Tools for Teachers
• What’s new?
• How to use tools to
improve practice?
• Engaging in student
work
• Trends
Tying Tasks to CCSS
2nd Gra de
Stu den t Task
C ore Ide a 1
O pe rations
an d
Algebraic
Thi nkin g
Task 1
Striped Fish
Use an interest ing context for describing and extending a patt ern.
Represen t an d solve problems involvi ng addi tion an d subtraction.
W ork with equal groups of obje cts to gai n fou ndations for
mul ti plication .
Solve problems i nvolvi ng the four ope rations, an d i denti fy an d
e xplai n patterns i n arith metic.
 Solve two-step problems using t he four operat ions. Represent
these problems using equa t ions with a let ter standing for the
unknown quan t it y. Assess the reasonableness of answers using
mental computat ion and est imat ion st rategies.
Mathe m atical C onstruct viable argum en ts an d cri ti que the argu men ts of other.
Practice s
Mathematical Practices
• Perseverance
• Reason abstractly and Quantitatively
• Construct viable arguments and critique the
reasoning of others.
• Model with mathematics.
• Use tools strategically.
• Attend to precision.
• Look for and make use of structure.
• Look for and express regularity in repeated
reasoning.
Students develop their ability to construct viable
arguments and to critique the reasoning of
others.
• What makes a good justification?
• What is involved in making it complete?
• What types of quantification can be used to
support claims or conjectures?
• What sentence structures help students
develop a logical sequence to ideas?
Defining Justification Through
Student Work
What qualities would you like to
see in a good justification?
• Work individually and then with a partner
to list what should go into a justification.
• Then look at student work.
– Which explanation did you like the best? Why?
– What were the qualities or arguments that
made it a good justification?
What qualities would you like to
see in a good justification?
A powerful technique from the Formative
Assessment Lessons is to use student work
and see how it could be improved.
• Then look at student work again.
– How could each statement be improved?
– What qualities or arguments are missing that
prevent it from being a good justification?
Attention to Precision- Marathon
Runners- 5th Grade
Looking at the Data
• 37% of the students omitted a vertical scale
• Making the scale was one of the two ramps
of the tasks.
Implications for Instruction
In planning learning trajectories or units around a big idea,
like understanding scale, there aren’t short cuts. The type
of thinking required to add data to a graph has a higher
cognitive demand than just reading information from a
graph. The ability to think between squares and what the
squares represent is challenging. Further, the cognitive
demands for planning a scale and putting equal-size units
on a graph is different from using a graph with a scale
already provided. Planning what size units will allow all
the data to be represented is one part. Understanding that
the scale is different from just a listing of the different
data points is another. Classroom activities and
discussions need to help students build these layers of
understanding.
Ideas for Action Research
How much did Sulian save in July?
Ideas for Action Research
How much did Sulian save in July?
How would you continue the
lesson?
• Looking at your own student work or work from the
toolkit, think about how to continue this lesson.
How can you use the student work to pose a
question?
• Besides graphing, what are some other important issues
that you want to address? Again find some pieces of
student work from the toolkit or from you student work.
How can you use it to pose questions for the class? How
does answering the question confront misconceptions?
How does it deepen the mathematics of the lesson?
Toolkits
• Help guide teachers in how to plan a reengagement lesson
• Resource for interesting student work which
will highlight big mathematical ideas and
concepts
Understanding Mathematical Modeling and
Deepening Understanding of Operations in
context
• Work the task: Sub Sandwiches (pg. 1 only)
• What does a student need to understand
about fractions and rates to make sense of
this task?
• What does a student need to understand
about multiplication and division?
• What is important in a model?
Re-engagement Lesson
Set the hook for the lesson:
• What do you know about multiplication?
• Where are the equal groups?
• How many equal groups?
Spend time on having students explain and
verbalize beginning of task, to build
foundation for rest of lesson.
How many tomatoes does she need?
“I saw a student make this diagram. Can you
help me figure out what the student is doing?
How do you think the student answered the
question?
“What do you think the student is doing? Does
this make sense?
“How is this the same as the drawing?
How is it different?”
“What is the student thinking? Does it make sense?
Why or why not?
How many sandwiches can Laura make with
5 tomatoes?
“Someone said there would be 5 sandwiches.
I drew buns with salami and 1 half tomato until
there 5 tomatoes in all.
What is this student doing? Does it make sense?
What qualities make it easier to use?
“Does this make sense? How it it like the
diagrams? What are the equal groups?”
Can this work? We just saw how multiplication
Helped to answer the question. So does this
Make sense? Why or why not?
