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
Lesson #8 • Revised and completed 12/1/20
- to understand
the benefits of mathematical investigation
- to know how discourse creates meaning for learners
Investigation is INQUIRY. It 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 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 as you will be
rewarded. Do not expect success on the first attempt but it may
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.
Empowering students to be self managed and to take more control of
their learning is being Future Focused. Live in the future, not
What is an Investigation?
An investigation involves exploring and problem solving just as
problem solving involves investigation and creativity. An
investigation is a broader or less focused task than problem
solving. Investigation means exploring many pathways and
discovering paths that may end or change. 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. Normalise the
expectation that students journal, every day!
Journaling in the C21st classroom may well involve online Google
Classroom or a similar blog. 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as being a bit outdated as the new media platforms popularise with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. I
remain a pencil and paper fan and will play Pictionary with
students to normalise cartooning and drawing.
This is the modern world we have 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 with 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 the internet to
find out other's thinking. Social media has serious issues!
Example student questions which have led to
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.
"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 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
This is an illustration of 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 be allowed to make mistakes! I know I did...many
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 is essential.
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
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) A number between 1.1
and 1.11 is 1.101, another is 1.1001.
Pentominoes (see Lesson 3) is an example of a
teacher directed investigation as few students would choose this
So tell me one number between 2:2 and 2:22. (All said by
(b) Give me a number between 1/3 and
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!]
How many different shapes can
you make using 5 squares? Side to side only and no corner only
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 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 are available 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
(b) The difference between the square numbers is very odd!
Make sense of this nonsense statement.
(a) Make a model of x, x^2,
x^3 and try to explain what can be added and otherwise
combined. (use Multi-link Blocks)
(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
(b) Use your cellphone to find to find the Girth of the Earth.
How many ways different ways could or can you do this using a
(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
(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"
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
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!
• Describe or find an investigation in each of the
strands that you use.
• Write down several questions you have been asked and how you
responded to them.
• List 10 competencies that are developed while investigating.