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Title: Assessment of Mathematics Unit 8


1
Assessment of MathematicsUnit 8
  • NC SIPII MATH

2
What teachers do What teachers think
  • True! What we think will effect what we learn.
  • Learning occurs instantaneously. T F
  • You can only practice what you know. T F
  • Proficiency builds gradually. T F

3
How do we build motivation?
  • If we give students easy tasks, are we really
    building success or self confidence?
  • Do learning styles exist?
  • LS Inventory assessment http//www.engr.ncsu.edu/
    learningstyles/ilsweb.html
  • Homogeneous Grouping- Is it good or bad? / Full
    Inclusion?
  • Student overachievement or low expectations

4
Failure - Who is to Blame
  • Failure to learn may stem from
  • Poor past experiences
  • Lack of prior knowledge
  • Lack of good instruction
  • What is instruction?
  • Good instruction is informed by assessment.
  • What about the things we cant control?
  • IQ
  • Learning style
  • Family home

5
If the student has not learned the teacher has
not taught!
  • Missing or erroneous prior knowledge may be your
    biggest barrier
  • Task difficulty is directly related to prior
    knowledge.
  • It is not the task it is the skill!

6
Lets Learn about Hacky Sack
  • Hacky Sack is the trademarked name of a type of
    footbag. The name "hacky sack" came from the
    inventors of the footbag, John Stalberger and
    Mike Marshall. Marshall suffered a fatal heart
    attack in 1975, however Stalberger continued the
    business. At a later date, Stalberger sold the
    title to Wham-O.

7
Moves
  • Inside Kick
  • Outside Kick
  • Toe Kick
  • Knee Kick

8
RULES
  • 1. No Hands (except when serving), No Arms -
    Shoulders are technically not allowed and are
    widely accepted among the average hack circle.
  • 2. Always serve the bag to someone else, unless
    of course you are alone. Footbag is traditionally
    a game of courtesy, hence "The Courtesy Toss" a
    light lob usually toward the receivers knee.
  • 3. Don't bogart that bag. - Don't always hog it
    'till you drop it because that is bad for
    everybody else. Being able to pass well is
    important to almost all footbag games.
  • 4. Don't say "sorry". Everyone makes mistakes,
    especially when learning, so sorries are
    unnecessary.
  • 5. Try not to give knee passes Passes from the
    knee tend to go straight to the ground.

9
Learning Process
  • Learning orientation vs. performance orientation
    (completing the task)
  • How can you show which is important in your
    classroom?

10
Fundamentals of Assessment
11
How is all of this connected to the RTI Model?
12
Tier IV
IEP
Consideration


Tier III

Student


Study
Team
Tier II

Consultation
With Other

REQUIRED TO MEET THE
AMOUNT OF RESOURCES
STUDENTS NEEDS
Resources
Tier I
Consultation
  • Intensive Interventions 1-7
  • Strategic Interventions 5-15
  • Core Curriculum 80-90


Between
Teachers

-
Parents


INTENSITY OF NEEDS
Needs
-
circles
-
pub

13
What happens in each tier?
  • Tier I- at least 3 times per year for
    benchmarking.
  • Tier I / Tier II (small group) start to move into
    progress monitoring to determine how your
    students are responding to your intervention.
  • Tier III (individualized) Progress monitoring
    continues.

14
Modified PSM
15
80-15-5 Rule
  • Apply the 80-15-5 rule to determine if the
    focus of the intervention should be on the core
    curriculum (and instruction), on subgroups of
    underperforming learners, or individual students
    (T. Christ, 2008)
  • Source Christ, T. (2008). Best practices in a
    problem analysis. In A. Thomas J. Grimes
    (Eds.), Best practices in school psychology V
    (pp. 159-176).

16
Application of the 80-15-5 Rule
  • If less than 80 of students are successfully
    meeting academic or behavior goals, the
    intervention focus is on the core curriculum,
    INSTRUCTION, and general student population.
  • If no more than 15 of students are not
    successful in meeting academic or behavior goals
    or have met the goals, the intervention is on
    small group treatments or interventions.
  • If no more than 5 of students are not successful
    in meeting academic or behavioral goals, the
    intervention focus is on the individual student.
  • Source Christ, T. (2008). Best practices in a
    problem analysis. In A. Thomas J. Grimes
    (Eds.), Best practices in school psychology V
    (pp. 159-176).

