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Mathematics Instruction for Students with Learning Disabilities or Difficulty Learning Mathematics: A Guide for Teachers

  • Instructional Research Group
  • Instructional Research Group


This guide for teachers is a companion piece to the meta-analysis from the Center on Instruction, "Mathematics Instruction for Students with Learning Disabilities or Difficulty Learning Mathematics: A Synthesis of the Intervention Research". Based on the findings of this report, seven effective instructional practices were identified for teaching mathematics to K-12 students with learning disabilities. It describes these practices and, incorporating recommendations from "The Final Report of The National Mathematics Advisory Panel" as well, specifies research-based recommendations for students with learning disabilities and for students who are experiencing difficulties in learning mathematics but are not identified as having a math learning disability. [To access "The Final Report of The National Mathematics Advisory Panel" see (ED500486) This publication was created by Instructional Research Group.]
A Guide for Teachers
Madhavi Jayanthi
Russell Gersten
Scott Baker
Instructional Research Group
A Guide for Teachers
The authors would like to express their appreciation to Becky
Newman-Goncher and Kelly Haymond for their contributions to
this publication.
This publication was created for the Center on Instruction by
Instructional Research Group. The Center on Instruction is
operated by RMC Research Corporation in partnership with
the Florida Center for Reading Research at Florida State
University; Instructional Research Group; the Texas Institute for
Measurement, Evaluation, and Statistics at the University of
Houston; and the Meadows Center for Preventing Educational
Risk at the University of Texas at Austin.
The contents of this document were developed under
cooperative agreement S283B050034 with the U.S.
Department of Education. However, these contents do
not necessarily represent the policy of the Department of
Education, and you should not assume endorsement by the
Federal Government.
Editorial, design, and production services provides by
RMC Research Corporation.
Preferred citation:
Jayanthi, M., Gersten, R., Baker, S. (2008).
instruction for students with learning disabilities or difficulty
learning mathematics: A guide for teachers.
Portsmouth, NH:
RMC Research Corporation, Center on Instruction.
To download a copy of this document, visit
Historically, mathematics instruction for students with learning disabilities and
at-risk learners has not received the same level of consideration and scrutiny
from the research community, policy makers, and school administrators as the
field of reading. A recent review of the ERIC literature base (Gersten, Clarke, &
Mazzocco, 2007) found that the ratio of studies on reading disabilities to
mathematics disabilities and difficulties was 5:1 for the years 1996–2005. This
was a dramatic improvement over the ratio of 16:1 in the prior decade. Even
though this is far from a large body of research, sufficient studies exist to
dictate a course of action.
Recently, the Center on Instruction conducted a meta-analysis on the topic
of teaching mathematics to students with learning disabilities (Gersten, Chard,
Jayanthi, Baker, Morphy, & Flojo, 2008). A meta-analysis is a statistical method
by which research studies on a particular method of instruction are summarized
to determine the effectiveness of that instructional method. A meta-analysis
helps combine findings from disparate studies to determine the effectiveness
of a particular method of instruction.
In the meta-analysis on teaching mathematics to students with learning
disabilities (LD), only studies with randomized control trials (RCTs) and high
quality quasi-experimental designs (QEDs) were included. In an RCT, the study
participants (or other units such as classrooms or schools) are randomly
assigned to the experimental and control groups, whereas in a QED, there
is no random assignment of participants to the groups.
Seven Effective Instructional Practices
Based on the findings of the meta-analysis report, seven effective instructional
practices were identified for teaching mathematics to K–12 students with
learning disabilities. In describing these practices, we have incorporated
recommendations from
The Final Report of the National Mathematics Advisory
(National Mathematics Advisory Panel, 2008) as well. This report
specified recommendations for students with learning disabilities
students who were experiencing difficulties in learning mathematics but were
not identified as having a math learning disability (i.e., at-risk). The seven
effective instructional practices in this document are supported by current
research findings. Other instructional practices may be effective, but there is, at
present, not enough high quality research to recommend their use at this time.
Some of the recommendations listed later in this document (e.g., teach
explicitly and use visuals) are age-old teaching practices. While there is nothing
new about these practices, research continues to validate them as effective
instructional practices for students with learning disabilities and at-risk students,
and continued use is warranted. Other instructional methods recommended
here, such as using multiple instructional examples and teaching multiple
strategies, have also been endorsed in studies that focused on reform-oriented
mathematics instruction in general education classes (e.g., Silver, Ghousseini,
Gosen, Charalambous, & Strawhun, 2005; Rittle-Johnson & Star, 2007). This
alignment of teaching methods between special education and general
education enables students with learning disabilities to learn meaningfully
from general education curricula in inclusive classrooms.
Mathematical Knowledge
Current mathematics researchers emphasize three areas of mathematical
abilities (e.g., Kilpatrick, Swafford, & Findell, 2001; Rittle-Johnson & Star, 2007;
Bottge, Rueda, LaRoque, Serlin, & Kwon, 2007). They are:
procedural knowledge,
procedural flexibility, and
conceptual knowledge.
Procedural knowledge
refers to knowledge of basic skills or the sequence
of steps needed to solve a math problem. Procedural knowledge enables a
student to execute the necessary action sequences to solve problems (Rittle-
Johnson & Star, 2007).
