Transcript 30 Term Value NEGATIVE ?? 20 - GAINS

Plenary 1
What’s important about the
Math we Teach?
A Focus on Big Ideas
Marian Small
www.onetwoinfinity.ca
Minds-On
• The third term in a linear
growing pattern is negative.
•The
th
30
term is 20.
What might the 20th term
be?
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Minds-On
Term
Number
Term
Value
1
2
3
NE
GA
TIV
E
…
20
??
…
30
20
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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 was the important
underlying idea?
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Characteristics of
Minds-On
• Would OR how would this
question force students to
deal with that underlying
idea?
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Would the student be
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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Would the student be
able to respond to…
• If the 100th term of a
linear pattern is relatively
small, then…..
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Teacher struggles
• My experience is that
setting lesson goals beyond
reciting an expectation or
simply using a topic name is
a struggle for teachers.
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For example…
• Instead of reciting this
curriculum expectation as a
goal: solve problems
involving percents
expressed to .. wholenumber percents greater
than 100%...
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For example…
• it could be:
If one number is less than
100% of another, the
second number is more
than 100% of the first. OR
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For example…
• it could be:
If a percent is greater
than 100%, its decimal
equivalent is greater than
1. OR
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For example…
• it could be:
The same strategies are
used to solve problems
involving percents greater
than 100% as problems
involving percents less than
100%. OR….
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Why you want to do
this…
• If you decide on the goal,
you are more likely to know
what questions to ask, what
activity to use,….
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Make it yours
• Even if you get a lesson
from a valued resource, you
have to make your OWN
decision about what to pull
out of that lesson.
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For example…
• Let’s look at this lesson
from Grade 7 TIPS.
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Maybe…
• Complete this:
The way someone figures
out terms 3, 4, 5, and 6
might be very different
from the way that person
figures out term 100
because….
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Big Ideas
• Randy Charles: A Big Idea
is a statement of an idea
that is central to the
learning of mathematics,
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Big Ideas
• Marian Small: ….one that
links numerous
mathematical
understandings into a
coherent whole.
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Big Ideas
• It is not a topic name nor
is it an overall expectation.
It is a statement
(sentence) that a student
could walk away with that
makes a fundamental
mathematical connection.
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Big Ideas for CAMPPP #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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It’s not…
• It is possible that, on
first blush, this may sound
like a definition, but it’s
not. It provides a new lens
in which to embed the
learning.
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Mathematical Processes
Representing
Communicating
Reflecting
Problem Solving
Reasoning and Proving
Connecting
Selecting Tools and
Computational
Strategies
Notice the processes
• Notice the
embedded processes
st
in the 1 big ideacommunication,
reasoning, connecting
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Big Ideas for CAMPPP #2
• Different representations of
relationships (e.g. numeric, graphic,
geometric, algebraic,
verbal,concrete/pictorial) highlight
different characteristics or
behaviours, and can serve different
purposes.
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• Which
processes
do you see
embedded
in this big
idea?
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Big Ideas for CAMPPP #3
• Comparing mathematical
relationships helps us see
that there are classes of
relationships and provides
insight into each member
of the class.
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Big Ideas for CAMPPP #4
• Limited information about
a mathematical relationship
can sometimes, but not
always, allow us to predict
other information about
that relationship.
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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 questions
that matched Big Idea 2
broaden a student’s sense
of what different
representations mean
and/or what their purpose
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Some questions about your
task
•How could the question
that matched Big Idea 3
broaden a student’s notion
of what a “class” of
relationships might be?
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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 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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Why use big ideas?
• It helps for students to
know what the big ideas
are so that the connections
to prior knowledge they are
making are more explicit.
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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 that
“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
•But it’s sure easier to see
the unknown value in one of
them.
•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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What does this mean for
consolidating the lesson?
•Agree or disagree: The
equation 5x – 4 = 17 + 3x is
really the equation x = 10.5
in disguise.
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What does this mean for
consolidating the lesson?
•Which equation would you
find easier to solve? Why?
5x – 4 = 17 + 3x
x = 10.5
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What does this mean for
consolidating the lesson?
•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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My goal today might be…
for students to 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 a
table of values.
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So I could ask…
Here are two tables of
values. You want to find out
if 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 that
same relationship gives
other insights into it.
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Consolidate
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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The Important Book
We would like to introduce
you to Margaret Wise
Brown’s The Important
Book.
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A Sample page
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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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