17
Tier I
18
Tier I
  • Universal Screening (both high and low achievers
    may need interventions)
  • Universal Screening is a general outcome measure
    used to identify underperforming students and to
    determine the rate of increase for students.
  • A Universal Screening will not identify why
    students are underperforming, rather it will
    identify which student is below / above the
    expected performance criteria for a given grade
    level in reading, writing, spelling, and math.

19
Primary PreventionFuchs Fuchs
  • Tier 1 CBM (Curriculum Based Measurement) is a
    tool used to screen all students in the class to
    identify those potentially at risk for poor
    outcomes at the end of the year.

20
Number Knowledge Test
  • http//clarku.edu/numberworlds

21
Name__________________________ Score _____
Date________Preliminary Lets
see if you can count from 1 to 10. Go
ahead.Number Knowledge Test Level 0 (4-year-old
level)____ 1. Can you count these Counters and
tell me how many there are? (Place 3 Counters
in a row in front of the child.)____ 2a. (Show
stacks of counters, 5 vs. 2, same color)
Which pile has more?____ 2b. (Show stacks of
counters, 3 vs. 7, same color) Which pile has
more?____ 3a. This time, Im going to ask you
which pile has less. (Show stacks of counters,
2 vs. 6, same color.) Which pile has
less?____ 3b. (Show stacks of Counters, 8 vs.
3, same color.) Which pile has less?____
4. Im going to show you some Counters. (Show
a line of 3 blue and 4 white Counters in a row,
as follows BWBWBWW.) Count just the white
Counters and tell me how many there are.____
5. (Pick up all the Counters from the previous
question. Show a mixed arraynot a rowof 7
white and 8 blue Counters) Here are some more
Counters. Count just the blue counters and
tell me how many there are. Go to
Level 1 if 3 or more correctIn order to receive
credit an item that has an a and b, both a and b
must be correct to be counted in the score.
22
Use data to develop a plan
Grade Computation Cut-Off for one minute
Grade 1 lt 5 digits
Grade 2 lt 10 digits
Grade 3 lt 10 digits
Grade 4 lt 10 digits
Grade 5 lt 15 digits
Grade 6 lt 15 digits
Other sources of data -Scores on the
NKT -One-on-one interviews -Analysis of student
work -Skill-based measures
Fuchs Fuchs Note These figures may change
pending additional RTI research.
23
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24
Tier I
25
Concepts and Applications
  • Student copy of a Concepts and Applications test
  • This sample is from a second-grade test.
  • The actual Concepts and Applications test is 3
    pages long.

26
Concepts and Applications
Grade Time limit
2 8 minutes
3 6 minutes
4 6 minutes
5 7 minutes
6 7 minutes
  • Length of test varies by grade.

27
Tier 1
28
Tier I
  • What does instruction in this classroom look
    like?
  • How is this instruction different from what we
    see in most classrooms?
  • Think about the Medical Model.

29
Tier I
30
(No Transcript)
31
Data to Consider
  • Continue Fall, Winter, and Spring benchmarks
  • Assess students who did not meet the first
    benchmark more frequently
  • Continue group progress monitoring
  • Analyze additional student work samples

32
Tier II
33
  • Student interviews
  • Conceptual understanding
  • Student work
  • Skill-based measurements

34
www.interventioncentral.org
  • http//www.interventioncentral.org/index.php/cbm-w
    arehousemathComputation
  • Here we can find
  • GOMs such as mixed computation probes
  • SBMs- such as specific calculation issues
  • MMs such as quantity discrimination and early
    numeracy fluency
  • NORMS for CDs
  • Instructional support ideas for different tiers


35
Performance Task
  1. Assess the mathematical errors in the student
    work.
  2. Design a plan for instruction for your student.

36
Connecting Forms of a Value
What fractional part of the figures are shaded?
1.
2.
Circle ¾ of the group of boxes.
3.
Put a ring around 3/8 of the group of triangles.
4.
Draw an arrow where the ½ should be.
0
1
2
5.
Put a circle on the number line where 2/3 should
be.
2
3
4
0
1
37
Addition of Fractions

1.
4.
2.
5.
3.
38
Tier II
39
Plan should consider
  • CONTINUING EFFECTIVE TIER I INSTRUCTION
  • TARGETED INSTRUCTION
  • INTERVENTION GROUPS (17)
  • SKILL-BASED INSTRUCTION

40
IES RecommendationsTier II and III
  • Instruction during the intervention should be
    explicit and systematic.
  • Provide
  • Models of proficient problem solving
  • Verbalization of thought processes
  • Guided practice
  • Corrective feedback
  • Frequent cumulative review
  • Interventions should include instruction on how
    to solve word problems that is based on common
    underlying structures.

