Mathematics instruction within a juvenile correctional school for committed youth
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Title: Mathematics instruction within a juvenile correctional school for committed youth
Series Title: Maccini, P., Gagnon, J. C., Cutting, C., & Leone, P. (2006). Mathematics instruction within a juvenile correctional school for committed youth. Journal of Correctional Education, 57, 210-229.
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Creator: Gagnon, Joseph
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Abstract: The current paper provides a description of instructional approaches for teaching mathematics to secondary students with learning disabilities (LD) and emotional disturbance (ED) within juvenile correctional schools. Recommendations for math instruction are based on a comprehensive review of the literature and examples are provided from an urban Juvenile correctional faciiity school for committed youth. Six key topics are presented: (a) advance organizers; (b) direct instruction (di); (c) use of technology and real-world problem solving tasks; (d) use of varied student grouping; (e) presenting information in a graduated instructional sequence; and (f) strategy instruction.
Acquisition: Collected for University of Florida's Institutional Repository by the UFIR Self-Submittal tool. Submitted by Joseph Gagnon.
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The Journal of Correctional Education 57(3) * September 2006


Math Instruction

for Committed Youth
within Juvenile Correctional Schools


Paula Maccini
Joseph Calvin Gagnon
Candace A. Mulcah
Peter E. Leon



Abstract
The current paper provides a description of instructional approaches for teaching
mathematics to secondary students with learning disabilities (LD) and emotional
disturbance (ED) within juvenile correctional schools. Recommendations for math
instruction are based on a comprehensive review of the literature and examples are
provided from an urban juvenile correctional facility school for committed youth. Six
key topics are presented: (a) advance organizers; (b) direct instruction (di); (c) use of
technology and real-world problem solving tasks; (d) use of varied student grouping;
(e) presenting information in a graduated instructional sequence; and (f) strategy
instruction.


The academic needs of youth with emotional disturbance (ED) and learning
disabilities (LD) in juvenile corrections are one of the most neglected areas in
practice and research (Browne, 2003; Coffey & Gemignani, 1994; Gagnon &
Mayer, 2003; Leone, 1994; Rutherford, Quinn, Leone, Garfinkel, & Nelson,
2002). Specifically, many correctional facilities provide unsatisfactory education
and special education support; which has resulted in class-action litigation in
more than 25 states (Leone & Meisel, 1997; Meisel & Leone, 2004). Of the
108,000 students committed to long-term juvenile correctional facilities
(Sickmund, 2004), approximately one-third have identified disabilities (Quinn,
Rutherford, & Leone, 2001). However, almost no information exists that
addresses instruction in juvenile correctional schools for committed youth. The

This research was supported In part by Grant #H324J990003 from the U.S. Department of Education.







The Journal of Correctional Education 57(3) * September 2006
Maccini, et. al. Math Instruction for Committed Youth

dearth of information regarding special education in juvenile correctional
commitment facilities contrasts with recent education and special education
reforms. Specifically, the Individuals with Disabilities Education Act (2004)
guarantees students with disabilities access to the same rigorous curriculum as
their nondisabled peers.
Similarly, the reauthorized Elementary and Secondary Education Act, also
known as No Child Left Behind (NCLB) (2001) promotes a high quality education
for all students and specifically targets student competence in math and
reading. Within this mandate, there is also a clear emphasis on using
empirically-validated instructional techniques. The calls for scientifically-based
interventions in NCLB are ultimately geared toward improving the performance
of all students in schools, including those students within juvenile corrections
(Leone & Cutting, 2004).
Competence in mathematics is a critical component for academic success,
graduation, and positive post-school outcomes. Secondary school-aged youth
with ED and LD need effective and efficient instructional approaches to pass
rigorous state assessments and earn high school diplomas. However, only
anecdotal information currently exists on teachers' use of validated math
instructional approaches within juvenile correctional schools (Gagnon 8 Mayer,
2004). Coffey and Gemignani (1994) noted that teachers' approach to math
within corrections commonly reflects an emphasis on worksheet-based drill and
practice of basic math facts. This contrasts with most states' alignment of
standards and assessments with the National Council of Teachers of
Mathematics (NCTM) Standards for teaching math (Blank 8 Dalkilic, 1992;
NCTM, 2000; Thurlow, 2000). Often, state assessments emphasize higher-level
problem solving tasks that focus on conceptual knowledge and real-world
application.
In addition to a dearth of research, other complications may exist in the
education of youth with special needs in these facilities (Gagnon, in press).
Common problems include an emphasis on punishment rather than
rehabilitation of juveniles, and competition for limited fiscal resources among
education, security, and physical plant maintenance (Leone & Meisel, 1997).
Leone and Meisel also noted that correctional education programs may not
have the autonomy necessary to ensure student academic needs are met. In
addition, security needs may require creativity and adaptation from teachers.
For example, the use of math manipulatives would require teachers to organize
and count materials at the beginning and end of each class.







