Cling to the main vine, not the
loose one.
Kei hopu tōu ringa ki
te aka tāepa, engari kia mau ki te aka matua
Thoughts on Teaching and
Learning of Mathematics
Case Study Examples
Introduction
Here I will try and describe a few projects and the interactions I
had, interventions, actions, observations, and developments that
helped cause student achievement. Like all social science it is
very difficult to pinpoint the actual cause to a shift in data.
Usually there are so many variables interacting simultaneously it
is impossible to be sure. Add also the complexity of human
behaviour when trying to assess learning and know that during a
test students are also learning to the extent that after the test
they are now better mathematicians than before it! It is a complex
stew or mess. I often say learning is messy. What was actually
measured?
In a post grad physics paper I sat I re-wrote the question and
answered the new question in preference to the one the professor
had presented. It did not seem odd at the time to do this and I
was rewarded with a Grade A, whatever that meant, and a huge
paragraph written by the prof complimenting me on the improved
question, a new approach to the problem, and a careful detailed
logical well reasoned solution. I knew what that paragraph meant.
There is a really good message in there and both he and I enjoyed
the event.
Later in my classes, the next 20 years or more, I returned to this
experience and...
- told my students they could rework the examination papers I
returned to them and they had this one period to talk to anyone
they like, use any text book, ask me questions and correct
anything they seem to have good wrong. I knew who my diligence,
top, and merit students were and I had a very strong grasp of need
and knowledge of each student in the group. The result was 60
minutes of feverish shared learning and 32 perfect scripts with
everyone getting top marks. I watched in astonishment at student
driven learning. Why am I teaching these kids? They can teach
themselves. I branched after that into more problem based
approaches. What a great lesson for me.
- in another case I told them they could each write there own exam
paper. It had to be 10 questions, cover most of the work we had
done and be about 60 minutes long. The result was 30 exam papers
and a wide range of questions all pretty much the same sort of
level, some easier and some harder questions, open and closed, and
some very creative innovation showing quite deep thinking and
pedagogical understanding. Questions that had more than one
answer, problems, unusual contexts, new applications, contexts
familiar only to that student, written and visual questions. The
many ways in which I had taught these students were reflected in
the questions they produced. They all had personal favorite
questions and this showed. Curiously not everyone scored the
perfect 100% or Grade A result. Some had set themselves tasks that
were quite difficult! No one had set 10 simple questions! I said
to them that I had looked at all their answer sheets, was
impressed by the integrity and quality of the questions from
everyone, and recorded my estimate of their math
understanding, how well they were going compared to all the
previous Year 9 students I had taught, compared their work to the
NZC L4/5 NZC displayed on the wall and added extra comment about
perseverence, attitude, creativeness and critical thnking and
other relevent competencies they displayed in more recent weeks.
They all loved that style of assessment and this also was a great
lesson for me and one that I used again in classes at most levels.
Case Studies
SCHOOL #1 2019 DEC
I had established the LOMAS Test data collection described in the
Measuring and Monitoring Chapter for all Year 7 to 10 students and
4 teachers during 2018 and this continued in 2019. We tested at
the 5/6 week stage in each term. I set up the spreadsheet and
monitored the cohort, classes and indviduals. The previous year,
Year 8 in this case, and the extra e-AssTTLe tests were included
to the data as checks. The data below is in broad
chronological order. I ran workshops on LOMAS, marking and
recording and normalised its use as a tool and not a teaching
device. Teachers were told they could run any unit tests they
liked but it was only this data that was needed.
The workshops and one on one teacher meetings during the year
revolved around the purpose of teaching, being multiplicative,
ways of teaching factors multiples, problem solving, lesson
starters, lesson structure, and journals. Quite often a teacher
would bring an idea and I would help develop and "fatten" it up.
We explored online resources and I encouraged teachers to use and
develop favorite sites. I always tried to support teachers in what
they are doing and add extra "dimensions" and "options" to
stretch, fill out and connect math ideas, activities and problems.
Building teacher PCK is a strong feature of my work. Building
Curriculum understanding of progressions of concepts is also a
feature. Deeply stressed is "all students becoming
multiplicative!"
Here is the EOY Year 9 report for the two class cohort.
This is from the End of Year Numeracy Report
Year 9
The success illustrated here represents a cohort improvement form
a little over NZC 2 to a little under NZC 4 in one year. The usual
progression expected is half of an NZC Level in one year. This is
based on official Ministry design having 4 Curriculum Levels
spread over 8 years (Y1 to Y8). These students as a cohort have
been accelerated past 0.5 to more like 1.6 (3.8 - 2.2) or 200%
acceleration (1.6-0.5)/0.5 expressed as a percentage. This is the
usual change/original% formula.
The pattern replicated itself across Year 7 to 10.
Year 7

Year 8
Year 9 (as above)
Year 10