1. technology and computing
2. computer certification

Leading effective math
instruction
Marian Small
April 2015
Talk to your neighbours
•  What does the teacher you believe is one
of the most effective math teachers in your
school do differently than others?
•  What does the teacher you believe is one
of the least effective math teachers in your
school do differently than others?
Do you believe….
•  that good math teaching is more or less
the same as good “any subject” teaching
or not?
Fundamentally,
I believe that effective math
instructionrequires
push and
support.
But what should we be pushing and
supporting?
4
Effective Math Instruction
A Balanced program
Requires attention to:
•  Concepts and skills
•  Processes
•  Big ideas
It recognizes
•  the importance and value of figuring things
out
•  the need for connections between ideas
•  the need for reflection
•  the need for purposeful practice
In Ontario
•  We have decided that much of this can be
accomplished using 3-part problem solving
lessons a fair bit of the time.
Lesson organization
• The 3 –part lesson includes:
• An activity/questions to engage students
at the start of the lesson (before/ minds
on/getting ready)
Getting ready
• It is engaging and might get kids ready for
the problem or might keep old topics alive.
During/action
• A significant activity where teachers
provide the opportunities and support, but
students drive the learning
Consolidation
• A meaningful consolidation
• Its focus is NOT just to share work.
• Its focus is to evoke the important ideas of
the lesson using the children’s work and
thinking.
Let’s consider an example
•  Grade 3 “place value” lesson
A good start
• Represent the number 225 in lots of ways.
• Which of your ways are most alike? Why?
Action
• Let’s all show 237 with 12 base ten blocks.
• What other numbers can you show with 12
base ten blocks?
• List lots of possibilities.
Success criteria
•  You listed only numbers that could be
modelled with 12 blocks.
•  You listed lots of numbers.
•  You made a generalization by looking at
what you noticed about your numbers.
•  You could explain why at least some of
your generalization made sense.
Expected values
66
570
390
1236
543
84
165
345
12
120
1200
21
30
300
111
201
Consolidating
• How might the number 21 have been
represented by 12 blocks? But how else
can you represent it?
• What does each way tell you about 21?
Consolidating
• Why is it okay to have more than 10 ones
blocks to represent 21?
• Why is it not okay to write 21 as 111 when
it was 1 ten rod and 11 ones?
And more
• Once you have 66, what would be a good
strategy to get another number that
works?
• What is the greatest 2-digit number you
can represent with 12 blocks? The
greatest 3-digit number?
• The least 2-digit? The least 3-digit?
• What’s the most important idea you
learned from doing this problem?
Fitting in practice
• Consolidation is NOT the same as
practice.
• Students need to consolidate the main
teaching activity by “debriefing” prior to
any practice activity.
It is essential that
• …. Teachers DO NOT RUN OUT OF
TIME to consolidate.
Be aware…
• Every lesson does need a beginning ,
middle and end, but not every lesson will
be a 3-part problem solving lesson.
• Some lessons might be games, but
teachers should choose games to elicit
both thinking and practice.
• Some lessons might be more directed, but
not that many.
Be aware…
• Some lessons might have 4 parts- with an
individual accountability piece brought in
at the end
• Or 5 parts with a checking part as kids
start their work.
So…
•  Are you seeing 3-part lessons?
•  What are they looking like?
•  Are you comfortable with what is
happening?
The role of technology
• Calculator issues
• The value of using tablets
• But the need to use them for the right
things
• Is the “flipped” classroom the answer?
THE MATH FOCUS
Curriculum
• The Ontario math curriculum mandates
attention to seven processes as well as
content.
• Teachers are professionally responsible
for BOTH.
Processes
• Problem solving
• Communication
• Reasoning and proving
• Reflecting
• Representing
• Connecting
• Selecting tools & strategies
Teachers should be able..
•  to tell you what processes they are
working on in a math lesson
For example…
•  I am working on selecting tools and
strategies if I provide multiple tools or
encourage different strategies and talk to
kids about why they chose the one they
did.
•  Let’s try a lesson focused on selecting
strategies.
Consider these questions
•  103 – 99
•  438 – 112
•  1003 – 428
•  Would you use the same strategy for
each? How would you choose?
1003 – 428
•  Have you ever seen these?
•  1003 = 999 + 4
– 428
428
571 + 4 = 575
Reasoning
•  I would specifically ask about why
something makes sense, e.g. why does it
make sense that 43 – 29 = 44 – 30?
Connecting
•  I might ask for a situation where you would
ever calculate 2 ½ ÷ 1/3.
