On day 6 (Saturday)
of ICME13, the plenary lecture was by Deborah Loewenberg Ball, whose work I
have referred to a lot in my own articles lately. She talked on
"Uncovering the special mathematical practices of teaching". As
usual, I was eager to get signs of her definition of "mathematical"
(see yesterday's blog). She organised her story in three parts: a journey,
getting lost, finding the way. At some point, many years ago, there was a
movement from wondering what mathematics teachers need in teaching to what they
use. The "getting lost" part was building tools to
"measure" teacher knowledge. On the positive side, ways of studying
outcomes of teacher education and professional development were developed, but
on the negative side we fell back from understanding practice to looking at
knowledge. Also, it may have contributed to separating aspects of teaching, for
instance equity. So in a way, we got microscope images of many aspects, but not
understanding it from the inside. She argued that we have to work on the
"mathematical work of teaching" mathematics. (Still not defining
"mathematical" or "mathematics", but at least referring to
previous talks this week.)

The work of teaching
is taking responsibility for maximising the quality of the interactions in a
classroom in ways that maximize the probability that learners learn. (This fits
quite well with Biesta, doesn't it?) Using "work" is to focus on what
the teacher DOES, not on the curriculum, what students are doing etc.
"Mathematical" listening, speaking, interacting, acting... are part
of the work of teaching. A fundamental part of the work is attuning to other
people and being oriented to others' ideas and ways of thinking and being.

She then showed us
examples from a grade 5 classroom with 29 pupils in a low-income community.
Showing us different sequences, she asked us to pinpoint what the teacher was
doing. (What I saw: Getting people to come up. Classroom management.
Monitoring. Steering discussion (sociocultural norms), respond to question,
saying " it doesn't have to be right") She pointed out how the
teacher had walked around the classroom, reading nearly 30 student answers,
choosing which to look into) Then Ball mentioned concepts from Cohen and
Lohan(?) on assigning mathematical competence, including developing competence.
The work of the teacher is partly noticing everything that is not the wrong
answer. Translation from learning goals to meaningful language for children.
The work of teaching is discursively intensive work: talking while monitoring
if anyone understands what you're saying and understanding the answers.

I liked the talk
very much. As I saw it, it was a step away from "the ball", which has
worked to privilege some kinds of knowledge and hide away others, not to
mention skills. The criticism of testing for MKT is also welcome. What comes
instead, though, is not clear. I'm afraid "the ball" is too simple
and pretty to be replaced by anything messier or more filled with doubt and
judgements...

The first invited
lecture I chose today was Ansie Harding's talk entitled "The role of
storytelling in teaching mathematics". Her talk was geared towards tertary
education, which is interesting, but I also hoped to learn things for primary
education. Of course, storytelling is also an important way of including
history of mathematics in primary school (re discussion group at previous
ICME), so also from that point of view it should be interesting, as well as for
my cooperation with the L1 teacher of my students is next academic year.

Harding teaches
400-500 students at a time at the University of Pretoria. Here, she first gave
the story of storytelling, from about 50.000 BC. (Already at that point someone
challenged the veracity of the story, which raises the question of the role of truth
in storytelling - which is also part of the discussion concerning history of
mathematics in teaching; which role does myths have in teaching?) In her story,
she included movies, TV and the internet (not including games) - but the
examples she used did not include movies - she stressed the value of low-tech
storytelling. Instead, her first story example was the story of the length of a
year, leap years and so on. (It is a bit difficult to see the difference
between her story and normal lecturing, so maybe normal lecturing should be
defined as well.) Features of storytelling: are essy to remember, are
compelling, are for all ages, embed value, travel, fuel conversation. A story
often has a character, ambition, problem, outcome as well as an emotional
connection.

The value of
storytelling in education is to entertain, inspire and educate. With tertary
students, stories can be used as an intro (but is a bit like having the dessert
before the main course), as a "by the way", as a commercial break, as
a reward (which is how she uses it). You need to get everyone involved and
create a "class feeling".