Procedural flexibility
refers to knowing the many different ways in which
a particular problem can be solved. Students with a good sense of procedural
flexibility know that a given problem can be solved in more than one way,
and can solve an unknown problem by figuring out a possible solution for
that problem.
Conceptual knowledge
is a grasp of the mathematical concepts and ideas
that are not problem-specific and therefore can be applied to any problem-
solving situation. Conceptual understanding is the over-arching understanding
of mathematical concepts and ideas that one often refers to as a “good
mathematical sense.”
It is reasonable to extrapolate from this small but important body of
research that such an emphasis would also benefit students with disabilities
and at-risk students. Recent studies have attempted to address reform-oriented
math instruction in special education settings. Researchers such as Woodward
(e.g., Woodward, Monroe, & Baxter, 2001) and Van Luit (Van Luit & Naglieri,
1999) have endeavored to address the issue of procedural flexibility in their
research by focusing on multiple strategy instruction—a recommendation in
this document, as mentioned earlier. Bottge (e.g., Bottge, Heinrichs, Mehta, &
Hung, 2002) has addressed the issue of procedural knowledge and conceptual
understanding by means of engaging, real-life, meaningful problem-solving
contexts; however the limited number of studies precludes any
recommendations at this time.
Effective Instruction at Each Tier
The current focus on Response to Intervention (RTI) as a tiered prevention and
intervention model for struggling mathematics learners also calls for evidence-
based instructional methods (Bryant & Bryant, 2008). While RTI models can
have three or more tiers (the most common being three tiers), they all share
the same objectives. For example, Tier 1 instruction, with an emphasis on
primary prevention, requires teachers to provide evidence-based instruction
to all students. Tier 2 focuses on supplemental instruction that provides
differentiated instruction to meet the learning needs of students. Tier 3
emphasizes individualized intensive instruction. The ultimate goal of the RTI
model is to reduce the number of students in successive tiers and the number
of students receiving intensive instruction. The groundwork for the success
of this model is the effectiveness of the instruction provided in Tier 1.The
evidence-based instructional strategies identified in this document need
to be part of the teaching repertoire of Tier 1 teachers. These validated
techniques, when implemented soundly, can effectively bring about student
gains in mathematics.
Our document guides K–12 teachers of students with disabilities and at-risk
students in their selection and use of effective mathematics instructional
methods. For each of the seven recommendations, we explain what works,
describe how the practice should be done, and summarize the evidence
supporting the recommendation.
Recommendation 1:
Teach students using
explicit instruction
on a regular basis.
Explicit instruction, a mainstay feature in many special education programs, includes teaching
components such as:
clear modeling of the solution specific to the problem,
thinking the specific steps aloud during modeling,
presenting multiple examples of the problem and applying the solution to the problems, and
providing immediate corrective feedback to the students on their accuracy.
When teaching a new procedure or concept, teachers should begin by modeling and/or thinking
aloud and working through several examples. The teacher emphasizes student problem solving
using the modeled method, or by using a model that is consonant with solid mathematical
reasoning. While modeling the steps in the problem (on a board or overhead), the teacher should
verbalize the procedures, note the symbols used and what they mean, and explain any decision
making and thinking processes (for example, “That is a plus sign. That means I should…”).
Teachers should model several problems with different characteristics (Rittle-Johnson & Star,
2007; Silbert, Carnine, & Stein, 1989). A critical technique is assisted learning where students
work in pairs or small groups and receive guidance from the teacher. During initial learning and
practice, the teacher provides immediate feedback to prevent mistakes in learning and allows
students to ask questions for clarification.
According to
The Final Report of the National Mathematics Advisory Panel
Mathematics Advisory Panel, 2008), explicit systematic instruction improves the performance of
students with learning disabilities and students with learning difficulties in computation, word
problems, and transferring known skills to novel situations. However, the panel noted that while
explicit instruction has consistently shown better results, no evidence supports its
for teaching students with learning disabilities and difficulties. The panel recommends that all
teachers of students with learning disabilities and difficulties teach explicitly and systematically on
a regular basis to some extent and not necessarily all the time.
Summary of Evidence to Support Recommendation 1
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 11 studies in the area of explicit instruction (10 RCTs and 1 QED). The mean effect
size of 1.22 was statistically significant (p<.001; 95% CI = 0.78 to 1.67).
National Mathematics Advisory Panel
The panel reviewed 26 high quality studies (mostly RCTs) on effective instructional approaches for
students with learning disabilities and low-achieving students. Explicit Systematic Instruction is
identified in
The Final Report of the National Mathematics Advisory Panel
as one of the defining
features of effective instruction for students with learning disabilities (National Mathematics
Advisory Panel, 2008).
RCT = Randomized control trial. QED = Quasi-experimental design. CI= Confidence interval.
Recommendation 2:
Teach students using
multiple instructional examples
Example selection in teaching new math skills and concepts is a seminal idea that is strongly
emphasized in the effective instruction literature (e.g., Ma, 1999; Rittle-Johnson & Star, 2007;
Silbert, Carnine, & Stein, 1989). Teachers need to spend some time planning their mathematics
instruction, particularly focusing on selecting and sequencing their instructional examples. The
goal is to select a range of multiple examples of a problem type. The underlying intent is to
expose students to many of the possible variations and at the same time highlight the common
but critical features of seemingly disparate problems. For example, while teaching students to
divide a given unit into half, a variety of problems can be presented that differ in the way the
critical task of half is addressed in the problems (i.e., use the symbol for half; use the word half,
use the word one-half; etc.) (Owen & Fuchs, 2002).
