NOT JUST ANOTHER
PROBLEM SOLVING
BOOK
Jim Hogan
Teaching and learning ideas, resources,
solutions, problems and some ideas
2015
Foreword
The time has come the walrus said
To speak of many things
O seas and ships
And sealing wax
And cabbages and kings
Of why the sea is boiling hot
And whether pigs have wings.
Lewis Carol
To all teachers of mathematics, you all need medals! If you have never read
“Alice in Wonderland” go and find a copy and read it. Likewise Mark Haddons
book called “The Curious Incident of the Dog in the Night”. Brilliant reads. My
favorite read is “And to My Nephew Albert I Leave the Island What I Won Off
Fatty Hagan in a Poker Game”. Google the name and buy it on Amazon. You will
collapse in laughter but please accept it as a gift in appreciation of endeavoring
to teacher students how to think mathematically and statistically.
I have been dithering about writing a book for a long time. What to write? I spent
20+ or so years, in a variety of classrooms where mathematics, physics, science
and computer technology were my contexts. Now when I reflect on the
experiences I am not sure what was taught or what was learned. Ex-students
stop me in the street and meet me in the work place and all talk enthusiastically
about the time we spent in school and no one ever mentions a mathematics
problem. Peter Grootenboer (Griffth University) told me about this curiosity in
2002. Google his name and read some of his work. Ahead of his time!
I know for the first few years it was I who was the student and I thank those
wonderfully patient students with whom I had the pleasure to be a learner. We
had fun. I learned where the edges were. We experimented, did problems, got
lost, found ways, explored and survived. That was May 1979. I arrived in the
secondary school to teach geometry and in the 1980 was teaching Year 13
physics, Year 12 physics, Year 9 Maths, Year 11 Maths, Year 10 Maths and was a
Dean and AV TIC. The next 20 years were a repeat of this concoction filling in
where needed. All good fun but there were some very tiring days.
After graduating university in 1976 with a double major in applied mathematics
and physics I trained as a weather forecaster and studied fluid dynamics.
Weather was rather human-less science and I had always wanted to be a teacher
so I fled windy Wellington and in 1978 engaged in the one year Dip Teach
program in Palmerston North.
My first appointment was in an intermediate School but as soon as I gained
registration I moved to Taupo and a huge pay increase. I was also escaping a
grumpy headmaster. My first class in secondary was May 1979 Form 5 (Year 11)
mathematics, geometry. Wonderful and I still have the lesson.
I learned during those early years that problems need to be modeled, made,
constructed, taken apart and viewed from different perspectives. I also learned
that I could learn. I learned that the more maths one knew the more maths one
used. I learned that discussing, talking, arguing, showing full proofs, exploring
hard maths, making mistakes and having fun caused learning. I had fun and I
made a lot of mistakes. We all did!
In 2002 I was appointed as a mathematics advisor with School Support Services,
Waikato University and helped with the introduction of the new NCEA, contacted
and supported teachers in schools and regional associations like BOPMA and
WMA, and the national association NZAMT. The learning curve was steep, I
travelled Waikato, BOP, Coromandel and Gisborne. I established contacts and
made a great many new friends. Dave Boardman had suggested I apply for this
job and he and I met up again in Gisborne many times. Dave and I had worked
together in Westland during 1983 to 1986. He was a problem solver and it was
he who started my journey as an advisor. Dave sadly died a few years ago.
The NZ Numeracy project had been underway for a few years as ENP, ANP. The
INP part was developing and the SNP was planned. I was given a crude
introduction to how students learn number and given a school to work in at INP
level. The following year the first major steps were taken for developing SNP and
in 2005 the project began. It was a lot of work and huge professional
development for most secondary schools and hundreds of teachers across NZ.
These resources, research and project are all stored by the Ministry of Education
on http://www.nzmaths.co.nz . I think we all did a good job.
It was expected that we would make a difference. Looking back we were told it
was “a professional development programme for teachers” but it would be
measured by student performance. It might have been wiser to have a control
group and focus on improvement in teacher knowledge and practice. We
measured, as had been confirmed for a few years that about 15% of students
entering Year 9 were multiplicative thinkers. The broad stages of increasing
complexity of thinking were very well researched and explored. This was
recorded in “The Framework”[see nzmaths].
Being a multiplicative thinker meant that a student would select a multiplication
related method or strategy as first choice when solving problems. They would
use factors, multiples and know facts. Being a “mult” thinker means you can be a
proportional thinker, if you learn how.
