By Thomas E. Hodges, Malisa Johnson, and Kyrsten Fandrich, Posted November 10, 2014 –   

Highly effective teachers know their students well. They can attend to students’ mathematical thinking, and then design instruction that capitalizes on what students know and are able to do. Our collaborative efforts often center on contextualized tasks, or story problems, where students have opportunities to think about and reason through problem-solving situations. Yet we know first hand how challenging this work can be.

The contexts and design of problem-solving tasks isn’t taken lightly. We’ve found there is much more than the mathematics content to attend to. As such, we designed the Preparing for Problem-Solving Interview, which has helped us unpack how students approach problem-solving situations. The protocol mirrors the Burke Reading Inventory, which provides insight into students’ beliefs about the reading process.

Try asking the following questions of your students, one-on-one, and see what they have to say about problem-solving. If you cannot ask all your students, try asking the questions of a few students you are particularly curious about.

1. Tell me the name of a student in our class or another class that you think is a good mathematician.
2. What do you think makes her or him a good mathematician?
3. Do you think he or she ever struggles to solve a story problem? If so, what do you think he or she would do?
4. When you are solving a story problem, what are some different ways you can find an answer?
5. If you tried a way that didn’t work, what might you do next?
6. If you get an answer that you think might be incorrect, what would you do next?
7. If you were helping another student solve a problem, what might you do to help?
8. Do you think you are a good mathematician? Why do you think that?

Students’ responses provide insight into their engagement with the Standards for Mathematical Practice (SMPs) (CCSSI 2010). In particular, how students make sense of problem situations (SMP 1) and their willingness to draw upon a variety of representations (SMP 2 and 4) and tools (SMP 5) to solve problems are often visible when asking these questions. Furthermore, we are able to observe how students position themselves academically in relation to others and how they come to value the knowledge that others bring to classroom learning experiences.

When asked about strategies, students often talk about using calculators, counting on fingers, asking other group members, using manipulatives, and decomposing large numbers. Some students talk about contextualizing problem situations, creating their own stories to fit problem scenarios.

What do your students say? Post responses from your students in the comments section. In our next post, we’ll provide more details on our own students’ responses and how we have designed problem-solving experiences out of these responses to deepen and enrich students’ mathematical proficiency.

A Task to Consider

As a follow-up to the interview, trying having students solve this derivation of the classic People of Freckleham problem (Treffers and Vonk 1987). Think about what students said and did during the interview to support their work and discussion of the task.

The people of Freckleham are interesting creatures. Every Frecklehammer is different from the other and has at least one freckle and one hair but no more than three freckles and three hairs.

Make a list of all of the different Frecklehammers.

The mayor of Freckleham decided to improve the manners of his townsfolk. He issued an order:

When two Frecklehammers meet, the one with the most hairs or freckles will greet the other and say, “I have more __________ than you have.” A Frecklehammer might say, “I have more freckles than you have,” or a Frecklehammer might say, “I have more hairs than you have.” Or a Frecklehammer might not be able to say anything at all.

At a town meeting of all of the Frecklehammers, the greeting “I have more _________ than you have” was heard many times. How many times?

Your Turn

We want to hear from you. Try the Problem-Solving Interview and/or the Freckelham task; then come back to this blog and post your comments. You may also share your thoughts on Twitter @TCM_at_NCTM using #TCMtalk.

Citation

Treffers, Adrian, and Vonk, H. 1987. Three Dimensions: A Model of Goal and Theory Description in Mathematics Instruction—The Wiskobas Project. Dordrecht: Reidel.

	Thomas E. Hodges is an assistant professor of mathematics education at the University of South Carolina. He teaches field-based mathematics methods courses, capitalizing on opportunities for preservice teachers, teacher educators, classroom teachers, and elementary students to learn with and from one another. He published on the field-based design in NCTM’s 2014 Annual Perspectives in Mathematics Education and regularly contributes manuscripts to Teaching Children Mathematics, Mathematics Teaching in the Middle School, and Mathematics Teacher.
	Malisa Johnson teaches a self-contained fourth-grade class at Oak Pointe Elementary School in Irmo, South Carolina. In her thirteenth year of teaching, Johnson often hosts mathematics methods courses in her classroom and collaborates with university faculty and other classroom teachers on mathematics education publications. She is interested in productive discourse and students’ use of representations in mathematics classrooms.
	Kyrsten Fandrich is a Master of Arts in Teaching candidate at the University of South Carolina, completing her internship experience in Johnson’s classroom. She is interested in learning alongside her fourth  graders through careful attention to students’ mathematical thinking.

Archived Comments

I had a conversation with some Fourth Grade students about, "Do you think he or she ever struggles to solve a story problem? If so, what do you think he or she would do?"
Students had good response such as, "draw a picture," "act it out", and "circle the numbers in the problem."
We then worked on some multi-digit addition and subtraction tasks and we drew on that conversation to help them think about how to enter and start tasks.
Posted by: DrewP_77482 at 11/22/2014 9:08 AM