The Journal of Correctional Education 57(3) * September 2006
Math Instruction for Committed Youth Maccini, et al.

Clearly, juvenile correctional facilities have some unique attributes and
require teacher flexibility. However, instructional strategies should be based on
research, rather than the setting in which these youth attend school (Gagnon, in
press). One promising approach is to rely on existing research that focuses on
effective instructional strategies for students with ED and LD in other settings,
as these youth are overrepresented in juvenile corrections (Foley, 2001; Quinn,
Rutherford, Leone, Osher, & Poirier, 2005; U.S. Department of Education, 1999).
In a review of math research for secondary students with ED and LD, Maccini
and Gagnon (2000) identified six general categories of empirically-validated
approaches. These areas can assist teachers in strengthening student basic
math skills and proficiency with open-ended problem solving questions. These
empirically-validated instructional approaches include: (a) advance organizers;
(b) direct instruction (di); (c) use of technology and real-world problem solving
tasks; (d) use of varied student grouping; (e) presenting information in a
graduated instructional sequence; and (f) strategy instruction. In the sections
that follow, we described the six instructional approaches, provide a specific
example of each instructional approach within the context of school
observations and the literature, and provide instructional implications for
teaching math to math to students with LD and ED committed to juvenile
correctional facilities.

School Description
The school of interest is located within a juvenile commitment facility within a
large metropolitan area. The school serves youth 12-18 years of age. At the
time of the observations, 80 committed youth were enrolled in the educational
the program, with 40% (n = 32) of the student population receiving special
education services. Additionally, the average length of stay was 6 months,
followed by a 6-week transition program. The school provides courses in
mathematics (e.g., basic, intermediate, advanced, GED preparation), science,
social studies, English/Reading, health, art (e.g., fine arts, graphics), life skills,
limited vocational training (e.g., technology) and GED preparation. Students can
earn academic credit at the school that is transferrable to public schools. Also,
students have the option of working toward a high school diploma or GED
(with some restrictions, such as age).
Math classes were divided into six skill levels including (a) level 1 (basic
skills, such as place value, fractions, pattern recognition); (b) level 2 (basic skills
continued, such as advanced fractions, decimals, percents; (c) level 3 (word
problems, ratios, geometry); (d) level 4 (Pre- GED, integers, equations,







The Journal of Correctional Education 57(3) * September 2006
Maccini, et al. Math Instruction for Committed Youth

