Transcript THE HAVING OF WONDERFUL IDEAS

phone:
212-766-2120
cell:
917-494-1606
Agenda
 Creating our Learning Community & Norms
 Setting Personal Goals and Identifying Themes
 Visualization through Quick Images’
 Math Routines that Develop Thinking
 Exploring Addition and Mental Math
 Lunch
 Games
Eight CCSS Math Practices
 Make sense of problems and persevere in solving
them.
 Reason abstractly and quantitatively.
 Construct viable arguments and critique the reasoning
of others.
 Model with mathematics.
 Use appropriate tools strategically.
 Attend to precision.
 Look for and make use of structure.
 Look for and express regularity in repeated reasoning.
Construct Viable Arguments and Critique
the Reasoning of Others
 Understand and use stated assumptions, definitions
and previously established results
 Make and explore conjectures using a logical
progression of statements
 Analyze arguments using cases, counterexamples,
data, contexts
 Compare effectiveness of two plausible arguments
 Distinguish correct from flawed reasoning
Model with Mathematics
 Apply mathematics to solve everyday, society, and
workplace problems
 Make assumptions and approximations to simplify a
complicated situation and revise as needed
 Map quantitative relationships in situations using
diagrams, charts, flow-charts, tables, etc.
 Analyze quantitative relationships to draw conclusions
 Interpret results in relation to the context determining
whether answer makes sense or model needs revision
Developing Addition Strategies
Today’s Session Will Explore …
 What does it mean to have computational fluency?
 How do we develop computational fluency in our
students?
Developing Addition Strategies
What is in our addition toolbox?
Developing Addition Strategies
39 + 68
Share your strategies in a small group.
Are your strategies the same? Different?
Developing Addition Strategies
39 + 68
What mathematical ideas make these strategies
work?
Developing Addition Strategies
Did you use the standard algorithm?
+
39
68
Developing Addition Strategies
+
1
39
68
107
Developing Addition Strategies
What mathematical ideas make the
standard addition algorithm work?
Developing Addition Strategies
What mathematical ideas make the standard addition
algorithm work?
 Knowing that place determines value
 Equivalence
 Associativity
 Commutativity
 Unitizing
Strategy
Mathematical Ideas
What it looks like …
Place determines value
39 + 68 = (30 + 9) +( 60 + 8)
Associative property
39 + 68 = (30 + 9) +( 60 + 8)
(9 + 8) +( 30 + 60)
Standard
Addition Commutative property
Algorithm
39 + 68 = 68 + 39
Unitizing
Saying, “3” + “6” in the
algorithm MEANS
3 (tens) + 6 (tens)
Equivalence
39 =
30 + 9 = 20 + 19 = 10 + 29
Developing Addition Strategies
1
39
+ 68
107
(8+9) + (60 + 30) =
17 + 60 + 30 = 10 + 7 + 60 + 30 =
90 + 10 + 7 = 107
Developing Addition Strategies
Other possible strategies for adding 39 + 68?
 Partial sums (splitting)
 Creating an equivalent problem (compensation)
 Keeping one number whole and adding in parts
 Using the ten structure of the number system to

