Csrqi Stw09 Presentation

The National
Forum
Mathematics
Improvement
Toolkit
Presented by:

Deborah Kasak
Sara Freedman
Emily Fagan
Anna McTigue
Stephen Best

Goals for this Presentation
• Provide background on the purpose of the
toolkit, and the teaching and learning needs
it was designed to meet
• Introduce the toolkit and its components
• Try out some of the actual PD activities
embedded within these tools
• Provide additional information for future use
and identify interested sites

What is the Mathematics
Improvement Toolkit?

What is the Mathematics
Improvement Toolkit?
 Joint venture of four groups to address
the needs of special populations
 Provides support for teachers,
professional developers, decision
makers, and students around middle
grades mathematics instruction
 Addresses speciﬁc instructional needs
that are often ignored.

Goal of the Project
 Create professional development
resources to address instructional needs
of:
English Language Learners
Students with Special Needs
Students and Teachers in Rural Settings
Communities and Families

Partners
 National Forum for Middle Grades Reform
 Turning Points
(Center for Collaborative Education)
 Talent Development
(Johns Hopkins University)
 Educational Development Center
 Middle Start
(Academy for Educational Development)

 Funded by the U.S. Department of Education
(Comprehensive School Reform program)

Common Ideas and
Considerations

Common Ideas and
Considerations
 Mathematics instruction need to focus on
building deeper conceptual understanding

Common Ideas and
Considerations
 Resources are designed for use with math
teachers and others supporting mathematics
learning for ALL students

Common Ideas and
Considerations
 Materials need to focus on getting teachers to
reﬂect on practice

Common Ideas and
Considerations
 Materials need to focus on getting teachers to
reﬂect on practice
 Effective professional development requires
extensive time and ongoing implementation

Teaching High-Level Mathematics
to English LanguageTool #1
Learners

Learners
 Issue: Teachers need support to ensure that
English language learners have access to and are
successful in learning high-level mathematics.

Learners
 Primary Resources:
Professional development workshops that include
videos, a participants’ packet and facilitator
materials.

Learners
Professional development workshops that include
videos, a participants’ packet and facilitator
materials.
 Combines a focus on English language learners with
general issues regarding deepening understanding
of concepts in mathematics

Tool #1
Learners

Tool #1
Learners
Who are the English language learners in
our schools today?

Tool #1
Learners
Who are the English language learners in
our schools today?
 English language learners are the fastest growing
segment of the school population. 1 out of 10 students
enrolled in public schools is an English language learner.

 Nearly 1 out of 3 students enrolled in urban schools is an
English language learner.

 The percentage of English language learners enrolled in
schools is increasing throughout the United States, in
suburban and rural, as well as urban, communities.

Tool #1
Learners

What do we know about their experience
in our schools?

Tool #1
Learners

What do we know about their experience
in our schools?
 English language learners have a strong desire to
receive an education. They have the highest daily
attendance rate of any segment of the school
population.

 English language learners have the lowest out of school
suspension rates of any segment of the school
population.

Tool #1
Learners

However...
 English language learners have the lowest standardized
test scores of any segment of the school population.

 English language learners have the highest dropout rate
of any segment of the school population.

Tool #1
Learners

However...
 English language learners have the lowest standardized
test scores of any segment of the school population.

 English language learners have the highest dropout rate
of any segment of the school population.

Why do you think this is so?

THINK WRITE PAIR SHARE

Letʼs look at a typical word problem

A certain construction job usually takes four workers six
hours. Today, one worker called in sick, so there are only
three workers. How long should it take them to do the job?

What speciﬁc challenges do you think an English
language learner in the middle grades might have in
trying to answer the question posed by this problem?

(Notice that you are NOT solving the problem; instead, you
are analyzing the difﬁculties raised for a diverse group of
English language learners as they approach the problem.)

WRITE: Use the handout to record
your responses

your responses

your responses
BEST PRACTICE:
PROVIDING an
ORGANIZING
TEMPLATE
• saves time
• focuses English
language learners’
attention on the
mathematical
concepts rather than
copying in a new
language
• creates expectations
about # and quality
of responses

Tool #1
Learners
What are the LANGUAGE challenges in this problem for English
language learners?

