Thinking Mathematically
I am going to consider each
session separately as I took away different things, and experienced different
emotions as learner in each.
On Saturday I arrived during
the intro and could feel the anticipation in the room. I grabbed a seat at a table with a few spare
with folks I didn’t know. This may not
have happened had I arrived sooner, but over the four sessions there was great
discussion with space to think and share.
After a gentle problem start,
I quickly felt in the place of a novice / student with the next problem which
was a good bit more challenging than the first…and than I initially realised.
I was faced with another in
the group who told me I was wrong, and kept trying to explain their thinking
while I was trying to reason further. I
grew frustrated, thinking “I’ll look at this later” ….which we were very soon
shown was part of the purpose of this session.
The silence which greeted the realisation that some of us had been in a
rush to share our thinking, that some of us had needed more space than we got,
immediately made me think that others were reflecting as I was on their own
classes and this type of situation.
Anne and John had clearly
planned this morning to a tee – playing on their immense experience and
skills. I was left marvelling at how
they got us to engage with the maths, but allowed us to be learners, while
delicately laying the pedagogy and overview across this – modelling as they
went.
What a dynamic duo, with a
partnership and ease to envy!
Takeaways from session 1:
Create space to learn…time and
opportunities to think.
Use concrete resources to make
problems real, and allow imagining
Give ‘permission’ to draw
diagrams, or say in own way
Numbers pop out and can take
over thinking
Remove numbers from situations
to get at problem structure
Get learners to create other
problems with same structure
The next problem was a
seemingly simple problem or two with many blank grids. We set about them in companionable silence
after the reflections from the first session, then started to share
observations in what felt a much more constructive manner! At our end of the table we talked around
number patterns and algebra, and made some assumptions about the sequences
increasing. However, we promptly broke
this assumption to play around with negatives and see what happened.
When presented with a visual
of the task – graphs – we rapidly fell into the same assumptions again….it
can’t go down, or fluctuate, it seems to stop at zero. Reinforcing the assumptions for each other,
somehow forgetting things we already knew.
At this point Anne quietly
appeared at my elbow and said “why has this stopped here?” “who said stop at zero?”
I sheepishly pointed at the
grids, which didn’t stop at zero, but the point was made…
Takeaways from session 2:
Beware assumptions from
student and teacher
Beware misconceptions, and
students’ mutual reinforcement of them
Always remember to consider
sense-making which fits to prior knowledge
Visuals are very powerful, use
them more, vary the mode/representation of info
When the third problem
Overflow was presented, featuring many words and numbers my brain rebelled, and
I struggled to engage with the task.
I was aware of being in
position of pupil for first time in a while, and how adeptly I was being made
to experience this and consider the impact on learners. I saw the title “overflow” as describing how
my brain felt, and probably how some of my learners feel at times when I talk
too much or present too much info at once.
John talked again about
stripping back the problem to the structure, removing the numbers, imagining
the situation, drawing a diagram, and as he led us through this is it was
brilliantly clear and powerful.
After I started to get my head
round the problem I started to work out capacities, while the physics (and
maths) teacher beside me patiently and repeatedly described the model and we
both drew diagrams and tried to make sense.
As the session evolved it became clearer that the problem was very
complex indeed, but by breaking it down and building it up again we were able
to get far with it.
I was repeatedly aware of the
time we were being given to wrestle with the problem, “space to learn”, and
although repeatedly stopped and prompted, given more time again. Something I increasingly fear we don’t do
enough of with pupils.
I have a fear from frequently
hearing teachers saying “but we don’t have time” (a pressure I feel too) that
we often worry about “covering the content” in courses, giving rules and
procedures, but perhaps don’t allow students to sense-make and experience
situations and reasons, to process and absorb, to appreciate the beauty of
maths. How often do we stop them or move
them on just as they are starting to engage or make progress? Then we wonder why students give up, or stop
engaging with the ideas we present them.
The groans that went up a few
times on being interrupted, but Anne then saying “finish your thought, make a
note of what you were doing”, was such a powerful experience. Give them time to engage and sense-make, to
experience the joy and wonder, and to continue to engage.
Again graphs were introduced
and added another dimension to the problem, varying the perspective, and
prompting us to ask each other repeatedly:
“how do you know?”.
Takeaways from session 3:
Strip back problems to
structure
Encourage visualisation and
drawing diagrams
Give them time….then give them
more time!
Give a variety of perspectives,
visuals and prompts
Then give them more time!
For the fourth problem I
quickly recognised it as a recurrence relation problem, but worked with the
numbers rather than jumping to formulae.
I noticed what was happening to the sequences, and tried the formulae to
check, but lacked confidence to say this definitely when asked what I thought
in the group – not sure why!
When someone else in the room
explained about the steps and fall backs in the problem becoming the same over
time, my colleague in the group said in surprise: “that’s what you said!”
Why did I not speak up
confidently? Was I scared of being
wrong? Again I found myself in a learner
experience and contemplating how to handle it in my classroom.
John spoke at one point in the
morning about how ideas need to be aired to be inspected and improved, but how
to encourage this, and how to make sure correct things proceed?
Takeaways from session 4:
Encourage voicing of ideas, to
examine and improve
Normalise ‘mistakes’ as part
of learning, and sense-making
Let students experience beauty
of maths, not rush to practice the next process
Overall takeaways for my own
classroom
Create space of learning, not
pace of learning
Use concrete resources,
visuals, diagrams and graphs more for other perspectives
Give ‘permission’ to use
concrete resources, diagrams and own words, to make mistakes while learning
Allow pupils to think about the
deep structure of problems
Use prompts to train pupils to
think mathematically – one at a time, repeatedly
Encourage pupils to reflect on
their work and purpose of it – again using one question often to embed it for
pupils
