Transcript Best Practices for Teaching Mathematics to Students with

Effective Instructional Strategies for
Correctional Education Programs
Joseph Calvin Gagnon, Ph.D.
George Mason University
Mike Krezmien, M.A.
University of Maryland
Richard Krause
1
Agenda


Technology
Graduated instruction and strategy instruction
2
Educational Reform:

Standards-driven reform is the primary
approach to assuring today’s high school
graduates are internationally competitive

Prompted by the public dissatisfaction
and poor performance by U.S. students
on international assessments
(McLaughlin, Shepard, & O’Day, 1995),
educators, curriculum specialists, and
national organizations have focused on
development of challenging standards for
over a decade.
3
Educational Reform:


Assuring all students achieve in math is a
national priority (IDEA, 1997; No Child Left
Behind Act of 2001)
Success in math is considered a gateway to
many educational and occupational
opportunities (Jetter, 1993)
4
Educational Reform:

Recent legislation has assisted these efforts
and assured that students with disabilities are
included, to the maximum extent possible

Central to this notion of reform is the
assertion that all students are, “entitled to
instruction that is grounded in a common set
of challenging standards” (McLaughlin, 1999,
p. 10)
5
Educational Reform:

Rigorous standards are especially crucial for
students with learning disabilities (LD) and
emotional disturbances (ED), who are commonly
included in the general education environment.

These students have historically been provided a
less rigorous curriculum with IEP goals that:
 Focus on computation (Shriner, Kim, Thurlow,
& Ysseldyke, 1993)
 Have minimal linkage to long-term general
education outcomes (Nolet & McLaughlin,
2000; Sands, Adams, & Stout, 1995; Smith,
1990)
6
Educational Reform:

Some states require students to pass
assessments with open-ended problem
solving tasks
7
Real World Problem Solving and Technology:


Computer/videodisc based
 Effective instruction and instructional design
variables
 Anchored instruction
 Web based instruction (WBI)
Calculators
 Standard
 Graphing
8
Real World Problem Solving and Technology:


Technology-based instructional approaches
can significantly affect student learning and
acquisition of higher-level math concepts;
particularly when embedded within real-world
problem solving tasks (Maccini & Gagnon, in
press)
This approach relies on the use of a
computer, calculator, or other specialized
systems as the mode of instruction (Vergason
& Anderegg, 1997)
9
Real World Problem Solving and Technology:

Technology-based instruction can:
 Assist teachers in moving away from a
focus on memorization and routine
manipulation of numbers in formulas and
toward instruction and activities embedded
in real-world problems (Bottge &
Hasselbring, 1993)
 Promote active student learning (Kelly,
Gersten, & Carnine, 1990)
10
Real World Problem Solving and Technology:

Embedding problem solving information
within a real world context helps:
 Activate student conceptual knowledge
when presented with a real-life problem
solving situation (Gagne, Yekovich, &
Yekovich, 1993)
 Improve student motivation, participation,
and generalization (Palloway & Patton,
1997)
11
Real World Problem Solving and Technology:
Teacher Quote:
“I incorporate fun activities such as timing a
wave, weighing bananas, and counting chips in
a cookie to acquire data. Students are included
in groups and usually have successful
experiences with others as they do the activities”
12
Real World Problem Solving and Technology:

Anchored instruction is one such example of
embedding problem solving situations in a
real-life situation via interactive video-disc
instruction (see Bottge & Hasselbring, 1993)

Students with special needs may need
additional support to benefit from enhanced
anchored instruction
 Additional review of the videodisc
 Cooperative learning strategies (Bottge et
al., 2002)
13
Real World Problem Solving and Technology:

However, “Rather than capitalizing on the
insights and motivation that students bring to
the classroom, schools may actually be
wasting valuable time by withholding more
authentic and motivating problems until
‘prerequisite’ skills are acquired” (Bottge et
al., 2001, p. 312)

It is effective to use videodisc-based
interventions that embed interesting and ageappropriate problem-solving situations
(Bottge, 1999; Bottge & Hasselbring, 1993;
Bottge et al., 2001; Bottge et al., 2002)
14
Real World Problem Solving and Technology:
Recommendations:
 Incorporate di (e.g., model, guided practice,
review, feedback) within technology-based
interventions

Incorporate effective instructional design
variables within technology-based instruction to
reduce student confusion and mathematical
errors
15
Real World Problem Solving and Technology:


Discrimination: Skills are introduced, practiced,
and mixed with other types of problems. Specific
instruction and remediation provide for
discrimination
Range of Examples: Students introduced to
fractions less than one, improper fractions, and
provided strategies for reading and writing both
16
Real World Problem Solving and Technology:


Explicit Strategy Teaching: Students provided
explicit problem solving strategies
Computer software should incorporate a wide
range of examples and nonexamples into
instruction for discrimination practice and
generalization
17
Real World Problem Solving and Technology:

One example of a series that uses technology,
the noted instructional design features, and
teaching practices recommended by NCTM is:
 The Systems Impact Direct Instruction
videodisc mathematics programs (DIV)
(Mastering Fractions, Mastering Decimals and
Percents, Mastering Equations, Roots, and
Exponents, Mastering Ratios and Word
Problems, and Mastering Informal Geometry)
(Scott Foresman, 1991)
18
Real World Problem Solving and Technology:
Recommendations:
 Incorporate technology-based tutorial
programs that embed basic math skills and
higher order thinking within problem-solving
situations
 This allows students to practice remedial
skills within context
 For example, it is recommended that
computers be available to students with LD
for tutorial assistance
19
Real World Problem Solving and Technology:
Example:
 The Hot Dog Stand and Geometric
SuperSupposer (both programs are available
from Sunburst Communications at
http://www.sunburst.com/index/html)
20
Real World Problem Solving and Technology:
Example:
 The Adventures of Jasper Woodbury consists
of 12 video-disc based adventures (plus video
based analogs, extensions and teaching tips)
that focus on mathematical problem finding and
problem solving

Each adventure is designed from the
perspective of the standards recommended by
the National Council of Teachers of Mathematics
(NCTM)
21
Real World Problem Solving and Technology:
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Each adventure provides multiple opportunities
for problem solving, reasoning, communication
and making connections to other areas such as
science, social studies, literature and history
(NCTM, 1989; 1991)
Jasper adventures are designed for students in
grades 5 and up
Each videodisc contains a short (approximately
17 minute) video adventure that ends in a
complex challenge
22
Real World Problem Solving and Technology:

The adventures are designed like detective
novels where all the data necessary to solve the
adventure (plus additional data that are not
relevant to the solution) are embedded in the
story

Jasper adventures also contain "embedded
teaching" episodes that provide models of
particular approaches to solving problems
23
Real World Problem Solving and Technology:


Geometry
 Blueprint for Success, The Right Angle, The
Great Circle Race
Algebra
 Working Smarter, Kim’s Komet, The General is
Missing
http://peabody.vanderbilt.edu/projects/funded/jasp
er/Jasperhome.html
24
Real World Problem Solving and Technology:

The Jasper series is currently being used in
classrooms in every state in the U. S., as well as
in classrooms in Canada and China. Jasper is
published and distributed by LEARNING, Inc., a
division of Lawrence Erlbaum Associates.
25
Real World Problem Solving and Technology:
Web-based Instruction:
 A promising approach is the TRIP project
(Christle et al., 2001), which:
 Extends the principles of anchored
instruction and contextualized learning
(Cognition and Technology Group at
Vanderbilt, 1990)
 Includes Web-based instruction (WBI)
within a universally designed environment
 Students work collaboratively
 Research is needed to fully realize the
effects of WBI, but the future appears
promising
26
Real World Problem Solving and Technology:

Limitations to the use of technology:
 The review was limited to 11 published
articles that met all criteria
 Although 73% (n = 8) of the studies
determined significant treatment effects,
three of the studies noted that the
proficiency levels of students with
disabilities fell below the established
criterion for learning of 80%
27
Real World Problem Solving and Technology:

Further, of the articles that obtained significant
findings, only 45% (n = 5) of the interventions
directly programmed for maintenance and 55%
(n = 6) programmed for generalization

The generalizability of the findings may also be
of concern because no information was
available on new technologies (e.g., DVD and
streaming video)
28
Real World Problem Solving and Technology:
Calculators:

In one study, calculator use was the most
prevalent adaptation noted by teachers
Maccini & Gagnon, 2002)

Consistent with Etlinger and Ogletree (1982),
teacher responses involved two primary
categories:
29
Real World Problem Solving and Technology:

The "practical" function: The use of
calculators to complete tedious
calculations, save time, increase student
motivation, and to decrease math anxiety

The "pedagogical" function: Relates to
similarities between calculators, textbooks,
and manipulatives in that each enhances
student understanding and competence in
mathematics
30
Real World Problem Solving and Technology:


These classifications are consistent with the
five primary functions of calculators as stated
by the NCTM
Within the practical classification, NCTM
identifies the use of calculators to
 Perform tedious computations that arise
when working with real data in problem
solving situations
 Concentrate on the problem-solving
process rather than calculations associated
with problems
 Gain access to mathematics beyond their
level of computational skill
31
Real World Problem Solving and Technology:


Teachers noted, Daily use of calculators to
eliminate arithmetic phobia (Maccini & Gagnon,
2002)
Use of calculators to increase motivation has
also been noted by researchers (Deshler, Ellis,
Lenz, 1996)
32
Real World Problem Solving and Technology:

The pedagogical function coincides with two
other uses identified by NCTM
 To explore, develop, and reinforce concepts
including estimation, computation,
approximation, and properties

To experiment with math ideas and
discover patterns
33
Real World Problem Solving and Technology:

Teachers noted using calculators as an aid
for students to solve problems,
 Students use calculators only after
attempting to solve problems

Students use calculators to help solve
problems, but still must show an
understanding by listing their steps
34
Real World Problem Solving and Technology:

Calculators can be used to enhance learning
by helping students to visualize connections
between symbolic and graphic solutions
(Milou, Gambler, & Moyer, 1997; Demana &
Waits, 1990), teachers noted,

Students use calculators or graphing
calculators for Algebra II
35
Real World Problem Solving and Technology:

Salend and Hoffstetter (1996) assert the
importance of:
 Training students to use calculators
 Using an overhead projector to teach this
skill
 Locating and describing the function of
each key to students
 Providing examples of calculator use
36
Real World Problem Solving and Technology:

Students should be provided opportunities to
practice calculations, including estimation skills
and reviewing answers obtained through
calculator use
37
Real World Problem Solving and Technology:

Another important category of teacher
responses related to teaching students to use
calculators,

I do extensive work with students on how to
use the calculator

I use an overhead calculator to assist with
VAKT [i.e., visual, auditory, kinesthetic, and
tactile learning]"
38
Real World Problem Solving and Technology:

Based on teacher responses, the literature,
and NCTM position statements (1998) the
following recommendations for teachers are
noted:
1.
Model calculator application
2.
Use calculators in computation, problem
solving, concept development, pattern
recognition, data analysis, and graphing
3.
Integrate calculator use in assessment
and evaluation
39
Real World Problem Solving and Technology:
4.
5.
Remain current with state-of-the-art
technology
Explore and develop new ways to use
calculators to support instruction and
assessment
40
Concrete-Semiconcrete-Abstract
Instructional Sequence (C-S-A)
Bruner’s structure-oriented theory of learning:
Enactive mode (e.g., the “doing” phase” - using
concrete objects to represent problems concrete representations)
Iconic mode (e.g., the “seeing phase”
visualizing representations of the problem semiconcrete representations)
Symbolic mode (e.g., using abstract symbols to
represent the problem - abstract
representations)
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C-S-A:
Research: Why teach for meaning?
Students with learning problems have problems with:
 classifying objects or ideas
 finding logical relationships/making logical
deductions
 generating hypotheses choosing among alternative
routes
42
C-S-A:

Empirical studies have validated CSA use with
students with high incidence disabilities for:
 Whole number operations
 Word problems
 Place value
 Introductory algebra skills
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C-S-A:
Recommendations for Practice:
Teach for meaning prior to abstract
representations
Examples:
1. Guidelines for using manipulatives - In the
Focus journal p. 11
2. Addition Sample Problem - In the TEC
journal p. 14
44
C-S-A:
Solve: 3a + 2a + 5 = 20
Concrete: Use paper plates and beans to solve:
1 bean = 1 unit
1 paper plate = variable “a” =
+
+••••• =••••••••••••••••••••
= •••••••••••••••
•••
•••
=3
•••
•••
•••
Source: Allsopp (1999, p. 78)
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C-S-A:
Solve: 3a + 2a + 5 = 20
Semiconcrete: Use pictures of plates and beans to solve
– = # less than 10; = tens
variable “a” =
+
+
+
+
–––
+
–––––
=
 
+ ––––– =  
+ ––––– =  ––––––––––
=  –––––
–––
–––
–––
–––
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C-S-A:
Solve: 3a + 2a + 5 = 20
Abstract:
5a + 5 = 20
5a + 5 - 5 = 20 - 5
5a = 15
a=3
Check:
5(3) + 5 = 20; 20 = 20 Yes, it checks.
47
Strategy Instruction:
Strategy instruction in math, such as a first
letter Mnemonic strategy, is used to help
students memorize and recall effective
problem solving skills

For example, the steps in the strategy can
help remind students to read the problem
carefully, to obtain a whole picture of the
problem (problem representation), to solve
the solution, and to check your answers
(problem solution)
48
Strategy Instruction:
Strategy Instruction Can Include:

Structured worksheets/cue cards to help
students remember problem solving steps or
strategies for solving problems

