Cling to the main vine, not the loose one.
Kei hopu tōu ringa kei te aka tāepa, engari kia mau te aka matua

123Thoughts on Teaching and Learning of Mathematics

Pics and links to do.
Investigations
and Lesson #8

- to understand the benefits of mathematical investigation
- to know how discourse creates meaning for learners 

Investigation is about building new knowledge and skills, making connections to  previous knowledge and most likely will involve collaboration.  Learning to investigate is like problem solving and should be normalized as regular event. Investigation is an inquiry so is led by student questions and answers. Groups are ideal situations where students create new knowledge and share learning. It is self paced and rewarding. Groups need some hard work to establish.

Allowing students to investigate their own questions means the teacher has to give some control to the student. This is a difficult idea to embrace if you run teacher directed lessons. See later for Team Building and Problem Based Learning. Empowering students to be self managed and to take more control of their learning is being Future Focused. See later lesson.

Example student questions which have led to Investigation.
"So why do you have to change fractions to have the same denominator Sir? You get the answer by just adding the numbers. EASY!" says student.
"Let's us look at what you say and see if it makes sense" says teacher "Write down a half plus a half and get your answer".
"I get 1/2 + 1/2 = 2/4" says student "See... it works." says student with conviction circling his answer on the white board so everyone could see.
" You could have added the 1 + 2 and the 2 + 1 and got 3/3" says teacher trying to cause a cognitive dilemma and internal questioning (critical reflection) by the student.
"3/3 is 1 and two halves should add to 1 so I like your way of doing that. That makes sense. Thanks Sir!"
"See if it works with other fractions first!" says teacher encouraging some structured investigation.

In your thinking groups investigate how fractions can be added"

This actually happened and it took a class discussion to unpack every misunderstanding this student had about fractions to try and remedy the issues.

The class learned a lot about collaboration, common understanding, numerator, denominator, what addition means and one of my favorite Axioms #1 was sited over and over again.

This student's issues of understanding was founded in his additive thinking and undeveloped ideas of multiplicative thinking. He did not understand the idea of 1 properly because it had never been explained or prioritised in previous learning. No only that but many students in the class exposed their misconceptions about fractions and we all decided to review everything we knew about fractions. That was a type of investigation. Relevant, co-constructed learning, in time and needed, mistakes encouraged, everyone to speak, discourse all present. All of these students eventually learned how to generalise fraction addition as a/b +c/d = (ad + bc)/bd. 

A secondary teachers job is not only about new conceptual learning and connections but about unraveling the confusions of concepts and ideas caused by earlier learning experiences, or lack of, and where the student has invented his/her own reality and sense making. This means, for the teacher, having a deep understanding of all the strands and progressions of concepts. Anything less promotes the confusion and loads up the issues for the next teacher. If you teach, teach for understanding. Know how but also know why.

Two Teaching Tools
1. Ask the Answer
2. Give me another, and another and now give me a problem.
"Ask the Answer" causes more calculation and thinking.
Task.
Ponder the difference in your thinking in answering these two questions.
(a) What is 4 x 6?
(b) List 24 problems that have an answer 24.

"Give me Another" probes for deeper understanding..the why!
A number between 1.1 and 1.11 is 1.101, another is 1.1001. So tell me one number between 2:2 and 2:22. (All said by the student).

Pentominoes (see Lesson 3) is an example of a teacher directed investigation as few students would choose this topic unassisted.
How many different shapes can you make using 5 squares? Side to side only and no corner only connections.
Draw all your solutions. Discuss and remove any rotations and reflections that are the same shape.
Do you have all the shapes? How do you know?

I make this a guided or directed investigation initially and use it as a first lesson in my mathematics classes for Year 9 students (See Lesson #3). I extend the pentominoes to n-ominoes, fat solutions, the 10x6 problem and then to 3d with the Soma cube. Students discover "mathematics" as distinct from "numbers" and begin to learn how to be curious and build confidence around making mistakes. I have them decorate solutions in their exercise book with colour and for faster students this is a natural and meaningful extension to a one hour class period.

Investigations in all Strands.
1. Number
(a) Start with any number, If the number is odd, add 1 and divide by 2, if the number is even divide by 2. Repeat with the answer. Do you always get to 1?
(b) The difference between the square numbers is very odd! Make sense of this nonsense.

2. Algebra
(a) Make a model of x, x^2, x^3 and try to explain what can be added and otherwise combined.
(b) Make a model of (x+1)^2

3. Geometry
(a) Is it more efficient to put a circle in a square or a square in a circle?
(b) Find out about the Lunes of Hippocrates.

4. Measurement
(a) Locate √2 on a number line.
(b) Use your cellphone to find to find the Girth of the Earth. How many ways can you think of.

5. Probability
(a) What is the probability that of you break a stick in two places you can form a triangle with the three pieces.
(b) How many people do you need in a room to have a 50% chance that two have the same birthday?

6. Statistics
(a) What is a good sample size?
(a) A sample size of 17 when ordered gives the 9th data point is the median, the 5th the LQ, the 14th the UQ and each is known exactly. What other sample sizes have this property?

Investigation is learning to be disciplined, getting organised and logical, exploring and being creative, communicating results, collaborating and sharing ideas. All this and my main vine is intact and my classes hum with thought and mathematical discourse.

Hence Lesson #2
Investigation and mathematics are the same thing. Investigation causes new learning and more questions. Problem solving and investigation are closely linked and develop many of the same skills. Investigations extend over time and involve others. Investigations support the main vine in every respect!

Teacher TASK
Describe or find an investigation in each of the strands that you could use.
Write down several questions you have been asked and how you responded to them.
List 10 competencies that are developed while investigating.

CHAPTER NAVIGATOR
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.
1. Intro and Relationships, L#1
2. The Main Vine, L#2
3. Beginning a Year, L#3
4. Number and Algebra, L#4
5. Geometry and Measurement L#5
6. Probability and Statistics L#6
7. Problem Solving L#7
8. Investigations L#8
9. Visual Mathematics L#9
10. Assessment and Learning L#10
11. Team and Problem Based Learning L#11
12. Engagement L#12
Planned
13. The Classroom L#13
14. Being a Teacher L#14
15. Being a Leader L#15
16. Managing the Principal L#16
17. The Importance of Whanau (Family)L#17
18. The Importance of the Student L#18
19. Math Topic A - Squares
20. Teacher Tools
21. LOMAS
22. Math Phobia