Problem Solving

in Mathematics Education - Intermediate Grades by Deb Keefe

Problem Solving in the mathematics classroom means:

  • Using Mathematical questions that do not need to be solved in one particular way
  • Providing opportunities for students to explore, create and prove mathematical thinking
  • Process for learning where students "discover mathematical relationships and pose questions of their own" (Rigleman, 2013, p. 417)
  • Using "problems that can promote students' conceptual understanding, foster their ability to reason and communicate mathematically, and capture their interests and curiosity."(Cai & Lester , 2010, p. 1)
  • Not something that is saved until the concept has been taught (Cai & Lester , 2010, p. 3)

YES examples

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Dan Meyer's "Sugar Packets" problem (above) can be found by

following this link.

Look at the images below. Why does each one not belong?

From the Which One Doesn't Belong website
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Below are some Qs from Taken from Dr Marian Small’s book Good Questions: Great Ways to Differentiate Mathematics Instruction

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Here is a question from More Good Questions by Dr. Marian Small and Amy Lin

Option 1: Describe two different ways to calculate 0.750 × 1.750.

Option 2: Describe two different ways to calculate 0.750 ÷ 1.750.

Using Problem Solving effectively means using "Rich Tasks"

According to researcher Jennifer Piggott, "Rich Tasks" have several of these qualities:


  • "are accessible to a wide range of learners
  • might be set in contexts which draw the learner into the mathematics either because the starting point is intriguing or the mathematics that emerges is intriguing
  • are accessible and offer opportunities for initial success
  • challenging the learners to think for themselves
  • offer different levels of challenge, but at whatever the learner's level there is a real challenge involved and thus there is also the potential to extend those who need and demand more (low threshold - high ceiling tasks)
  • allow for learners to pose their own problems
  • allow for different methods and different responses (different starting points, different middles and different ends)
  • offer opportunities to identify elegant or efficient solutions
  • have the potential to broaden students' skills and/or deepen and broaden mathematical content knowledge
  • encourage creativity and imaginative application of knowledge
  • have the potential for revealing patterns or lead to generalisations or unexpected results
  • have the potential to reveal underlying principles or make connections between areas of mathematics
  • encourage collaboration and discussion
  • encourage learners to develop confidence and independence as well as to become critical thinkers” (Piggott, 2011)

Challenges with using Problem Solving

  • Students' problem solving abilities often develop slowly
  • The teacher must develop a "problem-solving culture" in the classroom and make it a regular and consistent part of the classroom routines
  • Students have to recognize the importance of trying challenging mathematics problems and this needs to be an ongoing practice
  • Exposing students to problem solving has to be done at every grade level -- every day, if possible (Cai & Lester , 2010, p. 4)

What is the thinking about Problem Solving in Ontario?

  • Educators need to be aware of Big Ideas in the Mathematics Curriculum and use these Big Ideas to generate problems or Rich Tasks
  • Talk in the Mathematics classroom is important - students need to hear the thinking of others and consider various ways of "doing the math"
  • Wait time is important; so is asking students "what makes you say that"
  • Students have to be taught to listen to each other
  • Teachers interact with students to get feedback on what has been learned, not to get the right answer
(West, 2011)

  • Consolidating the learning is a very important phase that should be done with the students, not for them.

Dr Marian Small has done considerable work in building understanding of ON educators:

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The Art of Mathematics

"For me, the real goal is I want kids to want to learn, find it interesting; want to think, find it interesting... It is our main mission to want kids to be curious and want to figure things out. " (Small, 2014)

Math Currirulum: What Teachers Need to Know

I think the curriculum document has to do a little more for them in helping [teachers] see [the big ideas that go from K to 12], what to focus on, what to let happen but not really focus on? ...the word I've been using lately is intentions. I believe every teacher should teach a lesson with intention. This is what it's for. And I think the curriculum document should help them with what those intentions might be...I think is important is somehow to embed the process expectations more visibly with the content expectations. Right now, they sit as stand-alones. And we know that there are wonderful teachers who do embed them. We also know there are teachers who didn't even know there were those pages in the front...I think we need to do a better job of that... I think it needs to be integrated into the things you're doing." (Small, 2014)
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There are 7 Mathematical Processes identified in the Ontario Mathematics Curriculum:

See the "front matter" in the Curriculum Expectations (elementary and secondary) or the eduGAINS website for more information.(Education, 2016)

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Dr Mariam Small on the importance of teaching intentions and consolidating learning:

Teachers "teach a lesson for a purpose" and they pick a problem "to accomplish that purpose, and [the] consolidation will bring that purpose to the surface. And no child will leave that room not knowing what [the] lesson was about." (Small, 2014)

Thinking about Problem Solving and what it might look like in the classroom:

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Supporting Mathematical Thinking and Learning through Problem Solving

Resources to Get You Started

There are many resources available to help teachers incorporate Problem Solving in their classroom. Often there are plenty of resources available for Primary and Junior teachers. Here are some that also link to the Intermediate (and Senior) classrooms.
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