Applying Cognitive Science to Learning Mathematics
Many mathematics learners have reasonable factual and procedural knowledge; the ability to recall answers to basic calculations and follow a set of problem-solving instructions respectively. However, successfully learning mathematics requires 3 types of knowledge and it is the third where learners are falling behind¹.
Conceptual knowledge is understanding the why and how behind the maths. Whilst factual and procedural knowledge have been somewhat developed by the traditional algorithmic approach, instilling conceptual knowledge requires creativity and flexibility of thought. By neglecting to impart the 21st-century skill of ‘number sense’, a concept discussed in our previous article, students are unable to see how and where skills apply outside of the scope in which they are taught. Students may leave education with mathematics knowledge but they cannot always apply it in relevant contexts.
It is for these reasons that modern mathematics curricula should encourage learners to look at mathematics through a lens of creativity. Here, students can develop flexible thinking that is essential for cultivating conceptual knowledge. All three types of knowledge play a role in developing 21st-century mathematical skills, but how can we ensure that these content areas are successfully ‘learnt’ by pupils? In this article, we take a deeper dive into the cognitive science behind acquiring information and techniques that can be used for effective learning.
Our Cognitive Ability to Learn
Educational consultant, David Didau, defines learning as ‘the long term retention of knowledge and the ability to transfer it to different contexts²’. We successfully retain knowledge when it becomes stored in our long term memory (LTM), but first, it begins in our working memory².
This is where the mind processes moment to moment demands, however, it can only deal with around 3 to 5 thoughts at any one time³. Exceeding working memory’s capacity can lead to confusion and forgetfulness², which aren’t conducive to learning. By implementing effective strategies that cater to the optimal functioning of working memory, information can make the transfer into the well organised LTM, connecting to schemas or networks of existing knowledge². Once safely in the LTM knowledge isn’t erased, but it can be hard to locate.
Referred to as ‘retrieval strength’, the ease of remembering something in the present is balanced by ‘storage strength’, the ease of remembering something later². When we can't remember something we thought we knew, we have high storage strength but low retrieval strength. Conversely, research has found that the easier something is to learn now (high retrieval strength), the more difficult it is to recall in the future² (low storage strength). However, if tasks are made more challenging, we have more room to improve low retrieval strength, which in turn, enhances storage strength⁴. This is the concept of ‘desirable difficulties’, explaining how forgetting can sometimes be useful. But how can this be achieved in a classroom setting?
6 Cognitive Strategies That Promote Learning
Algorithmic thinking typically leads to students being unable to apply their ‘learnt’ skills in a variety of contexts. Essentially, their long term memory hasn’t connected the schema in a way that’s conducive to the flexible application of taught skills⁵.
Conceptual knowledge bridges these gaps, ensuring the how and why are a part of acquiring new knowledge. This allows for better learning to take place, with students taking onboard skills that serve them well for the 21st-century digital age. Aside from conceptual knowledge itself, mathematics curriculums should also consider the cognitive science behind learning.
In this section, we’ll explore 6 cognitive strategies that can be utilised in mathematics to create the necessary desirable difficulties for optimum learning.
Although common in educational settings, research has shown that preferred learning styles have no real impact on learning. Instead, how new information is presented should resonate with the information itself. For example, learning about shapes should involve plenty of visual aids. However, studies have also found that dual coding, pairing visuals with audio, is a universally effective strategy⁶.
Presenting new knowledge in this way improves the likelihood of it being stored long term⁶. This is because both representations, audio and visual, have separate systems when it comes to memory⁷. Therefore rather than interfering with one another and hindering the learning process, they work together to enhance knowledge retention by creating more connections in the long term memory⁶.
The ease of a task depends on two factors; the task demand and the available resources². Task demand can be further thought of in terms of quality and quantity, with the former referring to its difficulty and the latter to how long it takes. The harder or longer a task, the more likely it is to lead to cognitive overload. The resources available, externally or our internal knowledge, also impact on the ease or difficulty of a task².
Whilst tools can be useful, they also lower the cognitive demand and can impact the ability to store knowledge in the LTM². They should be used to complement learning, implemented in a strategic way to facilitate conceptual knowledge and mathematical understanding. Task difficulty and the availability of resources should be considered in mathematics learning, with alterations made to personalise the cognitive load to the individual learner. Instructional scaffolding is one way in which the cognitive load can be reduced for otherwise difficult tasks⁸.
It’s often said that the best way to lead is by example. This is also true when it comes to learning new mathematical concepts, with modelled solutions being a common teaching practice. However, this can run the risk of making the cognitive load too light, meaning students aren’t faced with enough challenge to solidify their knowledge. Didau suggests gradually withdrawing the amount of help to ‘promote internalising the support’. He states that providing extra help later on, when students are more comfortable with the topic at hand, can help to increase the cognitive load⁸.
Instructional scaffolding also provides an opportunity to bring context into mathematics. By demonstrating how to apply maths through ‘concrete examples’ students will be more likely to remember the information presented to them⁹.
Contrary to popular belief, multitasking is not more cognitively efficient than focussing on one task at a time. The time and energy spent going back and forth between tasks create a ‘switching penalty’¹⁰. However, strategically alternating can become an effective learning strategy.