Representing
•  I might ask students to represent 1000 in
different ways and tell me what each
shows about 1000.
•  For example, what does 10 x 100 show
me?
•  How about 998 + 2?
•  How about 10 000 ÷ 10?
Reflecting
•  I might reflect on whether an answer
seems reasonable or a strategy seems
reasonable.
•  For example, I might ask why an answer
of x = 1 for solving the equation
3x – 2 = 2.9x – 1.9 makes sense.
Content focus
• There are both overall and specific
expectations that guide instruction,
but…..
Content focus
• Each teacher should be thinking about
clustering the expectations they teach to
focus on Big Ideas.
• It is the Big Ideas that empower students
mathematically.
Big Ideas
• For example, if I know that subtraction is
always the “reverse” of addition, I can
use that information in grade 2 with
whole numbers, in grade 7 with fractions
and integers, and in grade 9 with
algebraic expressions.
Sample Big Ideas
• We gain a sense of the size of numbers
by comparing them to meaningful
benchmarks.
We might ask…
• What audience size would be a big one
for a Justin Beiber concert?
• a Toronto symphony concert?
• a school concert?
Sample Big Ideas
• How a shape can be dissected and
rearranged into other shapes helps us
attend to the properties of the shape.
(e.g. triangle as half of a rectangle)
We could ask..
How can you cut up this shape to more
easily figure out its area?
Sample Big Ideas
• The unit chosen for a measurement
affects the numerical value of the
measurement.
We might ask…
• Jeff said that the table is 20 rods long,
but Alison said it’s 10 rods long. Is it
possible that they are both right?
Big Ideas
• Big ideas are not the same as overall
expectations.
• They are different (in many cases, but
not all) from one strand to the next.
• There are several, although not a lot, of
variations on what these are, but
teachers could work collaboratively on
“tweaking” them to make them useful to
themselves. THIS IS VALUABLE PL.
In fact
•  Learning goals for lessons should derive
from big ideas.
For example
•  Instead of a learning goal being “can
multiply two 2-digit numbers”, the learning
goal would be “recognizes that operations
with big numbers can usually be
completed more easily if they are broken
up into pieces”.
For example
•  Instead of a learning goal being “uses the
formula for the volume of a rectangular
prism”, it might be “recognizes which
measurements of a rectangular prism are
and are not essential in calculating the
volume of a rectangular prism”.
Let’s talk
•  Are you seeing “deep” learning goals?
•  Are the processes being brought out?
•  Is the focus on either calculations or just
sharing answers to problems or is it on
deeper math ideas?
Success criteria
•  Rather than being a count of how many
ways a child does something or a laundry
list of things the child does “efficiently” or
“effectively” or…., these might help signal
to the student other things we value.
For example
•  Going back to our base ten block task
today, you saw criteria that reminded you
what to do, but expected more of you too.
OR
•  Suppose a task focuses students on
applying the formulas for areas of triangles
and parallelograms.
•  A task might ask students whether it is
always possible to create a triangle with
the same area as the area of a given
parallelogram.
Success criteria
•  Given a specific parallelogram, a triangle
with the same area is created.
•  Several examples are shown.
•  A clear explanation of why this has to be
possible is offered.
More on success criteria
• 
• 
• 
• 
They explain level 3, not level 4.
That is why some teachers prefer rubrics.
They cannot “give away the farm”.
They include a mix of “checklist” and
“value” statements.
Or
•  Represent relationships using unit rates
•  My learning goal might be: You realize that
any rate situation can be described by
more than one unit rate and why.
Task
•  You know that you can buy a 1.77 L
container of laundry detergent for $5.97. It
does 38 loads of laundry.
•  Calculate each of these unit rates. Decide
which you think is more useful.
•  How many loads/1 L?
•  How many loads/$1?
•  How many litres/$1?
•  Cost/1L?
Success criteria
•  You calculate each of the required rates.
•  You can describe when each might be
more useful.
•  You can explain why ANY rate situation
can be described by more than one unit
rate.
What does co-construction
mean?
•  Let’s talk about this.
PEDAGOGICAL FOCUS
Desirable pedagogy
• Students working in pairs/groups, maybe
even sharing chart paper on which to
report, but maybe not always
• Manipulatives and technology accessible
• A focus on encouraging personal
strategies (e.g. How might you solve 19 x
8?)
Desirable pedagogy
• Teaching through problem solving,
frequently with meaningful contexts
• Using guided groups where necessary
• Using effective questioning
Eliciting thinking vs procedures
• Which is greater: 29 or 92?
vs.
