Transcript CAAT Follow Up Meeting #1

Laurel County SSA
Day 4
Jim Moore
[email protected]
Jennifer McDaniel
[email protected]
Agenda
• 8:00-8:30 Sign In, Warm-Up Problems
• 8:30-8:45 Outline Goals and Expectations
• 8:45-10:45
Break Out Sessions
(Building Algebraic Thinking)
• 11:00-12:30
Lunch
• 12:30-1:30
Strategies to Engage Students
• 1:30-1:40 Break
• 1:40-2:30 Using Technology to Address the
Common Core Standards
• 2:30-3:00 Wrap-Up, Reflections, and
Evaluations
Summer Review
MEMORY BOX
Memory Box
• How might you use this strategy in your
classroom?
• Can you name five “go to” strategies that you
use already in your classroom to review
material?
SSA Goals
- Assist teachers with diagnosing needs specific
to learning goals in The Number System,
Expressions and Equations and Functions.
- Have teachers design instructional experiences
that incorporate technology, manipulatives,
vocabulary strategies, various representations,
communications and connections to engage and
motivate students of all learning styles.
- Connect the Mathematical Practice Standards to
the mathematical content
SSA Goals
- Teachers understand that strategies will also
be useful as interventions for struggling
students.
- Participants leave each training session with
the instructional materials used in training.
SSA Goals
Some Trainer Stepping Stones to
the Stated Goals:
- Creating familiarity with KCAS, Career and
College Readiness Standards and successful
EPAS performance.
To this end, trainers will work in partnership
with district leadership and MSLN leaders
to identify content focus and related
assessments as indicators of improvement
over the three year project.
SSA Goals
- give teachers experiences/strategies needed
to elicit higher level thinking skills that can be
incorporated into their instruction.
- give teachers first hand experience with
technology tools to use to support instruction
(e.g. GeoGebra, Graphing Calculators, TI
Nspire Teacher Edition Software, etc.).
- give teachers knowledge of and experience
with tools to create a classroom climate to
ensure that all students are engaged in the
learning of mathematics
Day One
• Stumbling Blocks -Prequisite Skills/retention/fear of
failure-past experiences; Lack of support from home,
motivation, waiting to dropout; Content: generalizations,
variables, moving beyond concrete stage, misuse of
technology
• Technology-Inspire CX - Introduction
• Characteristics of Highly Effective Teaching and
Learning (CHETL)
 Learning Environment
 States of Mind
• Integer Games and Number Line Activities
• Persistence in Problem Solving
 Connections to Mathematical Practice Standards
 Multiple Representations (NAGS)
Day 2
• Learning Styles & Student Engagement
 Task Rotations
 Question Museums
 Parallel and Open Tasks
 Conceptual Skills Practice Problems
 Examples:
•
Open Task:
You multiply two integers.
The result is 50 less than one of them.
What might the two integers be?
Day 2
• Parallel Task:
Choice 1: Describe what 10 colored
cubes you would put in a bag so that the
probability of selecting a red one is high, but
not certain.
Choice 2: Describe what 10 colored
cubes you would put in a bag so that the
probability of selecting a red one is 2/5.
Day 2
• Conceptual Skills Problem:
The average of four numbers is negative.
Explain your response to each:
(a) Can all four numbers be negative?
(b) Can all four numbers be positive?
(c) Can only two of the four numbers be positive?
(d) Can only three of the numbers be negative?
(e) Can only one of the four numbers be negative?
Day 2
• Writing Quality Questions
• Open Questions, Parallel Tasks and
Conceptual Skills Practice Problems
Day 3
• Modeling with Mathematics
• Data Analysis
 Comparing TI-84 & CX
• GeoGebra (Free Technology) – Introduction as
as dynamic geometry software, all types of
graphing and data analysis
• Grade level grouping for a first effort in writing
some open questions, parallel tasks and
conceptual skills practice problems.
