Chaffee J 1988 Thinking critically 2 (Boston: Houghton MifflinMA)
Scriven M, & Paul R 1987 The 8th Annual International Conference on Critical Thinking and Education Reform [Online: www.criticalthinking.org/pages/defining-critical-thinking/766]
Wallace B, Berry A and Cave D 2009 Teaching problem solving and thinking skills through science (Oxon: Routledge Abington)
Ennis R H 1986 Critical Thinking ed J B Baron and R G Strenberg (Philadelphia: Franklin Institute press) A taxonomy of critical thinking skills
Swartz R 1992 Thinking: The Second International Conference ed D Perkins, J Bishop and J Lochhead (Hillsdale, NJ: Erlbaum) Critical thinking, the curriculum, and the problem of transfer 261-284
Jaworski B 1996 Investigating Mathematics Teaching: A Constructivist Enquiry (No. 5) (London: Taylor & Francis Ltd.)
Quinnell L 2010 The Australian Mathematics Teacher66 35-40
Yeo J B W and Yeap BH 2009 Mathematical Investigation: Task, Process and Activity. [online: http://math.nie.edu.sg/bwjyeo/Publication/]
Creswell John W 2008 Educational Research: Planning, Conducting, and Evaluating Quantitative and Qualitative Research (New Jersey: Pearson, Merrill Prentice Hall)
Watson Peter 2016 Rules of thumb on magnitudes of effect sizes MRC Cognition and Brain Sciences Unit [online: http://imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/effectSize]
Carson C 2011 The effective use of effect size indices in institutional research [online: http://rlrw.bnu.edu.cn/uploads/2011420132315861.pdf]
Stacey K 2007 Progress Report of the APEC Project: Collaborative Studies on Innovations for Teaching and Learning Mathematics in Different Cultures (II) (Lesson Study Focusing on Mathematical Thinking) (Center for Research on International Cooperation in Educational Development (CRICED) University of Tsukuba) What is mathematical thinking and why is it important? 39-48
Yeo J B W and Yeap B H 2009 Mathematical problem solving: Association of Mathematics Educators Yearbook ed B Kaur, B H Yeap and M Kapur (Singapore: World Scientific) Solving mathematical problems by investigation 118-136
Aufmann R N, Lockwood J S, Nation R D and Clegg D K 2008 Mathematical Thinking and Quantitative Reasoning (Boston: Houghton Mifflin Company)
Higgins S, Hall E, Baumfield V and Moseley D 2005 A Meta-Analysis of the Impact of the Implementation of Thinking Skills Approaches on Pupils(Research in Education Library) (London: EPPI-Centre, Social Science Research)
Bensley D et al 2010 Teaching and assessing critical thinking skills for argument analysis in psychology (Education Resources Information Center) ERIC No. EJ883208
What is critical and creative thinking, and why is it so important in mathematics and numeracy education?
Numeracy is often defined as the ability to apply mathematics in the context of day to day life. However, the term ‘critical numeracy’ implies much more. One of the most basic reasons for learning mathematics is to be able to apply mathematical skills and knowledge to solve both simple and complex problems, and, more than just allowing us to navigate our lives through a mathematical lens, being numerate allows us to make our world a better place.
The mathematics curriculum in Australia provides teachers with the perfect opportunity to teach mathematics through critical and creative thinking. In fact, it’s mandated. Consider the core processes of the curriculum. The Australian Curriculum (ACARA, 2017), requires teachers to address four proficiencies: Problem Solving, Reasoning, Fluency, and Understanding. Problem solving and reasoning require critical and creative thinking (). This requirement is emphasised more heavily in New South wales, through the graphical representation of the mathematics syllabus content , which strategically places Working Mathematically (the proficiencies in NSW) and problem solving, at its core. Alongside the mathematics curriculum, we also have the General Capabilities, one of which is Critical and Creative Thinking – there’s no excuse!
Critical and creative thinking need to be embedded in every mathematics lesson. Why? When we embed critical and creative thinking, we transform learning from disjointed, memorisation of facts, to sense-making mathematics. Learning becomes more meaningful and purposeful for students.
How and when do we embed critical and creative thinking?
There are many tools and many methods of promoting thinking. Using a range of problem solving activities is a good place to start, but you might want to also use some shorter activities and some extended activities. Open-ended tasks are easy to implement, allow all learners the opportunity to achieve success, and allow for critical thinking and creativity. Tools such as Bloom’s Taxonomy and Thinkers Keys are also very worthwhile tasks. For good mathematical problems go to the nrich website. For more extended mathematical investigations and a wonderful array of rich tasks, my favourite resource is Maths300 (this is subscription based, but well worth the money). All of the above activities can be used in class and/or for homework, as lesson starters or within the body of a lesson.
Will critical and creative thinking take time away from teaching basic concepts?
No, we need to teach mathematics in a way that has meaning and relevance, rather than through isolated topics. Therefore, teaching through problem-solving rather than for problem-solving. A classroom that promotes and critical and creative thinking provides opportunities for:
- higher-level thinking within authentic and meaningful contexts;
- complex problem solving;
- open-ended responses; and
- substantive dialogue and interaction.
Who should be engaging in critical and creative thinking?
Is it just for students? No! There are lots of reasons that teachers should be engaged with critical and creative thinking. First, it’s important that we model this type of thinking for our students. Often students see mathematics as black or white, right or wrong. They need to learn to question, to be critical, and to be creative. They need to feel they have permission to engage in exploration and investigation. They need to move from consumers to producers of mathematics.
Secondly, teachers need to think critically and creatively about their practice as teachers of mathematics. We need to be reflective practitioners who constantly evaluate our work, questioning curriculum and practice, including assessment, student grouping, the use of technology, and our beliefs of how children best learn mathematics.
Critical and creative thinking is something we cannot ignore if we want our students to be prepared for a workforce and world that is constantly changing. Not only does it equip then for the future, it promotes higher levels of student engagement, and makes mathematics more relevant and meaningful.
How will you and your students engage in critical and creative thinking?