inequalities); (e) level 5 (GED, multi-step word problems); and (f) level 6 (pre-
college math and SAT preparation, including algebra, trigonometry, calculus).
Students are assessed and placed in math classes as a function of their reading
level (although not our current focus, this approach is not recommended).
Classroom observations included the lowest level math class, the highest level,
and an intermediate level math class. Teacher A taught the lowest level (i.e.,
basic skills) math class, which included 7 students. The class lesson objective
(i.e., greatest common factor) and date were written on the board prior to the
start of the lesson. The lesson consisted of whole group instruction and the
content of the lesson included: (a) review of factors; (b) introduction of greatest
common factor (GCF); (c) guided practice and review of GCF using a class
worksheet; and (d) a homework sheet on GCF.
The highest level math class (GED preparation) was taught by Teacher B
and included 7 students. The class lesson objective (i.e., volume of a cube,
rectangular solid, and cylinder) and date were written on the board at the start
of the lesson. During whole group instruction, the teacher collected homework,
and then introduced and provided guided practice for finding the volume of a
cube, finding the volume of a rectangular solid, and finding the volume of a
cylinder. Lastly, students were provided homework sheets.
Teacher C taught the intermediate math class consisting of 9 students. This
day's lesson consisted of a review of patterns (i.e., finding the missing numbers)
in a given sequence) and translating words into algebraic expressions. The
lesson objectives were verbally stated by the teacher at the beginning of class.
The content of the lesson included: (a) review of patterns using the dry erase
board to illustrate four problems; (b) introduction to unknown variables and
writing algebraic expressions using textbook problems; (c) guided practice and
review of writing algebraic expressions; and (d) review of multiples within a
class game format.

Instructional Approaches
The sections below describe the six research-based instructional approaches
and are illustrated with either teacher observations or the literature.
Implications for teaching are also provided.

Advance Organizers
Description. Students with LD and ED encounter a myriad of difficulties that
may impact their math performance. For example, students with social and
emotional difficulties may experience learned helplessness (e.g., student may







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Math Instruction for Committed Youth Maccini, et. al.

appear disinterested), impulsivity (e.g., learner may omit numbers, make
careless computational errors), and distractibility (e.g., learner may not finish a
multistep word problem) (Mercer, 1997). Use of advance organizers is effective
in helping students identify, organize, understand, and retain information (Lenz,
Bulgren, & Hudson, 1990). Specifically, advance organizers help to orient the
learner to the lesson, identify lesson objectives, connect current information to
a student's prior knowledge, and improve student motivation to learn the topic
(Miller, 1996).

Example. A teacher was observed incorporating use of an advance organizer
into instruction. Specifically, the teacher identified the lesson objective and
oriented the learners to the lesson, visually and verbally (i.e.., wrote the lesson
objective (GCF) on the dry erase board prior to the beginning of the lesson and
reviewed the objectives from the previous lesson).

Discussion. It is promising the teacher incorporated use of advance organizers
into math instruction via two different formats: (a) visual (i.e., written on the
board); and (b) verbal (e.g., eliciting student questions/information). Miller,
Strawser, and Mercer (1996) noted that use of advance organizers helps to
support strategic behavior (i.e., student use of math strategies) and student
attention to the lesson. Student attention can also be promoted by
incorporating prompts in the advance organizer (e.g., "What is the strategy we
learned yesterday for remembering the problem solving sequence?').
Moreover, Allsopp (1999) recommended that math lessons for secondary
students include an advance organizer to help orient learners to the current
lesson and to build connections between the new information and student's
prior knowledge. Teachers can use the mnemonic, LIP, to help them remember
to incorporate three critical features of advance organizers into instruction:
'(a) Link the lesson to previous learning or lesson; (b) Identify the target skill;
and (c) Provide a rationale for learning the skill' (Allsopp, 1999, p. 75). Teacher
B could have used the following advance organizer:

Yesterday we focused on identifying and drawing common geometric
solids including prisms, pyramids, and cylinders. (L). Today we are going to
learn to calculate the volume of prisms or the cubic units needed to fill the
space occupied by a solid (1). Knowing how to calculate volume will help us
when we go shopping, such as determining how much ice our cooler can







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hold for our lunch tomorrow. Who can tell me other reasons for
determining volume? (P) (Glencoe Mathematics, 1999).

