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
Beginning a Year
and Beginning as you mean to finish Knowing
to describe how you begin your journey with
a new class
to begin to see past mathematics to
the real learning
|1. Look for a Pattern
|2. Find a Rule
|3. Test the Rule.
I began all my Y9 classes with this mantra. This was the first
entry in our Math Journals. I said "We are going to be doing this
all year in everything we do" and I would revisit this statement
whenever I saw purpose. I wanted students to notice patterns and
to frame a rule around the pattern using words and algebra. My
Math Journals are to record important ideas and solutions. They
are colourful and something to treasure from school days. I still
have a few choice selections of my own. For me they are a place to
record formative comments. There are also write on books which are
great for practice. See Sigma for example.
Then after that first entry we went into the the world of
mathematics and statistics...and for Year 9 students into the
world of n-ominoes.
Lesson One in Hogan's class
Task 1 --- My students are asked to rule up a 2cm grid and cut
5 perfect 2cm side length squares.
This is an excellent first task to assess use of ruler, use of
scissors, following instructions, CCCC and self management!
I would discover many things while working my way around the room
and linking names to students. How astonished they were, each
year, when at the end of this first period I named each student as
they left the class. I also learned a lot about each student and
if each could measure or had "5 thumbs on each hand".
Task 2 --- Once each had acquired a tidy set of 5 squares
they create shapes by joining the squares.
Joining needed agreement. A join is a full side to side
connection. No halves or corner joins. No overlaps. The rules were
listed in their words. [in 2017 I did this using multilink
blocks" with Year 7 students.]
Students worked in small groups at tables of desks and soon
started to record shapes by drawing them. I used a space on the
whiteboard for students to record new shapes as they found them.
Reflection and rotation needed some discussion. Some shapes are
flips or rotations of others and we just need the one shape that
represents them all.
Task 3 --- A Problem, "Have we got all the shapes?" How can we
Pretty soon most shapes have been discovered and I challenge them
to be absolutely sure they have found all the shapes. We remove
the reflections and rotations and name the ones that are left with
How can we know that we have all the shapes? This is critical,
creative, logical, collaborative, communicative thinking all
wrapped in one. This is mathematical proof and students do do not
even know what they are really learning. I know they are learning
how to prove.
Day one, maths class we explore the elements of proof. Knowing.
This is the world of mathematics and "being a mathematician". That
is a good start to the year!
Does anyone have an idea?..... I suggest we "Take the Dog for a
Walk". It is time to tell. Telling is about guiding in the right
direction and not about doing the walking. Teacher "lust" often
takes control here and I see teachers taking the pencil, or the
scissors, or the squares, or the problem and suddenly the student
becomes a passive observer. Let the student do the walking and...
be patient! It is not your learning teacher!; it is the
The complete and named set of Pentominoe Shapes.
Task 4 --- Taking the Dog for a Walk.
Let us start with just the one square and walk another square
around each side and record all the different shapes that we make.
As a class we agree there is only one shape that can be made and
we shall call that shape a 2-ominoe or a "Dominoe". We know about
dominoes, and we will revisit them when we start algebra. I stick
with the Greek names for numbering.
Around the Dominoe we "walk the dog" and discover three new shapes
called the set of Trominoes.
And so a new list on the whiteboard begins.
And by building logically from one square we soon discover the 12
pentominoe shapes and know there are no others. A proof by "all
possibilities" or "exhaustion".
square 1shape Boring but
very useful, the Muppet Song - Mah Ną Mah Ną...fun
2 squares 1
shape A game
squares 3 shapes Curious
Quadominoe 4 squares ?
(left as ? for you the reader to rediscover)
Pentominoe 5 squares 12
Hogan's best discovery!
Hexominoe 6 squares ?
Heptominoe 7 squares ?
Octominoe 8 squares ?
Nonominoe 9 squares ?
Dekominoe 10 squares ?
In our first lesson we have done algebra and introduced the notion
of infinity, Greek names, and a problem solving method. We
have been creative, collaborative, critical and communicative. We
have been doing mathematics. The first lesson not surprisingly
runs out of time but students are surprised how fast that time
went. We enjoyed concentrating, persevering, contributing. We have
fun. I talked with each student.
At no time did we add or multiply, subtract or divide. But we did
maths! And we had fun.
Homework Challenge - Find all the hexominoes! I sometimes
found out what parents said when my students explored and
explained what they were doing. Amazingly many students willingly
did this task and returned the next day with the drawn list of all
Hence Lesson #3
After my first lesson the main vine is secure, they are all
thinking mathematically, we have looked for a pattern, found a
rule and we are testing it. We are developing the CCCC
competencies. The students are being mathematicians and learning.
In the first lesson above list the different forms of
thinking, communication, participation, contribution and self
management skills that have been build on.
Compare with your own first lesson. How was your Main Vine
Record and share what happens when you start a class for the
Intro and Relationships, L#1
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.
2. The Main
3. Beginning a
4. Number and
Geometry and Measurement L#5
Probability and Statistics L#6
Problem Solving L#7
Visual Mathematics L#9
Assessment and Learning L#10
11. Team and
Problem Based Learning L#11
12. Engagement L#12