Accessing Algebraic Curriculum
for Students with Learning
Difficulties
Brad Witzel, Ph.D.
Professor, Program Director of Intervention Education
Winthrop University
[email protected]
coe.winthrop.edu/witzelb
© Witzel, 2016
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In a game, exactly six inverted cups stand side by side in a straight line, and each has
exactly one ball hidden under it. The cups are numbered consecutively 1 through 6.
Each of the balls is painted a single solid color. The colors of the balls are green,
magenta, orange, purple, red and yellow. The balls have been hidden under the
cups in a manner that conforms to the following conditions:
The purple ball must be hidden under a lower-numbered cup than the orange ball.
The red ball must be hidden under a cup immediately adjacent to the cup under
which the magenta ball is hidden.
The green ball must be hidden under cup 5.
1.Which of the following could be the colors of the balls under the cups, in order from 1
through 6?
(A) Green, yellow, magenta, red, purple, orange
(B) Magenta, green, purple, red, orange ,yellow
(C) Magenta, red, purple, yellow, green, orange
(D) Orange, yellow, red, magenta, green, purple
(E) Red, purple, magenta, yellow, green, orange
2.If the magenta ball is under cup 4, the red ball must be under cup
(A) 1
(B) 2
(C) 3
(D) 5
(E) ©6 Witzel, 2016
2
A developer is planning a housing complex and uses exactly seven styles of
houses –Q, R, S, T, W, X, and Z. The complex will contain several blocks and
will put at least three styles on each block. The developer’s rules are as
follows:
Any block that has style Z on it must also have style W on it.
Any block adjacent to one that has on it both style S and X can have on it T or Z
No block adjacent to one that has on it both style R and Z can have on it either
Tor W
No block can have on it both style S and Q
Questions: 1) Which of the following can be the complete section of house styles
on a block? QRS, QSX, RTZ, SWZ, or TXZ?
2) Which of the following house styles must be on a block that is adjacent to one
that has on it only styles S, T, W, X, and Z?
Q, R, S, W, or Z?
© Witzel, 2016
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1
Sometimes you don’t know
what you don’t know
• http://www.youtube.com/watch?v=yZUVqyrctI4
Nationally, what do algebra teachers say? (NMP,
2008)
743 algebra teachers in 310 schools nationally responded to a
survey on algebra instruction and student learning in 2007.
Findings:
• The teachers generally rated their students’ background
preparation for Algebra I as weak. The three skill areas in which
teachers reported their students have the poorest preparation are
rational numbers, word problems, and study habits.
• Regarding the best means of preparing students, 578 suggested a
greater focus on mastery of elementary mathematical concepts
and skills.
• Teachers were less excited about how current textbook
approaches meet the needs of diverse student populations.
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More findings from the NSAT
• Use of calculators was quite mixed with 33% saying they
never use them and 31% use them frequently (more than
once a week).
• 60% use physical tools less than once a week and only 9%
use them frequently.
• The greatest challenge to teachers was #1 – “working with
unmotivated students.” This was chosen by 58% of the
middle school teachers and 65% of the high school
teachers. The next most frequent response was “making
mathematics accessible and comprehensible to all my
students,” selected by 14% of the middle school teachers
and 9% of the high school teachers.
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2
Teachers Have Power!
• “Differences in teachers account for 12% to 14% of total variability in
students’ mathematics achievement gains” (NMP, 2008, p. 35).
• Why?
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Today
• Algebra is considered a gateway to formal mathematics, quantitative
work skills, and graduation. However, the abstractness of algebra
makes learning difficult, especially for students who have not
mastered elementary level mathematics. Too often, students who
struggle early in math tend to struggle throughout their lives. In this
session, Dr. Witzel will present a way to scaffold stepwise algebraic
expressions and equations for students who struggle in math.
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Students Are Failing Algebra at Alarming
Rates. Why?