Multiple examples can be presented in a specified sequence or pattern such as concrete to
abstract, easy to hard, and simple to complex. For example, fractions and algebraic equations can
be taught first with concrete examples, then with pictorial representations, and finally in an
abstract manner (Butler, Miller, Crehan, Babbitt, & Pierce, 2003; Witzel, Mercer, Miller, 2003).
Multiple examples can also be presented by systematically varying the range presented (e.g.,
initially teaching only proper fractions vs. initially teaching both proper and improper fractions).
Sequencing of examples may be most important during early acquisition of new skills when
scaffolding is needed for student mastery and success. The range of examples taught is probably
most critical to support transfer of learned skills to new situations and problems. In other words, if
the teacher teaches a wide range of examples, it will result in the learner being able to apply a
skill to a wider range of problem types. Both of these planning devices (sequence and range)
should be considered carefully when teaching students with LD.
Summary of Evidence to Support Recommendation 2
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 9 studies on range and sequence of examples (all RCTs). The mean effect size of
0.82 was statistically significant (p<.001; 95% CI = 0.42 to 1.21).
National Mathematics Advisory Panel
The panel reviewed 26 high quality studies (mostly RCTs) on effective instructional approaches
for students with learning disabilities and low-achieving students. The panel recommends that
teachers, as part of explicit instruction, carefully sequence problems to highlight the critical features
of the problem type (National Mathematics Advisory Panel, 2008).
Recommendation 3:
Have students
verbalize decisions and solutions
to a math problem.
Encouraging students to verbalize, or think-aloud, their decisions and solutions to a math problem
is an essential aspect of scaffolded instruction (Palincsar, 1986). Student verbalizations can be
problem-specific or generic. Students can verbalize the specific steps that lead to the solution of
the problem (e.g., I need to divide by two to get half) or they can verbalize generic heuristic steps
that are common to problems (e.g., Now I need to check my answer). Students can verbalize the
steps in a solution format (First add the numbers in the
column. Write down the answer.
Then add numbers in the
column…) (Tournaki, 2003) or in a self-questioning/answer format
(What should I do first? I should…) (Pavchinski, 1998). Students can verbalize during initial
learning or as they are solving, or have solved, the problem.
Many students with learning disabilities are impulsive behaviorally and when faced with multi-
step problems frequently attempt to solve the problems by randomly combining numbers rather
than implementing a solution strategy step-by-step. Verbalization may help to anchor skills and
strategies both behaviorally and mathematically. Verbalizing steps in problem solving may address
students’ impulsivity directly, thus suggesting that verbalization may facilitate students’ self-
regulation during problem solving.
Summary of Evidence to Support Recommendation 3
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 8 studies in the area of student verbalizations (7 RCTs and 1 QED). The mean effect
size of 1.04 was statistically significant (p<.001; 95% CI = 0.42 to 1.66).
National Mathematics Advisory Panel
The panel reviewed 26 high quality studies (mostly RCTs) on effective instructional approaches
for students with learning disabilities and low-achieving students. The panel recommends that
teachers, as part of explicit instruction, allow students to think aloud about the decisions they make
while solving problems (National Mathematics Advisory Panel, 2008).
Recommendation 4:
Teach students to
visually represent the information
in the math problem.
Visual representations (drawings, graphic representations) have been used intuitively by teachers
to explain and clarify problems and by students to understand and simplify problems. When used
systematically, visuals have positive benefits on students’ mathematic performance.
Visual representations result in better gains under certain conditions. Visuals are more
effective when combined with explicit instruction. For example, teachers can explicitly teach
students to use a strategy based on visuals (Owen & Fuchs, 2002). Also, students benefit more
when they use a visual representation prescribed by the teacher rather than one that they self-
select (D. Baker, 1992). Furthermore, visuals that are designed specifically to address a particular
problem type are more effective than those that are not problem specific (Xin, Jitendra, &
Deatline-Buchman, 2005). For example, in the study by Xin and her colleagues, students first
identified what type of problem they had been given (e.g., proportion, multiplicative) and then
used a corresponding diagram (taught to them) to represent essential information and the
mathematical procedure necessary to find the unknown. Then they translated the diagram
into a math sentence and solved it.
Finally, visual representations are more beneficial if not only the teacher, but both the teacher
and the students use the visuals (Manalo, Bunnell, & Stillman, 2000).
Summary of Evidence to Support Recommendation 4
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 12 studies on visual representations (11 RCTs and 1 QED). The mean effect size of
0.47 was statistically significant (p<.001; 95% CI = 0.25 to 0.70).
National Mathematics Advisory Panel
The panel reviewed 26 high quality studies (mostly RCTs) on effective instructional approaches
for students with learning disabilities and low-achieving students. According to
The Final Report
of the National Mathematics Advisory Panel
visual representations when combined with explicit
instruction tended to produce significant positive results (National Mathematics Advisory
Panel, 2008).
