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DEVELOPMENTALLY APPROPRIATE PRACTICES IN
EARLY LANGUAGE, LITERACY AND NUMERACY
MATH
INTERVENTION

Move one stick to make the number
sentence correct.

Move one stick to make the sentence
correct.

Move three sticks to make the fish face
the opposite way.

Session I:
COMMON ERRORS
and ERROR ANALYSIS

Preview
• What causes difficulties in learning Math?
• What are the potential areas of difficulties
in learning Math?
• What information can we obtain from a
student’s work?

What causes difficulties in
learning Math?

Mathematics is a symbolic language used to:
• express relationships – spatial, numeric,
geometric, algebraic, and trigonometric, in both
real and imaginary dimensions ;
• communicate concepts through symbols;
• reinforce and practise sequential and logical
thinking.
(Clayton, 2003)

A. Nature of Math (Chinn & Ashcroft, 1998)
• Interrelated
Parts are learned that later
on build into wholes.
What are needed to
learn:
• place values?
• adding dissimilar
fractions?
• long division?

• Interrelated
Parts are learned that later
on build into wholes.
What will happen
when a student does
not learn some of
these parts?

• Sequential
The learning of higher
skills depends on the
learning of basic skills.

• Sequential

• Sequential
The learning of higher
skills depends on the
learning of basic skills.
What will happen
when the basic
skills are not
learned?

• Reflective
The meaning of concepts
expand as lessons
progress.

Polynomials
Fractions
Decimals
• Reflective
Wholes
What will happen
when the meaning of
concepts do not
expand?

B. Structure (Chinn & Ashcroft, 1998)
Math is learned
from concrete
to abstract
Levels of difficulty build
up as the lessons
progress.

B. Structure (Chinn & Ashcroft, 1998)
Implications:
1. If the basic levels are skipped or not well-taught, the
foundations of learning become shaky.
2. When foundations are shaky, learning becomes
segmented, thus the student has to resort to
memorization.
3. When lessons are simply memorized, more effort is
needed to learn higher-level lessons.

C. Skills and Processes
(DepEd Math Curriculum 2013)
• Knowing and understanding
• Estimating, computing, and solving
• Visualizing and modelling
• Representing and communicating
• Conjecturing, reasoning, proving, and
decision-making
• Applying and connecting

D. Characteristics of School Math
• There are rules but
they do not apply all
the time
• Answers are either
right or wrong
• Tasks require
concentration

E. Math Language
• Symbols + – x  =     
 A = r2
• Vocabulary Algebra, perimeter, sine
even, pound, table
• Syntax and
Semantics
seven more than one,
quarter of a half,
a difference of two

What are the potential areas
of difficulties in learning
Math? (Chinn & Ashcroft, 1998)

Preview
1. Direction and
sequence
2. Perception
3. Retrieval
4. Speed of working
5. Math language
6. Cognitive Style
7. Conceptual Ability
8. Anxiety, stress, self-
image

A. Direction and Sequence
1. Directional
confusion

A. Direction and Sequence
2. Sequencing
Problems
counting on vs. counting
backwards,
place values

B. Perception
3. Visual Difficulties

B. Perception
4. Spatial Awareness

C. Retrieval
5. Working Memory
and Short-term
Memory
6. Long-term
Memory

8. Math Language
• Vocabulary knowledge
• a symbol with different names
vs.
a name for different symbols

Solve
70
1540
The answer is 22

9. Cognitive Style (Chinn & Ashcroft,1998)
Analyzing and Identifying the Problem
1. Tends to overview,
holistic, puts together.
2. Looks at the numbers and
facts to estimate an
answer or restrict range
of answers. Controlled
exploration.
1. Focuses on the parts and
details. Separates.
2. Looks at the numbers and
facts to select a relevant
formula or procedure.
Grasshopper
Inchworm

Solving the Problem
Grasshopper
Inchworm
3. Answer orientated.
4. Flexible focusing. Methods
change.
5. Often works back from a trial
answer. Multi-method.
6. Adjusts, breaks down/ builds
up numbers to make an easier
calculation.
3. Formula, procedure orientated.
4. Constrained focus, Uses a
single method.
5. Works in serially ordered
steps, usually forward.
6. Uses numbers exactly as
given.
Cognitive Style (Chinn & Ashcroft, 1998)

Solving the Problem
Grasshopper
Inchworm
7. Rarely documents method.
Performs calculation
mentally.
8. Likely to appraise and
evaluate answer against
original estimate. Checks by
alternate method.
9. Good understanding of the
numbers, methods and
relationships.
7. More comfortable with paper
and pen. Documents
method.
8. Unlikely to check or
evaluate answer. If check is
done, uses same procedure
or method.
9. Often does not understand
procedure or values of
numbers. Works
mechanically.
Cognitive Style (Chinn & Ashcroft, 1998)

• IQ Score
• Abilities in the
Multiple
Intelligences

• Impact of Brain-based Condition(s)
• Social or behavioral skills-related
• Autism
• Asperger’s Syndrome
• Attention-Deficit/Hyperactivity Syndrome
• Communication skills-related
• Language Acquisition
• Receptive / Expressive Language
Difficulties