Move to landmarks

Take landmark jumps
Developing Addition Strategies
Other possible strategies for adding 39 +
68?
Partial sums (splitting)
(30 + 9) + (60 + 8) =
(30 + 60) + (9 + 8) =
90 + 17 = (90 + 10) + 7 =
100 + 7 = 107
Developing Addition Strategies
Other possible strategies for adding 39 +
68?
Compensation (creating an equivalent problem)
39 + 68 =
39 + (67 +1) =
39 + (1 + 67) =
(39 + 1) + 67 =
40 + 67 = 107
39 + 68 =
(37 + 2) + 68 =
37 + (2 + 68) =
37 + 70 =
37 + 70 = 107
Developing Addition Strategies
Other possible strategies for adding 39 +
68?
Keeping one number whole and adding in parts:
39 + 68 =
68 + (30 + 9) = (68 + 30) + 9
98 + 9 = 107
Developing Addition Strategies
Other possible strategies for adding 39 +
68?
Making or using landmark jumps:
39 + 68 = 68 + 39
68 + (20 + 10 + 9) =
(68 + 20) + 10 + 9 =
(88 + 10) + 9 =
98 + (10 – 1) = 107
Developing Addition Strategies
Other possible strategies for adding 39 +
68?
Moving to the nearest landmark:
39 + 68 =
39 + 1 + 67 =
40 + 67 = 40 + (60 + 7)
(40 + 60) + 7 =
100 + 7 = 107
Addition Strategies
What they look like …
partial sums
(splitting)
compensation
39 + 68 = (30 + 9) +( 60 + 8) = (30 + 60) +( 9 + 8)
keeping one number
whole
making landmarks
jumps
39 + 68 = 68 + (30 + 9) = 98 + 9 = 107
moving to the nearest
landmarks
39 + 68 = 39 + 1 + 67 = (40 + 60) + 7 = 107
39 + 68 = 39 + (67 +1) = (39 + 1) + 67 = 40 + 67
39 + 68 = 68 + 39 = (68 + 20) + 10 + 9 =
88 + 10 + 9 = 98 + (10 – 1) = 107
Developing Addition Strategies
It’s important to notice that many of the big ideas
underlying these addition strategies are the same:
Big ideas:
1. Equivalence 39 + 68 = 40 + 67
2. Commutative property: a + (b + c) = a + (c + b)
3. Associative property of addition: a + (c + b) = (a + c) + b
4. Place determines value (the “3” in 39 is 30 or 3 tens)
Developing Addition Strategies
All of these strategies can be used algorithmically.
The key is not just to have alternative mental-math
strategies, but to know when to use them.
Developing Addition Strategies
Why?
Developing Addition Strategies
One of the hallmarks of number sense is
flexible strategy use.
What does this mean?
In computation, it means looking to the
numbers to pick the most efficient strategy.
Developing Addition Strategies
HOW TO WE DEVELOP FLEXIBLE STRATEGY
USE IN OUR STUDENTS?
Developing Addition Strategies
One way to help students develop
important number relationships is
through computational mini-lessons.
Developing Addition Strategies
Guided Mini-lessons
“Strings”
Developing Addition Strategies
Strings are a series of interconnected bare number
problems which teachers design and modify ad hoc in
order to help students invent and/or use efficient mentalmath computation strategies.
Developing Addition Strategies
What mental–math strategy might a teacher using
this mini-lesson be developing?
43 + 20
62 + 30
62 + 39
54 + 48
Developing Addition Strategies
Developing Addition Strategies
To successfully use mental-math minilessons, one must consider
•The role of the student
•The role of the teacher
•The role (and power) of mathematical
models
Students are expected to
1. Find their own solutions to the problem
2. Share their thinking publicly
3. Listen to and make sense of the strategies of others
4. Find and pose questions when they don’t
understand or they need clarification
5. Try on new strategies
6. Practice until strategies become automatic
Why is talk so important to the development of
computational strategies?
1. Each time a strategy is discussed, students gain additional
insights through other children’s explanations.
2. Over time, both through listening and questioning,
students eventually make sense of the strategy and begin
to feel comfortable with the strategy.
3. The students then may attempt to use the strategy in
some situations.
4. Over time and with use, the strategy then becomes
integrated into students’ mental-math repertoires and is
used regularly when needed.
The Role of the Teacher
 Encouraging students to make sense of situations;
 Providing time for students to question each other’s thinking and
strategies;
 Connecting different strategies to help students understand each
other’s thinking;
 Highlighting efficient strategies;
 Using questions such as:
 How did you get your answer?
 Can you explain it another way?
 Did anyone do it the same way?
 Can you put in your own words _______’s thinking?
The Role of Mathematical Models
 Models become important tools to represent
student thinking.
 Models become important tools to connect and
juxtapose student strategies.
 Models become important tools for students to
think with.
Developing Addition Strategies
Addition Strings
What addition strategies are the strings on the
worksheet designed to develop?
Developing Addition Strategies
What strategy is the string trying to develop?
54 + 20
55 + 19
42 + 40
44 + 38
66 +30
69 +27
89 +73
Developing Addition Strategies
Choose one of the strings from the handout with a
partner to analyze for:
(1) potential student strategies and struggles;
(2) how to model or represent their thinking;
(3) what questions to pose to students given their
respective strategies or struggles.
Games
 Why use games to teach math?
 What do teachers need to do to ensure students are
actually attending to the math in the game?
 How can games be used to differentiate?
 What kind of record keeping system would help us
keep track of student learning when playing games?
 How might we organize games for the greatest student
autonomy and make sure they work with “just right”
games as determined by informal assessment?
Games
Games
 Choose a game and play it with a partner or in a small
group.
 Use the games analysis sheet to name the mathematics
in the game, compare with games you already
use,consider various ways to extend the game or make
it less challenging.
 Use the Games Continuum sheet to help you place the
game on the continuum and begin building your
games library.
Games Continuum
Counting
Comparing
Counting
Games That
On/Conserva Use 5/10
tion
Structure
Part Whole
Relations
and
Equivalence
Roll and
Record
Compare/Top
It/War
Racing Bears
Counters in a
Cup
Build It
Ten Frame
Match
Games Continuum
Addition Subracti
or Fact
on
Fluency Games
Money
Games
Combina Number
tions of
Line
Tens
Games
Games
Roll and
Record
Racing
Bears
Ten
Frame
Match
Build It
Compare
/Top
It/War
Counters
in a Cup
Addition Place
Games
Value
Games
Reflections
 What are you questions and takeaways from today’s
session?
 Thank you!