Transcript Math Facts - mtss-implementers

Digging Deeper in the Tiered System of
Support For Mathematics
Laura Colligan
Academic Consultant, Ingham ISD
[email protected]
517.244.1258
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Universal Screener: MCOMP, Now
What?
6 Areas of Demands and Difficulties for
Students
Identify research based strategies to
use with student emphasis on fluency
Individual reflect-and-write
Massachusetts Department of Elementary and Secondary Education
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Participants will…
 Be able to dig deeper into MCOMP
assessments to inform instruction
 Identify 6 Areas of Demands and Difficulties
for students in mathematics
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Identify research based strategies to use
with students emphasis on fluency
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Task:
Take 8 minute assessment
Score assessment
Small group discussion about MCOMP
Topics to ponder:
 Strengths
 Common skills across probes
 What could we take back to our schools
Directions: Solve the following basic facts. You have 1 minute to
complete this quiz. Please remember that the + symbol means
multiply, the - symbol means divide, the ÷ symbol means add, and the
x symbol means subtract.
8+2=
10 - 5 =
8x7=
14 ÷ 7 =
17 x 2 =
14 - 7 =
12 x 2 =
8÷4=
6x2=
10 - 2 =
4x3=
8+5=
6x5=
15 - 3 =
9-1=
9÷9=
9÷2=
8-4=
Massachusetts Department of
Elementary and Secondary
Education
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How did it feel to be in the place of
the quiz taker?
How might this experience translate
into ways in which students with
disabilities respond to typical
classroom learning experiences?
Massachusetts Department of
Elementary and Secondary
Education
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Chosen because they have an impact on
mathematics learning.
Memory
Attention
Organization
Language
Conceptual
Understanding
Visual/Spatial
Understanding
Massachusetts Department of
Elementary and Secondary
Education
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The impact of Memory on learning mathematics
includes:
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Difficulties storing and
retrieving facts
Math facts
Students’ lack fluency
and accuracy
Working memory
Impacts work on multistep problems
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Other theories: difficulties
with language of number
words or difficulties with
visual representations,
e.g. number lines
Difficulty holding
information in mind while
solving a problem
May be related to
difficulties inhibiting
correct answers
Sources: Gersten et. al., 2008; Mazzocco, 2007
Massachusetts Department of Elementary and Secondary Education
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Impact of Attention for learning mathematics
includes:
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Lack of focus on details
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Lack of routines to
follow
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Too much text on a page
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Finding key words or
phrases to solve
problems
Focus on only one
aspect of a problem
Source: Allsopp et al., 2003
Massachusetts Department of Elementary and Secondary Education
11
Impact of Organization for learning mathematics
includes:
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Aligning columns and
rows for computation
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Constant movement of
manipulatives
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Problem solving
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Creating graphs
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Ordering of numbers
and symbols
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Matching tables with
patterns
Source: Allsopp et al., 2003
Massachusetts Department of Elementary and Secondary Education
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Impact of Language for learning mathematics
includes:
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Reading Text
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Sharing ideas in groups
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Math Vocabulary
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Listening to instruction
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Writing math stories
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Writing explanations
Source: Allsopp et al., 2003
Massachusetts Department of Elementary and Secondary Education
13
Impact of Conceptual Understanding for learning
mathematics includes:
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Number sense
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Making generalizations
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Problem solving
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Applying strategies to new
situations
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Moving from concrete to
abstract, i.e. equations
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Reflecting on thinking—
metacognition
Source: Allsopp et al., 2003
Massachusetts Department of Elementary and Secondary Education
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Impact of Visual/Spatial Understanding for
learning mathematics includes:
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Reading tables
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Diagrams
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Visual examples
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Trouble following
graphs
May not line up
numbers correctly
Following patterns
from drawings
Source: Allsopp et al., 2003
Massachusetts Department of Elementary and Secondary Education
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What are the essential barriers that
students with these difficulties
experience?
What experiences have you had with
this area of demand with students or
with teachers?
Massachusetts Department of
Elementary and Secondary
Education
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There is a strong correlation between poor retrieval
of arithmetic combinations (‘math facts’) and
global math delays
Automatic recall of arithmetic combinations frees
up student ‘cognitive capacity’ to allow for
understanding of higher-level problem-solving
By internalizing numbers as mental constructs,
students can manipulate those numbers in their
head, allowing for the intuitive understanding of
arithmetic properties, such as associative property
and commutative property
Source: Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics
difficulties. Journal of Learning Disabilities, 38, 293-304.