Recommendation 5:
Teach students to solve problems using
multiple/heuristic strategies
Instruction in multiple/heuristic strategies is part of a contemporary trend in mathematics
education (e.g., Star & Rittle-Johnson, in press). Using heuristics shows some promise with
students with learning disabilities. Multiple/heuristic strategy instruction has been used in
addressing computational skills, problem solving, and fractions.
A heuristic is a method or strategy that exemplifies a generic approach for solving a problem.
For example, a heuristic strategy can include steps such as “Read the problem. Highlight the key
words. Solve the problems. Check your work.” Instruction in heuristics, unlike direct instruction,
is not problem-specific. Heuristics can be used in organizing information and solving a range of
math problems. They usually include student discourse and reflection on evaluating the alternate
solutions and finally selecting a solution for solving the problem. For example, in the Van Luit and
Naglieri (1991) study, the teacher first modeled several strategies for solving a computational
problem. However, for most of the lesson, the teacher’s task was to lead the discussion in the
direction of using strategies and to facilitate the discussion of the solutions provided by the
students. Each student was free to select a strategy for use, but the teacher assisted the children
in discussion and reflection about the choices made.
Similarly, in the Woodward (2006) study, students were taught multiple fact strategies. Daily
lessons consisted of introduction of new strategies or review of old strategies. Students were not
required to memorize the strategies. They were, however, encouraged to discuss the strategy
and contrast it with previously taught strategies. For example, students were shown that since 9
X 5 has the same value as 5 X 9, they were free to treat the problem as either nine fives or five
nines. They also were shown that this was equivalent to 10 fives minus one five, and that this
could be a faster way to do this problem mentally. Thus a variety of options were discussed with
the students.
Summary of Evidence to Support Recommendation 5
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 4 studies in the area of multiple/heuristic strategy instruction (3 RCTs and 1 QED).
The mean effect size of 1.56 was statistically significant (p<.001; 95% CI = 0.65 to 2.47).
Recommendation 6:
ongoing formative assessment data and feedback
to teachers.
Ongoing formative assessment and evaluation of students’ progress in mathematics can help
teachers measure the pulse and rhythm of their students’ growth and also help them fine-tune
their instruction to meet students’ needs. Teachers can administer assessments to their group
of students and then a computer can provide them with data depicting students’ current
mathematics abilities.
Providing teachers with information regarding their students’ progress in mathematics has
beneficial effects on the mathematics performance of those same students. However, greater
benefits on student performance will be observed if teachers are provided with not only
performance feedback information but also instructional tips and suggestions that can help
teachers decide what to teach, when to introduce the next skill, and how to group/pair students.
For example, teachers can be given a set of written questions to help them use the formative
assessment data for adapting and individualizing instruction. These written questions could
include “On what skill(s) has the student improved compared to the previous two-week period?”
or “How will I attempt to improve student performance on the targeted skill(s)?”
Teachers can respond to these questions and address them again when new assessment
data becomes available (Allinder, Bolling, Oats, & Gagnon, 2000). Teachers can also be provided
with a specific set of recommendations to address instructional planning issues such as which
mathematical skills require additional instructional time for the entire class, which students require
additional help via some sort of small group instruction or tutoring, and which topics should be
included in small group instruction (Fuchs, Fuchs, Hamlett, Phillips, & Bentz, 1994).
Summary of Evidence to Support Recommendation 6
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 10 studies on formative assessment (all RCTs). The mean effect size of 0.23 was
statistically significant (p<.01; 95% CI = 0.05 to 0.41).
National Mathematics Advisory Panel
The panel reviewed high quality studies on formative assessment and noted that formative
assessment use by teachers results in marginal gains for students of all abilities. When teachers
are provided with special enhancements (suggestions on how to tailor instruction based on data),
significant gains in mathematics are observed (National Mathematics Advisory Panel, 2008).
Recommendation 7:
peer-assisted instruction
to students.
Students with LD sometimes receive some type of peer assistance or one-on-one tutoring in
areas in which they need help. The more traditional type of peer-assisted instruction is cross-age,
where a student in a higher grade functions primarily as the tutor for a student in a lower grade.
In the newer within-classroom approach, two students in the same grade tutor each other. In
many cases, a higher performing student is strategically placed with a lower performing student
but typically both students work in both roles: tutor (provides the tutoring) and tutee (receives
the tutoring).
Cross-age peer tutoring appears to be more beneficial than within-class peer-assisted learning
for students with LD. It could be hypothesized that students with LD are too far below grade level
to benefit from feedback from a peer at the same grade level. It seems likely that within-class
peer tutoring efforts may fall short of the level of explicitness necessary to effectively help
students with LD progress, whereas older students (in cross-age peer tutoring settings)
could be taught how to explain concepts to a student with LD who is several years younger.
Interestingly, within-class peer-assisted instruction does appear to help low-achieving students
who have learning difficulties in mathematics (Baker, Gersten, & Lee, 2002). One possible
explanation seems to be that there is too much of a gap in the content knowledge between
students with learning disabilities compared to low-achieving students with learning difficulties,
thus enabling peer tutoring to be more beneficial to students at risk than to students with LD.
Summary of Evidence to Support Recommendation 7
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 2 studies in the area of cross age peer tutoring (both RCTs).
