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

123Thoughts on Teaching and Learning of Mathematics
Jim Hogan 2020

Introduction and Lesson #1
Revise 5/1/20
Introduction and Why...
- to explain my purpose
I have been wanting to write a book about the teaching and learning of mathematics for some time now. How do I start? How will it be organised? What do I say? What do I know that others do not? How can I share what I know? Will anyone take any notice? How long will it take? Stop asking questions and start!

2020 - Quite a few teachers now have been referred here to read what I say. There has been some good feedback. It is useful!

There are plenty of books about mathematics. Many about the history of number, geometry and more recently plenty of statistics resources. I learned using Durell's Algebra and still have a copy, PSSC physics, and likewise, still have a copy. [around 1965 to 1969]. As a teacher I used Andy Begg's "Moving with Math" Series, Bill Elwood's "It a Maths World" series and more recently the David Barton Series "Alpha, Beta, Gamma". The books are all great descriptions of how these guys see mathematics at school level and give deep insights through crafted explanation, diagram and graded exercises. Used appropriately, a text book like a work sheet, has its place in the classroom. I still like pulling out my fluid dynamics  and applied math texts and re-exploring something I  have learned.

I have met Andy, Bill and David. Great people, deep thinkers and my heartfelt gratitude for all their hard work, perseverance and the study that went into crafting the mathematics books that have helped thousands of students. For me Andy is remembered by the word "relationships", Bill by "Fibbonacci"and  David by "thoroughness and perseverance".

I do not need to repeat these masterpieces in text book form, even if I could. I am more interested in the interface between the teacher and the learner. I will use all texts available but it is the tenuous human interaction called relationships that allows learning to blossom. Ako; the bi-relationship of respect for knowledge. This interface needs to be understood and nurtured.

In the following chapters I try and unpack aspects that I see as important pathways and building blocks to help young people become persevering, collaborating, critical, communicative, creative, curious, respectful and appreciative using the Learning Area we call Mathematics and Statistics. I am linking them so the reader can choose in any order. The links are now on a navigation page on this website.

Another interest for me is understanding why so many people say "I was never any good at maths". I hear from students... "Mum says she was never any good at Maths so I am not either!" This is mathphobia and is a common mental condition which has been allowed to develop and promulgate itself from generation to generation in much the same way that poverty does. See Wikipedia...

Everyone and anyone can learn mathematics and actually have fun and enjoy the experience. I want students to say "Oh Cool! Maths again! I like maths" and after the lesson "Thanks Sir! That was fun!" If our classrooms are not fun and enjoyable we are contributing to the mathphobia syndrome. Make it a school rule to have every teacher say "I like maths"... every day.

I feel Math-phobia is a mental disease of the modern world and one, as math teachers, we must know more about to help prevent infestation. It is a disease embedded in society, upheld by teachers and mathematicians, scientists and politicians, parents and students, in many subtle and not so subtle ways. I think it is so severe it can manifest itself as discalulia which I see as another of these modern syndromes invented to help explain lack of achievement in mathematics. The brain is a very capable magician. How, from the thousands of variables involved in learning mathematics, is it that a person is so sure that the one variable called "being good at maths" was the exact factor that caused the lack of achievement? It is the magician, the brain, at work. The brain will create "truth" whenever it needs.

Now clockI start by beginning. Now. Here. (2017)

I organise it on a computer in .html so it can be accessed willy-nilly, forward and backward as need arises. This is how learning happens so I write in the same way. I write so I can edit and amend as I re-read and re-proof what I mean to say. I add and and change things. I respond to comment. I persevere. I struggle. I ponder and reflect.

I hope these pages inspire you to explore or revisit an aspect of teaching and learning in your own practice, or use in your department for workshops, and that you become a better practitioner as a result. I honour all the teachers who gave me their patience and all my colleagues who inspired me to learn more and who answered all my questions. I honour all my students who taught me so much and the many who now stop me in the street with "Hi Sir!" or "Hoagy!" and tell me about their journey. Thank you.

Above all, enjoy using mathematics.

Lesson #1
RELATIONSHIPS - KNOWING the student and discovering THEIR NEEDS.
At the start of each chapter [or page] is a couple of key outcomes I see as useful and achieveable.
- to increase your understanding of the need to establish a robust learning relationship with every student in your care.

- to interview two students to build the student-teacher relationship that will inform student learning needs
I  have spent the last 19 years as an advisor of mathematics and statistics visiting many schools, meeting hundreds of educators, running  workshops, answering and sending thousands of emails, driving more than 600,000km, organising and helping to organise conferences, creating many new resources and discovering many others, enjoying many hundreds of hours in the peaceful solitude of my own mind finding truth in mathematics, proving things I needed to prove and reading books I needed to read. It has been a very rewarding time, very satisfying and I can only recommend the pathway. Now in 2020 I do much the same work but more specific to a school's needs, always referenced in data and research, and flavoured with experience. I see my self more as a "dream maker" as I try and make the goals and ambitions of teachers come true. I do not tell teachers what to do but help them build their own ideas, knowledge and understandings so that their learning is embedded and retained. Too often I see facilitators tell and as as soon as they are gone, practice reverts to the same old.

