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
Jim Hogan 2017 • Teacher
Introduction and Lesson #1
Introduction and Why...
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!
There are plenty of books about mathematics. Many about number and
geometry and 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". They 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
might have learned before.
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
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 and knowledge. This interface needs to be nurtured.
In the following chapters I try and unpack aspects that I see as
important conduits and builders 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 go to anywhere they choose. The links are at the foot of
the each page (or chapter).
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...https://en.wikipedia.org/wiki/Mathematical_anxiety
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 "Thank's 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 think 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 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.
I start by beginning. Now. Here.
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 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 learn more
and 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 "Hogy!" and tell me about their journey. Thank you.
Above all, enjoy.
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.
I have spent the last 18 years as an advisor of mathematics
and statistics visiting many schools, meeting hundreds of
educators, running many workshops, answering and sending thousands
of emails, driving more than 500,000km, organising 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. Ask any
search engine "Jim Hogan NZ" and you will find my history and
story as it happened.
- to increase your understanding of the need to establish a
robust learning relationship with every student in your
- to interview two students to build
student-teacher relationship to inform student learning
Who am I?
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 that. 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 (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 at university and many
beers 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.
Computers were becoming smaller and more available. My physics,
app 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
I worked as a meteorologist for a few years after gaining a double
major in applied mathematics and physics and then spent 30 years
in a classroom 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 mathematics and physics then too. I
certainly was learning a great deal more than any of my students!
We learned together.
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
I now have a small vineyard and winery tempting fate and
challenging nature to produce a fine Pinot Noir in Taupo on the
cool Central Plateau at 345m above sea level. There are today 1200
bottles of a very drinkable Burgundy style Pinot Noir in my
cellar, all quite dark 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.
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 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 would a teacher intent on building a 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
• 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 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.
• The Benefits of Knowing Students as Learners
Developing an in-depth understanding of each learner enables
1. Create a psychologically safe environment for
2. Determine each student's readiness for
3. Identify multiple access points to the
curriculum to increase engagement and success.
This is to help look around my
pages. I have tried to make it consistent in all chapters.
The Planned chapters are only ideas at the moment.
SO YOU CAN READ IT!
All pictures on this site can be copied and enlarged.