Re-engagement Lessons
• Help to build from concrete to abstract
• Help to bridge the meaning of operations
and connections between multiplication and
division
• Show students strategies or new tools to use
• Help students understand how models can
be used and improved
What does it mean to know?
• Tool Possession: Is the tool in the toolbox?
• Too Understanding:Is the operation of the
tool understood?
• Tool Application: Can the tool be used in a
real-world context?
• Tool Selection: When given a big problem,
what tools does the student use?
Teaching for Transference
• What does a student need to understand
about variables and how they are used?
• What do students understand about setting
up equations? What clues do the equations
give them about solving the problem?
• What strategies do students have for solving
systems of equations algebraically?
Trends in Data
Declining Grades
• Geometry - Dropping from 78% in 2009 to
50% in 2011, a 2 year decline
• 8th Grade - dropped -.39 std units to 19.9%
meeting standards
• 7th Grade -dropped - .67 std units to 23.7%
from 54% in 2010
Improving Grades
•
•
•
•
3rd Grade - 70.7% meeting standards
4th Grade - 71.5% meeting standards
5th Grade - 76.5% meeting standards
6th Grade - 43.2% meeting standards up
from 34.5% in 2010
About the Same
• 2nd Grade - 78.9% meeting standards
• Algebra - 46% meeting standards up
from 39% in 2009
Looking at Middle Grades
•
•
•
•
•
6th Grade - 43.2%
7th Grade - 23.7%
8th Grade- 19.9%
Algebra - 43.2%
Geometry - 50%
Second Grade
• Students had the most difficulty with
Striped Fish. They had trouble with visual
discrimination and drawing the stripes on
the fish. This causes problems at later
grade levels.
Visual Discrimination - 3rd
26% thought the pentagon looked like
a house or an ice cream cone.
Visual Discrimination - 3rd
Students have trouble with the idea of
area because they aren’t seeing space
in terms of rows and columns.
Visual Discrimination - 4th/5th
Here are two of their patchwork quilt s.
A
B
G
C
E
H
D
F
2
3
Studen ts could name the squa re, the Studen ts st ruggles with how the square
rectangle, the two t riangles, and the and rhombus are alike. 6% said t he
hexagon.
rhombus was sideways. 13% said the
rhombus was a slanted or st retched
square.
Studen ts could name most of the
Studen ts st ruggled with how the square
shapes and explain how the square and rhombus are different. Many didn’t
and the rhombus are alike.
have the vocabulary to explain their
ideas. 20% said they are different shapes
or have different names. 17% said t hat
one shape was wider. 6% talked about
st retching.
Visual Discrimination - 6th
Grade
Students had difficulty maintaining shape
and scale. Only 24% of students met
standard on this task.
Visual Discrimination - 7th
Grade
S hape A
1.7 cm)
Shape B (The height of t he triangle is
2 in
1 in
2 in
1 in
2 in
6 in
2 in
6 in
2 in
2 in
2 in
Students can’t interpret the diagram to think
how to fold the shape or how dimensions
fit together.
3rd & 4th Grade Road Signs
and Spot the Difference
• Students don’t know how to compare and
contrast shapes.
• They struggle with geometric vocabulary.
• They only know a few attributes.
• They consider names and shapes to be
attributes.
5th Grade - Plum Jelly

Em ily has plum t rees in her bac kyard.
‰
Each year she uses the fruit to make jell y.
‰
‰
He re are the ingredients for her re cipe.
‰
4 poun ds of plums
‰
3 pounds of sugar
‰
one cup of wat er
This recipe makes 6 pounds of plum jelly.
Emily says that for 18 pounds of plum jelly she will need 10 pounds of sugar and
15 pounds of plums. Explain how you know she is wrong.
6th Grade
• Memory Game - Probability
– 32% scored no points
– Only 42% met standards
• Animal Pattern - Geometry
– 30% scored no points
– Only 24% met standards
• 100 People -Proportional Reasoning
– 20% scored no points
– Only 27% met standards
7th Grade
• Wallpaper - Geometry and Measurement
– 43.5% zero/ 16.8% met standard
• Million Dollars - proportional reasoning
– 21% zero/ 43% met standard
• Boxes - Geometry and Measurement
– 34% zero/ 18% met standard
• Population - Reading Graphs/Percents
– 16.5% zero/ 23% met standard
8th Grade
• Jane’s t.v. - Geometry/ Pythagorean
theorem & working with exponents
– 53% no points
– 27% met standard
• Matchsticks - Geometry/ Measurement
Volume
– 53% no points
– 14.8% met standard
Algebra- Understanding Graphs
Graph
Equation
Only 26.5% met standard.
Math fact