The following by Kelvin Smythe:
This posting supports mindset as expressed by the American academics Jo Boaler and Carol Dweck. I perceive mindset as a strategy to encourage holistic ideas into mathematics, in particular, a curriculum much in need of being repaired from its fragmented and perilous condition.
The historical problem of the teaching of school mathematics will be considered first. Then I comment on how Boaler and Dweck’s academic colleagues are turning mindset into a mess in that frenzied way we have come to expect North American academics to behave when an idea is presented that might be turned to professional advantage. American education has proved itself unworthy of mindset. A comparison will be made between two definitions from my File, and the nature of mindset as provided by Boaler and Dweck, which will demonstrate close similarities in the two ways of going about teaching and learning. Following that, using notes compiled by David McNair, formerly principal of Gordonton School, who visited Stanford University in 2017, I will set out in practical way the Boaler and Dweck principles of mindset.
Throughout much of the Western world, because of the way children are taught mathematics, there is a failure to make connection between the scatterings of mathematical ideas they are exposed to and the larger concepts able to give them coherence and meaning. Mathematics in classrooms is driven by specific objectives supposedly to end with something purposeful, but only spasmodically so – largely remaining a fragmentation as phonics is to reading.
I looked through the general North American academic response to mindset, it was education capitalism at work, every academic for him – or herself, an exploitation of a new source of career profit, with no sincere consideration of the effect of their behaviour on children. Now we have quantitative academics citing their appalling research and calling mindset a failure. What makes it a comfortable for the academics to be callous and uncomprehending is that most have not moved past the fragmentation phase in their understanding of mathematics. But the motivation for quantitatives to call the holistic mindset a failure is because fragmented learning gives power to them and publishers; while holistic learning gives power to qualitative academics and teachers. The writing by so many American academics on mindset has been simplistic, a concentration on learning carried out in short dashes to unsatisfactory effect.
For New Zealand, it would be advantageous, as a priority, if mindset was better ‘set’ in teacher’s minds before interpretation to children’s learning; and understood as a commonsense extension of the holistic.
There are four central components to mindset:
- The pivotal stimulus in learning is having the children affectively involved. That this is central to discussion rather than being so blindingly obvious is a commentary in itself. The more the affective is engaged in learning, the further and deeper, the teacher can take the learning. But we have allowed syllabuses, the pressure of testing, and small bits learning to hurry teachers along and confuse their judgement
- The interaction of knowledge with the affective is the key learning interaction: The two must act together in tight structural harmony
- Allowing children plenty of time and space to learn. This component is clearly connected with the components above
- Overcoming mathematics anxiety in children: many children, when facing anything other than routine mathematics, have the anxiety parts of the brain become agitated and the problem solving parts to close up. The way to solve this is set out in the components above.
Two definitions from the File:
- ‘The holistic philosophy is about the interaction of the cognitive and the affective; the combination of knowledges – teacher and academic (also other); teaching and learning being organised by dynamic broad aims (assisted by criteria that can be seen as converted objectives); dynamic broad aims being an expression of the essence of curriculum areas; learning being meaningful, exploratory, and challenging (hence the attention to discovery learning and problem solving); learning experiences having shape extending from an introduction, to gaining information, to using that information flexibly, to a conclusion; learning being coherent and organic not fragmented and desultory; children having significant control over their learning; observational evaluation being central; and that philosophy being fundamental to school education in a democracy.’
The holistic, which is a philosophy of education as a system, is closely compatible with mindset and encompassing of it. The holistic philosophy is based on the interaction of the cognitive with the affective in inseparable union. The holistic requires a dynamic main aim for the system (which is about preparing children for life in a democracy and to support and protect it), and a single dynamic main aim for each curriculum area (each, of course, connected to the system main aim). In mathematics, which is the curriculum under attention, the main aim always incorporates ‘problem solving’. The holistic main aim features the need for children to have ample time and space within in their learning. To ensure that the cohesiveness of learning is achieved: learning experiences need to have shape extending from an introduction, to gaining information, to using that information flexibly, to a conclusion. All these parts are about providing space and time within learning. There are further characteristics that the holistic and mindset share: that learning be organic; children have significant control over their learning especially when and how long learning occurs; and observational evaluation being the main form of evaluation.
- ‘In whatever the curriculum activity, the aim should be to provide an education experience of the sort that transforms (or is intended eventually to transform) children intellectually, aesthetically, ethically, and reflectively. Such experience takes children back to two fundamental cultural breakthroughs – the way things are, are not necessarily the way things are or should be; and an individual’s place in it is not predetermined: and from that the realisation of ‘I am’ and the capacity to think, imagine, appreciate, and appraise. In a sense, the individual becomes of the world he or she lives in and a step beyond it. Leading to three questions continually to be asked, wrestled with, and answered but never concluded, they are: Who am I? Why are we living this way? How can we make things better? It is this process that puts all curriculum areas, and the digital, into context, a democratic one – transforming the main purposes of everything that occurs to prepare children for life in a democracy and to support and to protect it.’
This definition is at slightly higher philosophical level than the preceding definition. But teaching, the aim should always be, whether mathematics or drama, to make the learning experience transformational; to have the children see the world in a different way.
The strong link between the New Zealand’s holistic in mathematics and mindset is demonstrated in a holistic mathematics series in the File (Attacks! 15, 28, 29) and in the quote that follows. Dan Murphy, principal of Winchester School and former maths adviser, has taken his teachers on an holistic mathematics journey.