The mean effect size of 1.02 was statistically significant (p<.001; 95% CI = 0.57 to 1.47).
1: Teach students using
explicit instruction
on a regular basis.
2: Teach students using
multiple instructional examples.
3: Have students
verbalize decisions and solutions
to a
math problem.
4: Teach students to
visually represent the information
the math problem.
5: Teach students to solve problems using
heuristic strategies.
6: Provide
ongoing formative assessment data and feedback
to teachers.
7: Provide
peer-assisted instruction
to students.
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... Because of these various entry and exit points, high-quality tasks are inherently differentiated as they provide multiple avenues for students with LD in mathematics and their typical peers to engage in the same task. This produces learning opportunities where various solving processes and strategies are modeled and shared (Jayanthi et al., 2008). ...
... Potential questions may include (a) "what strategy might you use?," (b) "what information is important in the problem?," or (c) "what might be your first step?." Monitoring can also include providing students with LD in mathematics or students who are struggling with more systematic and explicit instruction as needed (Gersten et al., 2009). Here, teachers can provide more concrete and specific instructional support such as, "maybe you could try drawing a picture to help you," thereby leveraging the effective teaching practices of explicit instruction and visual representations of problems (Gersten et al., 2009;Jayanthi et al., 2008). Monitoring also allows teachers the time and space to select strategies to share in the whole-group discussion and determine the most appropriate sequencing of said strategies (i.e., Practices 3 and 4; Smith & Stein, 2018). ...
... 3. Whole group work: Note students' understanding and providing support as needed. In addition to monitoring student thinking, it is important to consider how to keep track of their thinking and collect formative data to use for future instructional decisions (Gersten et al, 2008;Jayanthi et al., 2008). ...
Classroom instruction focused on discussion-based learning opportunities can provide productive and inclusive learning experiences for all students, including students with learning disabilities in mathematics and those without learning disabilities. Mathematical discourse allows students to share their ideas, justify their thinking, critique the reasoning of others, and refine their thought processes. While one might typically envision mathematical discourse happening during face-to-face instruction, meaningful discourse can also occur in online learning environments. This article presents a blended format of both synchronous and asynchronous learning opportunities, coupled with Smith and Stein’s (2018) “5 Practices” for productive mathematical discourse to support teachers in designing and facilitating lessons in which all students are actively engaged in the learning processes both for themselves and their classmates.
... To provide support, teachers can use explicit instruction along with visual representations linked to math schemas to facilitate a better understanding of mathematical concepts (Gersten et al., 2009). For example, teachers can link concepts in word problems to math equations using visual representations, model how to solve the problem, and provide worked examples with related equations (Jayanthi et al., 2008). ...
... Students who use metacognitive skills, such as self-evaluation, self-monitoring, and self-questioning, are likely to perform better in math problem solving (Rosenzweig et al., 2011). Research has shown positive effects for supporting students with and without MLD to think aloud and verbalize problem-solving strategies (Jayanthi et al., 2008). When attending to UDL, teachers can provide alternative modalities (e.g., write, speak, draw) for students with varying needs to express how they solve the math problem in an inclusive setting. ...
Mathematical problem solving is a complex cognitive activity, which poses difficulties for students with and without disabilities in inclusive learning environments. With a variety of functions, Learning Management Systems (LMSs) have the potential to enhance personalized learning to meet the diverse needs of all students. This paper provides teachers guidance on using LMSs to implement evidence-based practices for math problem solving in an online learning environment. This paper introduced multiple functions commonly available in most LMSs, such as quiz, multimedia content editor, Learning Tools Interoperability (LTI), and learning analytics. Guidance is provided to teachers to leverage these features to maximize student learning experiences, such as engaging in multimedia learning activities, interacting with the teacher and peers, and receiving tailored feedback.
... Lintuman and Wijaya (2020) also found that teachers often only focused on the material presented without allowing the students to express what they knew and understood. It can also reveal can give instructions to scaffold students' thinking (Jayanthi et al., 2008;Rasmitadila et al., 2020) when students still do not understand the concept. Hence, the teachers know what they should do or improve their teaching and learning. ...
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Remote learning that must be carried out by schools in Indonesia during the COVID-19 pandemic was a challenge for most teachers. This study aimed to investigate mathematical conversations maintained by the teachers in remote learning of mathematics during the COVID-19 pandemic. It was a descriptive qualitative study that was analyzed based on school time during the pandemic, online teaching platforms used, the way the teachers carry out remote learning of mathematics, and how the teachers and the students interact. The data were collected from three mathematics teachers and eight students in a public senior high school in Bekasi District, Indonesia. They were interviewed by phone, and then the results of the interview were complemented by several captures of the students' conversations when learning mathematics via WhatsApp. This study revealed that remote learning of mathematics was still dominated by a rigid or less interactive learning environment, which could be seen from how the teachers gave mathematics problems, questions, and instructions, as well as how they responded to student questioning.
