The document discusses common math difficulties and strategies for math intervention. It identifies several potential areas of difficulty in learning math, such as directional confusion, weak working memory, math language issues, and math anxiety. Error analysis of student work is presented as an important tool to identify specific difficulties. The document then covers principles of remedial instruction, such as building on student strengths, acknowledging different learning styles, using a concrete-to-abstract approach, and employing effective math vocabulary. A variety of teaching strategies are proposed, like using graphic organizers, mnemonics, showing patterns, and visualization.

1. Common Math Difficulties
Error Analysis
Introduction to Math Intervention
Garry L. Pangan, PhD
EPS – I Mathematics

2. Learning Objectives:
At the end of the session, you are
expected to:
1. Identify common math difficulties.
2. Determine strategies to address common
math difficulties.

3. Move one stick to make the number
sentence correct.

4. Move one stick to make the number
sentence correct.

5. Move one stick to make the number
sentence correct.

6. Move one stick to make the sentence
correct.

7. Move one stick to make the sentence
correct.

8. Move two sticks to make the sentence
correct.

9. Move two sticks to make the sentence
correct.

10. Move two sticks to make the sentence
correct.

11. Move two sticks to make the sentence
correct.

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

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

14. COMMON ERRORS
and ERROR ANALYSIS

15. 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?

16. What causes difficulties in
learning Math?

17. 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 practice sequential and logical
thinking.
(Clayton, 2003)

18. 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?

19. A. Nature of Math (Chinn & Ashcroft, 1998)
• Interrelated
Parts are learned that later
on build into wholes.
What will happen
when a student does
not learn some of
these parts?

20. A. Nature of Math (Chinn & Ashcroft, 1998)
• Sequential
The learning of higher
skills depends on the
learning of basic skills.

21. A. Nature of Math (Chinn & Ashcroft, 1998)
• Sequential

22. A. Nature of Math (Chinn & Ashcroft, 1998)
• Sequential
The learning of higher
skills depends on the
learning of basic skills.
What will happen
when the basic
skills are not
learned?

23. A. Nature of Math (Chinn & Ashcroft, 1998)
• Reflective
The meaning of concepts
expand as lessons
progress.

24. Polynomials
Fractions
Decimals
A. Nature of Math (Chinn & Ashcroft, 1998)
• Reflective
Wholes
What will happen
when the meaning of
concepts do not
expand?

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

26. 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.

27. 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

28. 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

29. 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

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

31. 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

32. A. Direction and Sequence
1. Directional
confusion

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

34. B. Perception
3. Visual Difficulties

35. B. Perception
4. Spatial Awareness

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

37. 7. Speed of Working

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

43. Solve
70
1540
The answer is 22

44. 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

45. 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)

46. 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)

47. 10. Conceptual Ability
• IQ Score
• Abilities in the
Multiple
Intelligences

48. 10. Conceptual Ability
• 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

49. 10. Conceptual Ability
• 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

50. 10. Conceptual Ability
• 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/)

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

52. Exercise:
ERROR ANALYSIS

53. ERROR ANALYSIS

54. ERROR ANALYSIS

55. ERROR ANALYSIS

56. MATH
REMEDIATION

57. Preview
1. Introduction to Remediation
2. Some Remedial Teaching Strategies
3. Principles of Remediation

58. 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?

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

60. 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

61. 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.

62. 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

63. SOME TEACHING
STRATEGIES

64. a. Rounding
1. Use landmark numbers
5
0 10

65. b. Solving
1. Use landmark numbers

66. a. Grids
2. Use graphic organizers

67. b. Tables (rows and columns)
2. Use graphic organizers

68. c. Grids and spaces for long division
2. Use graphic organizers

69. d. Guide
questions
and
spaces
2. Use graphic organizers

70. a. Order of operations
3. Use mnemonics

71. b. Parts of a subtraction sentence
3. Use mnemonics

72. c. Long division
3. Use mnemonics

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

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

75. 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

76. a. Unlock new terms
5. Teach math vocabulary

77. b. Teach word analysis
5. Teach math vocabulary

78. c. Tell the background story
5. Teach math vocabulary

79. 6. Visualize and verbalize

80. PRINCIPLES OF
INTERVENTION

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

82. 2. Acknowledge the student’s learning
style
 T’s best method might not work
 Let student discover the strategies that
work for him
Principles of Intervention
(Chinn & Ashcroft, 2007)

83. 3. Make math developmental
 Use the concrete-representational-
abstract progression
 Employ gradual transfer
Principles of Intervention
(Chinn & Ashcroft, 2007)

84. 4. Use the language that communicates
the idea
 Use the child’s language
 Use visuals, real objects, experiences
Principles of Intervention
(Chinn & Ashcroft, 2007)

86. 5. Use the same basic numbers to
build an understanding of each
process or concept
 Make instruction success-oriented
Principles of Intervention
(Chinn & Ashcroft, 2007)

87. 5. Teach ‘why’ as well as ‘how’
Principles of Intervention
(Chinn & Ashcroft, 2007)

88. 7. Keep a
responsive
balance in all of
teaching
Principles of Intervention
(Chinn & Ashcroft, 2007)
If the child does not learn the way you
teach, then you must teach the way he learns.
- Harry Chasty