If helping students consider multiple solutions to math problems is so important, why do so few teachers do it?
This was the question asked by Ed Silver and his colleagues at the University of Michigan in “Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom” (abstract only), published in the Journal of Mathematical Behavior. They worked with 12 experienced middle-grades math teachers for a year to identify some of the concerns and issues teachers had as they worked to become better at engaging students in discussions of multiple solutions. “It is nearly axiomatic among those interested in problem solving,” the authors write, “that students should have experiences in which they solve problems in more than one way.” Yet, as the teachers in this study revealed, it is hard to implement. It’s the kind of thing that math educators endorse with enthusiasm, but studies of the feasibility and benefits of this practice in classrooms are few.
What were teachers’ concerns? The majority worried about having enough time to devote to having students solve problems in more than one way and discuss the solutions. Many also were concerned that their lower-achieving students would be confused if they saw more than one way to solve a problem. One teacher confessed, “Sometimes I am scared to put even two strategies up there because [the students] are barely able to get one.” Other teachers believed that higher-achieving students would get bored if a lot of time was spent discussing a single problem. Teachers also had concerns about the limitations of their own knowledge.
As teachers grew to see the value of multiple solutions, they began to wonder about how to choose which solutions to have presented, the order in which they should be presented, and which ones to discuss in depth. One teacher noted that “Explaining is important, but which solutions you focus on have to be tied to the goals of the lesson instead of always sharing everything. I have not considered that before — who I want to share first.” Teachers were divided over whether to present incorrect approaches and how to deal with students’ mistakes. There is a lot of intellectual work involved in making decisions like these and no prescriptions exist. It’s easy to see how the exhortation alone to have students consider multiple strategies is insufficient support for successful implementation.
Beyond Show and Tell
As teachers grappled with these issues in the context of their own teaching, they began to think about how the purposeful selection of students’ solutions for presentation and discussion could help them “steer” the mathematics content.
Moreover, they noted, by attending carefully to the mathematical ideas embedded in students’ responses, a teacher could influence which ideas are likely to be discussed in class, and in what order, thereby improving their chances of meeting their mathematical goals for a lesson…. As one teacher stated, ‘I don’t usually allow multiple strategies, but I now see the possible benefits. [But if you do this], it is also very important to fully understand the mathematical goals of the lesson.’
Silver and his colleagues found that teachers’ instruction shifted over the course of the year to incorporate eliciting and discussing multiple solutions to problems. Teachers made the shift because, through their own discussions and reflections about promoting multiple solutions, they convinced themselves that it helped students’ learning.
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“Because, It’s Kind Of: Lower Achieving Elementary Students’ Participation in Problem-Based Mathematics
Researchers tend to agree that mid- to high-performing children benefit from instruction that emphasizes solving and discussing non-routine problems. When it comes to children who struggle, however, there is sharp disagreement and differing recommendations regarding how to teach them. Sometimes caught in a vicious cycle that requires that they know something before using it to solve problems, struggling students are taught basic skills and little else. This disconnect is evidenced in those children who have been required to memorize the multiplication tables, for example, but cannot use these facts to solve non-routine problems involving multiplicative situations…
To read more, download this paper, presented at the National Council of Teachers of Mathematics Research Presession, in St. Louis, as part of the symposium “Connecting Discourse, Teaching, and Curriculum.” Paper by Susan Empson, Erin Turner, Higinio Dominguez, and Luz Maldonado, April 2006.
Other papers presented in this symposium:
Teachers’ Negotiation of ‘the presence of the text’: How Might Teachers’ Language Choices Influence the Positioning of the Textbook? by Beth Herbel-Eisenmann, Iowa State University.
Difficulties with Scaling Up Reform: A Situated View of the Challenges of Transforming Classroom Discourse, by Jeffrey Choppin, University of Rochester
Tracing the Evolution of Pedagogical Content Knowledge as the Development of Interanimated Discourses, by Jennifer R. Seymour, Iowa State University and Rich Lehrer, Vanderbilt University.
Supporting Whole-Class Collaborative Inquiry in a Secondary Mathematics Classroom, by Megan Staples, University of Connecticut.
Developing Mathematics Students’ Critical Language Awareness, by David Wagner, University of New Brunswick, Canada
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Facilitating English Language Learners Participation in Mathematical Discussion and Problem Solving
A central tenet of the mathematics education reform movement is that mathematical discourse practices such as explaining thinking, conjecturing, analyzing strategies, and justifying solutions are essential component of what it means to do and learn mathematics… There is a paucity of research, however, focusing on English Language Learners’ (ELL) participation in these practices. At a time when 1 in 10 students is classified as an ELL (U.S. Department of Education, 2004), and 42% of all public school teachers have at least one limited English proficient (LEP) student in their classroom (U.S. Department of Education, 2003), we see this as a critical void in the literature….
To read more, download this conference paper, presented at the annual meeting of the American Educational Research Association in San Francisco, by Erin Turner, Higinio Dominguez, Luz Maldonado, and Susan Empson, April 2006. [Note: later version of paper uploaded May 2006]
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‘How did you figure that out?’
This question is probably the one I ask the most whenever I’m working with children. (Some people who aren’t children get irritated when I ask it. They take it as a sign that there is a flaw in their thinking. But usually, I’m just curious.) Today I ran across a passage from Choice Words, by Peter H. Johnston, that explains much better than I could how this question prompts children to assert their intellectual agency — the sense that “I am a person who does math,” for instance. This sense of agency is a powerful factor in students’ long-term success.
The question insists that a child respond with something like, ‘First I tried to …’ In other words, it requires the student to position himself as a story teller with himself as the protagonist in the story. ….Such a narrative invites a sense of agency as part of the child’s literate [or mathematical or insert-what-you-wish] identity.
This ‘how did you’ invitation to an agentive role is particularly important. We hear a lot about teaching children strategies, but we often encounter classrooms in which children are being taught strategies yet are not being strategic… Teaching children strategies results in them knowing strategies, but not necessarily in their acting strategically and having a sense of agency. …. Teaching for strategies requires setting children up to generate strategies, then reviewing with them, in an agentive retelling, the effectiveness of the strategies they generated, as in, ‘You figured out that tricky word by yourself. How did you do that?’ As children do this, they are in control of the problem-solving process and are asked to consciously recognize that control in an agentive narrative. (p. 31)
Don’t underestimate the power of the cumulative effects of asking this question of all of your students, as often as possible. The story of Billy, a first grader, illustrates it well.
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Do private schools do a better job teaching math?
This new study by Lubienski and Lubienski suggests the answer is a resounding, if not surprising, no. Past studies, including the most recent NAEP, have shown that private schools produce higher achievement than public schools in mathematics, even when the fact that private schools serve a different population of students is taken into account. However, those studies used a fairly crude measure of students’ socio-economic status (SES) based only eligibility for free or reduced lunch. Lubienski and Lubienski created a more sensitive measure of SES that included other factors such as parents’ education and income. Using this measure, they found that within all SES classifications (low, mid-low, mid-high, and high) that public school students’ achievement in math was higher than that of students in private schools, in fourth and eighth grades.
Paradoxically, overall mean achievement in math is higher for private schools than public, an example of Simpson’s paradox. This counterintuitive result appears to be due to a higher concentration of high SES students in private schools rather than better teaching. Lubienski and Lubienski’s data suggest that, in fact, public schools do a better job of teaching math than private schools.
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