As students enter the upper elementary grades, word problems become a main component of mathematics instruction, increasing in complexity as students advance through the curriculum. For students identified as emergent bilinguals with mathematics difficulty (MD), the linguistic complexity inherent in word problems may serve as a barrier to word‐problem proficiency. The current study investigated the potential relation between academic English proficiency and word‐problem outcomes for emergent bilinguals with MD. After analyzing data from 241 third‐grade students, results indicated students who participated in an evidence‐based word‐problem intervention outperformed students who did not receive the intervention. Moreover, students’ academic English‐language proficiency scores in the domains of reading and writing positively correlated with higher scores on a measure of word‐problem solving.
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ملخص: : هدفت هذه الدراسة الوصفية إلى الكشف عن مدى تضمين مستويات التفكير الهندسي في البرنامج التربوي الفردي من وجهة نظر معلمات برنامج صعوبات التعلم. وقد تكونت عينة الدراسة من 216 معلمة من مختلف المحافظات في سلطنة عُمان. حيث طُبق عليهنّ مقياس مستويات التفكير الهندسي الذي طُوّر وفق نموذج فان هيل. وقد تألف المقياس من 45 فقرة وزعت على خمسة مستويات (التصوري، والتحليلي، والاستدلالي غير الشكلي، والاستدلالي الشكلي، والتجريدي). وقد أشارت النتائج إلى أنّ مدى تضمين مستويات التفكير الهندسي في البرنامج التربوي الفردي جاء بمستوى متوسط على جميع مستويات المقياس. كما أشارت النتائج إلى وجود فروق ذات دلالة إحصائية في مستويات التفكير الهندسي الاستدلالي غير الشكلي، والاستدلالي الشكلي، والتجريدي وفق متغير المؤهل العلمي لصالح حملة البكالوريوس. كما أظهرت النتائج وجود فروق على جميع مستويات التفكير الهندسي وفق متغير الخبرة التدريسية لصالح فئة 10 سنوات فأقل. وقد أوصت الدراسة بتضمين موضوعات خاصة بالهندسة والتفكير الهندسي في البرنامج التربوي الفردي.
Araştırmanın amacı, mesleğe yeni başlamış ilköğretim matematik öğretmenlerinin lisans döneminde gördükleri özel eğitim dersi ve kaynaştırma yoluyla matematik eğitiminde yapılan uygulamalar hakkındaki görüşlerinin belirlenmesidir. Araştırmanın yöntemi nitel araştırma yöntemlerinden durum çalışması modelidir. Araştırmanın çalışma grubunu Kars ilinde bulunan meslekte en fazla 6 yıl çalışmış toplam 10 ilköğretim matematik öğretmeni oluşturmaktadır. Araştırmacı tarafından geliştirilen yarı yapılandırılmış görüşme formu ilköğretim matematik öğretmenlerine uygulanarak veriler toplanmıştır. Verilerin analizi, nitel veri analizi yaklaşımlarından içerik analizi ile yapılmıştır. Araştırma sonuçlarına bakıldığında ilköğretim matematik öğretmenlerinin lisans döneminde aldıkları özel eğitim dersinin teorik olarak kaldığı ve uygulama konusunda eksiklikler yaşadıkları görülmüştür. Ayrıca öğretmenlerinin kaynaştırma yoluyla matematik eğitimi konusunda uygulanacak yöntemler, etkinlikler konusunda ve sınıf içi uygulamalar hakkında desteğe ihtiyaç duydukları tespit edilmiştir.
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Executive summary/Abstract Adaptive teaching and individualization for k-12 students improves academic achievement: A meta-analysis of classroom studies Background The question of how to best deliver instruction to k-12 students has dominated the educational conversation, both in terms of theory and practice, since before 1960. Two predominant models have clashed: 1) traditional teacher-directed instruction (referred to here as teacher-centered T-C instruction), where there is little methodological adaptation for individual differences in ability, skills, interests, etc. among students; and 2) so-called student-centered instruction (referred to here as S-C instruction), deriving much of its theoretical justification and methodological intricacies from constructivist thought embodied in the works of Jean Piaget, Lev Vygotsky, Jerome Burner, and many others. While radical constructivism has never become dominant in k-12 schooling (except in a relatively small number of demonstration schools), there has been considerable interest in embedding some of the principles of constructivism into k-12 schooling. This is often referred to as individualized or adaptive instruction, meaning an operational concern for individual students, their abilities, interests, etc., which is nearly the opposite of T-C instruction. A great deal of research has demonstrated that approaches to individualism, such as mastery learning, collaborative and cooperative learning, problem-based learning, peer tutoring, and computer-based instruction, are effective in promoting achievement and attitudinal gains, as contrasted with T-C instruction, where mastery of content or subject matter is of the greatest concern, and the teacher is the ‘delivery mechanism.’ More recently, this has been extended to include video-based lectures often delivered through the internet, as proposed by proponents of blended learning and its variant the flipped classroom (e.g., Baepler, Walker, & Driessen (2014). Research has also demonstrated that T-C instruction is particularly useful in developing basic skills in areas such as reading, spelling, and math, (Stockard, Wood, Coughlin & Khoury, 2018). More recent theory and practice concerning T-C and S-C instruction suggests that neither perspective is entirely sufficient and that some combination of T-C and S-C instruction is possibly more productive. This notion of combined teaching methods (i.e., T-C plus S-C) is one of the defining characteristics of the flipped classroom (Baepler et al., 2014). Certainly, students need to acquire skills and knowledge, but they also need to develop their own personal preferences, creativity, problem-solving abilities, and evaluative and self-evaluative perspectives. The current meta-analysis aims to determine if the advantage endowed by S-C instruction also affects to content achievement (i.e., content achievement is the outcome measure in this meta-analysis). The current meta-analysis was designed to explore teaching and learning in k-12 classrooms and the achievement benefit that derives from more S-C versus less S-C classrooms. Several perspectives informed the basis for the research approach described here, but none more so than the words of Gersten et al. (2008) while exploring through