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
Introduction and Lesson #1 • Revised 3/1/19
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!
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
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
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 "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.
I start by beginning. Now.
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 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.
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 2019 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.
- 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? (The image behind the 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
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 30 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.
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 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, 13to 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. Last year, NZ Rocket Lab launched 3 orbital rockets and
launched 18 or so mini satellites for paying cutsomers.
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 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.
• 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
• 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.
• 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.
SO YOU CAN READ IT!
All pictures on this site can be copied and enlarged.