Not being identified early and receiving additional help
Lack of practice
Insufficient Prior Knowledge
Attention
___others__________________?
• http://education.seattlepi.com/problems-lead-students-fail-math2216.html
© Witzel, 2016
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3
One district’s plan to turn around Secondary
Math Achievement
More Review time (Students)
Poor study habits (Teachers)
More elementary math coaches (Administration)
Better collaboration among gened and sped (Administration)
Increased number of and improved professional development
(Administration)
__others_________________________________?
• https://www.washingtonpost.com/local/education/three-out-of-four-highschoolers-failed-algebra-1-final-exams-in-mddistrict/2015/07/22/d4ab97d0-2b0f-11e5-bd33395c05608059_story.html#comments
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What has algebra become?
• From a course to a goal
• Algebra vs. Arithmetic
• Bridge the Gap
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
Math Difficulties (Witzel)
Big Ideas (Allsopp, Ingen, Simsek, & Haley)
Core Algebra Instruction (Hughes)
Progressions (Powell & Witzel)
Computation (Woodward & Stroh)
Fractions (Witzel)
Problem Solving (Bouck & Bouck)
Progress Monitoring (Lembke, Strickland & Powell)
MTSS (Little & Dieker)
Students with Developmental Disabilities (Root,
Browder & Jimenez)
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State of Research: Why Algebra?
Moore and Shulock (2010)
• Community College dropout rates are very high, near 70%
• A student’s completion of a transfer level college math course within two years of enrollment
was more highly correlated to completion than English.
• However, over one-half of community college students reported that Algebra was the highest
math course they completed in High School and the average math placement by community
colleges was two levels below a college level course.
• When entering college, students are, on average, two math courses away from even receiving
credit. In conclusion, students must succeed early in advanced level mathematics in high
school
Adelman (2006) concludes,
“It’s not merely getting beyond Algebra 2 in high school any more: The world demands
advanced quantitative literacy, and no matter what a student’s postsecondary field of study—
from occupationally-oriented programs through traditional liberal arts— more than a
ceremonial visit to college-level mathematics is called for” (p.108).
Witzel, 2015
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Agenda
• Fully Worked Problems
• Task Analysis
• Differentiation
• HELPS: Field Dependent Learning
• CRA
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Panelists
• Jon R. Star (Chair)
Harvard University
• Anne Foegen
Iowa State University
• Matthew R. Larson
Lincoln Public Schools
• William G. McCallum
University of Arizona
• Jane Porath
Traverse City Area Public Schools
• Rose Mary Zbiek
Pennsylvania State University
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Three Recommendations (Star, et al., 2015)
1. Use solved problems to engage
students in analyzing algebraic
reasoning and strategies
2. Teach students to utilize the structure
of algebraic representations
3. Teach students to intentionally choose
from alternative algebraic strategies
when solving problems
© Witzel, 2016
Minimal ES
Minimal ES
Moderate ES
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5
Recommendation 1. (Star, et al., 2015)
Use solved problems to engage students in
analyzing algebraic reasoning and strategies.
a. Have students discuss solved problem structures
and solutions to make connections among
strategies and reasoning.
b. Select solved problems that reflect the lesson’s
instructional aim, including problems that illustrate
common errors.
c. Use whole-class discussions, small-group work, and
independent practice activities to introduce,
elaborate on, and practice working with solved
problems.
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Analyze Solved Problems (Star, et al., 2015)
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Video Modeling
https://www.khanacademy.org/math/probability/regression/regressio
n-correlation/v/regression-line-example
What is effective about this modeling?
Next Steps?
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6
Recommendation 3.
Teach students to intentionally choose from
alternative algebraic strategies when
solving problems.
a. Teach students to recognize and generate strategies
for solving problems.
b. Encourage students to articulate the reasoning
behind their choice of strategy and the
mathematical validity of their strategy when solving
problems.
c. Have students evaluate and compare different
strategies for solving problems.
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Recognize and Generate Strategies (Star, et al., 2015)
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Label Steps when Comparing Strategies
(Star, et al., 2015)
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Application
• Name effective ideas for modeling math problems.
• How many strategies should (or can) students learn at a time?
• When should you choose an alternative strategy?