We learned a lot during this project. One idea was that counting, adding
multiplication and then to proportional thinking was how we progressed in
increasingly complex thinking. I saw this repeated everywhere. A language for
conversation between primary and secondary was developed. A lot of
investment was made into mathematics. It was a huge time for math education.
Resources were created and trialed. Materials developed. Hundreds of them.
But in 2009/2010 we were still measuring 15% of all students entering Year 9
were multiplicative (CL4). I had expected that with the work being done in the
primary sector and certainly in the schools we were working in that a dramatic
improvement would be evidenced. That measure never changed.
Now 2014 I still measure 15% with some variation of course between 1 % to
80% in different schools as we did in 2004. The majority of students entering
secondary use addition as their most advanced form of problem solving.
Secondary teachers make a difference as this huge group improve from NZC
Level 3 to 5/6 in the first three years at secondary. There is variation of course
but pretty much the rate of progression through NZC levels doubles and I put
that down to specialist mathematics classroom teachers. Knowing and
appreciating mathematics is taught along side trigonometry (for example) and
that is probably more important.
The NZ NP was researched like no other project. It was reported and reported
and reported. Doctorates eventuated, masters programmes completed and
hundreds of workshops, yearly conferences and connections with MAV, NZAMT
and MERGA all studied aspects of education in mathematics. This is all recorded
on the nzmaths website.
I have wondered why no book had been produced about this massive injection of
effort into mathematics education. No record of the players and what happened
exists. It was all researched so there is a record and we do have the data. The
researchers had their names recorded.
I always contended that we did not emphasize multiplication enough. I can
remember long discussions with key players who maintained counting and
adding were essential and many strategies needed. These ideas were connected
to place value development and the way numbers work. Place value is however
heavily reliant upon the powers of 10 and that is very multiplicative!
No big change happened. People who did not really understand mathematics
before continued to do so and relied upon rules and skills. Our push for
understanding was a big ask. Perhaps a higher mathematical requirement for
teacher intake would be a good idea. I see a hope from key researchers who are
now suggesting that multiplicative ideas of grouping be introduced a lot earlier
in primary, Year 1 and 2, and the Ministry has embarked upon the ALIM project.
http://nzmaths.co.nz/accelerating-learning
In 2014 I now am seeing the revised NCEA standards implemented. These are
very clever documents with purposeful pathways and now we have a new
curriculum upon which it is all based. The new curriculum is rich in numeracy
ideas from the NZNP. It is important we all teach this curriculum and use the
NCEA standards to assess learning.
The new NCEA standards are focused around “solving problems”. This book aims
to help create a “culture of mathematics” in every school so that problem solving
is an expectation, an ongoing enjoyment and a challenge.
The key requirements to problem solving is a curiosity, perseverance and some
AHHA!. The more mathematics we know the more mathematics we will use.
Problem solving is a learned skill. It develops and one becomes better with every
problem attempted. George Polya (1945) wrote in his famous book “How to
Solve a Problem” and that “it is better to solve one problem five ways than five
problems one way.” Purchase on Amazon.com.
The “How to Learn Mathematics” course from Stanford University by Jo Boaler
was a brilliant experience. I completed the course on line along with 30,000
other teachers. Wow! This is the way to go. Jo emphased 7 key points.
KEY MESSAGES
1. Everyone Can Learn Math to the Highest Levels.
Encourage students to believe in themselves. There is no such thing as a “math”
person. Everyone can reach the highest levels they want to, with hard work.
2. Mistakes are valuable
Mistakes grow your brain! It is good to struggle and make mistakes.
3. Questions are Really Important
Always ask questions, always answer questions.
Ask yourself: why does that make sense?
4. Math is about Creativity and Making Sense.
Math is a very creative subject that is, at its core,
about visualizing patterns and creating solution paths that others can see,
discuss
and critique.
5. Math is about Connections and Communicating
Math is a connected subject, and a form of communication. Represent math in
different
forms eg words, a picture, a graph, an equation, and link them. Color code!
6. Depth is much more important than speed.
Top mathematicians, such as Laurent Schwartz, think slowly and deeply.
7. Math Class is about Learning not Performing
Math is a growth subject, it takes time to learn and it is all about effort.
Go the www.youcubed.com website and check it all out.
The structure of this book is around collections of problems into the
mathematics strands of Number and Algebra, Measurement and Geometry, and
Probability and Statistics, and Logic.
I have just updated my website and this file and created “X marks the Shop” so I
can sell copies and fund further development of such mathematics resources.
Jim Hogan
Taupo FEB 2015