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How much is 3 + 8?: Strategies to Solve…
Least efficient strategy: Count out and group 3 objects; count out and
group 8 objects; count all objects:
=11
+
More efficient strategy: Begin at the number 3 and ‘count up’ 8 more
digits (often using fingers for counting):
3+8
More efficient strategy: Begin at the number 8 (larger number) and
‘count up’ 3 more digits:
8+ 3
Most efficient strategy: ‘3 + 8’ arithmetic combination is stored in
memory and automatically retrieved: Answer = 11
Source: Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics
difficulties. Journal of Learning Disabilities, 38, 293-304.
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“[A key step in math education is] to learn the four
basic mathematical operations (i.e., addition,
subtraction, multiplication, and division).
Knowledge of these operations and a capacity to
perform mental arithmetic play an important role in
the development of children’s later math skills.
Most children with math learning difficulties are
unable to master the four basic operations before
leaving elementary school and, thus, need special
attention to acquire the skills. A … category of
interventions is therefore aimed at the acquisition
and automatization of basic math skills.”
Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs.
Remedial and Special Education, 24, 97-114.
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The three essential elements of effective student learning include:
1.
Academic Opportunity to Respond. The student is presented with
a meaningful opportunity to respond to an academic task. A
question posed by the teacher, a math word problem, and a
spelling item on an educational computer ‘Word Gobbler’ game
could all be considered academic opportunities to respond.
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Active Student Response. The student answers the item, solves the
problem presented, or completes the academic task. Answering the
teacher’s question, computing the answer to a math word problem
(and showing all work), and typing in the correct spelling of an item
when playing an educational computer game are all examples of
active student responding.
Performance Feedback. The student receives timely feedback about
whether his or her response is correct—often with praise and
encouragement. A teacher exclaiming ‘Right! Good job!’ when a
student gives an response in class, a student using an answer key
to check her answer to a math word problem, and a computer
message that says ‘Congratulations! You get 2 points for correctly
spelling this word!” are all examples of performance feedback.
Source: Heward, W.L. (1996). Three low-tech strategies for increasing the frequency of active student response during group
instruction. In R. Gardner, D. M.S ainato, J. O. Cooper, T. E. Heron, W. L. Heward, J. W. Eshleman,& T. A. Grossi (Eds.), Behavior
analysis in education: Focus on measurably superior instruction (pp.283-320). Pacific Grove, CA:Brooks/Cole.
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1.
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5.
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The student is given a math computation worksheet of a specific problem
type, along with an answer key [Academic Opportunity to Respond].
The student consults his or her performance chart and notes previous
performance. The student is encouraged to try to ‘beat’ his or her most
recent score.
The student is given a pre-selected amount of time (e.g., 5 minutes) to
complete as many problems as possible. The student sets a timer and
works on the computation sheet until the timer rings. [Active Student
Responding]
The student checks his or her work, giving credit for each correct digit
(digit of correct value appearing in the correct place-position in the
answer). [Performance Feedback]
The student records the day’s score of TOTAL number of correct digits on
his or her personal performance chart.
The student receives praise or a reward if he or she exceeds the most
recently posted number of correct digits.
Application of ‘Learn Unit’ framework from : Heward, W.L. (1996). Three low-tech strategies for increasing the frequency of active student
response during group instruction. In R. Gardner, D. M.S ainato, J. O. Cooper, T. E. Heron, W. L. Heward, J. W. Eshleman,& T. A. Grossi
(Eds.), Behavior analysis in education: Focus on measurably superior instruction (pp.283-320). Pacific Grove, CA:Brooks/Cole.
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Worksheets created using Math Worksheet Generator. Available online at:
http://www.interventioncentral.org/htmdocs/tools/mathprobe/addsing.php
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The student is given sheet with correctly
completed math problems in left column
and index card.
For each problem, the student:
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studies the model
covers the model with index card
copies the problem from memory
solves the problem
uncovers the correctly completed model to
check answer
Source: Skinner, C.H., Turco, T.L., Beatty, K.L., & Rasavage, C. (1989). Cover, copy, and compare: A method for increasing
multiplication performance. School Psychology Review, 18, 412-420.
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“Recently, some researchers…have argued
that children can derive answers quickly and
with minimal cognitive effort by employing
calculation principles or “shortcuts,” such as
using a known number combination to
derive an answer (2 + 2 = 4, so 2 + 3 =5),
relations among operations (6 + 4 =10, so
10 −4 = 6) … and so forth. This approach to
instruction is consonant with
recommendations by the National Research
Council (2001). Instruction along these lines
may be much more productive than rote drill
without linkage to counting strategy use.” p.
