Bernard Murphy, MEI
Elke Kurz-Milcke, University of Ludwigsburg
www.abcmaths.net
 The ABCmaths project
 What are the Big Ideas in Mathematics for you?
 Examples illustrating some Big Ideas
 The impact on our teachers
 Emotions and mathematics and Big Ideas
 Questions
The ABCmaths project
What are the Big Ideas in
Mathematics for you?
proof & argumentation
Our
working
list
multiple representations
doing & undoing
extending the domain
infinity (including limits/continuity)
exploring and extending understanding through questioning
teachers' awareness of existence of multiple strategies
dynamic imagery
specialisation & generalisation
functional dependency
modelling
using misconceptions and errors for learning
making connections within mathematics
using analogies
randomness & inference
Multiple representations
Why is it called
‘Completing the square’?
Extending the domain
Shorthand 2
3
↓
a b
Rules 2  2  2
a
b
↓
0
Definitions 2
↓
2
2 ?
2
4
2
1
2
Doing and undoing
What are the domain and range of
f  x   x  2x ?
2
Find a function with domain  x  ; x  0 ,
and range  y  ; y  0

2
0
f  x  dx  4
Give two similar examples
and another that is different.
Teachers' awareness of the
existence of multiple strategies
x
3
 3x  x  3   x  1
2
x3  3x 2  x  3
  x  1  ax  bx  c 
2
x 2 ...
x  1 x  3x  x  3
3
2
x3  3x 2  x  3
  x  1  x  ...
2

f  x  h  f  x
lim
h 0
h
Dynamic imagery
Exploring and extending
students' understanding
through questioning
Change one
aspect of
Change one
aspect of
2
2
2
3 y  x  4  x  4   y  2  9
y  2 x  3x  2
so that it is so that the circle
so that
perpendicular passes through
dy
1
to
dx
exactly three
y  2 x 1.
when x  2
quadrants.
Change one
aspect of
Give me an
Give me an
example of a example of two
curve with a simultaneous
maximum
equations with
point at
solution
(3,2)
x = 3, y=2.5
Give me
an example
of an
infinite GP
with sum 4.
Infinity
(including limits/continuity)
y
3
1
1 2 5 3
1

x

1

x

x  x  ...
 