meta-analysis the question of T-C versus S-C instructional practices in elementary mathematics instruction. In the final report of their study, the group stated: “The Task Group found no examples of studies in which learners were teaching themselves or each other without any teacher guidance; nor did the Task Group find studies in which teachers conveyed … content directly to learners without any attention to their understanding or response. The fact that these terms, in practice, are neither clearly nor uniformly defined, nor are they true opposites, complicates the challenge of providing a review and synthesis of the literature …” (p. 12). The current meta-analysis intends to investigate variations of more versus less S-C instruction and the four domains of the instructional process in which they are more or less profitable. Objectives (research questions) There are three primary objectives that this meta-analysis intends to address (research questions that this study explores): • Overall, does more S-C instructional practices lead to a significant advantage in the acquisition of content (subject matter) knowledge (i.e., measured learning achievement)? • Do any of the four primary (substantive) moderator variables (entered into multiple meta-regression), Teacher’s Role, Pacing, Adaptability, and Flexibility, predict an increase or decrease in achievement across degrees of S-C use (From less S-C to more S-C)? • Is there a difference in categorical levels of less S-C to more S-C for each of the dimensions of instructional practice listed above, tested in mixed moderator variable analysis? • Do any of the secondary (demographic) moderator variables interact with each other (i.e., combine) to produce more versus less S-C instructional practices? Search methods Following the guidelines of the Campbell Collaboration (Kugley et al., 2017), in order to retrieve a broad base of studies to review, we started by having an experienced Information Specialist search across an array of bibliographic databases, both in the subject area and in related disciplines. The following databases were searched for relevant publications: ABI/Inform Global (ProQuest), Academic Search Complete (EBSCO), ERIC (EBSCO), PsycINFO (EBSCO), CBCA Education (ProQuest), Education Source (EBSCO), Web of Knowledge, Engineering Village, Francis, ProQuest Dissertations & Theses Global, ProQuest Education Database, Linguistics and Language Behavior Abstracts (ProQuest). The search strategy was tailored to the features of each database, making use of database-specific controlled vocabulary and search filters, but based on the same core key terms. Searches were limited to the year 2000 to 2017, and targeted a k-12 population. Database searching was supplemented by using the Google search engine to locate additional articles, but principally grey literature (research reports, conference papers, theses and research published outside conventional journals). Selection criteria The overall set of inclusion/exclusion criteria (i.e., selection) for the meta-analysis contained the following requirements: • Be publicly available (or archived) and encompass studies from 2000 to the present; • Feature at least two groups of different instructional strategies/practices that can be compared according to the research question as S-C and T-C instruction; • Include course content and outcome measures that are compatible in the groups that form these comparisons; • Contain sufficient descriptions of major instructional events in both instructional conditions; • Satisfy the requirements of either experimental or high-quality quasi-experimental design; • Be conducted in formal k-12 educational settings eventually leading to a certificate, diploma, degree, or promotion to a higher grade level; • Contain legitimate measures of academic achievement (i.e., teacher/researcher-made, standardized); and • Contain sufficient statistical information for effect size extraction. Data collection and analysis Effect size extraction and calculation One of the selection criteria was “Contain sufficient statistical information for effect size extraction,” so that an effect size could be calculated for each independent comparison. This information could take several forms (in all cases sample size data were required): • Means and standard deviations for each treatment and control group; • Exact t-value, F-value, with an indication of the ± direction of the effect; • Exact p-value (e.g., p = .011), with an indication of the ± direction of the effect; • Effect sizes converted from correlations or log odds ratios; • Estimates of the mean difference (e.g., adjusted means, regression β weight, gain score means when r is unknown) • Estimates of the pooled standard deviation (e.g., gain score standard deviation, one-way ANOVA with three or more groups, ANCOVA); • Estimates based on a probability of a significant t-test using α (e.g., p < .05); and • Approximations based on dichotomous data (e.g., percentages of students who succeeded or failed the course requirements). Effect sizes were initially calculated as Cohen’s (Cohen, 1988) and then converted to Hedges’ (i.e., correction for small samples; Hedges & Olkin, 1985). Standard errors ( ) were calculated for and then converted to standard errors of applying the correction formula for . Hedges’ , and sample sizes (i.e., treatment and control) were entered into Comprehensive Meta-Analysis 3.3.07 (Borenstein, Hedges, Higgins, & Rothstein, 2014) where statistical analyses were performed. The effect sizes were coded for precision and these data were analyzed in moderator variable analysis. Statistical analyses Analyses were conducted using the following statistical tests: • Overall weighted random effects analysis with the statistics of , , , upper and lower limits of the 95th confidence interval, , and p-value; • Homogeneity is estimated using Q-Total, df, and p-value. I2 (i.e., percentage of error variation) and tau2 (i.e., average heterogeneity) is also calculated and reported. • Meta-regression (single and multiple) is used to determine the relationship between covariates and effect sizes; and • Mixed-model (i.e., random and fixed) moderator variable analysis is used to compare levels (categories) of each coded moderator