Extension
• What information do you need in order to choose your initial or base
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Agenda
• Fully Worked Problems
• Task Analysis
• Differentiation
• HELPS: Field Dependent Learning
• CRA
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Task Analysis
• “Task analysis for instructional design
is a process of analyzing and
articulating the kind of learning that
you expect the learners to know how
to perform" (Jonassen, Tessmer, &
Hannum, 1999, p.3)
• Task analysis builds an in-depth
understanding of what is to be taught
(Jonassen, et al., 1999)
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Five Functions to Task Analysis
• Classifying tasks according to learning outcomes
• Inventorying tasks – identifying tasks or generating a list of tasks
• Selecting tasks – prioritizing tasks and choosing those that are more
feasible and appropriate if there is an abundance of tasks to train.
• Decomposing tasks – identifying and describing the components of
the tasks, goals, or objectives.
• Sequencing tasks and sub-tasks – defining the sequence in which
instruction should occur that will best facilitate learning.
http://cehdclass.gmu.edu/ndabbagh/Resources/IDKB/taskanalysis2.htm
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Task Analysis
Breaking down a complex skill into manageable steps
1. Identify the target skill.
2. Identify the prerequisite skills of the learner and the materials needed to teach
the task.
3. Break the skill into small steps.
4. Confirm that the task has been completely analyzed by having someone follow
the steps verbatim. Adjust steps as necessary.
5. Determine how the skill will be taught (total, forward, or backward chaining).
6. Implement the task analysis and monitor student progress.
(Based on the work of Franzone, 2009)
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Linear Equations Example
2
• 3(5x – 6) = 3(6 – 3y), write the equation in standard form
• 15x – 18 = 4 – 2y
• 2y + 15x – 22
Rewriting equations require what skills?
• Whole number computation
• Rational number computation
• Distributive property
• Combining like terms
• Coefficient
• Equal sign
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Tasks Analysis
Linear Equations Example
Skill here
Steps to solve
Whole Number Computation
Rational Number Computation
Distributive Property
Combining Like Terms
Coefficient
Equal Sign
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Quadratics Task Analysis Example
Question. Find a quadratic equation whose graph has x-intercepts 5 and -3 and vertex
has a y-coordinate -8. Sketch the graph to check your answer (PA_A2.1.3.1.1 )
https://www.ixl.com/standards/pennsylvania/math/high-school?documentId=2007000304&subsetId=2007000590
Quadratics require what skills?
• Formulas
• Coordinators
• Distributive property
• Multiplication
• Equations when given x
• Like terms
• Substitution
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Task Analysis Example
• List the information from the task analysis in sequential order
• Consider cues for each step to justify the step
Skill here
Steps to solve
Formulas
Coordinators
Distributive Property
Multiplication
Equations when given x
Combine Like Terms
Substitution
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10
Grading based on Task Analysis
Student
Formulas
Ordinate
Plane
Distributi
ve
Equations
Like
Terms
Multiplica
tion
Substituti
on
Answers
Ann
Blake
Carla
Dwyer
Eduard
Frank
Gwen
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Algebra Standards (PaTTAN, 2016)
Identify the student expectation in either of these standards:
1) Extend the knowledge of arithmetic operations and apply to
polynomials.
2) Interpret the structure of expressions to represent a quantity in
terms of its context.
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Your Turn
• Task Analyze a math lesson skill / anchor
• Write the final skill that is needed on the pink line
Skill here
Steps to solve
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Agenda
• Fully Worked Problems
• Task Analysis
• Differentiation
• HELPS: Field Dependent Learning
• CRA
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Intro Differentiation (Blackburn & Witzel, 2013)
• A teacher’s response to the diverse learning needs of students
(Tomlinson, 1999).
• Differentiation types
• Content (what math is to be learned)
• Process (how math is taught)
• Product (how learning is evaluated)
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Differentiation Barriers
(Adapted from Bender, 2008)
Class Structure
• Group-based class design
Assessment
• Defining groups of students who need differentiation
Management
• Behavior management of student groups
Fairness
• Ignoring students not in a differentiated group
• Overcompensation for difficulties
Time
• Time for planning differentiation
• Time taken to wait for students to catch up
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Help this teacher
• Teaching factoring polynomials for the first time
8x4 – 4x3 + 10x2 = 2x2 (4x2 – 2x + 5)
• 20 students in the class
•
•
•
•
1 student cant yet multiply
3 have a hard time paying attention
5 have difficulty with English
2 show they already know how to do much of the work
• What adjustments might you make?