• []2 > []9
• What digits can go in the boxes? What
digits cannot?
Eliciting thinking vs procedures
• What is 22 x 12?
vs.
• Draw a picture to show what 22 x 12 looks
like and how much it is.
By the way
•  It looks like this:
Convergent vs. divergent
questions
• What is 252 ÷ 4 ?
vs.
• Which is easier for you to divide: 252 ÷ 4
or 240 ÷ 4? Why?
Convergent vs. divergent
questions
• How much is 13 x 12?
vs.
• What other products would help you figure
out what 13 x 12 is?
Convergent vs. divergent
questions
• Find the area of this triangle.
vs
5 cm
8 cm
• The area of another triangle is twice as
much as the area of this one. What are the
dimensions of the other triangle?
Thinking back..
Thinking about the questions you tend to
see in math instruction…..
• Do the questions tend to be convergent or
divergent?
Desirable pedagogy
• Attending to all students by appropriately
differentiating instruction
Differentiating instruction
• Teaching with a focus on big ideas allows
teachers to meet the needs of diverse
students at the same time
What is needed for DI
• A focus on big ideas
• Pre-assessment
•  Choice
Open questions
• One strategy we have been sharing is the
use of open questions.
• These can be addressed effectively by
students at many different readiness
levels.
An example
• The answer is 10%.
• What is the question?
Some possibilities
• What is 1/10?
• What is 0.10?
• How does 5 compare to 50?
• What is half of 20%?
• What might be a lowish tip at a
restaurant?
•What fraction of the world population lives
in Africa?
Do you notice?
•Do you notice how much more inclusive
the last question is than the typical
“percent” question?
• Can you see how open questions might
support building a culture of high
expectations?
Parallel questions
• We have also been using two or more
tasks founded on the same big idea but
meant for students with different readiness
levels.
For example…
"  Choose two
numbers between
20 and 30 that
you think are
easy to add. Why
are they easy to
add?
"   Choose two
numbers between
200 and 300 that
you think are easy to
add. Why are they
easy to add?
Common questions
• Which digit of the numbers did you
decide on first? Why?
• Did you think the other digits should be
small or large? What influenced your
decision?
• Would the task have been easier for you
or harder if the greatest value had been
higher?
Common questions
•  What numbers did you choose?
• How did you add them? Why was it
easy?
Notice
• All students, at whatever level, are still
focusing on problem solving and
communicating in math.
• They are building their confidence.
• The strugglers are not relegated to tedious
exercises.
Repeating vs Creating
• There is ample evidence that student
involvement in the development of
concepts is much more important, longterm, than their ability to recall rules. They
need to learn to think mathematically, not
just follow procedures.
So…
•  Are you seeing rich questioning?
•  Are you seeing DI?
TEACHER/STUDENT
INTERACTION
Ideally, the teacher
• does not dominate the conversation
• asks the student to create, not repeat
• does not over-scaffold
• listens intently to student responses and
provides specific feedback on it
Ideally, the teacher
• turns back a student’s request for answer
confirmation to the student
• not only encourages, but shows
confidence in her/his students verbally and
visually
•  centres the classroom around the
students
STUDENT/STUDENT
INTERACTION
Ideally,….
• Students spend a lot of time talking to
each other about math.
• Listen to each other’s thinking
• Politely challenge each other’s thinking.
• Support each other’s thinking
EFFECTIVE ASSESSMENT
Assessment of Learning
• Teachers will be gathering substantial data
using conversations and observations and
not just written products.
• Teachers will be using rubrics in
appropriate circumstances.
Assessment of Learning
• Will not be labelling students as 1, 2, 3, 4,
but work as 1, 2, 3, 4.
• Will be measuring performance on big
ideas and not just on “repeating” what was
shared
• All four categories of knowledge and
understanding, application, thinking and
communication are measured in
appropriate proportions.
Knowledge vs understanding
•  These are different.
•  We need more attention to and
assessment of understanding than we
often see.
For example
•  Knowledge: What is a common multiple of
4 and 6?
•  Understanding: Why two two numbers
have more common multiples than
common factors?
For example
•  Knowledge: What is 45 + 23?
•  Understanding: If you add subtract two
numbers and the result is just a tiny bit
less than what you subtracted, what do
you know about the relationship between
the two numbers?
They will…
• focus as much on assessing concepts as
procedures.
• ensure they are focusing on important
math in their assessment, although some
attention to skills is fine.
