In the August and September installments of my blog, I’ve been telling the story of Christian and our mathematical interactions with him. Christian is a second grader who came to us with mathematical skills that had been taught through his first years of schooling. He was bright, eager to work with us, and considered, by this teacher, to be good at math.
We gave him some addition problems. He applied his understanding of the procedures needed to solve these problems. For some of these problems, he arrived at the correct answer, but there were many more inaccurate answers that didn’t make sense without explanation. For example, answering that 27 & 9 added to 40, or 19 & 13 adding to 23. For Christian, his results did not cause him concern and he did not spend time worrying if it was right or wrong. He had a process for addition, used it, and arrived at an answer.
After the first week with Christian, our goal was to have him use counting as a method to solve the problem. The belief was that through his counting actions he would solve the problems and the results would make sense. As I wrote in my last blog, once he began to count, using his fingers as tools to monitor his counting of a second addend, he arrived at answers that were accurate and this surprised him.
The following week we were back with him once again presenting addition problems. After beginning the session using his “thinking in my head” method, his results were incorrect. After two problems he switched to counting to solve problems. It was tediously slow at first, as he worked each problem multiple times trying to get results he was confident were correct. As he became comfortable with this method, his speed increased, he was accurate, and his enthusiasm for working the problem increased. Using his counting became an efficient and effective method for him as he continued to work.
Some might have said that Christian did not have “good number sense” based on the results we experienced when we first visited him. I would say that his lack of having “good number sense” occurred through the prescribed methods taught in the classroom along with resistance to using fingers. Opportunities were missed which would have allowed him to build his number sense.
With continued work, I believe Christian could develop increasingly sophisticated ways to think about number along with operations he would be able to perform on them. The skills needed for mathematics are an outcome of the mental activities and understanding developed while engaging learners in ways that make sense to them. Mathematics education is thought to be a series of skills that students are to master, with previous skills laying the foundation for future work. The authors of the Common Core Standards included a different view. They have incorporated eight mathematical practices which have a focus on students’ mathematical thinking. Using student thinking to build on their mathematical knowledge allows for students to make sense of math, increase understanding, and develop proficiency and skills.