Direct Instruction (di)
Description. Limited research indicates that teacher expectations in juvenile
correctional schools are low and include a primary emphasis on drill and
practice of basic math facts via a worksheet-driven approach (Coffey &
Gemignani, 1994). However, Coffey and Gemignani asserted that there is no
justification for math teachers in juvenile corrections to approach math
instruction in this way. The need to establish high quality educational math
programs within corrections is especially critical in light of difficulties these
students commonly have with self-motivation, persistence, and concentration
(Bos 8 Vaughn, 1994). These traits may inhibit student success in an
individualized and worksheet-driven approach to instruction. Additionally,
students with ED typically obtain a percent correct rate between 20 and 76 on
independent seatwork (Guntner 8 Denny, 1998). Clearly, providing a textbook
or worksheet driven math program with little or no instruction provides daily
learning situations in which students may experience relatively little success.
One approach to assisting youth with ED and LD in math is to use direct
instruction (di) (Meisel, Henderson, Cohen, & Leone, 1998). The di approach
consists of five key components: (a) review of previously learned skills; (b)
teaching content using teacher modeling, guided, and independent practice; (c)
monitoring student performance and providing feedback to students;
(d) providing corrective feedback and use of review and reteaching when
needed; (e) use of cumulative review (Rosenshine 8 Stevens, 1986).

Example. Teacher A was observed using the first four components of di. For
instance, this teacher reviewed the divisibility rules/questions covered in the
previous class period. The rules provided a method for students to identify if a
given number was divisible by any of the numbers 2 through 10. Teacher A
then linked the divisibility rules/questions to the current lesson, which required
identifying the greatest common factor of two numbers. The teacher was also
observed modeling new concepts and providing students with guided practice.
For example, the teacher completed sample problems on the dry erase board
and "thought aloud' the processes as each step was demonstrated. Students
were then directed to complete problems from the textbook or worksheets, as
the teacher provided positive comments to students concerning their behavior,







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Math Instruction for Committed Youth Maccini, et al.

effort, and for correctly completing math problems. At times, the teacher
stopped the guided practice and provided additional explanation of concepts
and problems on the dry erase board. At the conclusion of the class, the
teacher also provided independent practice via a homework assignment.

Discussion. In light of available research supporting the use of di for secondary
students with ED and LD in math instruction (Maccini & Gagnon, 2000) teacher
use of di components in the current study is encouraging. In addition to the
observed components of di, it is also important to provide a cumulative review.
Cumulative review extends beyond review of the previous day's class and
supports maintenance of student skills. One common, appropriate approach to
cumulative review involves providing students with "warm-up' problems. These
problems are completed as students enter the room, providing for a smooth
transition between classes and an effective approach to cumulative review of
prerequisite skills (Rosenshine & Stevens, 1986).
It is also noteworthy that the teacher allowed time for students to complete
guided practice problems in class, monitored student performance, and
provided feedback. Adequately monitored practice is necessary to ensure
students complete problems correctly. Monitoring student performance and
providing assistance are particularly important for youth with ED and LD in
corrections, many of whom have a history of academic failure (Greenbaum et
al., 1996; Wagner, 1995; Wagner & Blackorby, 1996) may lack motivation, and
may become easily frustrated (Bos 8 Vaughn, 1994). As such, positive and
specific comments from the teacher may provide students the encouragement
to complete difficult problems. In addition to the actual teacher comments,
teacher circulation and use of proximity can decrease inappropriate behavior,
reduce student anxiety and frustration, and facilitate student attention to task
(Walker 8 Shea, 1999). The teacher also provided homework or independent
practice. The need for repetition of skills is critical for youth with LD and ED
who may have difficulties retaining information without sufficient practice
(Wilson, Majsterek, & Jones, 2001).