The principles followed were:
- Dispensing with ability grouping
- Dispensing with the levels and stages of the numeracy project
- Reducing assessment procedures to a minimum
- Basing programmes on problem solving
- Using contexts for learning that are real
- Having all children work on the same problem but in ways appropriate to them
- Having children learn in a social setting to encourage reasoning and discussion
- Paying close attention to the learning strategies of low achieving children
- Avoiding breaking learning into small measurable steps
- Basing programmes on a mix of commercial programmes and teacher developed mathematics units – the commercial programmes being, however, only a small part of the overall teaching programme, mainly serving as a model for the teacher.
The following parts are from the report brought back by David McNair from a presentation by Jo Boaler – my adaption of what David had to say:
The course opened with a discussion of organisational flexibility: Children talking in groups; sharing – no hands up; recording – take notes when they choose; time – working on problems for as long as they like.
Followed by mathematical openness: Different interpretations, methods, answers; seeing maths differently; being curious; valuing mistakes; posing questions, pursuing inquiries.
The role of the journal has grown in importance (from Kelvin: I have seen them in use in New Zealand classrooms and they are a real stimulus – the children had large sketch books in which thoughts were noted).
A journal can provide children with the opportunity to reflect on learnings; to record hunches; and to ponder frustrations and triumphs.
The American course listed the following about the use of journals: The journal was an important part of our camp (the use of journals originated and was developed at mindset camps); we wrote our thoughts and findings each night; teachers never wrote in them; comments and feedback were provided using post-it notes.
A key message we give our students is: ‘I am giving you this feedback because I believe in you.’
A principle: Brain differences at birth are eclipsed by many opportunities to grow and change brains.
With right teaching and messages, all students have the potential to study high level mathematics.
Jason Moser: Every time we make a mistake synapses fire.
Two possible synapses can occur: The first comes when we make a mistake; the second when we are aware we have made the mistake.
Less growth when answers are correct.
The next message is the need to dissociate mathematics from speed.
Laurent Schwartz Field’s Medal winning mathematician: ‘I am still just as slow … At the end of the eleventh grade, I took the measure of the situation and came to the conclusion that rapidity doesn’t have a precise relation to intelligence. What is important is to deeply understand things and their relation to each other. This is where intelligence lies. The fact of being quick or slow isn’t really isn’t relevant.’
The next key message is the need to move the orientation of mathematics from performing to learning. ‘Mathematics is too much answer time and not enough learning time.’ Mathematics tasks need to give students space to learn. What we tend to do is tell students the method before they have a go at solving it themselves.
Replace homework with reflection questions for journal:
- What was the big idea we worked on today?
- What did I learn today?
- What good idea did I have today?
- Where could I use the knowledge I learned today?
- What questions do I have about today’s work?
- What new ideas did my work today make me think about?
Visualisation is really important because seeing an idea in different ways, different forms, and different representations makes the mind work – not the repetition of one approach and near identical questions.
When we calculate different ways, different areas of the brain light up, including two visual pathways – the ventral and dorsal (with the dorsal visual pathway representing the knowledge of quantity).
What happens when we make all of these changes?
- From someone just doing maths … to someone of much higher potential
- From speed and procedures … to depth and creativity
- From one way, one answer … to multiplicity of ideas
- From numbers and calculations … to visualising and exploring
- From performance culture … to learning culture
- From focus on correctness … to value struggle.
What does it mean to debrief a problem?
Another idea issuing from the journal is debriefing a problem. That is, sharing what you felt, what you found out about the problem, and yourself and what you learned
A significant challenge for many at the conference was that you do not have to provide an answer to a problem. This allowed us to continue thinking about how it might be answered. (I am still challenged by this, not surprising, I suppose, given the years of mathematics being about the right answer.)
We were presented with the following problem but asked to look at in a different way, in ways set out by Boaler earlier, especially to do with visualising.
Rebecca went swimming yesterday. After a while she had covered one fifth of her intended distance. After swimming six more lengths of the pool, she had covered one quarter of her intended distance. How many lengths of the pool did she intend to complete?
We were then asked to construct this matrix and complete it with a variety of ways to present our understanding.
The title of this and any other problem would go in the middle of the diamond.
This variety of approach was a particular challenge to those who only saw it as an algorithm.
The next problem
Our next problem was of a type that is commonly referred to ‘Low Floor, High Ceiling’
‘Low Floor, High Ceiling’ problems are those that all children can access but can be extended to high levels. These problems are important because holistic classes are heterogeneous. ‘Low Threshold, High Ceiling’ problems are activities that everyone in a group can begin and then work on at their own level of engagement. But they also have lots of possibilities for the participants to do much more challenging mathematics.
The problem: Painted cube
Imagine that we paint a 4 x4 x 4 cube blue on every side.
How many of the small cubes have 3 blue faces?
How many have 2 blue faces?
How many have 1 blue face?
How many have not been painted at all?
How many of the small cubes would have 3, 2, 1, and no faces painted in a cube? Think visually.
This kept the group going for about 40 minutes as we made, modelled, drew, and finally agreed on a quadratic equation that could be used to solve any cube size like this one.
It is one that I’ve tried with students and the discussion has been amazing. If students don’t know about an equation to solve this, you might at some point over the week introduce ideas around equations and see where they take it. You do not give the formula as doing this takes away the creative thinking.
- How did you experience the freedom to discover?
- How did you feel about uncertainty and mistakes?
- How did you engage as a mathematics learner?
- How did the role of the sceptic play out in your team?
Reflection and the journal are an important interacting characteristic of mindset. Note the reference to the role of the sceptic. This is the person who is assigned to asking questions and who must be convinced that the solutions make sense and are logical.
A further report on mindset may follow. Your comments on your experiences with mindset and other forms holistic mathematics are welcome.