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Making inroads in math
teaching and learning
Marian Small
April, 2015
Our Ministry and boards
•  have paid a great deal of attention
to math, particularly how it should
be taught.
Step 1
•  It’s obvious that Step 1 is creating
the right classroom culture.
•  None of what I say later can work
without this, but I will say this is
only the start.
But
•  we also need to attend to what is
being taught.
•  It’s not just about the
expectations, but what we do with
the expectations.
How do we read them?
•  So many expectations can be
read with different perspectives–
a “do it” perspective or an
“understand important ideas
about it” perspective.
For example
•  Grade 4:
– add and subtract decimal
numbers to tenths, using concrete
materials and student-generated
algorithms
View 1
•  Make sure they can add and
subtract and give them some
concrete experiences and some
chance to invent their own
procedures.
View 2
•  Help them see how adding and
subtracting tenths really is adding
and subtracting whole numbers of
tenths and …
View 2
•  …that adding and subtracting
tenths apply to the same types of
situations as adding and
subtracting wholes.
Or Grade 9
•  express the equation of a line in
the form
y = mx + b,
given the form
Ax + By + C = 0.
View 1
•  Do the algebra to move from one
form of a line to the other
View 2
•  Recognize that any algebraic
relationship (including the
equation of a line) can be
described different ways and be
aware of when each way is more
useful and why
It makes a difference!
•  View 1 teachers bring out very
different highlights in a lesson
than View 2 teachers.
•  Which teacher do you want? I
know what I want, but I’m not sure
teachers all see the difference.
So
•  The classrooms of both teachers
could look like “problem solving”
classes but the walk-aways are
very different for the students in
those classes.
It’s about a view of math
•  Is math about solving problems?
Or
•  Is math about having a deep
understanding of a variety of
mathematical relationships?
Teachers are working to ..
•  use rich tasks.
•  But do they know what makes it
rich?
Is a task rich because
•  there is a context?
•  (I don’t think so.)
•  because it is open-ended?
•  (I don’t think so.)
Is it rich because
•  there are many strategies?
•  Many strategies is probably a
component.
•  But is there a mathematical
purpose beyond solving it?
•  Is there such a thing as a rich task
or does it depend what the
teacher does with it?
•  Can you tell if it’s a rich task if you
don’t know the math deeply?
•  Should a rich task usually take 40
minutes or so?
•  I’m not convinced it has to or
needs to or regularly should.
What would you think of this
task?
•  We were addressing the sorting of
quadrilaterals.
•  We had talked to kids about
various properties of
quadrilaterals in an earlier lesson.
Our activation
•  Use one phrase to describe one
of the triangles that will make it
easy for us to guess which one
you meant.
Our main lesson
Choose three of these phrases:
•  Some, but not all, equal side lengths
•  Some parallel sides
•  Symmetry
•  Right angles
•  Equal diagonals
•  4 equal side lengths
•  a very small angle
•  an angle bigger than a right angle
Then
•  Build a quadrilateral with your
three properties.
•  Choose three different ones and
go again.
•  Keep going until you run out of
ideas or time.
For example
•  Some parallel sides
•  Symmetry
•  Right angles
For example
•  a very small angle
•  an angle bigger than a right angle
•  symmetry
Consolidation
•  We would look at a quad and
guess which three properties kids
picked
•  (We were sometimes right and
sometimes wrong even though we
saw the three properties we did.)
Consolidation
Then we asked things like:
•  What properties would a square
and rectangle both have?
•  A kite and parallelogram? Etc.
Consolidation
•  What properties were easy to put
together?
•  What properties were hard to put
together?
Consolidation
•  What properties did most quads
have?
•  What properties were less usual?
What should be the focus of
consolidation?
•  Clearly, it relates to learning
goals.
•  But what are appropriate learning
goals?
Consolidation
•  They can’t be restatements of
expectations.
•  They can’t just be “topics”.
•  They need to be ideas.
For example
•  In a lesson on adding fractions,
my goal might be how adding and
subtracting fractions is like adding
and subtracting whole numbers
and why.
Consolidation
•  Does it have to involve whole
group sharing?
•  Is its function mostly about the
sharing?
Consolidation
•  Can you plan what to focus
consolidation on before you see
what the kids so?
You have read or will read..
•  a piece talking about 5 productive
dispositions for orchestrating
good math conversation.
The last one…
•  speaks to connecting responses.
•  I fully agree, but is it responses to
the problem or is it responses to
the ideas you chose the problem
to bring out?
Role of teacher beliefs
•  We have already discussed this a
little, but there’s more.
Many teachers…
•  really don’t have much faith in
their students and seek to give
them “simpler” problems.
•  Is this the way to go?
•  Let me tell you a story….
Myth/Attitude Barriers
•  Pictures, numbers AND words
•  Everything can and should be
levelled .
Myth/Attitude Barriers
•  A focus on solutions, instead of
thinking.
•  A focus on format/vocabulary in
communication.
Myth/Attitude Barriers
•  The purpose of consolidation
•  The belief that you need the
basics first before you can attack
a problem
Myth/Attitude Barriers
•  The belief that focusing on
templates and format will solve
problems.
•  A view of co-planning/co-teaching
or PL events as events.
Suggestions
Where I see the greatest need for
teachers:
•  Unpacking expectations, ideally
through a revised curriculum
document, but, if not, through
collegial discussion
Suggestions
•  Looking at consolidation in a
different way than they are; not
just about sharing strategies
Suggestions
•  Looking at consolidation in a
different way than they are; not
just about sharing strategies
Suggestions
•  Recognizing the critical role of
questioning and developing skills
to become better questioners
•  Content embedded within the
ideas above
Suggestions
Where I see the greatest need for
boards and the Ministry:
•  Putting a greater expectation on
principals as instructional leaders
(not just in name only but for real),
possibly by being somewhat less
demanding in other areas or allowing
others to oversee some other areas
Suggestions
•  Teaching principals, SOs and
coaches to focus on deep, not
surface, issues
•  Focusing less on “structure” and
more on “substance”