The following post comes to us from Patricia Franklin (’15) who taught 6th grade math in Jackson, MS. She read Children’s Mathematics: Cognitively Guided Instruction and reflects below on her implementation of what she learned from the book in her classroom.
I have always struggled with creating a deep understanding of number concepts for my students consistently finding that my students struggled with their “number sense” especially when applied to integers. In an effort to address this issue, I have completed a book study on Children’s Mathematics: Cognitively Guided Instruction by Thomas P. Carpenter, Elizabeth Fennema, Megan Loef Franke, Linda Levi, and Susan B. Empson. I hope that my findings will be useful to not only lower elementary teachers, but also upper elementary/ middle school teachers.
Purpose of blog post:
- Overview and Review of the book
- Show Classroom Applications
Children’s Mathematics focuses on explaining primary children’s intuitive understanding of addition, subtraction, multiplication, and division as well as development of mathematical thinking and classroom applications.
Elementary students choose to use different strategies based question type. The book starts by identifying types of questions for addition and subtraction: join, separate, part- part- whole compare, etc. The authors then show different strategies (such as direct modeling, matching, counting up, counting on to, counting down, pictorial representations, and number facts) to help children develop a deeper understanding of the question. The authors recommend that students be able to flexibly choose which strategy to use. As students advance similar strategies are offered for dealing with multiplication and division (i.e. multiplication, partitive division, and measurement division). The authors then give explanations using the strategies provided including counting, written representations, area representations, and number facts.
Overall I found the explanations of children’s intuitive thinking to be mirrored in my own classroom; however, as a teacher of6th grade math many of the particular strategies were not advanced enough for my students that have already developed addition, subtraction, division, and multiplication concepts. In order to develop more genuine mathematical understanding, I took the same conceptual thinking and applied it more advanced problems involving integers.
Here is an example:
-5 + 3 =?
Matching Strategy: “Students use one-to-one correspondence between two sets until one set is exhausted. Counting unmatched elements gives the answer.”
My students began with counting out 5 red bars for negatives and 3 green bars for positives. They then lined up the numbers and counted that there were 2 extra red bars. Thus, the answer was negative 2.
Counting On From First: “A child begins by counting on from first addend. The sequence ends when when the number of counting steps that represents the second addend has been completed.”
Me: You owe Jazmine 5 dollars. You pay her back 3 dollars. What integer represents how much you owe her now?
Me: You now have two dollars?
Caitlyn: no I owe two.
Me: Can you show me a way to get there using counting?
Caitlyn: -5 … -4, -3, -2… oh -2
My takeaways from the book and its application to middle level math is that students develop their understanding of integers in the same way that they developed addition and subtraction in their primary years.
Once the cognitive processes of the students are understood the book dives into teaching applications, specifically how to help students understand what problems are asking and how to support each other in the solving the problem.
The authors believe that when a class is “unpacking a problem,” the focus should be on the story comprehension before attempting to connect the story to math. Students should be expected to share their thinking, ask follow up questions, and use multiple strategies consistently with other students. The teacher should not impose their ideas. The problem solving needs to occur organically from students.
At this point my thought was… wait… my students do not do this all on their own… The authors agree that ‘organic’ discussion does require some teacher guidance and thus they have given stems to guide this process:
- Is your way the same or different?
- How are these two ways the same or different?
- Can [Student 1] explain [ student 2]’s work?
- Tell me one thing about your own work
- Tell me one thing about [student 1]’s work
- What’s next?
- Do you want to ask a question to a peer?
The fundamental take away from these questions is that it takes time and practice for students to critique each other. I modeled how I expected them to ask questions and answer questions at first. This seemed to help ease the transition. Additionally, by having higher performing students explain their reasoning and “easier methods” to lower performing students, my lower level students eventually started to solve problems with the same fluency as my higher performing students. Eventually as students become accustomed to this method the conversation does become ‘organic’.
Overall, I have found that this book has enhanced the way my understanding of student problem solving. Moving forward, I intend to use the questioning strategies and teaching practices in order to enhance my students conceptual understandings. Feel free to ask any questions or comments about the book.