41
IES RecommendationsTier II and III
  • Intervention materials should include
    opportunities for students to work with visual
    representations of mathematical ideas and
    interventions should be proficient in the use of
    visual representation of mathematical ideas.
  • Interventions at all grade levels devote about 10
    minutes in each session to building fluent
    retrieval of basic arithmetic facts.

42
IES RecommendationsTier II and III
  • Instructional materials for students receiving
    interventions should focus intensely on in-depth
    treatment of whole numbers in grades K-5 and on
    rational numbers in grades 4 through 8. These
    materials should be selected by a committee.
  • Monitor the progress of students receiving
    supplemental instruction and other students who
    are at risk.
  • Include motivational strategies in Tier II and
    Tier III interventions.

43
Tier II
44
Structures of Addition
  • Join and Part-Part Whole
  • There is something, and you get more of it?
  • There are two kinds, how many all together?
  • Start Unknown
  • Some are given away, some are left how many were
    there to start?
  • Compare--total unknown
  • - I know one amount and I have some amount more
    than that. How many do I have?

How many altogether?
?
What did I start with?
?
Taken Left
How many do I have?
?
Addition types Adapted from Carpenter, Fennema,
Franke, Levi and Empson, 1999, p. 12 in Adding it
Up, NRC 2001.
45
Structures of Subtraction
  • Ask yourself if a problem is a
    subtraction problem
  • Does it fit one of these three types?
  • The Classic Take away
  • (how many left?)
  • Comparison
  • (difference between? who has more?)
  • Deficit/Missing amount
  • (whats missing?)

Youve got some amount and take away from it
Whats left?
?
You compare to see Who has more or less?
?
?
You need some more to get where you want to
be What is the missing amount?
46
Multiplication Structures
  • Repeated Addition
  • Array/row-column
  • Counting Principle

47
Division Structures
  • Measurement/Repeated Subtraction
  • How many 2s can I get out of 10?
  • Partitive/Unitizing/Fair Shares
  • How many would one person get? or What would
    that mean in relation to 1?
  • Product/Factor
  • If I have an area of 10 and one side is 2, how
    long is the other side?

If I have 10 cups of beans and I give out 2 cup
portions, how many servings will that provide?
2
10
2
2
?
2
?
2
10
?
If 2 people find 10 how much will each person
get ?
10
2
?
48
Virtual Manipulatives
  • National Library of Virtual Manipulatives
  • http//nlvm.usu.edu/en/nav/vlibrary.html

49
Tier II
50
MULTIPLICATION 4-digit number times 2-digit
number with regrouping
  • Item 117 CD/17 CD Total
  • 7083x   57
  • 49581
  • 354150
  • 403,731

51

                                                
                                                 
52
On-line resources to help with data
  • http//www.thatquiz.org/
  • http//www.aimsweb.com/measures-2/
  • http//www.easycbm.com/
  • http//enumeracy.com/

53
Tier III
54
www.interventioncentral.org
  • http//ic.gorgeswebsites.com/index.php?optioncom_
    contentviewarticleid195
  • Here we can find
  • GOMs such as mixed computation probes
  • SBMs- such as specific calculation issues
  • MMs such as quantity discrimination and early
    numeracy fluency


55
  • Early numeracy probes
  • http//progressmonitoring.org/probes/earlynumeracy
    .html

56
Tier III
57
The Tier III plan should be
  • CONTINUING EFFECTIVE TIER II INSTRUCTION
  • MORE INTENSIVE
  • MORE INDIVIDUALIZED
  • Take into consideration the IES Recommendations

58
Implementation should be
  • MORE EXPLICIT
  • MULTI-SENSORY
  • MORE SYSTEMATIC
  • MORE DIRECT
  • CUMULATIVE

59
Tier III
60
On-line resources to help with
  • http//www.interventioncentral.org/index.php/acade
    mic-resources
  • http//enumeracy.com/

61
Tier III
62
  • At Tier III
  • PROGRESS MONITORING SHOULD TAKE PLACE 2-3 Times
    PER WEEK
  • WE SHOULD ONLY USE CBMs THAT ARE VERYSENSITIVE TO
    CHANGE AT THIS TIER

63
Case Study James-2nd Grade
  • James can identify the cardinal number of a set
    and can relate the numeral to that number.
  • He can skip count in series of 10, 2, 5, 9, 4
    without the use of manipulatives with
    automaticity.
  • He has developed automaticity with basic addition
    facts 1-6.
  • He understands the concepts of one more and one
    less.
  • He continues to use fingers to count up and count
    down for addition and subtraction. His fluency is
    weak.