Feedback
Technology – requests for TI 84 and TI Nspire,
GeoGebra, SmartBoard, Manipulatives, iPads
Hands on activities
Writing Quality Questions
Manipulatives
Follow Up
• Three Full Day On Site Sessions
 Sept. 28th
 Oct. 25th
 Nov. 29th
• Administration Support
 TBD
• Digital Presence
• Year 2 & Year 3
Laurel County SSA
Day 4
Middle School Breakout Session
Jim Moore
[email protected]
Jennifer McDaniel
[email protected]
Today’s Learning Targets
• I can use the border problem to build algebraic
thinking in my classroom.
• I can determine high and low cognitive tasks.
• I can identify appropriate mathematical
practices addressed during a high level
cognitive task.
• I can develop a stop doing/start doing list to
create high cognitive level tasks in my
classroom.
Connecting Mathematical Ideas
• Border Problem
The Border Problem
Without counting, use the information given in the figure
above (exterior is 10 x 10 square; interior is an 8 x 8
square; the border is made up of 1x1 squares) to determine
the number of squares needed for the border. If possible,
find more than one way to describe the number of border
squares.
• What about a 6 in by 6 in grid?
• What about a 15 in by 15 in grid?
• What about a 253 in by 253 in grid?
• What about an n inch by n inch grid?
• Create a verbal representation
• Use the verbal representation to introduce the
notion of variable
• If n represents the number of unit squares on
one side, give an algebraic expression for the
number of unit squares in the border.
• Develop understanding of function, variables
(independent and dependent) and graphing.
Border Problem Video
• Part One (Use printed transcripts to follow the
dialogue)
• As you watch the video, concentrate specifically
on the activity, the teacher, the students, and
the learning environment.
The Teacher’s Strategy
The teacher used the experience of the 10 by 10 border
problem to built algebraic understanding. She asked the
students to think about a smaller square, 6 by 6, and
asked the students to determine a set of equations of the
6 by 6 that matched the ways the students thought about
the 10 by 10 square. They had to write new equations in
the same manner that Sharmane, Colin and the others
had in the first problem. Next the teacher asked the
students to color a picture of the border problem, to match
each equation and also write the process to find each
total in a paragraph. Now she felt the students were
ready to use algebraic notation to generalize each
equivalent equation.
Video Discussion
• Why without talking?
• Why without writing?
• Why without counting one by one?
• Why not give them each a grid to facilitate their
thinking?
• Why did the teacher act as the recorder for the
arithmetic expressions?
Boaler, J. & Humphreys, C. (2005). Building on student ideas: The
border problem, part I. Connecting mathematical ideas: Middle school
video cases to support teaching and learning (pp.13-39). New
Hampshire: Heineman Publications.
The Border Problem
Sharmane:
4•10 - 4 = 36
Colin:
10+9+9+8 = 36
Joseph:
10+10+8+8 = 36
Melissa:
10•10 - 8•8 = 36
Tania:
4•9 = 36
Zachery:
4•8 + 4 = 36
Border Problem Video
• Part two (Use printed transcript to follow the
dialogue)
• As you watch the video, concentrate specifically
on the activity, the teacher, the students, and
the learning environment.
Student Equations
Generalizing For Any Size
Square
• 10+10+8+8=36
• Let x be the number of unit
square along the side of the
square.
• x + x + m + m = total
Introducing Algebraic Notation
Moving from the specific to the general
case.
Developing an understanding of variable
and its uses.
Tying abstract ideas to concrete
situations.
Fostering meaning to notation.
Developing the concept of equivalent
expressions.
Encouraging efficiency and brevity in
notation
Teacher Reflections
The Border Problem allowed for most (if not all) students to
develop an algebraic expression, which would calculate the
square units in the border of a square frame. What I found
is that many of the students did not naturally use a variable
in their expression. In the future, I would require students
to work with several different size square borders; then
have them present their expressions while I compiled a list
of correct ones. We would then look for similarities and as
a Part II, I would have the expectation that generalizations
be made, and that a variable represent the same “part” of
different sized frames.