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Expectations of Differentiation in Tiered
Interventions
• Group level interventions
• Student level accommodations
• If student-level modifications occur, then a plan is in place to
incrementally move away from those modifications when possible
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Extending the Quadratic Task Analysis Example
Question. Find a quadratic equation whose graph has x-intercepts 5 and -3 and vertex
has a y-coordinate -8. Sketch the graph to check your answer (PA_A2.1.3.1.1 )
https://www.ixl.com/standards/pennsylvania/math/high-school?documentId=2007000304&subsetId=2007000590
Quadratics require what skills?
• Formulas
• Coordinators
• Distributive property
• Multiplication
• Equations when given x
• Like terms
• Substitution
© Witzel, 2016
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Next Steps to Interventions
• Based on your task analysis, decide what is needed to help each student
• Use a pretest of similar questions to verify the needs
• Set-up a Formative Assessment Chart
Objective: Students will use three coordinates to write a quadratic equation
Skills necessary to solve the problem
Missing skills with a student
© Witzel, 2016
Accommodations or problem modifications
42
14
Student Needs Chart:
Match skills with student needs and accommodations
Objective: Students will use three coordinates to write a quadratic equation
Skills necessary to solve the problem
Missing skills with a student
Accommodations or problem
modifications
Know the quadratic formula and vertex
formula
8 students haven’t memorized the
vertex or quadratic formula
Provide those who need with the vertex
formula and show how it matches the
standard formula
Coordinates on an ordinate plane
1 student forgets positive and
negative direction
Provide positive and negative signs to cue
direction
Distributive property
3 students forget steps when using the Show a diagram with arrows to remind
distributive property
Writing equations when given the
answer (x=-7)
No students show this as a difficulty
n/a
Combining like terms
1 student shows an issue with
exponents
n/a with this problem
Multiplication
4 students are not consistently
accurate
Use numbers of calculations that have
been memorized
Substitution
3 students show difficulty with
substitution
Color-code example
© Witzel, 2016
Grading
Student
Formulas
Ann
Teacher
Provided
Ordinate
Plane
Distributi
ve
Equations
Like
Terms
43
Multiplica
tion
Substituti
on
Answers
Multiplica
tion sheet
Blake
Teacher
Cued
Carla
Dwyer
Teacher
Provided
Eduard
Teacher
Provided
Teacher
Cued
Color
coded
Frank
Gwen
Teacher
Cued
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Example: Class data – per student (PaTTAN, 2016)
Content: Calculation of expressions
Intervention group of 6
•
•
•
•
•
2 students lack multiplication/division computation accuracy (Malcom, Nate)
3 students lack consistency with integers (Nate, Penny, Quinn)
3 students struggle with exponents (Malcom, Quinn, Rachel)
6 students struggle with distributive property (all)
1 student has attention concerns (Quinn)
© Witzel, 2016
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Steps: Calculation of expression with
integers
-3x(4x-2) = ?
a)
b)
c)
d)
e)
f)
g)
Distribute calculations
(-3)(4x) and (-3x)(-2)
Fill in positive signs where needed (-3x)(+4x) + (-3x)(-2)
Calculate signs (-)(3x)(4x) + (+)(3x)(2)
Calculate numbers(-)(12)(x)(x) + (+)(6x)
Calculate symbols (-)(12)(x2) + (+)(6x)
Combine like terms – 12 x2 + 6x
Answer – 12 x2 + 6x
© Witzel, 2016
Student Needs Chart
46
Dependent
Transitional
Independent but prior cueing
Objective: Students will accurately calculate integers
Skills necessary to solve
the problem
Distributive Property
Missing skills with a student
Computation
Malcom and Nate lack the
multiplication need for accurate
calculation about the distribution
Nate, Penny, and Quinn are not
consistent with the integer
calculations
Malcom, Quinn, and Rachel
struggle with exponents
Integers (negatives)
Exponents
Combine Like Terms
Accommodations or problem
modifications
All 6 students
What
accommodations
would you use?