64
(No Transcript)
65
James Data
  • Baseline 13 digits per two minutes (2nd point)
  • Monitored for CDs for approximately 20 weeks
  • Last 3 data points ranged between 14 and 16- CDs
    (target should be about 18 to 20)

66
Baseline Data
  • Baseline
  • PM Day Date CDs per 2 min.
  • 1 5 January 2009 12
  • 2 5 January 2009 13
  • 3 5 January 2009 15

67
Tier I
  • Phase Change Tier I
  • PM Day Date CDs per 2 min.
  • 4 15 January 2009 12
  • 5 28 January 2009 12
  • 6 7 February 2009 13
  • 7 21 February 2009 13
  • 8 30 February 2009 14

68
Tier II
  • Phase Change Tier II
  • PM Day Date CDs per 2 min.
  • 9 8 March 2009 13
  • 10 16 March 2009 14
  • 11 20 March 2009 13
  • 12 25 March 2009 14
  • 13 29 March 2009 14
  • 14 5 April 2009 12
  • 15 7 April 2009 13
  • 16 12 April 2009 13
  • 17 17 April 2009 13

69
Tier III
  • Phase Change Tier III
  • PM Day Date CDs per 2 min.
  • 18 23 April 2009 14
  • 19 28 April 2009 15
  • 20 1 May 2009 16

70
(No Transcript)
71
Learning Task-Case Study
  • Using the information you already knew and what
    you learned today, work in groups to formulate a
    plan for instruction for James.
  • How will you evaluate his progress?

72
Possible interventions
  • Math Review Promote Mastery of Math Facts
    through Incremental Rehearsal (interventioncentral
    .org)
  • Subitizing
  • Number Lines/Hundreds Boards
  • Base Ten Instruction
  • The Structures of Addition and Subtraction

73
Tier IV--Intensive
  • Referral to Special Education
  • If the student has not shown progress even with
    targeted SBI instructional plans.
  • When do we make the referral?
  • TIERS I -III PROVIDE DATA SET FOR TIER IV
    EVALUATION

74
Be Careful----
  • We do not have the same correlations as we have
    with reading probes.
  • We need to be cautious about applying what we
    know about Reading to the world of Mathematics.
  • In Math, we must think about what we are
    measuring. We are not in the same place in Math
    as we are in Reading.

75
Web Resources for Assessment and Intervention
  • Yearly Progress Pro
  • http//www.mhdigitallearning.com/
  • AIMS Web
  • http//www.edformation.com/
  • Intervention Central
  • http//www.interventioncentral.org/
  • Number Worlds Home
  • http//clarku.edu/numberworlds/
  • John Woodward, Transitional Math
  • http//www2.ups.edu/faculty/woodward/home.htm
  • Progress Monitoring
  • http//www.progressmonitoring.net/
  • National Center on Student Progress Monitoring
  • http//www.studentprogress.org/chart/chart.asp

76
References
  • Assisting Students Struggling with Mathematics
    Response to Intervention (Rti) for Elementary
    and Middle Schools IES National Center for
    Education Evaluation and Regional Assistance,
    NCEE 2009-4060, U.S. Department of Education
  • Ball, Deborah (1992) Magical Hopes
    Manipulatives and the Reform of Math Education,
    American Educator, Summer 1992
  • Balls Website http//wwwpersonal.umich.edu/dbal
    l/
  • Fuchs, Lynn The Prevention and Identification of
    Math Disability Using RTI, September 18, 2008
    Presentation
  • Gersten, Russell, Clark, B, Jordan, N, Center on
    Instruction, Screening
  • for Mathematics Difficulties in K-3 Students
    2007.
  • Gersten, Russell, Jordan, N, Flojo, J., Early
    Identification and Interventions for Students
    with Mathematical Difficulties, Journal of
    Learning Disabilities, Volume 38, Number 4, July
    August 2005
  • Gickling, Edward, PhD, Instructional Assessment
    in Mathematics, March 2003, Presentation at
    Exceptional Childrens Conference
  • Griffin, Sharon. (2003). Mathematical Cognition,
    Royer, ed. Greenwich, CT. Infoage Publishing.
  • Ma, Liping (1999) Knowing and Teaching Elementary
    Mathematics. Edison, NJ, Lawrence Erlbaum
    Associates.
  • Mayer, Richard (2003). Mathematical Cognition,
    Royer, Ed.. Greenwich, CT. Infoage Publishing.
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