Interleaving turns the focus from one task to another to place a higher demand on retrieval, which paves the way for better storage strength. Switching too frequently turns interleaving into multitasking¹¹, but not switching often enough fails to create a desirable difficulty. This technique also provides an opportunity for learners to seek out similarities between the two different tasks, strengthening conceptual understanding¹¹.
It’s common in mathematics to focus on one specific topic at a time, sometimes even across the span of many lessons. Taking onboard new material in this way doesn’t always create enough challenge to store knowledge in the LTM, meaning students may not be able to recall this newfound information further down the line.
Spaced repetition involves leaving periods between learning or revising a topic. When students return to the topic again they may struggle with retrieval initially, but this recall will increase storage strength. By encouraging the brain to seek out information in the LTM schema is enhanced². In a classroom setting spaced repetition could take place by returning to newly taught topics regularly, such as during starter exercises.
Retrieval practice is simply the process of ‘bringing information to mind from memory⁹’. It’s about utilising the LTM stores and not solely relying on the working memory, with forgetting being a key component to the strategy’s effectiveness¹². In a classroom setting it may be used when revisiting a topic or during a test.
Cognitive psychological scientist and blogger, Megan Sumeracki, suggests students should write down everything they can remember on a blank piece of paper. She goes on to state that ‘The key is that they should bring the information to mind from memory. So, copying one’s notes would not be very helpful, but trying to summarize their notes from their memory would be very beneficial. The students do not need to remember everything and can check their notes or course materials after retrieval to fill in gaps⁹.’
Applying Cognitive Learning Strategies to Mathematics
Implementing a complementary blend of these strategies into mathematics education would help to create a learning environment and experience that is conducive to solidifying all types of essential knowledge, including conceptual. However, as beneficial as these techniques would be to a modern-day mathematics curriculum, there is a lack of awareness amongst educators about their effectiveness.
Further constraints include the time and effort required to create lessons and resources that utilise these approaches; particularly as individualisation is often required to ensure the desired difficulty is achieved for each student. Technology, however, is one medium through which these cognitive strategies could be effectively and efficiently implemented.
Digital tools can automate tasks and create tailored experiences that are conducive to learning. Education researchers Eady and Lockyer further endorse technology as a tool for dual coding. Their findings highlight how multimedia is more engaging for students, making use of their working memory, but also facilitating the transfer of knowledge to their long term memory¹³.
Technology should be a key consideration for 21st-century mathematics education; from generating cognitive-based resources to bringing modern-day context into the classroom. In our next piece, we’ll further explore how digital learning is an important part of democratising mathematics education and the role artificial learning could play in advancing the teaching of maths.
1. Willingham, D., 2019. Is it true that some people just can’t do Math. American Educator, [online] (Winter), pp.14-19. Available at: <https://www.aft.org/sites/default/files/periodicals/willingham.pdf> [Accessed 22 April 2020].
2. Didau, D., n.d. What Can Cognitive Science Tell Us About How To Design Learning Experiences?. [online] bbc.co.uk. Available at: <https://www.bbc.co.uk/teach/teacher-support/latest-theories-on-how-we-learn/zjwm92p> [Accessed 22 April 2020].
3. Cowan, N., 2010. The Magical Mystery Four. Current Directions in Psychological Science, 19(1), pp.51-57.
4. Yan, V., 2016. Retrieval Strength Vs. Storage Strength. [online] The Learning Scientists. Available at: <https://www.learningscientists.org/blog/2016/5/10-1> [Accessed 22 April 2020].
5. Oldridge, M., 2018. Teaching Mathematics Through Problem-Solving In K-12 Classrooms. Lanham: Rowman & Littlefield, p.53.
6. Brown, A. and Kaminske, A., 2018. Five Teaching And Learning Myths—Debunked: A Guide For Teachers. Routledge.
7. Thomas, Nigel J.T., "Mental Imagery", The Stanford Encyclopedia of Philosophy (Summer 2019 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/sum2019/entries/mental-imagery/>.
8. Didau, D., 2019. What Do Teachers Need To Know About Cognitive Load Theory?. [online] David Didau. Available at: <https://learningspy.co.uk/psychology/what-do-teachers-need-to-know-about-cognitive-load-theory/> [Accessed 22 April 2020].
9. Megan, S., 2019. Six Strategies For Effective Learning: A Summary For Teachers. [online] The Learning Scientists. Available at: <https://www.learningscientists.org/blog/2019/11/28-1> [Accessed 22 April 2020].
10. Kaufman, J., n.d. Cognitive Switching Penalty - The Personal MBA. [online] Personalmba.com. Available at: <https://personalmba.com/cognitive-switching-penalty/> [Accessed 22 April 2020].
11. Weinstein, Y. and Smith, M., 2016. Learn To Study Using…Interleaving. [online] The Learning Scientists. Available at: <https://www.learningscientists.org/blog/2016/8/11-1> [Accessed 22 April 2020].
12. Weinstein, Y. and Smith, M., 2016. Learn How To Study Using... Retrieval Practice. [online] The Learning Scientists. Available at: <https://www.learningscientists.org/blog/2016/6/23-1> [Accessed 22 April 2020].
13. Eady, M. J. & Lockyer, L. 2013, 'Tools for learning: technology and teaching strategies', Learning to Teach in the Primary School, Queensland University of Technology, Australia. pp. 71