# 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

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

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

## Below are some Qs from Taken from Dr Marian Small’s book Good Questions: Great Ways to Differentiate Mathematics Instruction

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

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

- 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:

## Math Currirulum: What Teachers Need to Know

## 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)

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

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

## Supporting Mathematical Thinking and Learning through Problem Solving

## Resources to Get You Started

## Dr. Marian Small's Website Dr. M. Small regularly posts "Good Questions" on her website. These are generally open ended questions that can have multiple entry points and answers. For instance: What other number do you think belongs with these: 0.4, 4/9, 42%
(Small, One, Two..Infinity, 2016) | ## Which One Doesn't Belong - Website " A website dedicated to providing thought-provoking puzzles for math teachers and students alike. There are no answers provided as there are many different, correct ways of choosing which one doesn't belong." (WODB, 2016) | ## Find a reason why each one does not belong. Here is one example of question posted by teacher contributor Jon Orr on the WODB Website. Consider having student provide reasons why each image doesn't belong as a way to encourage divergent thinking and to surface the what students know and understand...or even misunderstand. |

## Dr. Marian Small's Website

What other number do you think belongs with these: 0.4, 4/9, 42%

(Small, One, Two..Infinity, 2016)

## Which One Doesn't Belong - Website

## Find a reason why each one does not belong.

## Dan Meyer's Three Act Math Dan Meyer has created a series of videos that are designed to get students thinking. Each "problem" is asked using three steps (or Acts) | ## Act 1 After watching the first, brief video, students write down a question that comes to mind and share it with a peer(s) before sharing out to the class. Students are then asked to make a guess that is too high, and a guess that is too low. It doesn't take long before students try to narrow too high and too low amounts guessed. | ## Acts 2 and 3 Students are asked what information they need and provided some options. They learn to think about the variables they can consider to make the best guess and figure out the answer. Act 3 - students get a chance to return to their predictions and to see who came closest. |

## Dan Meyer's Three Act Math

## Act 1

Students are then asked to make a guess that is too high, and a guess that is too low. It doesn't take long before students try to narrow too high and too low amounts guessed.