Technology and Real-World Problem Solving Tasks
Description. Technology-based instructional approaches can significantly impact
student learning and motivation to learn higher-level math concepts;
particularly when embedded within real-world problem solving tasks (Maccini a
Gagnon, 2005). This approach relies on the use of a computer, calculator, or
other specialized systems as the mode of instruction (Vergason & Anderegg,







The Journal of Correctional Education 57(3) * September 2006
Maccini, et. al. Math Instruction for Committed Youth

1997). 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) that promote active student learning (Kelly, Gersten, 8
Carnine, 1990). Additionally, calculator use provides students access to math
concepts beyond their computational skills level (NCTM, 2000).

Example. During the observations, Teacher B incorporated technology and
linked instruction to real world activities. Specifically, the teacher allowed
students to use calculators for basic computational tasks as they calculated
volume. Also, prior to introducing the formula for volume of a cube, the teacher
asked the students, "Why do we need to know the formula?' and provided a
personal example of shopping and comparing prices by volume.

Discussion. Calculator use has minimal validation as an effective tool when
working with students labeled ED and LD (see Advani, 1972). However,
calculator use is widely accepted by researchers (Jarrett, 1998; Maccini &
Gagnon, 2005; Milou, 1999) and the NCTM (2000). Calculators may be used for
computation, error correction, problem solving, concept development, pattern
recognition, data analysis, and graphing (Jarrett, 1998; Maccini & Gagnon,
2002). For calculators to be effective during class activities and math
assessments, teachers should provide instruction to students on how to use
them. Instruction in calculator use should include all components of di (Maccini
& Gagnon, 2002). Additionally, Salend and Hoffstetter (1996) recommend use of
an overhead projector to model and teach calculator use and to provide
specific instruction to students concerning the function of each key.
The use of real-world math activities, or embedding instruction within a
context relevant to students can have a major impact on student interest,
participation, and generalization (Polloway & Patton, 1997). Additionally, by
embedding problem solving information within a real-world context, students
may activate their conceptual knowledge when presented with a problem
solving situation outside the classroom (Gagne, Yekovich, 8 Yekovich, 1993).
NCTM (2000) and researchers (Maccini 8 Gagnon, 2005) advocate use of real-
world activities to the greatest extent possible.
Both technology and real-world activities can be effectively combined to
teach math to adolescents with ED and LD. This combination has been
validated in other school settings including alternative schools where youth
typically have experienced a great deal of school failure (Bottge, 1999; Bottge 8







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Hasselbring, 1993; Bottge, Heinrichs, Chan, 8 Serlin, 2001; Bottge, Heinrichs,
Mehta, & Hung, 2002). Therefore, it is logical to assume that such strategies
would be similarly effective within a correctional classroom with appropriate
accommodations for student safety. Other critical features integral to
technology use include incorporating components of effective instruction with
extensive teacher involvement as facilitator and discussion leader (Maccini a
Gagnon, 2005). For example, researchers (Kelly, Carnine, Gersten, & Grossen,
1986; Kelly et al., 1990) noted the importance of computer software that
provides a wide range of examples and nonexamples, as well as pictorial
representations that enhance concept development (see Maccini 8 Gagnon,
2005 for examples of effective instructional design features that should be
included in software programs).

Student Grouping
Description. Youth in juvenile correctional math classes commonly work on
individual math assignments and there is little interaction among students
(Coffey & Gemignani, 1994). However, variations in grouping could include
whole group instruction, small group instruction, one-on-one support,
cooperative group activities, individualized instruction, and peer tutoring. Small
group instruction is particularly important in classes within juvenile correctional
schools due to the wide range of student abilities. Also, researchers (Meisel et
al., 1998) support the use of peer mediated instruction in juvenile correctional
schools. For example, classwide peer tutoring is an effective approach to
grouping that can strengthen student math skills (Allsopp, 1997) and may also
reduce occurrences of behavioral problems (Penno, Frank, 8 Wacker, 2000).
According to Allsopp, the approach is a, "systematic instructional strategy that
uses students within the same classroom to tutor one another" (p. 368).
Common components of classwide peer tutoring (CWPT) sessions includes
teacher-lead instruction of the new material, pairs working together during
scheduled tutoring sessions, reciprocal tutor and tutee roles, structured tutoring
formats for students to follow with correction procedures, teacher monitoring of
the tutoring sessions, and team competition (Olson & Platt, 2000).

