Embedding divergent thinking and creativity in mathematics

Embedding divergent thinking and creativity

in mathematics

To classify subjects as creative and non-creative is a contradiction. Like any other higher-order thinking skill, creativity is central to learning in any discipline (Miller 2015). The late, great Sir Ken Robinson put it best when he said, ‘creativity is as important as literacy, and we should treat it with the same status’ (Robinson 2006). As educators, we must endeavour to craft learning experiences that motivate our students to be imaginative and creative by employing divergent thinking skills, so that the classroom reflects the playground (Miller 2015). If we consider mathematics, there is a persistent societal norm the subject is dry, dull, and black-and-white, leading many to ‘hate it’ (Li 2019). Ideas about being creative in the subject can seem foreign because of the way maths is spoken about and taught in our schools. It is particularly troubling if we consider the vast difference in opinion a positive relationship with the subject at school can have. Schools seem to dichotomise students into those who think they are bad at maths always thinking so, and those who are good at maths becoming mathematicians, engineers, teachers, or academics. This is because the latter overwhelmingly agree that the subject can be as expressive as any of the fine arts under the right conditions, implying the subject is creative, but few teachers tap into this in their pedagogy (Taylor 2020). Consequently, if we want to stop disenfranchising our students from mathematics and change the ‘image problem’ the subject has, we must reconsider the way it is taught and look for opportunities to embed creative thinking.

The problem with convergent thinking

Convergent thinking is a broad term that encourages students to use their prior knowledge and learning experiences to ‘hone-in’ on one specific solution to standard questions (Cropley 2006; Nelson-Delaney 2020). It does not require significant creativity but is a successful strategy for school exams and standardised tests when the problems are linear and logical (Cropley 2006). However, a reliance on convergent thinking alone cannot help students when they encounter a cognitive hurdle that requires them to think around the problem in unfamiliar scenarios (Nelson-Danley 2020). In these circumstances, students need to tap into their creative thinking to work the problem; however, if they have never been exposed to such thinking their predicted success is low. This is unfortunately the case in many schools where teachers and students have placed an increased importance on convergent thinking strategies to be successful in mathematics. As an example, the breadth and pace of the Queensland senior syllabus does not lend itself to depth, and consequently teachers tend to favour pedagogical approaches that promote convergent thinking because that is the path of least resistance. Additionally, the pressures felt by students regarding short term metrics such as ATAR results emphasise the importance of this approach because convergent thinking is seen as the safer path toward success. However, the highest degree of difficulty on senior mathematics exams are called ‘Complex Unfamiliar’ questions that require students to adopt nonconvergent thinking styles. Only those students who have embraced creative thinking in mathematics are successful with them, and those who try and apply convergent strategies fail. If such students have a negative pre-existing attitude towards mathematics, this will only enhance their hatred for the subject, because they equate their failure with an inadequacy (Buckley 2013). This is a specific example, but it highlights the limitations of convergent thinking strategies and the problems educators face if they only endorse this type of linear style of thinking. This is because teaching mathematics for short-term success alone employs a finite mindset in the infinite game of education (Sinek 2020). According to Game Theory, playing an infinite game with a finite mindset will never be successful, and as such, teachers must embrace more opportunities for students to think creatively in mathematics (Sinek 2020).

Mathematical creativity and divergent thinking

Creativity looks different in every subject and in mathematics it is typically defined as ‘non-algorithmic decision making’ (Devlin 2019, para. 14). This implies that mathematical creativity is incompatible with convergent thinking strategies, because the latter only works when students apply a learned skill or worked example to arrive at a specific solution, like an algorithm. This is supported by the traits of mathematically creative individuals, typically characterised by ‘their understanding that more than one approach can lead to equivalent results and their ability to solve problems in different ways’ (Leikin & Lev 2007 in Kroesbergen & Schoevers 2017, p. 420). Furthermore, it is these students who embrace creativity that typically score higher on exams, because they can overcome the cognitive hurdles of the most complex and challenging problems (Kroesbergen & Schoevers 2017). It is for this reason competitions such as the Australasian Problem Solving and Mathematics Olympiads (APSMO) design their programs to ‘focus on the students’ ability to solve mathematical problems in a creative manner – as opposed SUNATA 35

SUNATA 36

to simply reaching a solution using a prescribed method’ (APSMO 2022, para. 1). As such, we can conclude that the differentiating factor that separates diligence from excellence in mathematics is the ability of students to think creatively. Consequently, as educators, we need to develop strategies that embed creativity in mathematics by getting students to think more in non-algorithmic ways. The opposite of convergent thinking is divergent thinking, which involves combining diverse types of information in familiar and unfamiliar scenarios to generate many possible solutions to a problem (Gregoire 2016; Madore, Jing & Schacter 2016; Nelson-Danley 2020). It typically emerges during nonlinear learning experiences when teachers cede control over the learning process to their students and empower them to think differently. For this reason, many of its advocates encourage mathematics teachers to embed opportunities for divergent thinking in their classes, because only in these situations does mathematical creativity flourish (Devlin 2019; Nelson-Danley 2020). As such, this supports the argument that mathematics teachers must embed divergent thinking opportunities in their lessons so that students can become more mathematically creative and successful with the subject.