301
Source: Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics
difficulties. Journal of Learning Disabilities, 38, 293-304.
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The student uses fingers as markers to find the
product of single-digit multiplication arithmetic
combinations with 9.
Fingers to the left of the lowered finger stands for
the ’10’s place value.
Fingers to the right stand for the ‘1’s place value.
99xx10
198765432
Source: Russell, D. (n.d.). Math facts to learn the facts. Retrieved November 9, 2007, from http://math.about.com/bltricks.htm
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“Students who learn with understanding have less to
learn because they see common patterns in
superficially different situations. If they
understand the general principle that the order in
which two numbers are multiplied doesn’t
matter—3 x 5 is the same as 5 x 3, for example—
they have about half as many ‘number facts’ to
learn.” p. 10
Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study Committee, J. Kilpatrick &
J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National
Academy Press.
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Students who struggle with math may find
computational ‘shortcuts’ to be motivating.
Teaching and modeling of shortcuts
provides students with strategies to make
computation less ‘cognitively demanding’.
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In this version of an ‘errorless learning’ approach, the
student is directed to complete math facts as quickly as
possible. If the student comes to a number problem that
he or she cannot solve, the student is encouraged to locate
the problem and its correct answer in the key at the top of
the page and write it in.
Such speed drills build computational fluency while
promoting students’ ability to visualize and to use a mental
number line.
TIP: Consider turning this activity into a ‘speed drill’. The
student is given a kitchen timer and instructed to set the
timer for a predetermined span of time (e.g., 2 minutes) for
each drill. The student completes as many problems as
possible before the timer rings. The student then graphs the
number of problems correctly computed each day on a
time-series graph, attempting to better his or her previous
score.
Source: Caron, T. A. (2007). Learning multiplication the easy way. The Clearing House, 80, 278-282
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Source: Caron, T. A. (2007). Learning multiplication the easy way. The Clearing House, 80, 278-282
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Here are two ideas to accomplish increased academic
responding on math tasks.
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Break longer assignments into shorter assignments with
performance feedback given after each shorter ‘chunk’ (e.g., break
a 20-minute math computation worksheet task into 3 sevenminute assignments). Breaking longer assignments into briefer
segments also allows the teacher to praise struggling students
more frequently for work completion and effort, providing an
additional ‘natural’ reinforcer.
Allow students to respond to easier practice items orally rather
than in written form to speed up the rate of correct responses.
Source: Skinner, C. H., Pappas, D. N., & Davis, K. A. (2005). Enhancing academic engagement: Providing opportunities for
responding and influencing students to choose to respond. Psychology in the Schools, 42, 389-403.
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The teacher first identifies the range of ‘challenging’ problem-types
(number problems appropriately matched to the student’s current
instructional level) that are to appear on the worksheet.
Then the teacher creates a series of ‘easy’ problems that the students
can complete very quickly (e.g., adding or subtracting two 1-digit
numbers). The teacher next prepares a series of student math
computation worksheets with ‘easy’ computation problems
interspersed at a fixed rate among the ‘challenging’ problems.
If the student is expected to complete the worksheet independently,
‘challenging’ and ‘easy’ problems should be interspersed at a 1:1
ratio (that is, every ‘challenging’ problem in the worksheet is
preceded and/or followed by an ‘easy’ problem).
If the student is to have the problems read aloud and then asked to
solve the problems mentally and write down only the answer, the
items should appear on the worksheet at a ratio of 3 ‘challenging’
problems for every ‘easy’ one (that is, every 3 ‘challenging’ problems
are preceded and/or followed by an ‘easy’ one).
Source: Hawkins, J., Skinner, C. H., & Oliver, R. (2005). The effects of task demands and additive interspersal ratios on fifthgrade students’ mathematics accuracy. School Psychology Review, 34, 543-555..
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Step 1: The tutor writes
down on a series of index
cards the math facts that the
student needs to learn. The
problems are written without
the answers.
4 x 5 =__
2 x 6 =__
5 x 5 =__
3 x 2 =__
3 x 8 =__
5 x 3 =__
6 x 5 =__
9 x 2 =__
3 x 6 =__
8 x 2 =__
4 x 7 =__
8 x 4 =__
9 x 7 =__
7 x 6 =__
3 x 5 =__
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Step 2: The tutor reviews
the ‘math fact’ cards with
the student. Any card
that the student can
answer within 2 seconds
is sorted into the
‘KNOWN’ pile. Any card
that the student cannot
answer within two
seconds—or answers
incorrectly—is sorted into
the ‘UNKNOWN’ pile.
