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maccss.ncdpi.wikispaces.net
Almost no student masters
something new after one or
two lessons or one or two
homework assignments.
The most effective strategies
for fostering mastery and
retention of critical
mathematics skills and
concepts is daily cumulative
review.
Number your paper one to six?
1. _
2. _
3. _
4. _
5. _
6. _
Mini-Math
1. 6 x 7
2. What number is 1000 less then 18,294?
3. About how much is 29¢ and 32 ¢?
4. What is 1/10 of 450?
5. Draw a picture of 1 2/3.
6. Estimate my weight in kilograms of any
average 6 foot man?
1. 6 x 7
• Taken affirmative action on fact mastery by
testing one of the more troublesome
multiplication facts
• Ascertained the number of students who still
don’t have a command of this fact
• Provided, if appropriate, positive reinforcement
about the progress your class is making
• Planted 9 x 6 into memory banks as
preparation for tomorrow.
2: What Number is 1000
Less Than 18,294?
• Broadened place value to an understanding of
10, 100, 1000, and 10,000 more and less than
a given number, and set the foundation for 0.1,
0.01, and 0.001 more and less than a number
• Recognized that topic taught one month earlier
needs periodic attention and reinforcement
• Supported and gradual development of number
sense for all students.
3. About How Much is 29 and 32?
• Reinforced a commitment to estimation and the
justification of all estimates
• Clearly communicated that there are many
correct approaches to arriving at an estimate
• Support a classroom culture that values the
development of number sense that transcends
a narrow focus on merely getting correct
answers to exercises.
4.
𝟏
𝟏𝟎
of 450
• Are you getting bigger or smaller? Why?
𝟏
(“Smaller because of a number reduces its
𝟏𝟎
magnitude.”)
• So does the decimal move to the left or the
right? Why? (To the left because you’re getting
smaller.”)
• And how many places must you move the
decimal point? Why? (Only one place because
it’s 10”)
5. Draw a Picture of
𝟐
𝟏 ?
𝟑
• Concretizing the mathematics by means of
pictures and visualization
• Amazingly informative, is clear
misunderstanding
6. My weight in Kilograms
• Measurement is often the lost strand of the
mathematics curriculum
Commit to helping students visualize
mathematics
• Another way to view the 5 to 8 minutes
allocated to this daily activity is to consider that
5 minutes x 180 days equal 900 minutes, or 15
hours!
• You know that you can change the world in 15
hours.
• Or think about how much can be accomplished
in 15 one-hour tutoring sessions
Easy to Modify
2nd grade example could look like this:
1. What is the difference of 9 and 5?
2. What number is the same as 5 tens and 7
tens?
3. What number is 10 less than 83?
4. Draw a four-sided figure and all of its
diagonals.
5. About how long is this pencil in centimeters?
Quick, focused, aligned with the curriculum,
reflective of what is coming on the grade 3
State Test, and wonderfully informative.
What more could we ask from the first few
minutes of a lesson?
Now your turn…
With someone in your grade level, create your own
mini-math questions.
1. review basic fact concept (challenging facts)
2. place value concept
3. estimation
4. multiplying and dividing numbers by 10, 100 &1000
5. concrete picture/visualization
6. measurement
In Summary
• A deliberate and carefully planned reliance on ongoing,
cumulative review of key skills and concepts
• Using cumulative review to keep skills and
understanding fresh, reinforce previously taught
material, and give students a chance to clarify their
understandings
• Classes that waste no time and begin with substantive
mathematics at the very start of every class
• The use of a brief review and whole-class checking of
“mini-math: questions as an opportunity to re-teach
when necessary
Accessible Mathematics
10 Instructional Shifts That Raise
Student Achievement
Steven Leinwand
”It's Instruction Stupid"
What questions do
you have?
maccss.ncdpi.wikispaces.net
Updates
• Games
• Put in examples! 
Contact Information
Kitty Rutherford
kitty.rutherford@dpi.nc.gov
Website:
maccss.ncdpi.wikispaces.net
F.A. Math Books….
Use individually or with PLC.
• Using Classroom Assessment to Improve
Student Learning. NCTM, 2011. Focus on
middle school
• Beyond the Bubble. Wickett & HendrixMartin, 2011. Focus on grades 4-5.
