I attended a one-day conference on the Gold Coast April 17 with this title. It was subtitled *Engaging Students in Mathematics Learning*.

The keynote – *The Psychology of Adolescent thinking and connections to Mathematics Learning* – was given by Andrew Fuller (see his website at www.andrewfuller.com.au) then there were presentations from Judy Hartnett (Lecturer – Queensland Uinversity of Technology), Judith Selby (Head Teacher at Cowra High in NSW), Julie Wright (Canobas High), Steve Flavel (Education Consultant, WA), Matt Skoss (NT) and Charles Lovitt.

Andrew Fuller challenged my thinking. He started with a list of what adolescent brains do well and what they do poorly.

**Do Well Do poorly**

Vigilance Transfer

Self-focus Self-Appraisal

Challenge Coping with anxiety

Taking risks Motivation in the face of ‘failure’

Forming new patterns Recognising patterns

Playing games (competitive element) Auditory processing

Sequencing

I had a few problems with some of these items – particularly how they relate to girls. In my experience of teaching girls for the last twenty-odd years, I have found that girls are pretty adept at self-appraisal and recognising patterns. I have also found that girls are not big risk-takers and sometimes don’t enjoy the competitive challenge of games. They also tend to have more of a social justice focus rather than self-focus. I did, however, agree with his inference as it applies to teaching: that it is useless asking “Does everyone understand that?” as some students may think they understand and not respond in the negative and some may say they don’t ‘get’ it as they always err on the side of under-achieving to avoid possible ‘failure’ if they say they understand then get ‘found out’. It is important that the teacher asks the right questions to elicit the right information about whether their students really understand the underlying concepts or not.

He then went onto say that it was a myth that students can multi-task and learn. They can do many things at the same time but not well enough to do so in a learning environment. Focus is needed to learn effectively. He said that learning:

- involved imitation and experiencing difference
- was designed to create meaning
- was motivated through mastery
- was best achieved in a social setting that allowed for interaction
- involved skills and habits, not just content

He followed this with a summary of how brain behaviour could positively affect the learning environment for adolescents and advocated immersing students in high-quality experiences that will train the brain to ‘skill up’. This, of course, is the same idea as what Martin Westwell from Flinders University has been saying for a while…and, to be perfectly honest, says much better than Andrew did. My notes from Matrtin’s address at last year’s ACEL conference follows.

*The way our brains are ‘wired up’ depends on the number and type of connections made. These connections are determined by the experiences we have. These lines of communication between brain cells consequently determine the learning formed. It is the interconnectivity of ideas between cells that transforms information into learning. Repetition of these experiences re-inforces the connections made. (Me: important, therefore, to ensure the connections made are those that produce quality thinking rather than regurgitating a learnt script). It isn’t important as to how the information gets into the brain but what the brain does with it when it receives it.*

*Anxiety (especially long term), just like how we are educated, changes the way we think…there is an emotional component. Anxiety can physiologically prevent us from achieving our potential, it inhibits learning. (Me: So intervening to improve learning means intervening when affective learning behaviours are not going to produce optimal learning as well as intervening when cognitive behaviours aren’t conducive)*

*An experiment done with a group of young black boys in the US produced the following. These boys were all given an IQ test. Half of them were just given the questions. The other half were first asked to tick a box to describe their ethnicity. Even though the groups’ ability make-up were very similar, the second group produced significantly less IQ points as the other. (Hattie’s “the best predictors of a child’s achievement are the child’s predictions”…if the child believes that they are going to perform badly then they will.) Students who think of their intelligence as fixed usually have achievements that decrease over the course of their schooling. Those who believe intelligence is malleable are more resilient, can come back from failure, don’t give up as easily and show a positive trajectory in terms of their achievements.*

*The executive functions of the brain that we should be encouraging and promoting in the way we teach are:*

· *Concentration*

· *Resisting temptation*

· *Delayed gratification*

· *Self-directed/interdependent learning (note to self: use these terms instead of independent/group work)*

· *Problem solving*

· *Creativity/Innovation*

*The environment we create in classes and schools can affect how students develop their intelligence.*

*Take, for example, the experiment done with mice who were deliberately injected with Huntington’s Disease…a disease that withers the brain. Huntington’s is a genetically inherited disease. If you have the gene, you develop it…or do you?*

*Only 20% of the infected mice who were placed in a rich environment full of wheels, crawl tunnels etc actually developed the disease. 100% of the infected mice, who were placed in an environment in which no stimuli were provided, developed the disease.*

*So..what is an enriched environment for schools? One that is multi-sensory, relevant, that has emotional content, interpersonal interactions, exercise, good nutrition and hydration and one that has sufficient blue light (eg sunlight)*

*Another automatic reaction of the brain (leftover from animalistic days when we needed to protect ourselves from harm) is its reaction to risk. This has huge implications for both teaching/learning and change agents of systems, such as education. We have impulsive preferences for certainty. This limits the potential for innovation. Our brains want us to ‘go back to what we know’…don’t risk the uncertainty. We see this whenever anything new is suggested or introduced. For example: technology. In the UK, when the internet meant that students were plagiarising their coursework component, the system reacted by making more assessment external and assessed by examination. As with anything new, however, the challenge is not to dismiss its existence in our reaction, but to be judicious and deliberate in our use of it to support, promote and encourage what is the essence of education: learning. The other mistake is to go overboard in its use. Not everything new is ‘good’ for learning – a lot of the educational technology games may lead to greater short term engagement but not to long term learning. Keep in mind the purpose. It’s not the technology per se that changes what and the way students think, it’s about you and what you do with it.*

Based on his knowledge of brain behaviour in adolescents, Andrew recommended the folowing strategies:

- High level of feedback
- Repetition
- Ample time for wondering, being intrigued
- Opportunities for challenge, experiencing difference
- Making meaning

Implications for teachers of mathematics? Mini maths review quizzes at the start of every lesson, more physicality in activities, more opportunities for student discourse and interaction, more problem solving. To expand their memory capacity, increase repetition of key ideas and processes, use new information deliberately to link into already-held knowledge and describing things in different ways. He also recommended teaching the ‘concept’ – but I don’t think he knew that a concept was different to a fact – before relational questions were asked, giving bullet-proof definitions then a series of examples and non-examples.