‘KNOWN’ Facts
‘UNKNOWN’ Facts
4 x 5 =__
2 x 6 =__
3 x 8 =__
3 x 2 =__
5 x 3 =__
9 x 2 =__
3 x 6 =__
8 x 4 =__
5 x 5 =__
6 x 5 =__
4 x 7 =__
8 x 2 =__
9 x 7 =__
7 x 6 =__
3 x 5 =__
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Step 3: Next the tutor takes a math fact from the ‘known’ pile and pairs it with
the unknown problem. When shown each of the two problems, the student is
asked to read off the problem and answer it.
3 x 8 =__
4 x 5 =__
2 x 6 =__
3 x 2 =__
3 x 6 =__
5 x 3 =__
8 x 4 =__
6 x 5 =__
4 x 7 =__
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Instructional Strategy (Hattie Effect Size)
1. Visual and graphic descriptions of
problems
Effect Size for Special Education Effect Size for LowStudents
Achieving Students
0.50 (moderate)
N/A
2. Systematic and explicit instruction (0.59)
1.19 (large)
0.58 (moderate to
large)
3. Student think-alouds (0.69)
0.98 (large)
N/A
0.42 (moderate)
0.62 (large)
5. Formative assessment data provided to
teachers (0.90)
0.32 (small to moderate)
0.51 (moderate)
6. Formative assessment data provided
directly to students(0.90)
0.33 (small to moderate)
0.57(moderate)
4. Use of structured peer-assisted learning
activities involving heterogeneous ability
groupings (0.72)
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Method of delivery (‘Who or what delivers the treatment?’)
Examples include teachers, paraprofessionals, parents,
volunteers, computers.
Treatment component (‘What makes the intervention
effective?’)
Examples include activation of prior knowledge to help the
student to make meaningful connections between ‘known’ and
new material; guide practice (e.g., Paired Reading) to increase
reading fluency; periodic review of material to aid student
retention. As an example of a research-based commercial
program, Read Naturally ‘combines teacher modeling,
repeated reading and progress monitoring to remediate
fluency problems’.
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Interventions. An academic intervention is a
strategy used to teach a new skill, build
fluency in a skill, or encourage a child to
apply an existing skill to new situations or
settings.
An intervention is said to be research-based
when it has been demonstrated to be
effective in one or more articles published in
peer–reviewed scientific journals.
Interventions might be based on commercial
programs such as Read Naturally. The
school may also develop and implement an
intervention that is based on guidelines
provided in research articles—such as Paired
Reading (Topping, 1987).
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Modifications. A modification changes the
expectations of what a student is expected to
know or do—typically by lowering the academic
expectations against which the student is to be
evaluated.
Examples of modifications are reducing the
number of multiple-choice items in a test from
five to four or shortening a spelling list. Under
RTI, modifications are generally not included in
a student’s intervention plan, because the
working assumption is that the student can be
successful in the curriculum with appropriate
interventions and accommodations alone.
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Accommodations. An accommodation is
intended to help the student to fully access the
general-education curriculum without changing
the instructional content. An accommodation
for students who are slow readers, for example,
may include having them supplement their
silent reading of a novel by listening to the
book on tape.
An accommodation is intended to remove
barriers to learning while still expecting that
students will master the same instructional
content as their typical peers. Informal
accommodations may be used at the classroom
level or be incorporated into a more intensive,
individualized intervention plan.
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 PALs
Strategy
 Rocket Math
 Origo Education Program
 Websites and online
resources

On your candy bar personality/feedback
sheet give feedback about session and
needs or wants for future professional
development.
Thanks for coming today! 
Laura Colligan
Academic Consultant, Ingham ISD
[email protected]
517.244.1258
Candy Bar Personality Test
What’s your favorite bar say about
you?
Milk Chocolate
Krackel
Mr. Goodbar
Special Dark
You're an all American who loves baseball, Mom & apple pie. You're a cheerleader
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also kind, thoughtful, and always remember everyone's birthday. You are nurturing,
dependable, loyal, and help others to "shine". Others often turn to you for help.
You're creative, optimistic, always see the cup as half full. You're messy (messy desk
or classroom) but organized (eventually find a missing item or believe you will). You
like to be a hands-on person. You're a little off-beat, ditzy, funny, friendly and
outgoing person who is always will to help. You like the surprising things in life, the
"krackel". You like situations that allow flexibility, change and growth.
You're analytical and logical. You gather data first before giving an opinion, play the
devil's advocate at meetings, tend to see all the possibilities and drive people crazy
by sharing all the "what ifs". You hate deadlines and put off starting things; you're a
procrastinator. You like to be the expert but in your own time frame. You can
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