Example. In each of the three classrooms observed, teachers combined whole-
group instruction with opportunities for students to work individually on a
textbook or worksheet assignment. At times, spontaneous student collaboration
existed. However, this took the form of a single student asking another student







The Journal of Correctional Education 57(3) * September 2006
Maccini, et. al. Math Instruction for Committed Youth

how to complete a specific problem. The teachers did not reprimand students
for helping each other, nor promote a formal process for student collaboration.

Discussion. The whole group approach employed by each teacher is one
appropriate method of math instruction at the secondary level. Large group
presentations can be effective due to time efficiency and student preparation
for postsecondary setting demands. However, in addition to whole group
instruction, teachers should consider use of peer tutoring. This approach may
be particularly effective for independent practice of math skills (Allsopp, 1997).
If properly structured and monitored, peer tutoring can increase student
learning and provide needed socialization (Franca, Kerr, Reitz, & Lambert, 1990).
However, use of peer tutoring should include student training in a specified set
of steps or procedures. Such structure is important for youth in corrections,
many of whom may have limited socialization and self-regulation skills.
Allsopp (1997) noted the positive effects of a classwide peer tutoring
(CWPT) approach. In CWPT, the entire class participates in a 30-minute program
that can be implemented 2 - 5 days per week. Within a given session, each
student acts as a tutor for 10 minutes and tutee for 10 minutes. Students take
the last five minutes of each session to record their individual points. Each
student may earn points for correct responses, error correction, and following
the tutoring procedures. To increase self-management and positive student
interactions, the tutor may record points for correct tutee responses and
correcting errors. The teacher is free to award points to students for adhering to
the CWPT procedures. At the end of each week, teams that meet a specified
criteria level may earn a special reinforcing activity. To implement CWPT, it is
important for the teacher to identify specific procedural steps and expectations
for students to follow (see Allsopp, 1997). Students should then be taught the
procedures through a process based on the previously noted components of di.

Strategy Instruction
Description. Teaching learning strategies helps learners to: (a) comprehend,
organize, and remember vast amounts of content knowledge required in most
secondary classrooms; and (b) become more active and self-regulated learners
(Ellis & Lenz, 1996). Strategic instruction involves, 'guidelines for how to think
and act when planning, executing, and evaluating performance on a task and
its outcomes' (Ellis & Lenz, p. 30). Research (Huntington, 1994; Hutchinson,
1993; Montague, 1992; Maccini 8 Hughes, 2000; Maccini & Ruhl, 2000)







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Math Instruction for Committed Youth Maccini, et. al.

supports the use of learning strategies by secondary students with math
difficulties. Math strategies for students with LD include explicit teacher
modeling in cognitive (i.e., the steps necessary for solving a problem) and/or
metacognitive steps (i.e., helping students instruct, monitor, and assess, their
problem solving performance via self-questions) (Montague, 1997).
Example. In the second level or intermediate math class, Teacher C reviewed a
previously introduced strategy to help students remember division: divisor --
down -, door. For modeling how to translate "half of a number' into a math
expression, the teacher wrote 1/2 x 10 = 10/2 on the board and stated, "leave
the "2' at the "door' and the "10' inside the 'house' while writing the following:

