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

Lesson #3 • Revised 9/1/20

Beginning a Year and  Beginning as you Mean to Finish • Knowing your Purpose
• to describe how to begin your journey with a new class
• to begin to see beyond 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 diagram stating what mathematicians do. This was the first entry in our Math Workbooks. I said "We are going to be doing this all year in everything we do" and I would revisit this statement whenever I saw the connection. I wanted students to notice patterns and to frame a rule around the pattern using words and algebra. My target...nth term, a key indicator of being a multiplicative thinker. More on this later.

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 and by students. 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.

10x6 Pent
          PuzzleN-ominoes Lesson One in Hogan's class
Task 1 --- My students are asked to rule up a 2cm grid and cut out five perfect 2cm side 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"and if their ears were "painted on"!

Task 2 --- Once each had acquired a tidy set of 5 squares they create shapes by joining the squares.

Joining squares needed agreement and was our first discussion. 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 or 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 as well. It is always surprising what students know. 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 know?

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 alphabet clues.

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 about proof. Knowing. This is the world of mathematics and "being a mathematician". That is a good start to the year! On Day 1 we are doing some serious maths.
Set of 12 Pentominoes
The complete and named set of Pentominoe Shapes.
Does anyone have an idea on how we can prove there are only 12 shapes? It appears we can not make more but that is not a proof.....

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!; that belongs to the student.

Task 4  --- Taking the Dog for a Walk - Explanation
Let us start with just the one square, which is the trivial or obvious case, 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.
  The N-Ominoes
Monominoe        1 square    1shape    Boring but very useful, the Muppet Song - Mah Ną Mah Ną
Dominoe            2 squares    1 shape    A game called Dominoes and worthy of a lot of maths investigation.
Triominoe          3 squares    3 shapes    Curious
Quadominoe    4 squares    ? (left as ? for you the reader to rediscover)
Pentominoe     5 squares    12 Hogan's best discovery! How many make an open box? Connections.
Hexominoe    6 squares    A Homework problem for students. How many fold to make a closed box.
Heptominoe    7 squares    ?
Octominoe    8 squares    ?
Nonominoe    9 squares    ?
Dekominoe 10 squares    ?
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".

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 the time went. We enjoyed concentrating, persevering, contributing. We struggled. We have fun. I talked with each student. Each student talked to me. The relationship has begun developing.

At no time did we add or multiply, subtract or divide. But we did maths! And we had fun. My main vine is secure.

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 36 hexominoes. 

Hence Lesson #3
After my first lesson and 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.

          reflectionTeacher TASK
• 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 nurtured?
• Record and share what happens when you start a class for the first time.

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