Tool #1
Learners
What are the LANGUAGE challenges in this problem for English
language learners?
Small Group
discussion
GROUP #1 #2 #1 #2
Get into groups of four. Assign
one person to chart the responses
to the ﬁrst question, one at a time.
Take turns listening to each others’
➟ ➟
responses.

➟
➟
As each person speaks, ask any
questions or make comments that
help expand their comments
further. ➟ #4
➟ #3
#4 #3

Focus 2: Students with Special
Universal Design Curriculum Modules
Learning Needs

Learning Needs
 Issue: Curriculum materials do not support
students with special learning needs.

Learning Needs
Modiﬁed curriculum resources, student
materials, and instructional practices based
on Universal Design for Learning principles

Learning Needs
Modiﬁed curriculum resources, student
materials, and instructional practices based
on Universal Design for Learning principles
 Resources need to be comprehensive in
nature to have full impact on learning.

Focus:
Students with

Special Learning Needs

Focus:
Students with

 Students come into a
class with varying
levels of
understanding

Focus:
Students with

 Students come into a
class with varying
levels of
understanding
 Some students need
explicit instruction to
get to a functional
level

Focus:
Students with

 Students
need
support for
visual,
auditory,
attention,
and
memory
functions.

Tool #3Collaboration and Co-Teaching
Learning Needs

Learning Needs
 Issue: Mathematics and Special Educators are
sometimes paired to co-teach without speciﬁc
professional development and preparation

Learning Needs
Video, a PowerPoint presentation, and a
facilitator guide for a workshop to implement or
strengthen co-teaching.

Learning Needs
Video, a PowerPoint presentation, and a
facilitator guide for a workshop to implement or
strengthen co-teaching.
 Teachers beneﬁt from seeing and discussing a
video example of co-teaching

Letʼs try a task...
• Watch a video clip from a lesson taught
by co-teachers

by co-teachers
• As you watch, jot down your ideas
about the questions:

by co-teachers
• What roles did the co-teachers take?

by co-teachers
• What roles did the co-teachers take?
• What actions did they take to support
student learning?

Co-teaching Roles
• Work with a
partner and
brainstorm roles
that co-teachers
could take to
beneﬁt students.

Co-teaching Roles
• Work with a
partner and
brainstorm roles
that co-teachers
could take to
beneﬁt students.
• Record your
ideas on the
handout.

Focus 2:Language in the MathSpecial
Students with Classroom
Learning Needs
Tool #4

Learning Needs
Tool #4
 Issue: Sometimes difﬁculty with language presents
an obstacle to learning mathematics

Learning Needs
Tool #4
Video, a PowerPoint presentation, and a facilitator
guide for a workshop exploring the language
demands and challenges in mathematics and
offering vocabulary and writing strategies to address
these challenges.

Learning Needs
Tool #4
Video, a PowerPoint presentation, and a facilitator
guide for a workshop exploring the language
demands and challenges in mathematics and
offering vocabulary and writing strategies to address
these challenges.
 With instruction and support in communication skills,
students can more deeply develop and express their
mathematical ideas.

Learning Needs
Tool #4
Language Module topics:
✦ Demands and challenges of language
✦ Instructional strategies
✦ Planning for vocabulary instruction
✦ Writing strategies for mathematics

Collaborative Online
Tool #5
Focus 3: Rural Education
Professional Development

Tool #5
 Issue:
Access to quality mathematics PD

Tool #5
 Issue:
Online professional development program,
PD materials focusing on depth of
understanding and appropriate instruction

Tool #5
 Issue:
Online professional development program,
PD materials focusing on depth of
understanding and appropriate instruction
 High quality PD in mathematics education
requires reﬂection on practice and sample
tasks and cases.