Ingredients: maths
connection, human element, tale (start, flow, end), humour. Her second example
was "How mathematics burned down the houses of parliament".

With engineering
students, she uses the last ten minutes every week (which contains 200 minutes
a week) for story telling. She showed a list of examples, much of which the
usual history of mathematics "stories". Telling stories takes some
effort (you must make it your own), it is a culture that you foster, there are
unexpected rewards.

5-10 students leave
before the stories all the time. She asked students for comments, and 30
students responded. Categories: emotional impact, reward, motivation, subject
impact, appreciation (more approachable), bigger picture (more to maths than
this course).

She ended her talk
with two stories, one on De Moivre and one called "The story of one, five,
seven" on van Gogh's sunflowers.

For me, this talk
was an inspiration to dare telling stories more in my own teaching, and to put
stress on them, not to tell them apologetically and quickly. However,
"making it my own" will probably mean making sure they are relevant
to the mathematics and often the history of mathematics that we are working on.

After a break, I was
back to TSG47, where there were three talks:

Caroline Lajoie
talked on "Learning to act in-the-moment: prospective elementary teachers'
roleplaying on numbers". The observation was made in one lesson in teacher
education. Her key concept was knowing to act in-the-moment. (See Mason and Spence
1999.) in this context students take
role of the teacher and 1-3...students, in front of the whole class. Role play
involves introduction time, preparation time (everybody prepares all roles),
play time and discussion time. She gave examples on how students, even though
prepared, need to improvise. One took a risk in a situation where she was not
certain of the answer, while another chose to switch to an explaining mode when
he saw that he didn't know what would happen. This came up in the discussion,
but that was because students were willing to discuss and "criticize"
each other.

Pere Ivars on
"The role of writing narratives in developing pre-service primary teachers
noticing". "Narratives" in this talk refers to stories about
classroom interactions. The hypothesis is that writing narratives will help
noticing. Students wrote two narratives, with feedback after the first
narrative (and some detail of the feedback given were provided here). The
development from the first to the second narrative was more evidence of
students understanding in more detail (it is important also to notice that
students wrote the first narrative based on observation, the other based on
what they themselves taught).

Lara Dick on
"Noticing and deciding the "next steps" for teaching: "a
cross-university study with elementary pre-service teachers." Noticing is
a skill teachers need and that they need to develop. Much work have been done
with using video to work on noticing, but little on use of student work (such
as the previous one). The current project was a three-hour lesson on
multiplication. The students saw student examples and analyzed one each and
made posters. The students were asked to attend to the mathematics, interpret
what they see and make an instructional decision - based on looking at all the
posters. Four major themes: gravitation towards traditional teaching ideas,
vague decisions, desire for written number sentences, focus on strategy
progression. (This seems like a lesson it would be useful to try to copy with
my students as well.)

(About TSG
proceedings: there is a format suggestion from the ICME , but it is unclear how
much flexibility there is. The suggestion was few but extended papers, while
the TSG organizers would prefer shorter but more papers.)

These talks remind
me that key concepts to meet for new teacher students are noticing and building
on what students know instead of figuring out what they don't know.

After an extended
break, I attended, of course, the HPM meeting. Luis Radford gave an
introduction to the HPM group, its
research and publications, while Fulvia Furinghetti presented the history of
the group. Then, Kathy Clark, the new chair of the HPM, gave a presentations
with her own background and some views on the HPM. She mentioned two projects:
ÜberPro project (Germany; Übergangsproblematik) and TRIUMPHS (US; transforming
instruction in undergraduate mathematics primary historical sources).

(The proceedings of
the 2016 HPM conference are online here: http://www.mathunion.org/fileadmin/ICMI/files/Digital_Library/History_and_Pedagogy_of_Mathematics_Proceedings_of_2016_ICME_Satellite_Meeting.pdf
)

This concluded the
last whole day of the conference. However, discussions on mathematics education
and other interesting topics continued for many more hours...

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