October | 2018 | Morgan Watt- Eportfolio

According to Kahneman (2011), our thought process is determined by two different systems – impulsive/automatic and thoughtful/deliberate – also considered as system 1 and system 2. He makes apparent that for the majority of the time we should be using our system 2, however, this is not what happens. In fact, the reason we don’t use system 2 all of the time is because it is more exhausting, therefore, we resort to using system 1 (Kahneman, 2011).

To show how these systems work my lecturer used a very easy but clear example. He put this on the board:

We’ve used our system 1 process and immediately thought the answer is £10 – which is wrong. The answer in fact is £5- HOW?? Due to our system 1 thinking it automatically creates the calculation of £110 – £100 = £10, however, if and when we use our system 2 thinking –our thoughtful and deliberate thinking – we can see where we are going wrong. If we used system 2 thinking from the beginning, this is how we would work out the answer:

This isn’t “hard and complex” maths which makes us get the wrong answer. This has then made me wonder is it in fact Kahneman’s (2011) system 1 and system 2 approach to the way we think which affects the answer we first give?

So what?

The ideas that Kahneman’s (2011) Thinking Fast & Thinking Slow give, really make up the whole spectrum of human thought, action and behaviour.

After getting my head around the “Thinking Fast & Thinking Slow” approach (Kahneman, 2011), it led me to further thinking on how, the way we think and process what we read, can affect the answers we give and its not nesessaraly because we do not know the answer. This got me thinking about exams within maths, in particular, and how the reason we give the answer may not be due to if the maths is ‘easy’ or ‘hard’. It could be caused by using only our system 1 (Kahneman, 2011) thinking.

Furthermore, still thinking about exams, this time multiple choice, our counter-intuitive thinking becomes apparent. This is when you put an answer but then go back and think it might be a different answer. Do you stick with the first answer you put or do you change to the other answer you think it might be?

Brownstein et al (2000) make apparent that those who go ahead and change their answer actually change it to the wrong answer. However, Kruger et al (2005) creates an alternative argument to Brownstein et al (2000) and highlights that over 70 years of research which has been focused on changing answers and this belief that if you change your answer your more likely to change to the wrong answer, that this is not the case. In fact, Kruger et al (2005) make clear that people who have this counter-intuitive thinking, and who then go on to change their answer, majority of which end up changing their answer from the wrong one to the right one. This then improves their overall result/mark.

So why is this the case? Well my lecturer presented a problem to prove how this can mathematically work. He asked us this:

Straight away, using our system 1 approach (Kahneman, 2011), we think the answer is 50/50. He then showed us the mathematical proof that there is more chance of winning the car if you switch doors.

After my lecturer explain why this works, making me use my system 2 approach, it got me thinking even more about Kahneman’s (2011) Thinking Fast and Thinking Slow approach in that is our counter-intuitive thinking really just questions tricking our system 1 thinking so we then use our system 2 thinking. This meaning that when it comes to changing your answer, really we should change it because as Kruger et al (2005) state, there is more a chance of us changing it to the correct answer and there is more of a chance this is us then using our system 2 thinking which Kahneman (2011) would argue is the system we should be using all of the time. Finally, this has highlighted to me that throughout teaching mathematics we should really consider this Thinking Fast & Thinking Slow (Kahneman, 2011) approach as this could be a major factor which affects a child’s answer being wrong and not actually how difficult the mathematics is.

References

Kahneman, D. (2011) Thinking Fast & Slow. New York :Farrar, Straus and Giroux

Brownstein, Wolf, and Green, Barron’s ‘How to Prepare for the GRE: Graduate Record Examination’, 2000, p. 6 as cited in Kruger, Wirtz & Miller, 2005, p. 725

Kruger, J., Wirtz, D. & Miller, D. (2005) ‘Counter factual thinking and the first instance fallacy’, Journal of Personality and Social Psychology, 88(5), pp. 725-735.

Discovering maths has really opened my mind and has got me thinking about how I want to teach maths in my future classroom and what kind of teacher in maths I want to be.

After one of the discovering maths lectures it allowed me to begin to ask myself a series of questions around the way maths is perceived within schools and the way in which maths is taught. I now ask myself, can we raise attainment within maths, not by doing more maths but by doing different maths?