Tool #5
 Online
community

Tool #5
 Online
community Modifying a Task: Task 1
 Focus on mathematics
The Old Farmer’s Almanac
tasks as a lens to suggests that you can tell
the temperature outside by
examine teaching counting the chirps a cricket
practice and student makes in 14 seconds and
adding 40 (to get the
understanding temperature in degrees
Fahrenheit). Use this to ﬁnd
how many chirps the cricket
makes when it is 72 degrees.

middlestart

Tool #5
 Online
tasks as a lens to The cost of a taxi in the city
of Boston is $2.50 for using
examine teaching the cab, plus $0.30 for every
ﬁfth of a mile. What is the
practice and student cost of a two mile ride?

understanding What is the cost of an N-mile
ride?

middlestart

Tool #5
 Online
tasks as a lens to
examine teaching Write an expression to x
represent the area of the
practice and student ﬁgure at right. x
understanding 5 x

middlestart

Tool #5
 Online
Which of the following is true about the
tasks as a lens to equation at the right?

examine teaching a) If you cancel the x’s, the equation says 1/2
= 1, which is impossible, so there is no x+1
= 1
practice and student
solution.
x+2
b) It might have solutions, because n could
understanding be a fraction or negative number.

c) It can’t have any solutions, because x + 1
can’t be equal to x + 2, so that ration can’t be
equal to 1.

middlestart

Tool #5
 Online
What type of sequence is shown in the ﬁgures
tasks as a lens to at the right? Explain.

examine teaching a) Linear
b) Quadratic 1 3 6
practice and student c) Exponential

understanding d) None of the above

10 15

middlestart

Tool #5
 Online
community
tasks as a lens to
examine teaching
understanding

Shade 6 of the small squares in the rectangle shown
below. Using the diagram, explain how to determine
each of the following:

1. the percent area that is shaded
2. the decimal part of
the area that
is shaded
3. the fractional part of
the area that
is shaded.

Tool #5
 Online
community
tasks as a lens to
examine teaching
understanding
 Review student work

Tool #5

Tasks as
Tasks as
enacted
they Tasks as
by
appear in set up by
teachers
curriculum teachers Student
and
materials learning
students

Tool #5

Mathematical Task Framework
(Stein & Smith)

Tasks as
Tasks as
enacted
they Tasks as
by
appear in set up by
teachers
curriculum teachers Student
and
materials learning
students

Tool #5
 Online middlestart
community Module 1 - Case 1: David Orcutt

Focus on mathematics
This mini-case provides an introduction to the use of cases as a reflective professional development tool, and is not intended

 for sustained use. This also uses student work examples to explore understandings and misconceptions around fractions,
percents, and decimals.

tasks as a lens to
INTRODUCTION AND CONTEXT
David Orcutt is one of two 7th grade mathematics teachers in the lone junior high school for this
district. The district serves students from a largely rural agricultural and recreational area which
includes two villages. The school is a 7-8 school in a small school building next to the district’s

examine teaching
high school. In fact, a number of teachers are on the faculty of both schools to provide appropriate
coverage for topic areas. David has four classes among his other duties as the 7th grade advisor
and a track coach.

In his three years of teaching, he has learned that students coming in from the two K-6 schools in

the district (as well as a small but growing migrant labor population that is becoming a more
permanent fixture in the area) often have varying skills and understanding in mathematics. To
understand each of the student’s abilities and conceptions about basic topics, he has devised a
two week introduction to his course which addresses a different topic from the grade 4-6

understanding
standards each day or two, and uses this to establish norms for classroom participation, work
expectations, etc. The following sample of classroom interaction starts by asking students to take
out the homework task from the previous day, which was really a pre-assessment of sorts to
understand student knowledge of decimals, percents, and fractions.

CLASSROOM ACTIVITIES

David starts class by greeting all students at the door as they come in, and has a problem on the
board, which he reminds students to get a paper out and copy the problem down after they have
taken their homework out from the previous day. Meanwhile, he checks attendance and missing
assignments from the previous day, and then begins wandering through the aisles to see what
students are doing with the problems on the board, and whether they have their homework out.