3
9
81
 1  x  1
y
1
x
−1
1
Proof and argumentation
Prove that if you
multiply any triangular
number by 8 and add
1, the answer is
always a square
number.
n  n  1
2
8
 1   2n  1
2
The impact on our teachers
Our discussion of Big ideas gave me time to reflect about which ones I
had really been putting into practise. … as I plan my lessons always
thinking about the visual concepts, how can I use ICT to demonstrate,
how can I question and designing activities so I am assessing
effectively throughout, and I'm thinking about these across my lesson
planning at all stages, but particularly at A-level. The extent to which I
am successful I am unsure but I feel my understanding of the topics is
being enhanced, so perhaps my students are gaining that wider
perspective as well…
I can’t say that some are more useful than others but I would say that
doing and undoing and multi-representation can be far easier to
implement and also give an instant improvement to the level of
understanding and learning going on with the pupils
The use of writing sinx as a power series was possibly one of the most
powerful and unexpected links I have ever seen in mathematics. For
someone with a first class degree this procedural instrumental women
knew nothing!
For me, effective questioning is questioning that responds to and
challenges issues/questions/misconceptions that students raise in
lessons, which you can prepare for to an extent but there will always
be surprises. I think the skill of an outstanding teacher at KS3 and KS4
is to be able to answer these questions (or better still, encourage other
students to answer these questions) and lead the students to develop a
particular line of thought or enquiry, or deal with the issue that arises
and then re-direct or re-focus them back to the lesson objective. At
KS5 I think the latter is less important as the concepts (or topics?) are
fewer and more closely linked, so any question that arises is more likely
to have relevance to the theme of the lesson and will have merit in its
own right in terms of mathematical thinking and discussion.
The presentation from the group members and the paper on the use of
ICT was enlightening. It generated lots of ideas about how ICT can be
used to promote thinking about a topic, rather than just demonstrate or
project the learning. A theme was the effect that dynamic software has
on expounding ideas, raising questions and discussing concepts.
The questions that were presented in the session seemed to draw on
many ideas from Mason, including ideas of doing/undoing and giving
freedom and constraint on a where a question might lead. The ideas
and questions that each initial question presented engaged us as
learners to think carefully about the topic and expand on the
mathematical themes.
I am feeling more and more enthused to go and try some of the
activities we have looked at in these sessions as I'm sure you
intended...but there is a growing feeling in me of excitement about
changing the way I teach in general and encouraging others to do the
same. I feel like I am bursting with ideas and opinions but can't imagine
when I will have the opportunity, or indeed energy, to share them and
CONVINCE others to try them out too!!!
The 'Big ideas' of multiple representations and developing
understanding through discovery seem so instrumental (no pun
intended) to effective teaching and learning, yet the demands of our
curriculum restrict their use to a minimum. And the notion that we are
SCARED to try for fear of scolding by our peers and superiors instills
fear and anger in me in equal measures.
As teachers we are under all sort of pressures (and fears) that can
prevent us from being able to produce lessons that provide relational
understanding…Again I come away with a lot of ideas. I do just want to
be able to rip up the national curriculum and schemes of work and start
afresh. If I ever get this opportunity I fear that all my ideas will slowly
evaporate under the pressures and fears that then engulf you from
external sources. However, if I can make things work now on a
classroom level, I can move them to a curriculum area and then the
department area.
Emotions and mathematics
and Big Ideas
Bernard Murphy, MEI
Elke Kurz-Milcke, University of Ludwigsburg
www.abcmaths.net
Emotions and Mathematics
Does the emotional experience of mathematics point to Big Ideas?
Report by Elke Kurz-Milcke, University of Education, Ludwigsburg, Germany
Here I give a brief report on how we approached the relevance of emotions to mathematics
and the mathematics classroom.
It is quite common for people to have a general attitude towards mathematics, either generally
positive or negative. In this session the focus was not so much on this general attitude towards
mathematics but on the emotional states that are experienced while engaging in mathematics
as a set of practices. That people experience emotions, sometimes vigorously so, when doing
mathematics is undoubtedly the case. The couple, both professional mathematicians, who in
their daughter’s memory never had a quarrel about domestic issues but door-slamming
controversies about mathematical proofs is but one of the more exceptional variants. Each of
us having attended a mathematics classroom has surely experienced and observed emotions.
Bernard Murphy showed quotes from teachers participating in a TAM course designed around
Big Ideas and their potential for informing teaching. The quotes let us in on some of the
emotional aspects of learning mathematics, of teaching, and of improving teaching. In
addition to the impression that all of the quotes convey emotional engagement on some level,
the teachers explicitly described feelings of uncertainty, of fear, of pressure, but also
mentioned to be enthused, to be surprised, to feel energetic and their understanding to be
enhanced.
In our session at the MEI conference we asked the participants to each contemplate for
themselves the emotions that they themselves had experienced or that they had observed with
others while they or others were doing mathematics at some level. We also distributed a
leaflet describing 63 emotional states by an icon showing a facial expression and giving a
corresponding adjective . The participants were asked to note one emotional state of their
choice on an index card. These index cards were collected and one “emotion” was randomly
drawn from the stack. We then discussed where and how the people in the room had possibly
experienced this emotional state while engaging in mathematical problem-solving and how it
related to what they or someone else was trying to achieve at the time. The emotional states
that we “collected” in this fashion were:
- alienated
- angry
- apathetic
- excited
- ecstatic
- happy
- helpless
- mischievous
- satisfied
We discussed three of these states and saw that for each the group was able to describe a
number of situations in which it had occurred with a protagonist, i.e., themselves, a student, or
a pupil. We had allocated a brief time slot for this reflection on mathematics and emotions,
which was but an initial approach to the question of how awareness of emotions in
mathematics classrooms might square with an awareness of Big Ideas. In closing, it appears
that mathematics and emotions sometimes share the aura of being among the more solitary,
private experiences. Communication about emotions and mathematics also show us how this
is but an image.