Cling to the main vine, not the
Kei hopu tōu ringa kei te aka tāepa, engari kia
mau te aka matua
Thoughts on Teaching and Learning of Mathematics
Pics and links to do.
Investigations and Lesson #8
- to understand the benefits of mathematical
- 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
"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
Two Teaching Tools
"Ask the Answer" causes more calculation and thinking.
1. Ask the Answer
2. Give me another, and another and now give me a problem.
Ponder the difference in your thinking in answering these
(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
Pentominoes (see Lesson 3) is an example of a teacher
directed investigation as few students would choose this topic
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.
(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.
(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
(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.
(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.
(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?
(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!
Describe or find an investigation in each of the strands that you
Write down several questions you have been asked and how you
responded to them.
List 10 competencies that are developed while investigating.
1. 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.
The Main Vine, L#2
Beginning a Year, L#3
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
Team and Problem Based Learning L#11
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
22. Math Phobia