It was this last set of recommendations that made me realise the message was getting lost due to the fog created by his lack of understanding about the nature of mathematics and the learning of mathematics. He, like many others, including a substantial number of mathematics educators, believe that mathematics is a set of definitions, rules and processes to be memorised and practised rather than a set of ideas and understandings and his examples demonstrated this.

In fact, I was a little disappointed, truth be known, in the conference as a whole. I got the distinct impression that this was the underlying mindset of most of the presenters – that mathematics is a set of skills that define the content and that the best we can do as teachers is prepare engaging activities (like the Maths 300 ones that were used extensively as examples) that use these skills in interesting ways or provide insight into what ‘working like a mathematician’ means….activities that our students enjoy and thus engender a positive relationship to the discipline.

I believe, we, of course, can do so much better than this.

I really found it difficult to justify separating classroom experiences into ‘problem solving’ and ‘skills’. As readers would know, I believe it is possible, desirable, no; imperative, to teach mathematics for understanding. Start with purpose and carefully consider what it is we want our students to be able to do (skills), understand and know for every lesson in every topic, then deliberately align all learning activities to those learning objectives and assess for our students’ learning at regular intervals. The Maths 300 lessons are great but sometimes go off on tangents, in my opinion, that take students (and teachers!) away from the concept one is trying to instil. Here I agree with Andrew Fuller – adolescents need repetition of the ‘big idea’, they need to be focused on that big idea to develop understanding and create their own meaning. That’s the job of the teacher: to decide what the big idea is and design learning activities that deliberately address and continually re-inforce this big idea.

As an example, take the Radioactivity Maths 300 activity. I use this as my introduction to exponential functions in Year 10. We start with 1/6 as the rate of decay, we do the die activity, we plot the class results as a graph. My learning objective is to develop the understanding that multiplying by the same constant for each successive value of the indepedent variable, always results in a similar curve that we call exponential decay (or growth) and that the magnitude of this constant determines the rate of decay, or growth, of these curves. Hence, my next step in my lesson plan is to stop and discuss the shape of the graph obtained. Why isn’t it linear? (Opportunity here to repeat the conditions that result in a linear graph) Why is it coming down? We talk about half life. We do it again with a different rate of decay. We stop and compare. What’s the same as before? What’s different? Why? What if we ‘grew’ rather than decayed? What if the number of atoms doubled each year? etc. I have a clear purpose in mind for the lesson. I keep bringing back my students’ focus to that learning objective. I would then transfer that understanding to some skills-based questions on exponential rules and graphing.

How is this different to what was recommended at the conference? Skills seemed to be done in isolation and prior to ‘using’ them. With the lesson plan above, I started with ‘the whole game’ – as David Perkins would say – then drew out the understanding I wanted and then used this as a justification for learning about the related skills. This is how I think mathematics is best taught. With understanding comes engagement. Students can ‘work like mathematicians’ every single lesson!

I did come away with a couple of lovely ideas I will put into practice this coming term. I hadn’t seen the Maths 300 Soft Drink Cans activity. It was recommended as a ‘strategy lesson’ at the conference – to teach students about breaking a problem down into manageable chunks, considering one aspect at a time. For those who don’t know it, students are given a 6×4 rectangular grid and 18 cubes. The problem is to place the cubes onto the grid in such a way that every column and every row has an even number of cubes in it. This proves to be quite tricky and frustration tolerance is tested. After about 5 to 10 minutes, the class is stopped and the teacher brings everyone around one student’s table. He/she asks the student to arrange the blocks on the grid so that every row has an even number, forgetting about the columns for the time being. When this is done, the teacher says…”Now, slide cubes along the rows – but keeping the rows intact in total – to make the columns all even” The problem becomes incredibly easy with this simple instruction. The Maths 300 lesson goes onto extend the problem in various ways BUT this won’t help students transfer knowledge of this strategy into other situations. I immediately thought of its application to solving simultaneous equations. The whole idea of solving simultaneous equations is to deal with one variable first then come back and find the other one. I’m going to try doing this little acitivity first then, in the same lesson, start simultaneous equations and try and embed the strategy in some skills work straight away so that their brains can make the connection and thus strengthen the procedural link.

I’ll let you know how it goes. Good luck for Term 2.

Linda

Thanks for taking the time to write this reflective piece.

I use Maths 300 in the same way you do. I use it to set the context out of which I draw the key topic and skill set to be worked on.

Did you get to spend time with Charles Lovitt? I am usually bored at PD events. Most of it is pretty mediocre. But Charles. he seems to know his stuff and have maintained a strong connection with actual teaching. Quite rare.

Anyway. Thanks for the post. I actually read it to the end.

Have you seen the Conceptions of Mathematics inventory developed by Crawford et al. (1998)? It sounds like it will test some of what you’re interested in.

Thanks for your careful reading and comments, Russel and Maria. Glad to hear that you use the Maths 300 lessons in a similar way, Russel. I feel somehow validated!! Yes, I did get to spend some time with Charles (and he was our guest presenter at the faculty retreat last year).

Maria – thanks for the information re the Conceptions of Mathematics study (1994). I’ve accessed the abstract and will follow up with the full article. It sounds very interesting and, yes, just up my alley.