 and then turned the cards over to verify his results. His current mathematics allowed him to be successful in some situations but not in others.
Christian has what I described as a prescribed method of mathematics. I use the term prescribed because it is not a method he developed, rather one that has been given to him. He has an understanding of an addition procedure he needs to carry out, but it is limited to a specific class of addition situations. The results were not meaningful for him as seen in his satisfaction with his initial response to the problem of adding 19 and 13.
So how can we help Christian to make sense of these problems? Understanding how is exactly what our work here at AIMS is about. Join me in my next blog as we guide Christian towards understanding and meaningful mathematics.