Review brief case studies
He quickly scans the homework for each student, noting whether they have all twenty problems

 done, and whether they have them numbered, the problem written down, and the answer
underlined for each. Most do, which results in him writing a “10” on the top of the page, but a
couple did not finish, receiving 5 and 7 points respectively, and three others had 3 points deducted
from these for not organizing their work properly. For these, David underlined a few of the answers

to encourage reflection
they had in their work that were not already underlined, and had jotted down the words “show your
steps” on some of these papers. While doing this, he marked on a copy of a grade sheet the
points for the homework assignment for each student.

Following this fairly quick review (which took four minutes from the time he started moving around
the room), he told the students they would review the answers of the homework. He circled the
room as he called out problem numbers, and would look around the room to see who was looking
at him (or not) and would call out the names of students to state what their answer was. Once one
student gave the answer, he would call on two other students and ask if they came up with the

Tool #5
 Online
community same answer as the original student, or if they had something different. At every problem in which
all students agreed on the answer, he would quickly ask if any other answers were out there, and
unless a quick response came, he would say “correct” and repeat the problem number and
answer and move on. When students disagreed, he would quickly survey students in the room to
see which of the stated answers other students got, or, what other answers people came up with,

and unless it seemed that one was an outlier, would note that problem number of the whiteboard,
so that the class could go through it after checking homework. Six of the problems were noted on
the board, and he they asked, problem by problem, if there were any volunteers to go to the board
and do the problem. Two of the problems had no volunteers, so he asked one student what

tasks as a lens to
answer they got for the problem, then asked if anyone had a different answer, and had both (or
more if several different answers arose) go up to the board to write their explanation or procedures
for the problem.

One of the two problems that had contested answers was the following:

examine teaching
! Emma was asked to order the following numbers from smallest to largest: .43, 8%, and .7
! Emma’s order was: .7, 8%, .43
! Is she correct? Why or why not?
Two students wrote their answers on the board initially as shown below.

practice and student Student D: No because .43 is just about half and .7 is almost full and 8% is like 8 1s. .43 .7 8%

Student F: She is correct because 7 is the smallest and 43 is the biggest

understanding The following dialog is taken from this activity:

DO: “So, what do we think everyone. We have two answers here. What do we think?”

Student H: “[D] is right. Emma didn’t get the right answer.”

 Review student work DO: “And why is that?”

H: “Well, sort of right. Emma didn’t get the right answer, but [D] didn’t get it right either.”

DO: “[F], what you you think? You said Emma got the right answer. Explain what you said.”

 Review brief case studies F: “Well, the numbers get larger, um, in Emma’s order, and, um, the dots and percents are the
same cause you can change from dots to percents and so I, um put them in order, and so, um, 7
is smallest, then 8, then 43.”

to encourage reﬂection H: “But they aren’t the same. Dots are two places different.”

DO: “[D], what do you think? You said Emma wasn’t right, just like [H], but she said you weren’t
either. What do you think?”

D: “I was just trying to see what they are close to, and .43 is close to .5, which is a half. .7 is
bigger. It is nearly a whole thing, and deﬁnitely more than half. The percents don’t have the
decimals, so I thought 8% is like 8 whole things. But I think [H] is kinda right, um, ‘cause you have
to do move the dot two places.”

Tool #5
 Online
community DO: “Let’s see what someone else says. [G], how about you? What did you say?”

G: “I said Emma was wrong. It should be 8%, .43, .7 in that order because I put them all in
percents.”

DO: “Aha. There we go. You put them all in percents. All in the same units. That is exactly what
we want to do when we have decimals and percents together is put them in the same units. [H], is
that what you meant? Is that what you did?”

tasks as a lens to
H: “Yeah, I made them all the same, but I didn’t do percents. I changed percents to fractions, so
they were all some part of 100.”

DO: “Excellent. There we go. We want to change them all to the same, and the best way is to
change them to fractions. Since we have percents, we should change them to parts of 100. That

examine teaching is what percents really are. They are parts of 100. So, when you have all of your test right, for
instance, you have 100%. You get everything out of 100. So, how do we want to change these to
fractions of 100?”

C: (called on after raising hand) “If it is one place. like .7 was, that is 7 out of 10, because the first
place is tenths. Then hundredths. so we could add a zero to the end of that, because .7 is the
same as .70, and that is seventy out of a hundred.”

understanding
DO: “Great. That’s exactly it. Are we okay? Can we move on?”

