I have spent time reading this informative and thought provoking book by MikeAskew (@mikeaskew26). Below are some of the key points that resonated with me.
Contrary to popular opinion, most children rise to the challenge of ‘hard’ mathematics rather than shy away from it…dependent upon a particular style of classroom ethos, close attention to the mathematical challenges presented and support for children in their efforts.
The nature of teaching is, and always will be, an adaptive challenge, rather than a technical problem…adaptive challenges require solutions that have yet to be found. We need to work with a view of mathematics teaching as an adaptive challenge. That means trying out new ways to teach and in particular allowing pedagogies to emerge rather than imposing them.
Current practices establish norms about different abilities…Society at large also labels people.
To meet the challenges of mathematics teaching, ways of working in classrooms need to emerge through the joint activity of teachers and children. Learning does not only happen in the minds of individual children – classrooms are learning systems. By attending to how the classroom community grows and learns (teacher and children together) it is possible to create classrooms where children: engage with meaningful mathematics; learn that they can learn mathematics; develop socially and emotionally; realise the importance of inter-dependency.
Thinking about learning
“Learning…is more of a reaching out than a taking in. It is participation…The agent’s activity and identify are inseparable from his, her or its knowledge. Knowing is doing is being.”
It is the combination of the following 3 together that makes mathematical teaching powerful:
- Learning as a collective activity
- How learning involves becoming as well as acquiring
- Integrating the maths that ‘emerges’ as children work on rich problems and investigations with pre-planned learning intentions
Being meaningful is not merely about relating contexts to their ‘real’ lives; it is a meaningful context they can ‘mathematise’. (E.g. Place Value in Market Stalls example)
Teachers often say that the children can understand the mathematics but cannot apply it, rather than question the assumption that there is a logical connection in going from the abstract to the application. Starting from realistic contexts and mathematizing these may help the reapplication of this mathematics to other contexts later.
How do we encourage a classroom community that is a co-operative collective rather than a collection of individuals?
Thinking about curriculum
There is little point in teaching something if 3 months later the children show no understanding of it.
Most skills in maths only make sense in relation to other ones, so picking the off in isolation isn’t the most sensible way to address them…Outside of school most learning comes about through engaging in whole activities rather than learning discrete actions or behaviours.
Reflective teaching needs to focus on the activity, the experience, of the learner, not on the actions of the teacher.
“Learning originates in the experiences of the learner, not those of the teacher.” (Bernie Neville)
If we want children to engage in maths then teaching has to be based around collective problem solving, more closely matching the communities of practice that learners are engaged in outside of school.
One of the biggest difficulties in teaching maths is the assumption that what the child brings is not significant.
Thinking about teaching
Teaching and Learning is somewhat mysterious and unpredictable and we need to accept and work with that rather than behave as though it were completely controllable.
Complicated systems: particular actions determine particular, predictable results
Complex systems: effects of particular actions are much harder, if not impossible, to predict. Gardens are typical examples as they involve multiple feedback loops that dynamically change the whole structure. The results of actions depend upon the actions, but they do not uniquely determine them.
Teaching and learning is a complex system: learning is dependent upon teaching but cannot be completely determined by it. Accepting this complexity is liberating. It means teachers accept things as they are and work from that reality, rather than wish things were different.
The role of a teacher is to optimise experiences of learning, and in doing so maximise the likelihood of learning.
Mathematical activity: mindful or fluent?
“Trying to solve a maths problem in a way dictated by the teacher is different from attempting to test one’s own hypothesis. The teacher who tells students to solve a problem in a prescribed manner is limiting their ability to investigate their surroundings and to test novel ideas.” (Langer)
We need to pay more attention to the process of coming to know rather than the end results.
Successful performance depends on engaging in co-constructing emergent mathematical activity.
Is our focus more on finding answers to calculations or more on becoming mindful of the underlying mathematics?
Do the children need to learn their tables? The point of being fluent in addition and multiplication bonds is not as an end in themselves, but how they free up working memory when tackling more interesting and engaging pieces of maths.
As learners become more fluent and confident so they become more engaged and involved in maths lessons.
It is the lack of experience that limits what children can do.
Asian children adopt addition strategies based on partitioning (e.g. 6 + 8 as 6 + 4 + 4) sooner than international peers.
VT provides a framework for thinking about how to maximise the likelihood of learning.
“Exposure to variation is critical for the possibility to learn, and that what is learned reflects the pattern of variation that was present in the learning situation.” (Runesson)
Directing the children to look for and think about possible connections.
Transforming the learner
“The primary aim of every teacher must be to promote the growth of students as competent, caring, loving and loveable people.” (Noddings)
Research shows that attending to relationships in maths lessons helps to raise standards.
Inequality of attainment in the primary years may be more a result of children’s different experiences than their ‘innate’ mathematical ability.
You find yourself in the flow with an optimum level of challenge that stretches your capabilities.
Build a classroom culture that emphasises listening to each other, working together and trying things out rather than waiting for the teacher to provide help.
Any worthwhile mathematical experience is going to lead at time to some difficult emotions: frustration, confusion and irritation. Confusion is a necessary part of learning mathematics and can never be removed from the process.
Building mathematical community
“A sense of belonging, of continuity, of being connected to others and to ideas and values that make our lives meaningful and significant – these needs are shared by all of us.” (Sergiovanni)
Mathematical communities needs to promote: trust, friendliness, inclusion as well as resilience, perseverance and curiosity, and be inviting, engaging and welcoming.
Work towards building and creating shared goals and values, rather than imposing rules and regulations that create an orderly class but not a community.
Learning needs periods of incubation – over more time.
‘Sharing’ needs the vital component of ideas and solutions being built upon by other learners.
Tasks, Tools, Talk
Maths not based on procedural fluency but involves understanding means learners are active constructors of knowledge, not passive recipients of it.
Setting up tasks with a certain amount of uncertainty is a way to make learners engage mindfully and bring their sense making to the activity.
Introducing models (10 frame, numberline, arrays…) takes time. Learners will only appreciate them through repeated exposure, and it takes them different time to take them on as tools for thinking…they need to be part of the pedagogical furniture of the classroom.
Talk is central to maths lessons…it mean mathematical vocab becomes part of classroom discourse.
Making sense of problems by explaining them to someone else, putting them in your own words and comparing your answer with others all help meaning to emerge.