Cling to the main vine, not the loose one.
Kei hopu tōu ringa kei te aka tāepa, engari kia mau 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

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