Should you show students how to solve problems?
For many people, the answer is “obviously, yes.” But when and how? Research on young children’s mathematical thinking has shown that children can invent strategies to solve problems that are posed within their zone of understanding. Asking children to invent strategies aids the growth of understanding; how a child solves a problems can tell you a lot about what the child understands. Yet certain tools for representing and supporting understanding, such as numerals or number lines, must be shown to children, because they are conventions or because inventing them from scratch is inefficient. But there is a large gray area here where situational variables and teachers’ beliefs and experiences come into play and decisions are not clear cut.
How does a teacher decide when to show and when to not show students how to solve problems? I asked two veteran teachers when, if ever, they show young children how to solve problems. They have been involved in Cognitively Guided Instruction (CGI) for over 15 years each. Both now work as math specialists for the district. Before that, they were classroom teachers. Carrie Valentine taught upper elementary and Mazie Jenkins (sometimes known as Ms. J in CGI writings) taught primary. Both have worked a lot with kids from all kinds of backgrounds, including low income and low achievers in math. It’s certainly a question worth thinking hard about.
Here’s what they had to say:
MJ: The famous question always gets asked. The answer and how you go about getting students to exhibit different strategies is very complex.
Teachers have to learn how to get students to reflect on the strategies that are shared. For example, how is Susan’s strategy like Megan’s strategy? How are they alike and how are they different? Who else solved their problem like Susan’s? Who solved their problem like Megan? Who has a completely different strategy? How is it different than Susan’s? Ask specific questions for the strategy, how are tens used in this strategy? Was this a good way? How do you know? Have we seen this strategy before? When? What kind of problem did we see this strategy used? Did anyone use numbers to solve this problem?
I usually do not show a strategy. I might talk about how I have seen another student solve a similiar problem. (Deborah Ball demonstrates this on a videotape).
I listen and observe to find students who are solving a problem in a different way and build upon that. I ask students to solve problems in at least two different ways (direct modelers show two ways of direct modeling – but they do not know this)
Teachers need to have a good understanding of the development of strategies to know what is developmentally appropriate.
CV: I think it’s possible but requires a great deal of expertise. Skill in problem posing and questioning is essential. I wonder what is meant by strategies? I think of them from a cognitive perspective. Is there something going on internally different in the math understanding? But, I think most people think about the representations that their students use. In that case I definitely see utility in showing ways to represent to hasten the learning. Kids love ‘tools’ such as the empty numberline, arrow language, and ratio tables when they are ready. I would introduce them after some level of understanding emerges. The empty numberline after counting on emerges, arrow language after decomposition and facts of ten are known, and ratio table after kids can double using base ten. There are other ‘tools’ but these seem to capture kids’ attention and are efficient ways to record their thoughts and later to help them solve problems.
You get the gist. It’s actually a complicated question and deserves a nuanced response.
SE: What about upper grades?
CV: Same way. First what is meant by a strategy vs. a tool if there is a difference. Then talk about ways to record thinking.
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I work with pre-service secondary math teachers in California and I had one of them ask me an interesting question, somewhat related to the comments about “showing students how to solve problems.” I thought I would post it to see what comments folks had.
QUESTION: How do we balance a “problem-solving” approach in high school math with English learners? It seems that the recommendations from the EL literature favor a lot of language scaffolding, which might dilute the kind of grappling we want students to do with math ideas. Also, how are beginning English learners suppose to communicate their complex thinking?
Thanks!
Jack
Comment by Jack Dieckmann — January 4, 2006 @ 12:11 am
HI Jack. Thanks for your comment. I’m hoping someone who knows the ELL literature better than I do will comment, but in the mean time, I’ll post my thoughts.
I see two main things to consider in supporting ELLs’ meaningful engagement in math. One is that students understand a problem well enough to generate a solution. Two is that they communicate about their solution to others. It’s hardly an option to forego either of these types of engagement, in my view, because they are so central to learning math with understanding and confidence. So what can a teacher do?
Some suggestions for posing problems students can make sense of without heavy langauge scaffolding: use students’ first language (I know this is not an option in many places for various reasons); use problems that are not heavily language dependent (e.g., that involve algebraic relationships, dynamic graphs, reasoning about shapes).
To support students’ communication about what they have done, teachers can help students represent their reasoning (not just their answer) using numbers, diagrams, graphs, and other notations that are not dependent upon a specific language; enlist the aid of bilingual students to help translate from student’s first language to English. The important thing here is for the teacher and students to hear and be able to respond to an EL learner’s thinking.
Finally, I think teachers need to consider whether their goal for students is to learn English or learn math. Both are important long-term goals for students. But it’s my sense that these goals can actually compete and so a teacher needs to be clear about what he/she is hoping to accomplish in a given instructional episode.
Comment by Susan Empson — January 5, 2006 @ 9:03 am
I showed my 8 year old daughter how to solve multidigit multiplication problems. I am fairly confident of her (age appropriate) grasp of both place value and multiplication as a concept. I showed her an organized way to write out the partial products and then add them up–without mysterious “carrying” and emphasizing that 2 in 23 represents 20. She “got it” and uses it successfully.
Could she have been led to create this? Could she have come up with her own strategy? Certainly yes to both. But sometimes she just wants to know “how you do it” and I think I should honor that desire.
And…there’s a potentially infinite number of mathematical topics to explore, and multidigit multiplication may not be the most profound or productive one to dwell on.
And…it’s my personal experience that working through and understanding someone else’s algorithm or method is a challenging and worthwhile mathematical experience.
All that is about working with my own kid, of course. I think I’d work similarily with a group of kids. (Though my teaching experience is secondary.) One difference is that in school, they have to spend so much time on these algorithms that they probably are THE place to support kids’ independent reasoning.
So certainly kids can construct their own strategies for solving “operation” problems in elementary school. My question is should we always require them to do so?
AND
What’s the role of “standing on the shoulders of giants” in the elementary school classroom?
Comment by Jennifer Knudsen — January 5, 2006 @ 4:07 pm