# Exploring Classroom Cocoons

For this blog I’ve chosen to speak about Ron Berger and Austin’s Butterfly as well as on the book relational understanding and instrumental understanding’ by Richard Skemp.

I believe Ron Berger uses Austin’s Butterfly to emphasize that learning is non-linear and that it’s OK to make mistakes. Indeed this isn’t immediately clear to many students especially since the education system expects nothing less than correct answers. The process or individual journey of learning is overlooked in standardized testing and so it can become easy for a student to lose focus on their progression and dwell on mistakes. By allowing the students to give feedback on Austin’s work and put themselves in the teacher’s shoes they will hopefully become less resistant to feedback as they will see it as motivation and a chance to improve and hone their skills; keeping the final draft of Austin’s butterfly in mind when improving their work.

To go back to the idea of non-linearity. It is significant that in the Austin’s butterfly example the third draft is less butterfly like because students will not always be working at a consistent grade. It can be very demotivating for a student to have gotten an A on one piece and a B on the next, especially since consistency is expected. Indeed some students might be quick to assess that they are simply “Not as good” on that topic, instead of striving to develop their understanding.

Skemp’s ideas on relational mathematics were profoundly interesting to me because I can remember different teachers who favoured either relational or instrumental teaching of mathematics. In cases where I was shown both approaches for a topic I always favoured the relational learning and the instrumental method. That is to say I liked learning why it worked but I liked having a quick and reliable method for solving problems which I think might not be dissimilar to most students. At the end of the day though teachers have to detach from what worked for them and consider the differing needs of students. Indeed a relational method of mathematics, while more broad and adaptable, might not be as accessible to lower sets who aren’t motivated enough or able to explore further into a topic.

I also really enjoyed Skemp’s analogy of Mathematics in relation to directions. In this case a relational approach would be his consolidating knowledge of the town and other streets, whereas an instrumental knowledge would be knowing a path to get from A to B. On the one hand it is always better to have a relational knowledge because we are more adaptable and can have several tools at our disposal for solving a problem (As Skemp says) but on the other hand teachers have a limited time with students to prepare them for exams and it may not always be realistic to dive into every topic. Skemp concedes that teachers will have to make a judgement on this matter and think about when students need an instrumental understanding and when we should focus on a relational understanding and I agree with him.

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## 3 Comments

## Stephen

4 November 2015I like your thoughts on the inconsistencies in students grade and how it can be demotivating, everyone has good and bad days yet we expect students to always be on top form. I was wondering whether you thought that as in the butterfly example whether relational understanding allows for a non-linearity in mathematics. If for example we are aware of similar situations in other branches of mathematics it allows us to compound our learning by drawing links between those branches or do you think the non-linearity is dependent on the feedback of other students? Just a few thoughts that came to mind.

## Emma

8 November 2015I found your piece thought-provoking and interesting. I was intrigued by the way in which you linked the relational learning to instrumental methods, in your real life experience. I like your last statement, and wonder how teachers are to make these decisions, in so far as, what factors would dictate whether, instrumental or relational learning was more beneficial?

## pepsmccrea

9 November 2015Good to see you trying to make sense of the video and article. I’m not 100% sure exactly what you are referring to by non-linearity in this context, but it sounds interesting!

Next time, try to see what connections you can make between the articles, and with previous ones explored.