• INFORMative Assessment. Joyner and
Muri. 2011. Grades K-6.
Common Core College and
Career Readiness Standards
1. Mathematical Practice
Proficient students expect mathematics to make
sense. They take an active stance in solving
mathematical problems. When faced with a nonroutine problem, they have the courage to plunge in
and try something, and they have the procedural
and conceptual tools to carry through. They are
experimenters and inventors, and can adapt
known strategies to new problems. They think
strategically. See … www.corestandards.org
Using NCDigIns (Part 4)
• Help the students grow in self and
peer assessment. It is a cultural shift.
• Teacher feedback is most effective for
common errors. (one to many)
• Self and peer assessment expands
feedback … many to many
Build your own …
• Develop a classroom culture of peer and
self descriptive feedback along the format of
NCDigIns.
• This is a great way to find out the learning
progressions of your students, which often
do not match the logic of textbooks.
Is Drilling Worth it? Yes, but ..
• “Think of as many ways as you can to
practice the really crucial skills, but
remember that students can still get practice
in the basics while they are working on
more advanced skills.” p.125
Great resource by a cognitive scientist: “Why
Don’t Students Like School?” by Daniel T.
Willingham.
FA -> CCSSO Definition
Definition: “Formative assessment is a
process used by teachers and
students during instruction that
provides feedback to adjust teaching
and learning to improve students
achievement of intended instructional
outcomes.”
CCSSO 2008, adopted by N.C. Board of
Education
Squirrels:
Problem Solvers?
Squirrels:
Problem Solvers?
Squirrels:
Problem Solvers?
A challenge!
NCDigIns, Problem Solving and
Squirrels
CCSA Conference
March 2014
Mike Gallagher
Math Test and Measurement Consultant
Accountability Services, NCDPI
michael.gallagher@dpi.nc.gov
Overview
• NCDigIns and “deconstructed” math
problems.
• Students need to grow in problem
solving strategy use.
• Achievement Gap. What do
mathematicians look like?
A challenge!
NCDigIns, Problem Solving and
Squirrels
CCSA Conference
March 2014
Mike Gallagher
Math Test and Measurement Consultant
Accountability Services, NCDPI
michael.gallagher@dpi.nc.gov
Overview
• NCDigIns and “deconstructed” math
problems.
• Students need to grow in problem
solving strategy use.
• Achievement Gap. What do
mathematicians look like?
Squirrels:
Problem Solvers?
A challenge!
NCDigIns, Problem Solving and
Squirrels
CCSA Conference
March 2014
Mike Gallagher
Math Test and Measurement Consultant
Accountability Services, NCDPI
michael.gallagher@dpi.nc.gov
Overview
• NCDigIns and “deconstructed” math
problems.
• Students need to grow in problem
solving strategy use.
• Achievement Gap. What do
mathematicians look like?
August 1948: Maria Goeppert
Mayer and the Nuclear Shell
Model
William Schumacher Massey
wrote several textbooks,
including A Basic Course in
Algebraic Topology
1. NCDigIns
• NCDigIns were created by NC teachers and
test developers as a pilot project for grades
4, 7 and Algebra 1.
• The North Carolina sample and released
test questions (based on the 2003 Standard
Course of Study) were “deconstructed”.
NCDigIns
• For each of the three pilot grades there are
about 150 questions, arranged by 2003 SCS
Standard.
• The first 1/3 of the questions are from the
previous grade.
• Lots of these questions do align with the
Common Core State Standards.
♥
NCDigIns
Formative Assessment
• Assessment for (not of ) Learning
• Descriptive feedback to the student is
key, rather than a score in the grade
book.
• Feedback can be from teacher, peer,
or self.
NCDigIns - - Example
Let’s look at an example which is
aligned with a Common Core State
Standard for Mathematics (CCSSM).
-----------------------------------------------------------
Grade 4 DigIn Number 11, Question 2
“Folding papers”
• 2003 SCS Objective 1.02
• CCSSM 4.OA.3
Format of NCDigIns
1. Question on a single page. (Not
intended for use as a quiz.)
2. Feedback hints on common errors.
3. Feedback hints on Problem-Solving
Strategies.
