This document provides information on instructional and learning strategies for mathematics. It discusses using concrete models, representations, and abstract concepts to teach mathematical skills and concepts (CRA instruction). It also recommends using mnemonics and graphic organizers as learning strategies to help students organize and retain mathematical information. The document provides examples of instructional strategies like direct instruction and computer-assisted instruction, as well as learning strategies like study skills and mnemonics. It emphasizes applying these strategies when teaching skills like problem solving, representing problems visually, and self-monitoring.

1.  Background Information
 Instructional and Learning
Strategies
 Application of Strategies
1

2. Pair off with another person, count the
number of triangles, explain the process,
and record the number.
2

3.  Language through which most of
mathematics is communicated
 Required course for high school
graduation
 Gateway course for higher math
and science courses
 Path to careers – math skills are
critical in many professions
3

4.  Learn to value mathematics
 Become confident in their ability to do
mathematics
 Become mathematical problem solvers
 Learn to communicate mathematically
 Learn to reason mathematically
4

5. “Teachers must be given the
training and resources to
provide the best
mathematics for every
students”
-NCTM
5

6.  Translate word problems into mathematical
symbols (processing)
 Distinguish between patterns or detailed
information (visual)
 Describe or paraphrase an explanation (auditory)
 Link the concrete to a representation to the abstract
(visual)
 Show fluency with basic number operations
(memory)
 Maintain focus for a period of time (attention deficit)
 Show written work (reversal of numbers and letters)
6

7.  Solving problems
 Visually representing problems
 Processing problem
information
 Self-monitoring
7

8.  Language and communication
 Processing
 Attention
 Organizational skills
 Math anxiety
8

9.  Instructional Strategies are
methods that can be used to
deliver a variety of content
objectives.
Examples:
Direct Instruction
Computer Assisted Instruction
9

10.  Learning Strategies are
techniques, principles, or rules
that facilitate the acquisition,
manipulation, integration, storage,
and retrieval of information across
situations and settings
Examples: Mnemonics, Graphic
Organizers, Study Skills
10

11. 1. Pretest
2. Describe
3. Model
4. Practice
5. Provide Feedback
6. Promote Generalization
11

12. Instructional Strategy: Concrete-
Representational-
Abstract (CRA) Instruction
Learning Strategies: Mnemonics
Graphic Organizers
12

13.  CONCRETE: Uses hands-on physical
(concrete) models or manipulatives to
represent numbers and unknowns.
 REPRESENTATIONAL or semi-
concrete: Draws or uses pictorial
representations of the models.
 ABSTRACT: Involves numbers as
abstract symbols of pictorial displays.
13

14. 14
Add the robots!

15. 15
+
+
=
=
2 1 3

16. Tilt or Balance the Equation!
16
3 * 4 = 2 * 6
?

17. 3 * 4 = 2 * 6
17

18. 3 * + = 2 * - - 4
18
Balance the Equation!

19. 3 * + = 2 * - 4
19
3 * 1 + 7 = 2 * 7 - 4
Solution

20.  Discover the sign
 Read the problem
 Answer or draw a
representation of
the problem using
lines, tallies, or
checks
 Write the answer
and check
20

21.  D iscover the variable
 R ead the equation, identify
operations, and think about the
process to solve the equation.
 A nswer the equation.
 W rite the answer and check the
equation.
21

22. 4x + 2x = 12
Represent the variable "x“ with circles.
+
By combining like terms, there are six "x’s."
4x + 2x = 6x
6x = 12
22

23. Divide the total (12) equally among the circles.
6x = 12
The solution is the number of tallies
represented in one circle – the variable “x."
x = 2
23

24. The steps include:
 Search the word problem;
 Translate the words into an
equation in picture form;
 Answer the problem; and
 Review the solution.
24

25.  Search: read the problem carefully,
ask questions, and write down facts.
 Translate: use manipulatives to
express the temperature.
 Answer the problem by using
manipulatives.
 Review the solution: reread and
check.
25

26. More than & less than (duck’s mouth open
means more):
5 2
5 > 2
(Bernard, 1990)
26

27.  O bserve the problem
 R ead the signs.
 D ecide which operation to do
first.
 E xecute the rule of order
 R elax, you're done!
27

28. Solve the problem
(4 + 6) – 2 x 3 = ?
(10) – 2 x 3 = ?
(10) – 6 = 4
28

29. PRE-ALGEBRA: ORDER
OF OPERATIONS
 Parentheses, brackets,
and braces;
 Exponents next;
 Multiplication and
Division, in order from
left to right;
 Addition and
Subtraction, in order
from left to right.
29
Please Excuse My Dear Aunt Sally

30. (6 + 7) + 52 – 4 x 3 = ?
13 + 52 – 4 x 3 = ?
13 + 25 - 4 x 3 = ?
13 + 25 - 12 = ?
38 - 12 = ?
= 26
30