Embedding divergent thinking in mathematics 1. Develop subject expertise

Good teaching begins with understanding the content, and educators who know their subject well stand a better chance embedding meaningful learning experiences in their lessons than teachers who don’t (Gregoire 2016; Thompson 2021). However, knowing the content alone is not enough as the research in John Hattie’s Visible Learning for Teachers (2011) showed. He found that teachers’ subject-matter knowledge was important but not critical to student achievement, because there are too many other factors that influence the metric (Hattie 2011). However, the research did show that expert subject-teachers can use their advanced knowledge to introduce new and difficult concepts more effectively than those who are not (Hattie 2011). Additionally, subject experts can implement the correct balance of convergent and divergent thinking because they know which topics and concepts are suited to each thinking technique (Gregoire 2016). Simply put, to embed divergent thinking and allow creativity to flourish, mathematics teachers need to know their stuff.

2. Think deeply about simple ideas

One of the keys to learning is to spark curiosity, even if it is about simple ideas. Enabling students to think deeply about fundamental concepts in mathematics can help develop the cognition to do so with more complex ideas later. Not only that, but by sparking curiosity, students tend to develop an interest in the subject matter and proactively engage in learning more about the subject (Gregoire 2016). A simple example of this is to get students to think about the language of mathematics they use every day and where the notation comes from. Mathematics is a language like any other and only with a fluent understanding of the syntax will students appreciate it as such.

3. Explore misconceptions and their consequences

When a student demonstrates a misconception, such as dividing by zero, never just say ‘no’ (Finkel 2015). Take the time to explore their misconception such that if what they said were true, it would lead to a contradiction. I have found this not only to be beneficial for students but for my own professional development too. It has allowed me to collect common misconceptions about subject matter, which has helped me identify them sooner during subsequent lessons. Additionally, misconceptions don’t have to always be wrong, just out of context. Students should be rewarded with an explanation, because questioning misconceptions is how new mathematics is discovered (Finkel 2015). For example, the angles in a triangle do not always sum to 1800 if we consider non-Euclidean geometry; a square can have three sides if we draw it on a sphere; and 2 + 2 is not always 4 if we consider modular arithmetic. The reason why certain concepts remain undefined in mathematics is because we have not defined them yet, so what better way to get students thinking creatively than exploring how we could possibly define dividing by zero.

4. Use technology, but do not rely on it

A lot of mathematics can be abstract, and there are many interactive tools and pieces of software out there. These tools can help students visualise otherwise difficult concepts and ideas and help develop their creative and divergent thinking skills. For example, the ease with which I can graph a function quickly on GeoGebra during learning has been an important development of my pedagogy. However, educators must remember that technology can amplify good teaching, but good technology cannot replace bad teaching (OECD 2015). As such, it is recommended teachers embrace the power of digital technology to enhance student understanding but not rely on it alone as a means to induce divergent thinking.

5. Use structured ‘multiple solution tasks’

Pose a problem and offer what the answer should be, but don’t provide the route. This gives students control over their learning, and they must adopt a divergent thinking mindset to arrive at a solution in their own way. This promotes cognitive fluency, flexibility, originality, and resilience (RycroftSmith 2018). An example I use in class is when dealing with Mathematical Proof, by using the analogy of driving to the airport. There are many routes you can take, and yes, the better you understand the road networks, the quicker you will get there, but any road you take will lead to the airport eventually. A caveat on this recommendation, though, is to ensure students have all the foundational skills beforehand, so they don’t drive around in circles while trying to prove something.

6. Reiterate that learning takes time

In this age of immediacy, reassuring students that learning does not happen instantly is crucial. In every other aspect of their lives, gratification is more or less immediate, thus reducing the potency of patience as a favourable virtue. This is reflected in the classroom when we see students give up if they don’t understand something quickly and unfortunately equate this with being inept. As educators, we must emphasise patience with the learning process and allow students time to struggle with ideas and concepts, because perseverance in the face of difficulty is necessary for cognition (Finkel 2015). Mathematicians can spend years, even decades, grappling with a problem, and so teachers need to give students time to think a problem through, so they can be creative and explore multiple solutions. This is particularly important for creativity too, as only by pondering a problem or idea for a while can new ideas flourish.