4. Concepts/Skills for Improvement
5. Where to go next (resources)
Another Example
Often it is more useful to remove the
answer choices. This can be useful to
prepare for grid-in questions.
----------------------------------------------
Grade 4 DigIn Number 11, Question 3
“Bowling pins”
• 2003 SCS Objective 1.03
• CCSSM 4.NF.3.a
NCDigIns – How to find ‘em
• Go to NC Education
https://center.ncsu.edu/nc
• Click on the teeny “login” upper right.
• Sign in or go get an account (free)
• Lower right … click on link to NCDigIns.
• Scroll down to Math and CHOOSE
2: Grow in Problem Solving
• Students need to grow in problem
solving strategy use.
Adapted from SAT Celendar
www.collegeboard.com
If x < y < 0, which of the following is
greatest?
(A) -x
(B) -(x+y) (C) x (D) 0 (E) -y
Problem solving is important in SAT!
… p.s. How great would it be if most NC
students tried and talked about the SAT
and/or ACT “problem of the day” daily?
What Strategies would help
students with the Question?
• Understand the problem
• Make a drawing
• Try a simpler problem
• Check your answer
………………………..……. and more!!
Problem Solving Strategies
in ... (drum roll)
Just 1 Sheet!
Adapted from G. Polya How to
Solve it. (1957)
… at the NCDigIns site.
Why “Just 1 Sheet?”
• Students are more likely to retain and use a
small and constant set of strategies.
• Makes it easier to provide the descriptive
feedback so important for formative
assessment.
• No problem adding to the strategies and
personalizing them.
Build your own NCDigIns!
• Use your own sample questions, one
at a time.
• Help students see how each problem
connects to a learning target.
• Keep reinforcing that now is the time
to make mistakes and learn from
them.
Build your own …
• Develop a classroom culture of peer and
self descriptive feedback along the format of
NCDigIns.
• This is a great way to find out the learning
progressions of your students, which often
do not match the logic of textbooks.
Is Drilling Worth it? Yes, but ..
• “Think of as many ways as you can to
practice the really crucial skills, but
remember that students can still get practice
in the basics while they are working on
more advanced skills.” p.125
Great resource by a cognitive scientist: “Why
Don’t Students Like School?” by Daniel T.
Willingham.
FA -> CCSSO Definition
Definition: “Formative assessment is a
process used by teachers and
students during instruction that
provides feedback to adjust teaching
and learning to improve students
achievement of intended instructional
outcomes.”
CCSSO 2008, adopted by N.C. Board of
Education
How best to use …
…math questions in the formative
assessment process.
• Give sufficient time
• Students show work and rationale
• Usually not graded
• Descriptive feedback is key … from
the teacher, a peer, or self.
F.A. Math Books….
Use individually or with PLC.
• Using Classroom Assessment to Improve
Student Learning. NCTM, 2011. Focus on
middle school
• Beyond the Bubble. Wickett & HendrixMartin, 2011. Focus on grades 4-5.
• INFORMative Assessment. Joyner and
Muri. 2011. Grades K-6.
Common Core College and
Career Readiness Standards
1. Mathematical Practice
Proficient students expect mathematics to make
sense. They take an active stance in solving
mathematical problems. When faced with a nonroutine problem, they have the courage to plunge in
and try something, and they have the procedural
and conceptual tools to carry through. They are
experimenters and inventors, and can adapt
known strategies to new problems. They think
strategically. See … www.corestandards.org
Using NCDigIns (Part 4)
• Help the students grow in self and
peer assessment. It is a cultural shift.
• Teacher feedback is most effective for
common errors. (one to many)
• Self and peer assessment expands
feedback … many to many
R We Marketing Education?
• Target Audience
• Simple Message
• Unexpected
• ……
• Resource: Made to Stick – Why Some Ideas
Survive and Others Die. By Chip Heath and
Dan Heath
NAEP items …
• Grade 4, 1999
• M10 #14
• M5 #13
Problem Solving
“I’d unravel any riddle, for any individ’le
In trouble or in pain .. Oh, I would tell
you why, The ocean’s near the shore. I
could think of things I never thunk
before …” If I only had a brain.
Scarecrow in The Wizard of Oz.
Co-Creating NCDigIns
• The NCDigIns have been developed by
other teachers and provide a good starting
point.