None have shown difficulty with
terms
© Witzel, 2016
Student Needs Chart
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Dependent
Transitional
Independent but prior cueing
Objective: Students will accurately calculate integers
Skills necessary to solve
the problem
Distributive Property
Missing skills with a student
Accommodations or problem
modifications
All 6 students
Computation
Malcom and Nate lack the
multiplication need for accurate
calculation about the distribution
Nate, Penny, and Quinn are not
consistent with the integer
calculations
Teach acronym of steps; draw lines
for student with attention difficulty
Calculator
Integers (negatives)
Post integer rules
Exponents
Malcom, Quinn, and Rachel
struggle with exponents
Make simple exponents (squares)
Combine Like Terms
None have shown difficulty with
terms
--
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Formative Assessment Recording
Student Distribut Fill-in Calculate Calculate
e
Signs
Malcom List steps
Nate
List steps
Omar
List steps
Penny
List steps
Quinn
Teacher
draws
lines
List steps
Rachel
Numbers
Calculator
Post
integer
rules
Calculate
Symbols
Combine
Like
Terms
Total
Simple
exponents
calculator
Post
integer
rules
Post
integer
rules
Simple
exponents
Simple
exponents
© Witzel, 2016
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Your Turn
Choose a Math Anchor
1) Review the task analysis.
2) Review your students.
1) Who would have trouble with certain steps?
2) What will you do to prevent the difficulties?
3) Set-up a Formative Assessment Recording chart.
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Your Class data – per student
Content: _________________________
Skills lacking per student
A.
B.
C.
D.
E.
F.
G.
H.
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17
Common steps for ________________
𝑒𝑥𝑎𝑚𝑝𝑙𝑒:
Step 1)
Step 2)
Step 3)
Step 4)
Step 5)
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Student Needs Chart
Dependent
Transitional
Independent but prior cueing
Objective: Students will ________________________
Skills necessary to solve
the problem
Missing skills with a student
Accommodations or problem
modifications
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Formative Assessment Recording
Dependent
Transitional
Independent but prior cueing
Student
Answer
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Debrief on Differentiation
• Why is it important to differentiate?
• How might this approach help?
• What are some barriers to this approach?
• What is your next step?
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Agenda
• Fully Worked Problems
• Task Analysis
• Differentiation
• HELPS: Field Dependent Learning
• CRA
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Field Dependent Learning(Witzel & Little, 2016)
An option for core / standards instruction during remediation of a
related skill
Prepare a fully worked problem
Then:
a) leave some steps blank and ask students to fill in the blank
b) once students appear to master the steps, write errors within
an example and have student analyze the error
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Analyze Incomplete Problems (Star, et al., 2015)
© Witzel, 2016
•
•
•
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Intervention Option:
Field-dependent practice
Have partially completed problems
Students fill in blanks
This approach provides student examples and inherent
cueing Solve the problem 12a – 7(6x – 1a + x)
12a – 7(7x – 1a) + x
Calculate within parentheses
19a – 49x + x
Evaluate the expression by
each variable
19a – 48x
Evaluate the expression by
each variable
19a – 48x
Answer
© Witzel, 2016
•
•
•
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Intervention Option:
Field-dependent practice
Have partially completed problems
Students fill in blanks
This approach provides student examples and inherent
cueing Solve the problem 12a – 7(6x – 1a + x)
12a – 7(7x – 1a) + x
Calculate within parentheses
12a – 49x + 7a + x
Distributive prop of
multiplication
19a – 49x + x
Evaluate the expression by
each variable
19a – 48x
Evaluate the expression by
each variable
19a – 48x
Answer
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20
Intervention Option:
Field-dependent practice
•
•
•
Have partially completed problems
Students fill in blanks
This approach provides student examples and inherent
cueing
Solve for x, 15 – 3x = - 4x + 32
15 – 3x = - 4x + 32
15 – 3x = - 4x + 32
+4x +4x
15 + 1x = 0 + 32
“Move the little guy”
1x = 17
1
1
“Divide by the coefficient”
x = 17
Answer
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Intervention Option:
Field-dependent practice
•
•
•
Have partially completed problems
Students fill in blanks
This approach provides student examples and inherent
cueing
Solve for x, 15 – 3x = - 4x + 32
15 – 3x = - 4x + 32
15 – 3x = - 4x + 32
+4x +4x
15 + 1x = 0 + 32
“Move the little guy”
15 + 1x =
+ 32
-15
-15
0 + 1x = 17
“Naked numbers to the other
side”
1x = 17
1
1
“Divide by the coefficient”
x = 17
Answer
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Analyze Errors within Problems (Star, et al., 2015)
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21
Common Algebra Errors (Dawkins, 2006)
• Where is the error made?