7. Play

One of my favourite quotes from Einstein about education was ‘play is the highest form of research’. The only reason we have discovered anything in mathematics is because people have had the courage to play with it. It is not just about following rules and established principles, because these too had to be discovered at some point. Mathematics teachers who allow their students to play with mathematics give them the gift of ownership (Finkel 2015). It is this recommendation I see as the most important for embedding divergent thinking, because play is inherently creative and speaks to the deeper meaning of mathematics. If we all agree the purpose of music is deeper than reciting the works of others, or that literature is more than a summation of all the books you have read, then surely we can agree that the purpose of mathematics is more important than completing endless exercises and regurgitating worked examples on an exam.

Conclusion

A shift from absolute convergent thinking practices in the mathematics classroom is necessary if educators are to combat the image problem the subject has with creativity. Embedding opportunities for divergent thinking allows mathematical creativity to flourish, because it is non-algorithmic and speaks to the deeper meaning of the subject. If we can accomplish this as educators, we would see more students empowered by mathematics and see breakthroughs in the field that help us better understand the nature of the universe.

References

Australasian Problem-Solving Mathematical Olympiad (APSMO) 2022, About APSMO, APSMO, viewed 31 May 2022, https://apsmo.edu. au/about-us/ Buckley, S 2013, ‘Deconstructing Math Anxiety: Helping students to develop a positive attitude towards learning Math,’ ACER Occasional Essays, pp. 1-3. Cropley, A 2006, ‘In Praise of Convergent Thinking’, Creativity Research Journal, vol. 18, no. 3, pp. 391-404. Devlin, K 2019, What is mathematical creativity, how do we develop it, and should we try to measure it? Part 2, Mathematical Association of America, viewed 21 March 2022, https://www.mathvalues.org/ masterblog/2019/1/26/what-is-mathematical-creativity-how-do-wedevelop-it-and-should-we-try-to-measure-it-part-2 Finkel, D 2015, 5 ways to share math with kids, online video, Ted, viewed 25 May 2022, https://www.ted.com/talks/dan_finkel_5_ways_ to_share_math_with_kids Gregoire, J 2016, ‘Understanding creativity in mathematics for improving mathematical education’, Journal of Cognitive Education and Psychology, vol. 15, no. 1, pp. 24-36. Hattie, J 2011, Visible learning for teachers, Routledge, London. Kroesbergen, E & Schoevers, E 2017, ‘Creativity as Predictor of Mathematical Abilities in Fourth Graders in Addition to Number Sense and Working Memory’, Journal of Numerical Cognition, vol. 3, no. 2, pp. 417-440. Li, C 2019, Why We Hate Math: Three Reasons, online video, Ted, viewed 24 March 2022, https://www.ted.com/talks/carr_yijin_li_why_ we_hate_math_three_reasons Madore, K, Jing, H & Schacter, D 2016, ‘Divergent creative thinking in young and older adults: Extending the effects of an episodic specificity induction’, Memory & Cognition, vol. 44, no. 1, pp. 974-988. Miller, D 2015, ‘Cultivating creativity’, The English Journal, vol. 104, no. 6, pp. 25-30. Nelson-Delaney 2020, How to teach divergent thinking, Teach Hub, viewed 25 May 2022, https://www.teachhub.com/teachingstrategies/2020/07/how-to-teach-divergent-thinking/ OECD 2015, Students, Computers and Learning: Making the Connection, PISA, OECD Publishing. Robinson, K 2006, Do schools kill creativity?, online video, Ted, viewed 31 May 2022, https://www.ted.com/talks/sir_ken_robinson_do_ schools_kill_creativity Rycroft-Smith, L 2018, Building mathematical creativity, Cambridge Mathematics, viewed 21 March 2022, https://www.cambridgemaths. org/blogs/building-mathematical-creativity/ Sinek, S 2020, The infinite game, Portfolio Penguin, London. Taylor, P 2020, ‘Mathematics is about wonder, creativity and fun, so let’s teach it that way’, The Conversation, 18 July, viewed 21 March 2022, https://theconversation.com/mathematics-is-about-wondercreativity-and-fun-so-lets-teach-it-that-way-120133 Thompson, S 2021, ‘1 in 4 Australian year 8s have teachers unqualified in maths — this hits disadvantaged schools even harder’, The Conversation, viewed 21 March 2022, https://theconversation.com/1in-4-australian-year-8s-have-teachers-unqualified-in-maths-this-hitsdisadvantaged-schools-even-harder-161100 SUNATA 37