• But teachers and students will find out what
are the … strengths, … misconceptions … &
weaknessess for students in this class.
There’s a Problem …
… with Problem Solving
Grade 4 Question from NAEP 2003
Estella wants to buy 2 notebooks that cost
$2.79 each, including tax. If she has onedollar bills and no coins, how many onedollar bills does she need?
A 3
B 4
C 5
C 6
Nationally, 31% of the students correctly
selected D. And 41% chose A.
nces.ed.gov/nationsreportcard/itmrls/
Co-Creating NCDigIns (Part 2)
Using the NCDigIns Template
What classroom discussion would
provide feedback to students on
Estella’s problem? Link to specifics on
the Problem Solving Strategy Sheet.
--------------------------------------------------Check out: The Housekeeper and the Professor by
Yoko Ogawa
Formative Assessment
“Toto, we’re not in Kansas anymore.”
Formative: Assessment for Learning
Summative: Assessment of Learning
FA -> Key User = Student!
When it comes to learning, students are the
key assessment users: “Am I succeeding?
Am I improving over time? What should I do
next to succeed? Does my teacher think I
am capable of success? Do I think I am
capable of success?”
…from Student-Involved Classroom Assessment, 3rd
Edition. Richard J. Stiggins, 2001, Prentice-Hall.
FA –> 3 Big Questions
Help students answer three questions:
• Where am I going?
• Where am I now?
• How do I get there?
Formative Assessment helps students
take more responsibility for their
learning.
FA –> Practices:
Descriptive Feedback w/o Grades
•
Provide descriptive feedback without
grades in the gradebook.
•
Put more emphasis on self and peer
feedback, not more teacher grading.
NOTE  Formative Assessment involves a
cultural change … it doesn’t happen
overnight, but it persists!
FA –> Practices:
Classroom Dialogue
Classroom discussion and questioning.
• Prepare questions which get to
complex thinking. Give students
sufficient “wait time” to do the thinking.
Great resource by a cognitive scientist:
“Why Don’t Students Like School?” by
Daniel T. Willingham. (e.g., pp. 103104 on classroom question posing)
FA –> Practices:
Where Am I Going?
• Tell students what the goal is: to
become great math problem solvers!
• Have a class discussion on what great
problem solving looks like. Use
examples.
• Ask students to critique the quality of
problem solving of some examples.
Design and Use - continued
• Content: start year with previous grade
questions and mix in current grade
standards
• Source: NCDigIns, Released items*, sample
items*, NAEP releases, other?
• Scoring: Nope. Descriptive feedback from
teacher, peer, and self. Use Polya’s
problem solving strategy.
*www.ncpublicschools.org/accountability/testing/
Closing Note:
Summative Assessment can
present motivational issues
You cant win Song from The Wiz:
“You can’t win, chile,
you can’t break even,
and you can’t get out of the game.”
Motivation and attribution, see Learned
Optimism by Martin Seligman
We need a balance of FA & SA
What does the student believe is the
cause of his/her success (or failure) on
a summative test?
• “I passed because I’m smart.” “I failed
because I’m not smart.”
• “I passed because I worked hard.” “I
failed because I didn’t work.”
Be careful what our tests teach!
A Message from the President! *
“NCTM’s Assessment Principle indicates that
assessment should not be done to students;
rather, assessments are for students and
should be used to guide and enhance their
learning …” (continued)
*Francis (Skip) Fennell, past President of the National
Council of Mathematics Teachers (NCTM)
President’s Message (cont)
“Formative assessment involves using
classroom-based assessments to
collect feedback that can be used to
improve teaching and learning.”
From NCTM News Bulletin (December
2006) Complete message on the web
at http://www.nctm.org/ … Go to the
“President’s Corner” -- Just one page
and well worth reading!
Puzzles
“Ever since I can remember, my father has
given us little problems or puzzles. … These
puzzles challenged us and encouraged our
curiosity … they taught us how to reason
and think for ourselves.” Sarah Flannery In
Code – a Young Woman’s Mathematical
Journey. Algonquin Books of Chapel Hill,
2002. Ms. Flannery’ high school project on
cryptography won Ireland’s Young Scientist
award in 1999.
For all you do for our students!