1. a=b
2. ab=a2
3. ab-b2=a2-b2
given
multiply both sides by a
subtract b2 from both sides
4. b(a-b)=(a+b)(a-b)
5. b=a+b
6. b=2b
factor both sides
divide both sides by a-b
recall the given
7. 1=2
divide both sides by b
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Introducing HELPS
Example:
H- The goal is to understand and perform matrix addition
E- The student is receiving intervention for integer computation
L- Use premade steps and explanations for students to reorganize
P- Students arrange steps and explanations and verbally answer with
partners
S- grade according to steps, explanations, verbal reasoning, and
accuracy; continue to build independence with the task through
practice
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HELPS – Matrices example
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HELPS – Matrices example
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HELPS Order of Ops example: Step 1
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HELPS Order of Ops example: Step 1
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HELPS Order of Ops example: Step 2
Swap blank pieces
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HELPS Order of Ops example: Step 2
4(25) + 9
exponents
Students write on
blank pieces
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HELPS Order of Ops example: Step 3
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HELPS Order of Ops example: Step 3
4(25) + 9
180 + 9
exponents
multiplication
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HELPS Order of Ops example: Step 4
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HELPS Order of Ops example: Step 4
4(5)2 + 9
parentheses
4(25) + 9
exponents
180 + 9
189
multiplication
addition
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HELPS – Your Turn
• Choose a multi-step math skill
• Task analyze the skill per procedural step
• Fully work a sample problem
• Write steps and answer on cards
• Write reasoning on cards per step (if applicable)
• Prepare which steps you will fade
Report out to the group
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What are some benefits from HELPS?
• Scaffolded support can vary per student
• Scaffolds are faded
• Student must understand steps
• Computation does not have to be an initial deterrent
Roadblocks.
• Time to develop these
• How do I differentiate HELPS for some students and
not all?
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Agenda
• Fully Worked Problems
• Task Analysis
• Differentiation
• HELPS: Field Dependent Learning
• CRA
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26
Use Clear Algebraic Representations (Star et al., 2015)
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Use Visual Organizers to help with stepwise
learning (http://www.purplemath.com/modules/synthdiv2.htm)
• 2x4 – 3x3 – 5x2 + 3x + 8 ÷ x – 2
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Using Graphic Organizers for Problem Solving
(Fulton, 2010)
Proportional Reasoning
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Organizers for Procedural Work
Clues
5/
6X
Undos
+ 4 = 8,
solve for X
Multiply by 5/6
Add 4
Divide by 5/6
Subtract 4
Algebraic - The student describes, analyzes, and generalizes a wide variety of patterns,
relations, and functions. (MA.D.1.3)
Operations - selects the appropriate operation to solve problems involving addition,
subtraction, multiplication, and division of rational numbers, ratios, proportions, and percents,
including the appropriate application of the algebraic order of operations. (MA.A.3.3)
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Use Visuals to Explain the Why
Pythagorean Theorem
http://math.pgseducation.com/history.html
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Concrete – Representational – Abstract
Sequence of Instruction (CRA)
(Gersten et al, 2009; NMP, 2008; Riccomini & Witzel, 2010; Witzel, 2005)
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CRA approach
• CRA is the Concrete to Representational to Abstract sequence of
instruction.