2)10


2 at door; 10 inside the house

Discussion. The teacher incorporated strategy instruction that focused on a
method for solving a particular problem. It is also recommended that instructors
teach a few rules (e.g., multiplication rule, divisibility rule) that are applicable to
many problem types. Ellis and Lenz (1996) also recommend more generalizable
strategies that are applicable within and/or across content areas. The
researchers asserted that three features are necessary within strategy instruction
to help facilitate independent learning. The first feature involves the content of
the strategy and refers to the specific steps needed to cue the learner to use
cognitive and metacognitive strategies, rules/or actions. Metacognitive
strategies are necessary to help students monitor and regulate their processes
while performing a task. Students can ask themselves questions while problem
solving (e.g., "Did I follow the correct procedures?'; 'Did I check the
reasonableness of my answer?').
The second feature, design, addresses the packaging of a strategy that
facilitates learning and ease of use. Such design features include the use of a
memory device, such as the first letter mnemonic strategy. Strategies should consist
of seven or fewer steps and have simple wording for ease of recall. For example,
one empirically validated memory device is the first-letter mnemonic, SOLVE (Miller,
1996, p. 353) in which each letter represents a specific step necessary for
successfully solving basic math facts: (a) See the sign; (b) Observe and answer; (c)
Look and draw; (d) Verify your answer; and (e) Enter your answer.
The third feature, usefulness, refers to the transferability of the strategy
across time, settings, and contexts. For example, the STAR strategy, is a first-







The Journal of Correctional Education 57(3) * September 2006
Maccini, et. al. Math Instruction for Committed Youth

letter mnemonic that cues students with disabilities to memorize and recall
effective problem solving skills. The strategy has been shown to help secondary
students across contexts (e.g., with integer numbers and problem solving) and
results have been maintained over time (Maccini & Hughes, 2000; Maccini 8
Ruhl, 2000). Specifically, the STAR steps prompt students to: (a) Search the word
problem (e.g., read the problem carefully); (b) Translate the problem into a
picture form (e.g., represent the problems via use of manipulatives or pictorial
display); (c) Answer the problem (e.g., solve for the solution); and (d) Review the
solution (e.g., check the reasonableness of their answer). The strategy steps and
substeps are also listed on a structured worksheet to help facilitate both
problem representation (i.e., the first two steps of STAR) and problem solution
(i.e., the last two steps of STAR).

Graduated Instructional Sequence
Secondary students with learning problems commonly experience difficulty with
problem solving tasks and perform at about the fifth grade level (Cawley &
Miller, 1989). Youth with LD and ED encounter difficulty with problem solving,
such as: (a) selecting which operations to use when solving for the solution; (b)
deciphering important from erroneous information embedded in word
problems; and (c) and actively participating in problem solving tasks (Miller,
1996). The graduated instructional sequence is a systematic approach to
teaching concepts and skills to ensure student understanding (Mercer, 1997).
The sequence includes three phases for assisting students as they move from
concept development to skill acquisition: (a) concrete (e.g., representing the
concept via objects); (b) semiconcrete (e.g., drawing pictures of the objects); and
(c) abstract (e.g., using numerical representations).

Example. The graduated instructional sequence was not observed in the school
observation. However, one example of how the concrete-abstract-abstract (CSA)
could have been applied is if teacher C had used the sequence with the lesson
objective on translating verbal phrases into math expressions in the
intermediate math class. For example, suppose the students were asked to
translate the following sentence:

Candy is saving money for an MP3 player. She has $24 in her savings
account on January 14 and deposits $3 per week. Write an expression that
gives the amount of money that Candy has after N weeks. How much
money will she have after saving for 6 weeks?







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At the concrete level, students can use the algebra tiles to help them to
represent the problems in pictorial or 3-dimensional form by representing the
number or constant and variable amounts:


ALGEBRA TILES:

L= 1 unit E = 5 units = 25 units I l = N (# of weeks)


I I -- I I Write the expression (24 + 3N)
I I D I I
I I E


After 6 weeks of saving money: N or E = I I then:

I D II
| i I l 24 + 3N = 24 + 3(6) = $42 saved after 3 weeks
I 1 I I
I I 0


At the semiconcrete level, students draw pictures of the representations.
Finally, at the abstract level, students write numerical representation: 24 + 7(3)
and then solve the problem Students practice a variety of problems per phase
until they reach mastery (80% or greater) prior to advancing within the
instructional sequence.