• Impact of Brain-based Condition(s)
• Cognitive/learning skills-related
• MR/ Intellectual Disability
• Learning Disabilities
• Long and Short-term Memory Deficits
• Physical or sensory skills-based
• Visual Impairment
• Hearing Impairment

• Dyscalculia
• Dyscalculia is usually perceived of as a
specific learning difficulty for mathematics,
or, more appropriately, arithmetic.
(http://www.bdadyslexia.org.uk/)
• Dyscalculia is a brain-based condition that
makes it hard to make sense of numbers
and math concepts.
(https://www.understood.org/)

11. Anxiety,
Stress, and
Self-image
• Effect of
experiences and
environment
• Attitude towards
Math

ERROR ANALYSIS
(Refer to Worksheet)

Preview
1. Introduction to Remediation
2. Some Remedial Teaching Strategies
3. Principles of Remediation
4. The Remedial Plan

The commonly accepted idea of
remediation as a careful effort to
reteach successfully what was not well
taught or not well learned during the
initial teaching. (Glennon & Wilson, 1972)
What is remediation?

Students who show lags in math
performance that are unlike his or her
potential or performance in other
academic areas
Who needs math remediation?

Do not allow children who may have special
needs to go from one grade to another
without a professional team assessing the
student for eligibility for services and
supports. "Waiting" is NOT an effective,
educational practice. Although the process of
referral can be cumbersome, it is well worth it
when it identifies needs that can be met
during the educational life of the child.
– Barbara T. Doyle, Johns Hopkins School of Education

The Remediation Process
1) Identify the concepts, skills, procedures to be
retaught.
2) Collect supporting information, such as
anecdotes, work portfolio, and assessment
reports.
3) Select appropriate re-teaching methods and
strategies.
4) Provide remediation.
5) Evaluate and determine next steps.

1) Structure of Mathematics
2) The student’s strengths and
difficulties
• Error analysis
• Formal Testing
• Diagnostic Testing
3) Remedial instruction strategies
What a Remedial Math Teacher
Needs to Know

a. Rounding
1. Use landmark numbers
5
0 10

b. Solving
1. Use landmark numbers

a. Grids
2. Use graphic organizers

b. Tables (rows and columns)

c. Grids and spaces for long division

d. Guide
questions
and
spaces

a. Order of operations
3. Use mnemonics

b. Parts of a subtraction sentence
3. Use mnemonics

c. Long division
3. Use mnemonics

a. Properties of addition and
multiplication
4. Show patterns and
properties

b. Breaking numbers down /
decomposing
4. Show patterns and
properties

c. The hundreds chart
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100

a. Unlock new terms
5. Teach math vocabulary

b. Teach word analysis

c. Tell the background story

1. Build on what the child knows
 Show interconnectedness of lessons
 Promote reasoning
Principles of Intervention
(Chinn & Ashcroft, 2007)

2. Acknowledge the student’s learning
style
 T’s best method might not work
 Let student discover the strategies that
work for him

3. Make math developmental
 Use the concrete-representational-
abstract progression
 Employ gradual transfer

4. Use the language that communicates
the idea
 Use the child’s language
 Use visuals, real objects, experiences

5. Use the same basic numbers to
build an understanding of each
process or concept
 Make instruction success-oriented

5. Teach ‘why’ as well as ‘how’

7. Keep a
responsive
balance in all of
teaching
If the child does not learn the way you
teach, then you must teach the way he learns.
- Harry Chasty

REMEDIAL PLANNING
(Refer to Worksheet)

Why does the student need to undergo
remediation?
Who made the referral?

What do we know about the child, in relation to
math learning?

What do we know about the student, in relation to
math learning?

What behaviors did the student show – during and
outside math sessions?
What do we want to do about these behaviors?

What behaviors did the student show – during and
outside math sessions?

What is the student’s most recent Math
performance?

What do we do now? / What’s the plan?
An
overview

What do we do now? / What’s the plan?

Session III:
PLANNING FOR
INTERVENTIONS
(Workshop)

REFERENCES:
Bley, N.S. and Thornton, C.A. (2001). Teaching mathematics to students with
learning disabilities, 4th ed. USA: Pro-Ed.
Chinn, S. and Ashcroft, J. (1998). Mathematics for dyslexics: A teaching
handbook, 2nd ed. UK: Whurr.
Chinn, S. and Ashcroft, J. (2007). Mathematics for dyslexics: Including
Dyscalculia, 3rd ed. England: John Wiley and Sons.
Doabler, C.T., et.al. (2012). Evaluating Three Elementary Mathematics
Programs for Presence of Eight Research-Based Instructional Design Principles.
Learning Disability Quarterly, 35(4), 200-211.
Lalley, J.P. and Miller, R.H. (2002).Computational Skills, Working Memory, and
Conceptual Knowledge in Older Children with Mathematics Learning Disabilities.
Education, 126(4), 747-755.
Mabbott, D.J. and Bisanz, J. (2008). Computational Skills, Working Memory,
and Conceptual Knowledge in Older Children With Mathematics Learning
Disabilities. Journal of Learning Disabilities, 41(1), 15-28.
Miles, T.R. and Miles, E, Eds. (1992). Dyslexia and mathematics. USA:
Routledge.

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