31. A graphic organizer is a tool or
process to build word knowledge by
relating similarities of meaning to the
definition of a word. This can relate
to any subject—math, history,
literature, etc.
31

32.  Assist students in organizing and
retaining information when used
consistently.
 Assist teachers by integrating into
instruction through creative
approaches.
32

33.  #1 works with the figures
 #2 asks questions
 #3 records
 #4 reports out
33

34.  Differentiate the figures that
have like and unlike
characteristics.
 Create a definition for each
set of figures.
 Report your results.
34

35.  GOs connect content in a meaningful way to
help students gain a clearer understanding
of the material (Fountas & Pinnell, 2001, as
cited in Baxendrall, 2003).
 GOs help students maintain the information
over time (Fountas & Pinnell, 2001, as cited
in Baxendrall, 2003).
35

36.  Heighten student interest.
 Should be coherent and consistently
used.
 Can be used with teacher- and
student- directed approaches.
36

37. 1. Provide clearly labeled branch
and sub branches.
2. Have numbers, arrows, or lines
to show the connections or
sequence of events.
3. Relate similarities.
4. Define accurately.
37

38.  Teacher-Directed Approach
 Student-Directed Approach
38

39. 1. Provide a partially incomplete GO for
students
2. Have students read instructions or
information
3. Fill out the GO with students
4. Review the completed GO
5. Assess students using an incomplete
copy of the GO
39

40.  Teacher uses a GO cover sheet with
prompts.
 Example: Teacher provides a cover
sheet that includes page numbers
and paragraph numbers to locate
information needed to fill out GO.
 Teacher acts as a facilitator.
 Students check their answers with a
teacher copy.
40

41.  Framing the lesson
 Previewing
 Guided practice
 Independent practice
 Check for understanding
 Peer mediated instruction
 Simplifying the content or structure of the
GO
41

42.  Hierarchical diagramming
 Sequence charts
 Compare and contrast charts
42

43. 43

44. 44
Algebra
Calculus Trigonometry
Geometry
MATH

45. 45
Category
Subcategory Subcategory Subcategory
List examples of each type

46. 46
Algebra
Equations Inequalities

47. 47
Category
What is it?
Illustration/Example
What are some
examples?
Properties/Attributes
What is it like?
Subcategory
Irregular set
Compare and Contrast

48. 48
Positive Integers
Numbers
What is it?
Illustration/Example
What are some
examples?
Properties/Attributes
What is it like?
Fractions
Compare and Contrast - example
Whole
Numbers Negative Integers
Zero
-3, -8, -4000
6, 17, 25, 100
0

49. 49

50. 50
Prime Numbers
5 7
11 13
Even Numbers
4 6
8 10
Multiples of 3
9 15 21
3
2
6

51. 51

52. 52
TRIANGLES
Right Equiangular
Acute Obtuse
3 sides
3 angles
1 angle = 90°
3 sides
3 angles
3 angles < 90°
3 sides
3 angles
3 angles = 60°
3 sides
3 angles
1 angle > 90°

53. Word = Category + Attribute
= +
Definitions: ______________________
________________________________
________________________________
53

54. Word = Category + Attribute
= +
Definition: A four-sided figure with four equal
sides and four right angles.
54
Square Quadrilateral
4 equal sides &
4 equal angles (90°)

55. 55
1. Word: 2. Example:
3. Non-example:
4. Definition

56. 56
1. Word: semicircle 2. Example:
3. Non-example:
4. Definition
A semicircle is half of
a circle.

57.  Divide into groups
 Match the problem sets with the
appropriate graphic organizer
57

58.  Which graphic organizer
would be most suitable for
showing these relationships?
 Why did you choose this
type?
 Are there alternative choices?
58

59. Parallelogram Rhombus
Square Quadrilateral
Polygon Kite
Irregular polygon Trapezoid
Isosceles Trapezoid Rectangle
59

60. Counting Numbers: 1, 2, 3, 4, 5, 6, . . .
Whole Numbers: 0, 1, 2, 3, 4, . . .
Integers: . . . -3, -2, -1, 0, 1, 2, 3, 4. . .
Rationals: 0, …1/10, …1/5, …1/4, ... 33, …1/2, …1
Reals: all numbers
Irrationals: π, non-repeating decimal
60

61. Addition Multiplication
a + b a times b
a plus b a x b
sum of a and b a(b)
ab
Subtraction Division
a – b a/b
a minus b a divided by b
a less b b) a
61

62. Use the following words to organize into
categories and subcategories of
Mathematics:
NUMBERS, OPERATIONS, Postulates,
RULE, Triangles, GEOMETRIC FIGURES,
SYMBOLS, corollaries, squares, rational,
prime, Integers, addition, hexagon,
irrational, {1, 2, 3…}, multiplication,
composite, m || n, whole, quadrilateral,
subtraction, division.
62