Discussion. To help students understand underlying concepts and become
active participants in their learning process, instruction should include a
graduated sequence. Incorporating CSA into math instruction is effective in
helping secondary students with LD and ED solve algebraic word problems
involving integer word problems, relational word problems, and complex
equations (Huntington, 1994; Maccini & Hughes, 2000; Maccini & Ruhl, 2000;
Witzel, Mercer, & Miller, 2003). The process helps learners understand the
fundamental math concepts prior to advancing to generalizations of rules, facts,
or algorithms.







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Maccini, et. al. Math Instruction for Committed Youth

Implications for Practice
Based on classroom observations and a review of validated teaching practices,
the following math instructional approaches are recommended to help meet
the unique needs of youth with LD and ED committed to juvenile corrections:

1. Include the use of advance organizers to help orient learners to the lesson
and to link the lesson to student interest and prior knowledge;
2. Incorporate principles of direct instruction (di) when designing and
implementing lessons;
3. Provide lessons and activities that use technology and embed mathematics
in real-world situations to foster student understanding of mathematics and
promote generalization beyond the classroom;
4. Integrate calculators within instruction and include training for using
calculators effectively;
5. Provide a variety of instructional groupings including whole group
instruction, small group instruction, collaborative and structured peer
activities, and individual work;
6. Incorporate use of a graduated instructional sequence that progresses from
use of concrete objects, to semiconcrete, to abstract representations to
promote student generalization;
7. Incorporate strategy instruction in mathematics to facilitate independent
student learning during mathematics representation and problem solution.

Summary
The instructional approaches described are effective ways for educators in
correctional settings to assist learners in mathematics. This is essential in light
of legislation advocating empirically-validated instructional practices for all
students (NCLB, 2001). Additionally, due to the high percentage of students that
are classified as ED and LD within juvenile correctional schools, it is imperative
that teachers rely on empirically-validated instructional math practices to ensure
students gain access to the general education curriculum and the full range of
educational opportunities available to students in public schools (Gagnon &
Mayer, 2003).

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Biographical Sketches
PAULA MACCINI, PH.D., Associate Professor, Department of Special Education, University
of Maryland, College Park, MD. Dr. Maccini's research and teaching interests focus on
teaching mathematics to secondary students with learning disabilities and
emotional/behavior disorders. Dr. Maccini is particularly interested in methods for
teaching math in alternative settings to students with mild disabilities.







The Journal of Correctional Education 57(3) * September 2006


JOSEPH CALVIN GAGNON, PH.D., Assistant Professor, Graduate School of Education,
George Mason University, Fairfax, Virginia. Dr. Gagnon's research interests Include
effective mathematics instruction and technology-based practices for students with
emotional disabilities and learning disabilities. Dr. Gagnon also conducts research on
curriculum, assessment, and accountability policies in day treatment programs, residential
schools, and Juvenile corrections.

CANDACE A. MULCAHY, M.ED., is a doctoral candidate In Special Education and works as
a faculty research assistant at the National Center on Education, Disability, and Juvenile
Justice (EDJJ) at the University of Maryland. Her research Interests include education
policies that apply to youth in corrections, provision of appropriate education and special
education services to marginalized youth, and effective reading and mathematics
instruction for youth with disabilities in public schools and for youth in corrections. She
has visited numerous education programs In juvenile and adult corrections facilities and
has consulted with state agencies on the provision of education services to incarcerated
youth.

PETER E. LEONE, PH.D., is a Professor in the Department of Special Education at the
University of Maryland. Dr. Leone is a leader in the field of special education and has
dedicated his time to youth with EBD and those involved In the juvenile justice system.
Dr. Leone has authored a number of book chapters, research reports, and monographs
related to students with disabilities and is currently director of the National Center on
Education, Disability, and Juvenile Justice.







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