Lately I have been reflecting on how much my understanding mathematics has changed. How much I now appreciate the beauty of mathematics and how much now I understand that the way I presented maths previously totally undermined the multi-dimensional nature of maths and encouraged children to see it as a narrow 'subject' full of rules to follow and correct answers.
I spent far too much time focusing in on knowledge (in fun ways) and lots of practice that unfortunately reinforced to children that for every question there was one right answer. I used devices as fun follow ups for this knowledge, thinking that this was engaging, but once again I was just reinforcing speed and the idea that for everything there was one answer.
Don't get me wrong, I used materials, we played active games, we sang songs, maths was fun...but the one common factor that my teaching was reinforcing to my children was that maths was largely about getting the correct answer.
There is absolutely a place for knowledge and 'the right answer' however the way we present it to children can affect the whole way they see maths.
When we started maths PLD I struggled to believe that presenting problems to new entrants would be beneficial at all, I in fact admit to thinking it would be an absolute waste of time and akin to herding cats. To be honest for a while I basically proved my own hypothesis, as we often do when we are quite negative about something. Problem solving felt clunky, but I persisted. Slowly I began to open up to the new ideas, incorporating talk moves, starting to use more visual problems, incorporating 'maths eyes' into my programme and really focusing in on the talk. Reading Jo Boaler and Youcubed assisted my journey and continues to do so.
Over time my sessions became less focussed in on the knowledge and practice (although this still had a place) and more open, incorporating more visual tools, more open ended images to encourage talk and things like dot talks. Throughout this time I felt myself changing, seeing maths everywhere, constantly thinking about maths, truly appreciating the multi-dimensional nature of maths. Throughout this time I also started to truly appreciate the power of talk, hearing children naturally bring in their maths talk when looking at visual images, truly growing in their understanding of mathematics.
As we have progressed I have noticed the number sense of children growing, as children faced authentic problems they began to see a real purpose for this understanding and so the knowledge came along for the ride.
From Youcubed "Mathematics is a beautiful, open, creative, and multi-dimensional subject. But school mathematics is often uninspiring, procedural and one-dimensional – it is all about memorizing methods and procedures."
As my maths eyes were opened I started to see the mathematical potential in everything around me, constantly taking photos to use within our sessions...I mean look at these photos below, now I see potential discussion....before I wouldn't have been able to see the maths in these. Just like the children my maths eyes have been opened. I know I am only just at the beginning of my journey, but feel like I have already come so far!
Taken from Extract from Chapter 1. What is Math? from What’s Math Got To Do With It: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (Penguin). pages 25, 27 & 29 Jo Boaler
"Problem solving is at the core of mathematicians’ work, as well as the work of engineers and others, and it starts with the making of a guess. Imre Lakatos, mathematician and philosopher, describes mathematical work as “a process of ‘conscious guessing’ about relationships among quantities and shapes.”13 Those who have sat in traditional math classrooms are probably surprised to read that mathematicians highlight the role of guessing, as I doubt whether they have ever experienced any encouragement to guess in their math classes. When an official report in the UK was commissioned to examine the mathematics needed in the workplace, the investigator found that estimation was the most useful mathematical activity. Yet when children who have experienced traditional math classes are asked to estimate, they are often completely flummoxed and try to work out exact answers, then round them off to look like an estimate. This is because they have not developed a good feel for numbers, which would allow them to estimate instead of calculate, and also because they have learned, wrongly, that mathematics is all about precision, not about making estimates or guesses. Yet both are at the heart of mathematical problem solving. After making a guess, mathematicians engage in a zigzagging process of conjecturing, refining with counterexamples, and then proving. Such work is exploratory and creative, and many writers draw parallels between mathematical work and art or music. Robin Wilson, a British mathematician, proposes that mathematics and music “are both creative acts. When you are sitting with a bit of paper creating mathematics, it is very like sitting with a sheet of music paper creating music.”15 Devlin agrees, saying that “Mathematics is not about numbers, but about life. It is about the world in which we live. It is about ideas. And far from being dull and sterile, as it is so often portrayed, it is full of creativity.”
"We cannot keep pursuing an educational model that leaves the best and the only real taste of the subject to the end, for the rare few who make it through the grueling years that precede it. If students were able to work for at least some of the time in the ways mathematicians do—posing problems, making conjectures using intuition, exploring with and refining ideas, and discussing ideas with others—then they would not only be given a sense of true mathematical work, which is an important goal in its own right, they would also be given the opportunities to enjoy mathematics and learn it in the most productive way."