Author Archives: David Pearce
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.
In my previous blog I introduced Christian. He had a prescribed method for solving addition tasks, but many times his answers were not accurate. In our second session with Christian our primary goal was for him to use a counting strategy when adding two numbers. We began by presenting cards to him with the numerals… Continue Reading
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… Continue Reading
In the Common Core State Standards for Math, counting-on is considered “a strategy for finding the number of objects in a group without having to count every member of the group.” Counting-on is an efficient way to add and we want children to count-on. Yet, many young children begin by counting-all. For example: Teacher [placing… Continue Reading
My 4-year-old grandson and I have been spending a lot of time together. Without him realizing, we have been doing and talking more mathematics around the house. The other day I was rebuilding a gate and while I was measuring a board to cut, he picked up the tape measure wanting to help out. So… Continue Reading
I recently attended the Annual Conference of the National Council of Teachers of Mathematics (NCTM) in San Antonio and came away invigorated and hopeful about our children’s future in math education. The creativity and passion on exhibit within the many sessions and workshops was impressive. I had numerous conversations with awesome teachers that eagerly shared… Continue Reading
Recently, while working with students, we offered up a situation where nineteen counters were placed under a cloth. Seven of the counters were pulled out and the students were asked how many remained under the cloth. One child extended ten fingers, pulled them back, and then re-extended nine. He pulled back seven fingers, one by… Continue Reading
We at the AIMS Center have been digging into research on how children come to know number. What I realize is that knowing the journey from perceptual to conceptual which children must take is important for teachers to understand, in order to effectively help children on their path to understanding number. We have discussed the… Continue Reading
The latest results from international testing in mathematics have been released and results show the United States finished behind Kazakhstan. Now I don’t know much about Kazakhstan, except this is the country Borat hails from. I’m sure Kazakhstan is a beautiful country, but it doesn’t jump out with me as a math powerhouse. What does… Continue Reading
“What’s wrong with the way I learned math?” Seriously, we all made it through school, a few of us even did well, and some even liked math. So, what’s wrong with the way math has been taught for years? The Common Core Standards have focused national attention again on math education. But the truth is… Continue Reading