Who am I? (The image on the background to this page.)

I grew up on a farm in the Manawatu near a little place called Maewa on the main trunk line. Our family donated the 9 acres of native bush now called Maewa Bush to the QE2 National Trust. Maewa means "a quiet walk" and the bush is just that. Overgrown now after being locked up for 20 years but still a quiet place. I learned many things on the farm about life, surviving and food. I became resourceful, curious, creative, critical and communicative. I learned to collaborate and worked with diverse groups mixing with old, wealthy, poor and young. I did not learn about investment or shares. My family used their hands as did most of my ancestors from what I can tell.

I enjoyed school at Lytton Street School (1956-1964) and Feilding Agriculural High School (FAHS) as it was known then (1965 - 1969), gained one prize in Applied Maths and Chemistry in Sixth Form (Y12) and was a Prefect in my final year. I had many magnificent teachers and paid for most of my studies and many beers at university by becoming a #1 Chain Slaughterman at Borthwicks Freezing Works and working there for about 8 seasons in the 1970's. 

I grew up in the Cold War, Sputnik and Apollo Moon times. New Zealand was developing fast and recovering from the WW2 sacrifice too many made.  Computers were becoming smaller and more available. My physics and applied math teacher Jock Tennant, ex Sapper in RA, built a computer that could add a couple of binary numbers and explained how it did so. Who could not become interested in mathematics at that time! I later became a Ham Radio operator, ZL1AJQ, and built many gadgets, clocks, radios and power supplies. I once invented a cloud counter! All a bit old fashioned now when I look at the computing power of the latest iPhone XS.

I worked as a meteorologist for a few years after gaining a double major in applied mathematics and physics and then spent 23 years in a classroom learning and teaching maths, science, physics and computing (when computing was interesting). Like most teachers I had many jobs to play out as a teacher including being a Dean, HOD, House Leader and ran clubs for Tramping, Ham Radio, Fishing and Pythagorus (mathematics). I learned to teach when I become a teacher! I think I learned to understand mathematics and physics then too. I certainly was learning a great deal more than any of my students in the early days! We learned together.

Lake Taupo
I now live in Taupo beside our Great Lake Taupo. It is a large puddle with 62 cubic kilometers of water. This is enough for every person on Earth to have a crystal clear glass of potable water. I have traveled to Italy a few times, France, UK, USA, Australia and several Pacific Islands. I am glad I traveled the world when I did. New Zealand is a very nice place to be and I enjoy every day right here.

I now have a small vineyard and winery tempting fate and challenging nature to produce a drinkable Pinot Noir on the cool Central Plateau at 345m above sea level. There are today 1200 bottles or so of a very drinkable Burgundy style Pinot Noir in my cellar all quite dark, 13 to 14.5% ALC and full of berry flavours.  My favorite tools include a fishing rod, a chainsaw, a hammer, a spade, a computer, a radio and a vehicle. I play golf, walk the dog and shoot possums that stray near the fruit trees. I learned how to do cryptic crosswords in the NZ Herald over the last few years and usually get them out within and  hour or so. I love learning new things. I dreamed of the WWW internet, "beam me up Scotty" technology, and space travel when I was a kid. I loved the TV programme Thunderbirds which let my imagination loose and was an avid Dr Who fan. It is now all here. In 2017, NZ Rocket Lab launched 3 orbital rockets and launched 18 or so mini satellites for paying cutsomers. They are up to double figures now and teh US has launched the Space Force - roll on Thunderbirds!

Ce moi! I learn by doing. I learn in many ways. I strive to be a better teacher.

I have shared myself with you in the same way you must share yourself with your students. Having your students answer a letter about yourself is a good way to get to know them. Welcome them in and give them a letter to read and a letter to write!

Hence Lesson #1 and a Task.
The first thing a teacher must do is to create a relationship with every student by sharing themselves as a person. So must each student for each other and for each teacher. As I have done above so must a teacher intent on building a connected learning environment do so for each class situation. People like to be connected and when this happens the platform for sharing and learning takes place. The Māori word for this is ako, a mutual understanding of respect for learning.

Know studentsTeacher TASK
Some questions...
How do you build relationships in the classroom with your students?
What deliberate actions or activities do you engage in to promote robust learning relationships?
How do you have students share who they are?
and an Action...
Interview two students and collect all the knowledge you have about them. The wheel in the picture gives a few sample questions. There are some examples interview sheets around but start with school records, genealogical data, previous math tests, ask them about sports, family, ambitions, reading, learning, friends and let them talk. This is not prying; it is getting to know them.

A SELF CHECK - List all the students in one class and write ten things about each one... all from memory. Do this during the year to check.

Reading The Benefits of Knowing Students as Learners
Developing an in-depth understanding of each learner enables teachers to
1.    Create a psychologically safe environment for every learner.
2.    Determine each student's readiness for learning.
3.    Identify multiple access points to the curriculum to increase engagement and success.

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know the student