By Thomas E. Hodges, Malisa Johnson, and Kyrsten Fandrich, Posted November 24, 2014 –
Although Teaching Children Mathematics is
building a great collection of rich tasks on this blog, we thought it might be
a good time to step back and examine one way to support students as they tackle
the tasks. One way to do so is with the Preparing for Problem Solving Interview.
In our previous post, we discussed the
Preparing for Problem Solving Interview as a way to attend to students’
mathematical thinking. In
this follow-up blog, we share some of our take-aways from conducting the
Preparing for Problem Solving Interview. We then showcase solution strategies
to the People of Freckleham task and discuss how students’ interview answers
can be used to support implementation of the task.
We observed
patterns in our fourth-grade students’ responses that honored procedures, tips,
or tricks versus those with responses that honored processes and problem-solving
ideas. For example, Maya said that she “looks for words that tell me what to
do. Like when it says ‘altogether,’ that
means add.” Her use of word clues is of limited value. Maya would likely
benefit from opportunities to solve problems that contain no such clues or from
problems that include the phrase altogether
but do not involve additive reasoning.
In response to being
asked for ways that Cameron would try to help another student solve a problem, he
said he would rewrite the problem using a context familiar to the student,
using his or her name. Cameron’s ability to re-contextualize problem-solving
situations is an important problem-solving practice that he could share with
others in the class. Below is a summary of some of the responses we received
and the kinds of actions that could be taken.
| Student |
Reponse |
Action to be taken |
| Maya |
Said she uses certain words to tell her which operation
to use: “I look for words that tell me what to do. Like when its says,
‘altogether,’ that means to add.” |
Purposely composing and presenting problems where the
phrase altogether does not involve additive reasoning will show Maya
that clue words do not always result in the same operation being used.
Presenting problems where word clues are absent will force Maya to chose
an operation on the basis of what makes sense in the context. |
Ella
Cameron
Sophia |
Talked about finding a context to fit the problem. Ella
said, “Some problems are like the Windows and Tower problems. You have
to find a pattern.” Cameron referred to a measurement problem for which
he created a context that was familiar to him. |
When given problems
that are not in context, it helps students to think of a context in
which the numbers and operations make sense. This is a strategy that
others could benefit from. Building students’ fluency with creating
problem contexts can be accomplished through problem-posing activities. |
| Beau |
Said he could check his answer with a neighbor to see if
it was right. Asked if there was anything else he could do, he said
no. |
Beau would benefit from using multiple representations
in problem situations, alongside opportunities to see relationships
among operations (e.g., inverse operations) that would allow him
opportunities to use various representations and operations to ensure
accuracy. |
| Jamiel |
Focused his responses on the end product (answer). Asked if there
was anything he could do to see if he was on the right track while
working through the problem, he said no. |
The idea that mathematicians are checking their progress throughout
the problem-solving experience is a new idea to many. Asking questions
throughout problem solving to look for comprehension supports students
as they self-monitor their work. This is similar to the strategy of
monitoring for understanding while reading. |
| Danielle |
When asked about strategies, said, “Counting on fingers,” but she was reluctant to share beyond that. |
Honoring all strategies when having whole-group conversations is
going to be important for Danielle to feel as though her strategies have
merit—even those strategies that we hope she will move away from.
Moving away from finger counting necessitates access to more efficient
methods, likely methods that her peers are using. Having open
discussions of those strategies and making connections between them is a
critical part of our follow-up discussion in problem solving. |
Implementing the People of Freckleham Task
The Freckleham
task from our previous blog entry provides
an interesting demonstration of students’ problem-solving strategies. First,
the task has no word clues that students like Maya can rely on. By drawing all
the Frecklehammers first, students like Beau are scaffolded into representing
the problem situation in productive ways. One student’s representation of the
Frecklehammers looked like this:
Whether students
draw Frecklehammers, use ordered pairs, or create some other representation, a
systematic structure to accounting for all Frecklehammers is critical. We see
this structure in both students’ solutions above. We often ask the question,
“How do you know you have listed all the possible Frecklehammers?” Students who
systemically list all possibilities can provide sound responses. In reference
to the problem-solving interviews,
Jamiel
benefits from considering this intermediate step in problem-solving situations.
Additionally, Beau benefits from the opportunity to see other students
represent Frecklehammers in multiple ways.
Another useful representation, particularly
when considering how many times the greeting was said at the town meeting,
involves the use of a table. Presenting the Frecklehammers in this manner
allows for an opportunity to account for the number of statements, 54 in all.
| Freckles |
1 |
2 |
1 |
2 |
2 |
2 |
3 |
3 |
3 |
| Hairs |
1 |
2 |
3 |
1 |
2 |
3 |
1 |
2 |
3 |
| "I have more freckles" |
0 |
0 |
0 |
3 |
3 |
3 |
6 |
6 |
6 |
| "I have more hairs" |
0 |
3 |
6 |
0 |
3 |
6 |
0 |
3 |
6 |
Similarly, one
student drew upon the ordered pairs representation to arrive at the same
solution:
Students like
Danielle may be tempted to account for the number of statements said by
counting one by one. Encourage these students to look for patterns in solutions. For instance, after students
determine the number of statements said for, “I have more freckles than you,” they
could be encouraged to consider how the solution to freckles could be used to
determine the solution to hairs.
Students similar
to Ella, Cameron, and Sophia draw on contexts to make sense of problem-solving
situations. As such, they may benefit from acting out the problem or
personalizing it in ways that allow for even greater identification. Creating
space in classroom discussions for students to consider alternate contexts,
strategies, and representations is essential in furthering students’ abilities
to see themselves as mathematicians capable of solving the sorts of challenging
mathematics tasks that deepen and extend mathematical proficiency. The use of
the interview protocol allowed us to better design tasks and scaffold students’
work. We hope that this protocol provides others with insights into students’
views of problem solving and ultimately informs the implementation of problem-solving tasks.
Your Turn
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 |
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. |