• Three stages of learning
• C = Learning through concrete hands-on manipulative objects
• R = Learning through pictorial forms of the math skill
• A = Learning through work with abstract (Arabic) notation
• www.rtitlc.org
• fcit.usf.edu/mathvids/strategies/cra.html
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(4.3)(2.4) using CRA
Tenths
times ones
are tenths.
There are 6
tenths.
Ones times
ones are
ones. There
are 8 ones.
Tenths times
tenths are
hundredths.
There are 12
hundredths.
Total = 8 ones; 22 tenths; 12
hundredths
Ones times
tenths are
tenths.
There are
16 tenths.
8.0_
2.2_
0.12
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2016
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86
(4.3)(2.4) using CRA
Ones times
ones are
ones. There
are 8 ones.
Ones times
tenths are
tenths.
There are
16 tenths.
X
X
X
X
X
X
X
X
X
x
x
x
x
x
X
X
X
X
X
x
X
x
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
x
x
x
X
x
X
Total = 8 ones; 22 tenths; 12
hundredths
8.0_
2.2_
0.12
10.32_
© Witzel, 2016
Tenths
times ones
tenths are
tenths.
There are 6
tenths.
x
Tenths times
tenths are
hundredths.
There are 12
hundredths.
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29
(4.3)(2.4) using CRA
multi
4
.3
2
8
0.6
.4
1.6
0.12
Total = 8 ones; 22
tenths; 12 hundredths
8.0_
2.2_
0.12
10.32_
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(3x+5)(2x+2) using CRA
Ones times
x is x. There
are 10x.
x times x is
x2.There are
6x2.
Total=
6x2
Ones times
x is x. There
are 6x.
One times ones
are ones. There
are 10 ones.
6x + 10x
10
6x2 + 16x + 10
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(3x+5)(2x+2) using CRA
x times x is
x2.There are
6x2.
Ones times
x is x. There
are 6x.
X
X
X
+1
+1
+1
+1
+1
X
X2
x2
x2
X
x
x
X
X
X
x2
x2
x2
X
X
x
X
X
+1
X
X
X
1
1
1
1
1
+1
X
X
X
1
1
1
1
1
Total=
Ones times
x is x. There
are 10x.
One times ones
are ones. There
are 10 ones.
6x2
6x + 10x
10
6x2 + 16x + 10
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(3x+5)(2x+2) using CRA
multi
3x
+5
2x
6x2
10x
+2
6x
10
Total =
6x2
10x
6x
10 =
6x2 + 16x + 10
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A CRA Connection
National Training Network
http://www.ntnmath.com/video%20index/Index%20Videos/Polynomia
ls/lesson%20112.htm
CRA
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(Gersten et al, p. 35)
31
Concrete Misconceptions
• Manipulative objects teach children
• Any concrete object transfers to abstract understanding
• All students will achieve higher gains in math scores when taught
using concrete objects
• All math can be taught through the use of concrete objects
• Math teachers are united in their belief in manipulative instruction
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Your Turn
a) What visuals do you use to help students make sense of
mathematics?
b) When would CRA be most effective?
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Conclusion – 3 – 2 – 1
3 things you learned
2 things you can implement
1 thing you still need
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Resources
Blackburn, B., & Witzel, B. S. (2014). Rigor for students with special needs. Larchmont, NY:
Routledge.
Gersten, R., Clarke, B., & Witzel, B. (2008). Onwards to algebra: The case for mathematics
interventions for struggling students in the intermediate grades. Compass
Learning. [Available online at http://www.compasslearning.com/files/Mathemail.pdf].
Hughes, E. M., Witzel, B. S., Riccomini, P. J., Fries, K. M., & Kanyonga, G. (2014). A metaanalysis of algebra interventions for students with learning disabilities and struggling
learners. Journal of the International Association of Special Education, 15(1), 36-47.
Witzel, B. S. (Ed.). (2016). Bridging the gap between arithmetic and algebra. Arlington,
VA: Council for Exceptional Children.
Witzel, B. S., & Little, M. E. (2016). Teaching elementary mathematics to struggling learners. New
York: Guilford.
Witzel, B. S., & Riccomini, P. J. (2011). Solving equations: An algebra intervention. Boston, MA:
Pearson.
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