MATH
Thinking Mathematically
Allowing students to solve problems strategies, “In which of the strategies did mathematical thinking have higher achieving
in ways that make sense to them the students start by combining the tens?” students.
CGI research shows that it is critical to create Sharing mathematical thinking allows
an environment in which students can solve students to learn from each other. About the author
problems in a way that makes sense to them. Megan Franke is a professor at the Graduate
This requires knowing that not all students Probing students’ mathematical School of Education and Information Studies,
will use the same strategies at the same time, thinking University of California, Los Angeles.
and ensuring that students have the tools CGI teachers ask students not only, “How did
they need to solve the problem. Teachers you solve that?” but also specifi c follow-up
need patience and knowledge of what questions, “Where did the ten come from?”
Reference
strategies to expect and how to scaffold tasks in ways that lead students to detail their 1. CGI is an elementary level mathematics
to help students see what they do know. thinking. We know that supporting students professional development program
to get complete and detailed explanations on developed at the University of Wisconsin,
Structuring classrooms for shared the table is productive for their learning. This Madison by Elizabeth Fennema, Tom
thinking means that beyond the initial question the Carpenter, Megan Franke, Linda Levi,
CGI teachers offer students opportunities follow-up questions matter. Specifi c follow-up and Susan Empson along with a number
to share their thinking. They create a set questions that are related to what the student of graduate students and teachers. The
of norms and expectations that encourage said or did help students the most. Questions original research was conducted by
students to explain their thinking as they like, “I see you started counting at 9, can you Fennema, Carpenter & Peterson.
learn mathematics. They do this by providing tell me why you did that?” or “I see three tens
time for students to work together to solve there, what did you do with those?”
problems and share solutions. Students
Further reading
are always asked how they worked a Conclusion
Carpenter T P et al (1999) Children's
problem out, and the teacher listens to CGI researchers continue to study the
Mathematics: Cognitively Guided
their misunderstandings as well as their development of students’ mathematical
Instruction. Portsmouth, NH: Heineman
understandings. They make sure that thinking, as well as how knowing about
Carpenter T P et al (1999) Using Knowledge
students explain their responses, and ask the development of student mathematical
of Children's Mathematics Thinking in
other students what they heard and if they thinking shapes teachers’ practice and
Classroom Teaching: An Experimetal Study,
understand. A teacher might ask: “Can supports student learning. CGI research
American Educational Research Journal,
anyone tell us what they think Anna is consistently demonstrates that teachers
26: 499-531
doing?” They might ask students to compare who know the details of their students’
fall 2009 Better: Evidence-based Education 17
Better(US) Fall09.indb 17 14/10/09 13:05:23
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