They will…
• allow students to show what they can do in
a variety of ways
Differentiating..
• Teachers will provide alternative forms for
assessing student learning as needed
• Teachers will ensure at least some
questions are open enough to allow
students to show as much as they can
about their knowledge on the relevant
topic, perhaps using open or parallel
tasks.
So…
•  Are you seeing broad based assessment
of learning that uses multiple sources of
data, that allows for flexibility in approach
and that is not focused on just getting
answers?
Your challenges
•  One real challenge is that elementary
teachers are insecure about what math to
teach and their own ability to teach it
•  Secondary teachers are less insecure but
their comfort is often with procedures and
not with a deeper understanding of math
Your Role
• Your role is to expect teachers to improve their
teaching and to support and coach teachers
trying to do that.
• By and large, elementary teachers seek
approval for their instructional approaches.
You might…
• participate in a classroom as an observer
or as a co-teacher with students or work
with a particular small group of students as
they work through a problem
You need to think about…
• what the kids say and do--- are they risktakers? If not, why not?
Are they willing, or even eager, to solve
problems? If not, why not?
You need to think about…
• Are some kids missing important
prerequisite chunks? (You can find out
later what the teacher is doing about those
children.)
You also need to think about
the teaching environment
• What is the teacher doing or not doing to
bring about those behaviours?
Is s/he creating independent confident
learners or insecure learners?
Teachers need to be..
• working on building their pedagogical
content knowledge.
• This includes reading, studying or working
with colleagues to learn to know what
math is important, be aware of different
approaches to that math, know how
students might respond differently, and
prepare for those responses.
You might
• set up common planning times and make
sure the focus is on broadening, and not
narrowing, instructional approaches
How are you using your PLCs?
•  Is it just an exchange of opinion or are you
having teachers read and talk about what
they read?
•  Are you having teachers challenge each
other with different opinions on issues and
have rich, deep conversations?
How are you using your PLCs?
•  Are you ensuring that there are between
PLC activities and that teachers cannot
opt out, but are responsible for bringing
work, reporting, etc.?
•  Are you using board support to ensure that
someone with math expertise can provide
answers you might not have at your
fingertips?
How do you push?
• You should be able to ask a teacher, in
passing, what new idea s/he is trying and
s/he should have an answer
You should be able to..
• Ask teachers to identify a single idea (not
skill) they want students to be able to
articulate as a result of the learning in a
lesson.
You might look for..
• how teachers balance requests for written
and oral communication
• how much teachers have students build on
other students’ thinking
You might look for...
• how comfortable students are asking
questions
• how often students are asked to explain
their thinking
You might ask..
• How are you getting students to listen to
each other’s strategies?
• How are you getting students to share
their thinking?
• How are you handling situations where the
student suggests ideas that are not clear
to you when they are first offered?
Are teachers...
• using ongoing assessment for gathering
data to plan their instruction?
You might look for...
• how often teachers are changing
instructional plans based on prior
assessment
• how often teachers change instructional
plans based on comments students make
• how often teachers ask questions that
really expose student thinking
You might ask..
• How are you gathering data on your
students’ prior knowledge?
• How are you using that data to change
your instructional plans for individual
students (or groups)?
Are teachers...
• building
MEANINGFUL success for all
learners?
You might ask…
• How
do you decide when to use open
or parallel questions?
• How often are you focusing on the
same big idea at different levels?
•  How do you make sure ALL of your
students are asked meaningful, higher
level questions?
You might ask…
• How
are you teaching your students to
self-scaffold?
• How likely is it that you show an
alternate method when students
struggle?
• How are you encouraging students to
persevere?
Acting as an advocate
•  Ultimately, you are the advocate for your
students, and, often for their teachers.
•  You have to work at getting the resources
needed to do the best job, but often it is
not money– it is just a commitment.
Getting started
• Become aware of what’s available, e.g.
Gap Closing. CLIPS; participating in
PRIME workshops, CIL-M sessions, etc..
Your role is…
•  To set the tone
•  To expect, but to help
•  To advocate, but for what’s best for your
students
•  To encourage some consistency in the
school, but individuality too.
Your role in learning teams…
• 
• 
• 
• 
• 
To facilitate, not necessarily to “run”
To ask meaningful questions
To probe more than criticize or tell
To continue to encourage growth
To encourage some level of consistency
Questions
•  What other questions do you have?
Something to try…
•  To put these ideas into practice….
•  What will you try before we meet again?
Download at
www.onetwoinfinity.ca
AvonMaitlandAdmin