Creative and critical thinking is a cross-curriculum priority in Australian schools and features in each subject area.
Reasoning tasks promote critical and creative thinking in maths
Most primary teachers think of problem solving, one of the four mathematics proficiencies where children inquire into real world problems or solve open tasks. However mathematical reasoning, the fourth proficiency in the mathematics curriculum, is often overlooked by primary teachers but fits very neatly with creative and critical thinking.
- Mathematical reasoning
- mathematical thinking that creates new knowledge and understandings in mathematics and tests and validates conjectures or solutions
- Developing “an increasingly sophisticated capacity for logical thought and actions" (source: Australian Curriculum)
In our research with primary teachers and students we have focused on three key reasoning actions:
Not only do these actions embrace almost all of the other actions listed in the curricula definition of reasoning but they match neatly with the ideas of creative and critical thinking.
How can creative thinking be provoked by maths?
“Creative thinking involves students learning to generate and apply new ideas in specific contexts.” (ACARA)
In order to generalise students need to first analyse the problem to notice things that are the same or different, notice things that stay the same and things that change, or order examples to notice patterns. Expressing the common property or pattern noticed is generalising.
How teachers can use prompts to help students
You might use an open problem such as “A rectangle has a perimeter of 24m. What might be its area? Find as many different rectangles as you can.”
Along with questions and prompts to elicit students’ understanding of area, perimeter, rectangles and squares as they develop a range of examples that fit the constraint you can also use analysing and generalising prompts.
For example, “What do you notice about the area of your different rectangles?” And the student might say “They are different.” You would then prompt them to record their observation as a conjecture or rule, and prompt them to test it by asking “I wonder does that happen for other rectangles with a given perimeter? What if it was 40m?”
Alternately you might ask a different analysing prompt “I wonder if there is a pattern. How could we order your examples to see if there is a pattern?” Students who organise them by length or area will then notice and express a generalisation such “as the sides get closer to each other the area gets bigger.” Different students will notice and express this relationship between area and perimeter differently, such as “a square is the biggest” or use the terms length, width, perimeter and area in their generalisation .
When students discover this relationship, it’s a real “Aha” moment.
We have noticed that students can become very engaged and want to try out other cases to test their conjectures.
Open tasks help students discover new ideas
Open tasks along with generalising prompts enable students to discover a range of properties in number, geometry and measurement.
Likewise “What else belongs?” and “What does not belong?” tasks also require students to analyse the numbers or shapes to identify the common property, such as common factors, or a common geometric property. Students not only need to draw on their knowledge but need to express the common property using drawings, words or symbols. Tasks about patterns and growing patterns also provide opportunities for students to generalise and think creatively. Sometimes these tasks also call on students to use visualisation skills.
For example when students are trying to work out how many matchsticks are needed to make 5, 8 or 100 conjoined squares. For this problem we are looking for students to form a rule, part of the early algebra curriculum.
When students form conjectures about common properties or relationships, the challenging prompts for students is “Why does your rule work?” This involves justifying and hence critical thinking.
How asking your students to convince you, helps them become critical thinkers
26 + 47 = 613 Is it true? Justify
503 – 47 = 456 Is it true? Justify
1248 / 6 = 28 Is it true? Justify
(5 x 7) + (2 x 7) = 7 x 7 Is it true? Justify
For these examples students should be encouraged to use multiple methods to prove and disprove these claims. They can use estimation, diagrams, materials as well as mental strategies to prove or disprove to provide a logical argument.
Is Nathan correct?
Another task for measurement that also concerns the relationship area perimeter is:
Nathan said: “When you increase the perimeter of a rectangle, the area always increases.” Is this statement true for all cases? Explain why Nathan might be correct or incorrect.
Year 5 students who worked on this task generated many examples to test this conjecture. Some students organised their drawings and calculations in a table. They found counter examples and patterns. One group of students wrote that they learned that “when you increase the perimeter of a rectangle, the area always increases is wrong. It was proven by me, Xuen and Ms [teacher ]”.
Another student went further. They wrote “I learnt that if you get a rectangle and increase both sides the area will increase, but if you increase one side but decrease the other the area will not increase.”
Useful tools for primary teachers
Primary teachers have found the use of analysing, generalising and justifying prompts very useful for providing opportunities for students to reason and think creatively and critically during mathematics learning. Teachers have noticed the way in which the activities have provided practice with skills, and drawn on and developed their students’ understanding of content and illuminated the need for students to develop knowledge of mathematical terms to use when communicating their reasoning.
Further examples of tasks, student’s reasoning, teacher prompts, and a rubric to formatively assess students is available at ReSolve.