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

Lesson #8 Revised and completed 5/1/19


- 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 hard work to establish so persevere. You will be rewarded.

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 a later chapter later for comments in Problem Based Learning. Empowering students to be self managed and to take more control of their learning is being Future Focused. See a later lesson for more explanation here as well. Live in the future, not the past!

What is an Investigation?
An investigation involves exploring and problem solving just as problem solving involves investigation and creativity. An investigation is a broader and less focused task than problem solving. Investigation means exploring many pathways and discovering paths that end and change, paths that lead to new paths and paths that stop. This also means time and patience, persevering and being open minded. It might involve asking good questions and doing some research for new learning.

An investigation needs record keeping or journaling. Journaling in the C21st classroom may well involve online Google Classroom or similar. There are some excellent places students can store images, write and connect all this together. I like Microsoft Powerpoint as a media but I am starting to see this as being a bit outdated as the new media platforms popularise with the newer generation. I am a firm believer of using a pencil and a piece of paper to investigate but every phone now has a camera, every school has internet available, iPads and other tablets are common so students have a lot of online technology to use.

This is the modern world and the "Now" generation. I call it the "Me, Me, Me, Now, Now, Now" generation! The concentration time of this new generation after growing up with instant response technology is pretty low and undeveloped. Instant food, instant movies, T20 cricket, games available witrh instant responses, instant, instant instant, Perhaps this is the "Instant Generation". Whichever, they need to learn to invest some time to investigate an idea or curiosity and not look up teh internet to find out what someone else's thinking says.
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 unpacking and meaning making is learning in action. The engagement (see a later chapter!) is established because the students asked the questions, the relationship and safety established by the teacher creates the rich environment, and the experience and understanding of what learning is really about by the teacher feeds all the investigation and permanent learning for students. They do not need to be tested after doing an investigation like this! 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. Not 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 investigation in action. 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. Investigation takes time and was often, for my classes, an unplanned event. I think this is OK if the teacher has a deep mathematical understanding and strong pedagogical practice to allow exploration to lead discussion. It is handy also to know when to stop and time to tell. Only experience guides this and all teachers should bhe allowed to maek mistakes! I know I did!

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 (Liping Ma)
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.
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) 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).

(b) Give me a number between 1/3 and 1/4
Give me a number between 1/4 and 1/6
Give me a number between 1/2 and 2/3

Now you give me a question like this!
3x to the student and once back to reverse the question. [You learn when you teach!]
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 call 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 "exact" 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 #8
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.