Skip to content

Rich mathematical tasks

See also: teaching-mathematics, mathematical-mindsets, big-ideas-in-mathematics

Sources to explore:

Rich, relevant, and engaging tasks#

Six characteristics of rich tasks

  1. Accessibility - invite all learners to engage in and contribute
  2. Real-life connections - authentic contexts for students will increase interest and creativity in solutions and strategies
  3. Multiple approaches and representation - the more avenues for engagement, the more opportunities for success
  4. Collaboration & discussion - encourages students to engage in collaborative groups when they have reached an independent conclusion, encouraging reasoning and new perspectives
  5. Engagement curiosity & creativity - when interested in the task learners demonstrate perseverance and work through challenges that arise
  6. Opportunities for extension - all students engaged with challenges and extensions for those ready without pressuring others to hurry up

Developing rich, relevant and engaging tasks#

  1. Start with a closed version of the problem

    $ \dfrac{1}{12} + \dfrac{5}{12} = \fbox{ } $

  2. Open up the problem by removing or adapting parameters

    $ \dfrac{\fbox{ }}{12} + \dfrac{5}{\fbox{ }} = \dfrac{\fbox{ }}{2} $

  3. Add complexity

    Two fractions add up to one half. What might those two fractions be?

  4. Introduce a requirement for reasoning

    Two fractions add up to one half. What might those two fractions be? How many possible answers are there? Do you notice any patterns?

Boaler (2015) - 6 questions to enrich learning tasks#

Boaler (2015) provides the following 6 questions as a way of enriching mathematical learning tasks. A related idea are mathematical-thinking-tasks

1. Can you open the task to encourage multiple methods, pathways and representations?#

  • Add a visual requirement
  • Ask students to make sense of their solutions

e.g. \(1 / \frac{2}{3}\) has extra requirements added

  • make sense of your answer
  • provide a visual proof

2. Can you make it an inquiry task?#

Rather than reproduce a method, ask students to think about ideas and use a procedure.

Examples

  • Rather than find the area of 12 x 4 rectangle, ask them how many rectangles they can find with an area of 24.
  • Rather than name quadrilaterals with different qualities, ask them to come up with their own
  • Ask students to make all the numbers between 1 and 20 using four 4s and any operation

3. Can you ask the problem before teaching the method?#

Echoes of productive-failure

4. Can you add a visual component?#

Diagrams or physical objects - multilink cubes, algebra tiles

Examples

  • Have students show functional relationships in many forms: expression, picture, words, and a graph. One school colour coded function components. i.e. the \(x\) always in red.
  • Parallel lines and traversal - colour code different types of lines

5. Can you make it low floor and high ceiling?#

  • lower - ask students how they see a problem
  • higher - ask students who finish to write a new question that is similar but more difficult

6. Can you add the requirement to convince and reason?#

  • ask students why the used a method and why it made sense
  • three levels of convincing: yourself, a friend, a skeptic

Example - paper folding, working in pairs students must fold a piece of paper to meet some criteria (e.g. a square exactly ¼ the area of the origin) and then convince their partner it is correct.

Suggested resources#

References#

Boaler, J. (2015). Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching. John Wiley & Sons, Incorporated.