When I think back to my experience of maths in both primary and secondary school they are very similar – sitting at a desk, the teacher talking a new concept to you, you then completed some examples and then you were assessed on your understanding … or more like memory. Nothing ‘fun’, no enthusiasm or creativity presented. It was JUST maths.

Jo Boaler’s (2009) study has really begun to form a foundation to my thinking of my future teaching of mathematics. The study carried out was on 2 schools, Amber Hill which was a traditional school which followed set methods and procedures, distinct topics and closed mathematical problems and Phoenix Park which was a progressive school where there was a degree of choice, openness and mathematically rich experiences. From a series of assessments Amber Hill pupils performed worse than Phoenix Park. Amber Hill pupils had a broad understanding of facts, rules and procedures however they found them difficult to remember over time. Whereas Phoenix Park pupils were flexible and adaptable and they were able to see their knowledge in different situations.

Going back to my question of can we raise attainment within maths, not by doing more but by doing different maths, Amber Hill pupils had not learned any less maths than those at Phoenix Park, but different maths. Jo Boaler concluded from her study that transmitting mathematics is less helpful, which those at Amber Hill experienced, than classrooms where pupils are “apprenticed into a system of knowing, thinking and doing”(Boaler, 2009), which those at Phoenix Park experienced. Therefore, this has allowed me to reflect on the teaching of maths I have experienced and has really opened my eyes to some of the ‘could be’ causes of maths anxiety amongst both pupils and teachers.

Through Jo Boaler’s findings it has become apparent that Amber Hill pupils demonstrated instrumental understanding and Phoenix Park pupils demonstrated relational understanding(Skemp,1989). The best way to think about these concepts is that those pupils who demonstrated relational understandings are like a chef, if they are missing an ingredient to their recipe they know what they can use as a substitute to make what their making taste the same. Whereas those pupils who demonstrated an instrumental understanding are like recipe followers, meaning that if they are missing an ingredient they would have no idea what they could use as a substitute, therefore would be unable to complete the recipe. In a more mathematical example, mathematics is hidden in our everyday lives which some people are unaware of (Haylock, 2010, p13) such like telling the time, counting money, reading bus timetables as they forget that this is ‘maths’, therefore unconsciously allow themselves to overcome their confusion/fear of maths with no realisation that they are doing it, because it is relevant at that time. However, as soon as maths is disclosed, it causes fear and uncertainty across many (Haylock, 2014, p4). Therefore, again implying that when people have that basic understanding, know where maths can be used in their own lives and are able to use their knowledge in different situations, it begins to erase this maths anxiety (Haylock, 2014).

From this it has highlighted to me that we need to create ‘chefs’ within maths in schools, where pupils are enthused, engaged and enjoy maths because it should not just be about learning a concept, having to memorise it, just to get the answer correct when completing a test because asHaylock (2010, p5) goes on to describe, a lack of confidence within maths is formed from thinking you have to get the answer correct. This is where from the discovering maths lectures, already have got me thinking about how I can have an influence in reducing the fear and anxiety of maths amongst pupil. I am now aware that in the classrooms today we need to create an understanding, which allows for wrong answers and allows children to focus on the methods and thinking behind the answer (Hansen et al, 2017, p3) and move away from this ‘traditional’ maths and begin to teach ‘different’ maths.

Overall, I think maths anxiety can be destroyed within the younger generation and I believe that a raise in attainment within maths can be created by doing ‘different’ maths, other than just the traditional set methods and procedures, distinct topics and closed mathematical problems. I think that if we teach maths to create basic understandings where choice, openness and experiences are present and we create ‘chefs’ within the classroom, children will have a much better chance of having flexibility and adaptability within their learning, being able to transfer their knowledge to different situations, therefore meaning that when they do come across concepts/problems that are more difficult and that they haven’t seen before, the pupils can begin to look at them with a much more open mind set.

References

Boaler, J. (2009) The Elephant in the Classroom: Helping Children to Learn and Love Maths. London: Souvenir Press Ltd.

Skemp, R. R. (1989) Mathematics in the Primary School. London: Routledge.

Hansen, A., Drews, D. and Dudgeon, J. (2017) Children’s Errors in Mathematics. 4^thedn. London: Learning Matters

Haylock, D. (2010) Mathematics Explained for Primary Teachers. 4^thedn. London: SAGE Publications Ltd.

Haylock, D. (2014) Mathematics Explained for Primary Teachers. 5^thedn. London: SAGE Publications Ltd.