No responses, so they go on to the next question. Shortly thereafter, David moves through the
other answers, and to the boardwork task. This task is written on the board already. It was
modified by David from a task he had seen in a workshop focusing on differentiation, which was
addressing visual learners. The original task from the workshop is below.

 Review student work Shade 10 of the small squares in the rectangle shown below. Using the diagram, explain how to determine
each of the following: a) the percent area that is shaded, b) the decimal part of the area that is shaded, and
c) the fractional part of the area that is shaded.

 Review brief case studies
to encourage reflection David’s modified version that is on the board is the following:

Shade 10 of the boxes in the rectangle shown below (same rectangle). Find the percent area that is shaded.

David says that, in the interest of time, he is going to go through it, and asks students to watch.
He shades in 10 of the rectangles, picking them at random, and shading individual rectangles.

Tool #5
 Online
community DO: “So, it really doesn’t matter which ones I pick, it will be the same. What I really care about is
how many total ones we have. [A], how many total boxes are there?”

A: “40”

 Focus on mathematics DO: “And how did you get that?”

A: “I counted ten across, and there are four rows, so it was four times ten.”

tasks as a lens to DO: “Exactly... or you could count everyone of them if you didn’t ﬁgure that out. So, what next
(looking at A)?”

A: “Well, it is a quarter. There are 10 out of 40, and if we write that as a fraction (DO pauses A with

examine teaching a hand gesture and writes this on the board as the fraction 10/40, and then motions for him to
proceed)... so yeah, that’s it. And then you can cross out the zeros, cause 10 out of 40 is like 1
out of 4, and that’s a quarter. And a quarter is always 25%.”

DO: “Exactly. Does everyone see that? Once [A] got it to a fraction, he could easily change it to a
percent. If it was a fraction you didn’t know already, like... suppose we had 12 shaded boxes
instead? You could make it 12 out of 40, and then cross multiply to ﬁgure out the number out of
100 (as he draws on the board ‘12/40 = n/100’ and then proceeds to write, ’12 x 100 = n x 40’),

understanding
and so in this case you could multiple 12 and 100...[A], what is that?”

A: “Twelve and a hundred? That’s one thousand two hundred.”

DO: “and divide that by 40 and we would get 30. Thirty percent... if it was twelve out of 100.” Do
you all see that?

 Review student work The class seems to agree quietly, and David moves on to the next part of class...


Tool #5
 Online
community
tasks as a lens to
examine teaching
understanding
 Teachers share
examples, observations,
and reﬂections on their
own and others practice.

Tool #5
 Initial module: Developing Student Understanding of
Mathematics
 Content modules: Issues in the instruction of...
 Ratio and Proportion
 Patterns, Functions, and Algebraic Reasoning
 Measurement and Geometry
 Skill and strategy module: Issues in the instruction of
Problem Solving and Use of Inquiry

Tool #6
Engaging Families and Communities

Tool #6

 Issue: Schools struggle with this in general
and many mathematics issues for students arise
from parent/community misunderstandings,
stereotypes, and attitudes toward math.

Tool #6

Online PD tools for schools and teachers that
guide them through family engagement
Resources to guide communication with parents

Tool #6

Online PD tools for schools and teachers that
guide them through family engagement
Resources to guide communication with parents
 Audience for these resources needs to be broader
than mathematics teachers alone.

Focus #6Family Engagement
Tool 4:

 Needs assessment
and introductory
activities

Tool 4:

and introductory
activities
 Sample discussion
materials (big picture)
and communications

Tool 4:

and introductory
activities
and communications
 Strategies to provide an awareness of
approaches to learn mathematics

Tool 4:

and introductory
activities
and communications
 Strategies to provide an awareness of
approaches to learn mathematics
 Discussion of deeper issues and research

For more information…
 Visit:
http://www.middlegrademath.org

to ﬁnd out more about the tools...

 And come back to the site again in
August, when all of the resources will
be available.

Csrqi Stw09 Presentation

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