Comparing Two
Mathematical Tasks
 Martha’s Carpeting Task
 The Fencing Task
Comparing Two Mathematical
Tasks
How are Martha’s Carpeting Task
and the Fencing Task the same and
how are they different?
Martha’s Carpeting Task
Martha was re-carpeting her
bedroom, which was 15 feet long
and 10 feet wide. How many
square feet of carpeting will she
need to purchase?
The Fencing Task
• Ms. Brown’s class will raise rabbits for their spring
science fair. They have 24 feet of fencing with
which to build a rectangular rabbit pen to keep the
rabbits.
 If Ms. Brown’s students want their rabbits to have
as much room as possible, how long would each of
the sides of the pen be?
 How long would each of the sides of the pen be if
they had only 16 feet of fencing?
 How would you go about determining the pen with
the most room for any amount of fencing?
Organize your work so that someone else who
reads it will understand it.
Comparing Two
Mathematical Tasks
• Think privately about how you would go
about solving each task (solve them if you
have time)
• Talk with you neighbor about how you did or
could solve the task
Martha’s Carpeting
The Fencing Task
Solution Strategies:
Martha’s Carpeting Task
Martha’s Carpeting Task
Using the Area Formula
A=lxw
A = 15 x 10
A = 150 square feet
Martha’s Carpeting Task
Drawing a Picture
10
15
Solution Strategies: The Fencing
Task
The Fencing Task
Diagrams on Grid Paper
The Fencing Task
Using a Table
Length
Width
Perimeter
Area
1
11
24
11
2
10
24
20
3
9
24
27
4
8
24
32
5
7
24
35
6
6
24
36
7
5
24
35
The Fencing Task
Graph of Length and Area
40
35
30
Area
25
20
15
10
5
0
0
1
2
3
4
5
6
7
Length
8
9
10
11
12
13
Comparing Two
Mathematical Tasks
How are Martha’s Carpeting Task
and the Fencing Task the same
and how are they different?
Similarities and
Differences
Similarities
• Both are “area”
problems
• Both require prior
knowledge of area
Differences
• The amount of
thinking and
reasoning required
• The number of ways
the problem can be
solved
• Way in which the
area formula is used
• The need to
generalize
• The range of ways to
enter the problem
Similarities and Differences
Similarities
• Both are “area”
problems
• Both require prior
knowledge of area
Differences
• The amount of thinking
and reasoning required
• The number of ways the
problem can be solved
• Way in which the area
formula is used
• The need to generalize
• The range of ways to enter
the problem
A Critical Starting Point for
Instruction
Not all tasks are created equal, and different
tasks will provoke different levels and kinds
of student thinking.
Stein, Smith, Henningsen, & Silver, 2000
The level and kind of thinking in which students
engage determines what they will learn.
Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997
• There is no decision that teachers make that
has a greater impact on students’ opportunities
to learn and on their perceptions about what
mathematics is than the selection or creation of
the tasks with which the teacher engages
students in studying mathematics.
Lappan & Briars, 1995
Task Analysis
• Use Cognitive level Handout and printed tasks
To Do List
To STOP Doing
To START Doing
Laurel County SSA
Day 4
High School Breakout Session
Jim Moore
[email protected]
Jennifer McDaniel
[email protected]
The House that Jack Built
• Building Algebraic Thinking
Lunch
11:00-12:30
Ideas for engagement
• Turnover Cards
• Card Sorts
• Math Strings
Technology
• Human Wave/Pass the ball
Regression activity for graphing calculator
How does the CX compare to the TI-84?
Solving Equations using the
graphing calculator.
• Standard (A-REI.1) Explain each step in solving
a simple equation as following from the equality
of numbers asserted at the previous step,
starting from the assumption that the original
equation has a solution. Construct a viable
argument to justify a solution method.
New Ideas: The hidden potential
in QR Codes
Shared Resources
• Content Network Updates
• NROC website