Transcript Plenary1 SI v6 GAINS - GAINS

Summer Institute
Thunder Bay
MaryLou Kestell
John Rodger
Wendy Telford
Plenary 1
What’s Important About the
Math we Teach?
A Focus on Big Ideas
Marian Small
www.onetwoinfinity.ca
[email protected]
MaryLou Kestell
Minds-On
• The 3rd term in a linear
growing pattern is negative.
• The 30th term is 20.
What might the
th
20
term be?
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Minds-On
th
20
• Could the
term be
either positive or negative?
Why is that?
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Characteristics of
Minds-On
How does this minds-on
engage students?
How is it open?
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Characteristics of
Minds-On
What do you think the
important math underlying
idea is?
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Characteristics of
Minds-On
How would this question
prompt students to deal
with that underlying idea?
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Might lead to students
being able to respond to…
• What makes a pattern
linear is…
• There are a lot of linear
patterns that include the
same term because…
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Or…
• If the 100th term of a
linear pattern is
relatively small, then…
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What are the Big Ideas?
Randall Charles: A Big Idea is
a statement of an idea that is
central to the learning of
mathematics.
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What are Big Ideas?
Marian Small: A Big Idea is
one that connects numerous
mathematical understandings
into a coherent whole.
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A Big Idea
• is NOT a topic name nor an
overall expectation.
• is a statement (sentence) that a
student could walk away with that
makes a fundamental mathematical
connection.
• provides a lens in which to embed
new learning.
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Big Idea #1
Algebraic reasoning is a
process of describing and
analyzing (e.g. predicting)
generalized mathematical
relationships and change
using words and symbols.
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Mathematical Processes
Representing
Communicating
Reflecting
Problem Solving
Reasoning and Proving
Connecting
Selecting Tools and
Computational
Strategies
Notice the processes
Big Idea #1 Algebraic
reasoning is a process
of describing and
analyzing (e.g.
predicting)
generalized
mathematical
relationships and
change using words
and symbols.
• communicating
• reasoning
• connecting
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Big Idea #2
Comparing mathematical
relationships helps us see
that there are classes of
relationships and provides
insight into each member of
the class.
Which processes do you
see embedded in this 2nd
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big idea?
Big Idea #3
Different representations of
relationships (e.g. numeric,
graphic, geometric, algebraic,
verbal, concrete/pictorial)
highlight different characteristics
or behaviours, and can serve
different
Which processes do you
purposes.
see embedded in this 3rd
big idea?
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Big Idea #4
Limited information about a
mathematical relationship
can sometimes, but not
always, allow us to predict
other information about
Which processes do you
that
see embedded in this 4th
relationship. big idea?
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Getting a feel for the
big ideas
Two sets of questions will
be circulated which are
designed to bring out the
big ideas.
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Getting a feel for the
big ideas
Choose one of those sets
of questions. Match each
question to the big idea it
is most likely to elicit.
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Some questions about
your task
Which big idea did you
find easiest to match
first?
Which did you find
hardest to match first?
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Some questions about
your task
Which of the questions
did you like best?
Why?
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Some questions about
your task
How do the questions that
matched Big Idea #1 show
the notion of
generalization?
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Some questions about
your task
How do the questions that
matched Big Idea #1
show the notion of
describing or analyzing
relationships or change?
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Some questions about
your task
How could the question
that matched Big Idea
#2 broaden a student’s
notion of what a “class”
of relationships might be?
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Some questions about
your task
How could the questions that
matched Big Idea #3 broaden
a student’s sense of what
different representations
mean and/or what their
purpose is?
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Some questions about
your task
Can you think of other examples
that you’ve used in the past
(with or without realizing it) to
make students see Big Ideas #4
that from limited information
you can get more?
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You just experienced a
parallel task
We will talk more about these, but
these two very related tasks were
adjusted to meet your needs but
treated together in our
consolidation.
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Why use big ideas?
• to build connections
students need in order to
learn both through grades
and within grades
• to prioritize instructional
goals
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Sharing big ideas
with students
When students articulate big
ideas, it becomes easier for
them to make connections to
prior knowledge.
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Building lesson goals
You can use a big idea to
hone in on an appropriate
lesson goal.
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For example…
Consider the expectation:
“Solve first degree
equations with nonfractional coefficients
using a variety of tools
(e.g. 2x + 7 = 6x – 1).”
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What is my lesson goal
I am going to propose
that it is not “students
will use a balance to solve
a linear equation”, but…
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What is my lesson goal
maybe:
“recognizing that solving an
equation means determining
an equivalent equation
where the unknown value is
more obvious.”
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What I mean
These equations are
equivalent:
x = 4
2x – 7 = 1
3x + 7 = x + 15
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What I mean…
It certainly is easier to see
the value of the unknown in
one of these equations.
These equations are equivalent:
x = 4
2x – 7 = 1
3x + 7 = x + 15
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What does this mean for
consolidating the lesson?
I need to ask a question
or two that gets RIGHT
to my goal.
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Consolidation option
Agree or disagree?
The equation
5x – 4 = 17 + 3x
is really the equation
x = 10.5
in disguise.
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Consolidation option
Which equation would you
find easier to solve?
Why?
5x – 4 = 17 + 3x
x = 10.5
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Consolidation option
Why might someone say
that solving an equation is
about finding what easier
equation is being
disguised?
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One more example
The curriculum expectation
reads: construct tables of
values and graphs using a
variety of tools to represent
linear relations derived from
descriptions of realistic
situations
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Possible goal
Students will see that it is
useful to write the table of
values where the independent
variable increases in a
consistent way, but that’s not
required for all tables of
values.
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Related consolidation question
Here are two tables of
values.
Determine whether or not
they represent linear
relationships. Which table
makes it easier to tell?
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x
y
x
y
2
4
2
4
5
13
5
13
8
22
16
44
11
31
20
58
14
40
19
55
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and…
Could you use the other
table too, if you wanted
to?
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Or…
My goal could have been,
instead, to ask students to
consider how a graphical
representation of a
relationship described verbally
gives other insights into the
relationship.
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That consolidating could
have been …
What characteristics of the
relationship did the graph
make easy to see that were
not so obvious before?
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Consolidate today’s work
Think/pair/share:
What is the difference
between an expectation and
a big idea? OR
What’s so big about big
ideas?
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In preparation for our next
time together:
Could you think about what
you consider important
issues in developing lesson
goals?
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The Important Book
We would like to introduce
you to Margaret Wise
Brown’s, The Important
Book.
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We will
use this book throughout the
week as a way for you to
consolidate what you explore
in our CAMPPP.
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