Mnemonics to help students recall problem
solving steps or important facts
Research:
Strategy instruction that incorporated a first-letter
mnemonic and structured worksheets helped
students with LD learn prealgebra skills and
concepts (Maccini & Hughes, 2000)
49
Strategy Instruction:
Recommendations for Practice:
Refer to the “STAR Strategy Steps” as an
example of how to teach both problem
representation and problem solution
using a first letter mnemonic strategy
Example:
1. STAR - See article in TEC journal, p.
10 and instructional steps, p. 12
50
Strategy Instruction:
Example of a Structured Worksheet:
STAR Steps
Search the word problem
a. Read the problem
carefully
b. Ask yourself, “What
facts do I know?”“What do
I need to find?”
c. Write down facts
Translate the words into
an equation in picture
form
a. Represent the
problem (use Graphic
Org.) and draw a picture
Answers
____________________________________
____________________________________
____________________________________
____________________________________
____________________________________
51
Strategy Instruction:
Answer the problem
a. Look for patterns:
1) What is the
difference
between frames?
2) Look for recursive
patterns and write
numbers under GO
Review the Solution
a. Reread the problem
b. Ask, “Does the
answer make sense?
Why?
c. Check answer
____________________________________
____________________________________
____________________________________
____________________________________
____________________________________
52
Strategy Instruction:
Example 2: CAP Strategy (for Abstract
Application)
C = Combine like terms
A = Ask yourself, “ How can I isolate the
variable?”
P = Put the value of the variable in the initial
equation, and check if the equation is
“balanced”
53
Strategy Instruction:
Solve: 3a + 2a + 5 = 20
C = 5a + 5 = 20
A: “To isolate the variable, I need to subtract 5
from both sides” 5a + 5 - 5 = 20 - 5; 5a = 15;
a = 3.
P: 5(3) + 5 = 20; 20 = 20 Yes, it checks.
Allsopp (1999)
54
Strategy Instruction
Strategy instruction in math, such as a first letter
mnemonic
strategy, is used to help students memorize and recall
effective problem solving skills.

For example, the steps in the strategy can help
remind students to read the problem carefully, to
obtain a whole picture of the problem
(representation), to solve the solution, and to
check your answers.
55
Strategy Instruction
Recommendations for Practice:
 Guidelines for using a first-letter mnemonic
strategy:
 Model how to use the strategy and the purpose
of the strategy.
 Model each letter and relate it to the math task
(e.g., solving equations)
56
Strategy Instruction



Model application of the strategy and relate
students’ prior understanding of the math task(s).
Provide cue cards of the strategy to aid memory,
posters on the wall, etc.
Provide rapid-fire-verbal-rehearsal (randomly call
on students, increase pace of questioning students).
57
Strategy Instruction
Example: CAP Strategy (for Abstract Application)
C = Combine like terms
A = Ask yourself, “ How can I isolate the variable?”
P = Put the value of the variable in the initial equation, and
check if the equation is “balanced”
Solve: 3a + 2a + 5 = 20
C = 5a + 5 = 20
A: “To isolate the variable, I need to subtract 5 from both
sides” 5a + 5 - 5 = 20 - 5; 5a = 15; a = 3.
P: 5(3) + 5 = 20; 20 = 20 Yes, it checks.
58
STAR Strategy Steps and Sub-steps
59
Instructional Strategy Steps
Step 1: Provide an Advance Organizer
Provide an advance organizer to: a) connect
new information to previously learned skills,
b) state the new skill to-be-learned, and c)
provide the rationale introducing the new
topic.
60
Instructional Strategy Steps
Step 2: Provide Teacher Modeling
Provide teacher “modeling” via two methods.
First, “think aloud” to students while
introducing a strategy. Then, fade teacher
prompts while involving students in application
of the strategy. For example, following the
teacher model, students answer questions and
write down their responses using the graphic
organizers or structured worksheets.
61
Instructional Strategy Steps
Step 3: Provide Guided Practice
Provide opportunities for students to practice the
new strategy with teacher assistance. Fade
teacher assistance until students can perform the
task independently.
62
Instructional Strategy Steps:
Step 4: Provide Student Independent Practice
Assess student mastery of the skills by
providing problems without teacher
prompts/assistance.
Step 5: Provide Feedback
Provide positive and corrective feedback
throughout the lesson via five steps:
a) document student performance (e.g., calculate
the percent correct)
b) target error patterns/incorrect answers
63
Instructional Strategy Steps
c) reteach if necessary
d) provide student practice with similar
problems and monitor student performance
e) close with positive feedback
Step 6: Program for Generalization
Provide prompts or questions to promote
generalization to other:
a) problem solving situations
b) content areas
c) real-world situations
64
Structured Worksheets/Prompt Cards, and
Graphic Organizers
Structured worksheets, prompt cards, and graphic
organizers with the strategy steps are used as visual
prompts and organizers for students to use as they
solve problems (until they can recall the steps
without the card).
65
Recommendations for Practice:
Provide teacher modeling of the self-questions that
are on the structured worksheets/prompt cards.
 Have students practice the self-questions while
solving problems. Provide corrective and positive
feedback and fade assistance.
Refer to the “Growing Patterns”, “Structured
Worksheet” for examples of activities

66