variable. Q-Between, df, and p-value are used to make decisions about the significance of each categorical variable. Results The results are presented here in relationship to the four research questions previously described. • Question 1: Overall, does more S-C instructional practices lead to a significant advantage in the acquisition of content (subject matter) achievement (i.e., measured learning). • Result: Answering the basic question, more S-C instructional conditions (i.e., the treatment described above) outperform less S-C to a moderate extent. The average effect, = 0.44, k = 365, z =4.56, p < .00, SE = 0.03, Q = 3,095.89, I2 =88.22, tau2 = 0.27, between the mean of the > S-C treatment and the < S-C control, suggesting that teachers who promote and enact active, student-based classroom processes (more S-C instruction), can expect to see better student achievement than in classrooms where teachers employ less-student-based (less S-C) instruction. Also, a linear trend was found in meta-regression when Hedges’ was regressed on degree of S-C instruction ( = 0.04, SE = 0.02, z = 2.41, p = .032). The distribution remains significantly heterogeneous. • Question 2: Do any of the four moderator variables (entered into multiple meta-regression), Teacher’s Role, Pacing, Adaptability, and Flexibility, predict an increase or decrease in achievement across degrees of S-C use (From less S-C to more S-C)? Result: In meta-regression, Teacher’s role produces a significant linear trend ( = 0.06, SE = 0.04, z = 4.42, p < .001) and Pacing ( = -0.14, SE = 0.04, z = 3.18, p = .002). Adaptability, and Flexibility are not significant (p > .05). However, the trend for Teacher’s role and Pacing is opposite (note the opposite signs on ). Teacher’s role is significantly positive (i.e., more S-C instruction produced higher achievement), while Pacing produces the reverse (i.e., a significantly negative trend). For Pacing, more S-C methods produce lower achievement. Question 3: Do any of the moderator variables interact with each other (i.e., combine) to produce more versus less S-C instructional practices? • Result: Yes, Teacher’s Role compared to two dimensions added to the Teacher’s Role produce significantly different results (Q-Between = 7.76, df = 3, p = .02: Teacher’s Role and Teacher’s Role plus Adaptability significantly outperformed Teacher’s Role plus Flexibility. • Question 4: Is there a difference in categorical levels of less S-C to more S-C for each of the dimensions of instructional practice listed above, tested in mixed moderator variable analysis? Result: Only one of five moderator variables produced a significant differentiation among levels. Among four moderator variables (i.e., grade level; STEM vs. Non-STEM subjects; individual subjects; and ability profile) only ability profile significantly differentiated among levels. Special education students demonstrated significantly higher achievement compared to the General population of students. Authors’ conclusions This meta-analysis provides strong evidence that S-C instruction leads to improvements in learning with k-12 students. Not only is the overall random effects average effect size of medium strength ( = 0.44), but there is also a demonstrated (subtle but significant) linear relationship between more S-C classroom instruction and effect size (p = .03). Taken together, these results support the efficacy of allowing students to engage in active learning or other forms of S-C as part of a comprehensive educational experience
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The purpose of this meta-analysis was to synthesize findings from 42 interventions (randomized control trials and quasi-experimental studies) on instructional approaches that enhance the mathematics proficiency of students with learning disabilities. We examined the impact of four categories of instructional components: (a) approaches to instruction and/or curriculum design, (b) formative assessment data and feedback to teachers on students' mathematics performance, (c) formative data and feedback to students with LD on their performance, and (d) peer-assisted mathematics instruction. All instructional components except for student feedback with goal-setting and peer-assisted learning within a class resulted in significant mean effects ranging from 0.21 to 1.56. We also examined the effectiveness of these components conditionally, using hierarchical multiple regressions. Two instructional components provided practically and statistically important increases in effect size–teaching students to use heuristics and explicit instruction. Limitations of the study, suggestions for future research, and applications for improvement of current practice are discussed.
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The acquisition and refinement of cognitive strategies are described as a collaborative effort between teachers and students that is facilitated by scaffolded instruction. Although dialogue does not currently have a preeminent role in our classrooms, it can promote the kinds of opportunities necessary for the teacher to provide scaffolded instruction. To support and illustrate this point, a program of research investigating the use of dialogue to teach comprehension strategies is reviewed with particular attention to its extension to first-grade students at risk for academic difficulty. Transcripts from this research are presented to capture the quality of dialogue that fosters scaffolded instruction.
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This study investigated the effects of process mnemonic (PM) instruction on the computational skills performance of 13- to 14-year-old students with mathematics learning disabilities. Two experiments are described. In Experiment 1, 29 students were randomly assigned to one of four instruction groups: PM, demonstration-imitation (DI), study skills (SS), or no instruction (NI). In Experiment 2, instructors with no vested interest in the outcomes of the study were employed to teach 28 students who were assigned to PM, DI, or NI groups. Both PM and DI students made significant improvements in addition, subtraction, multiplication, and division. However, improvements were often greater for PM students. More importantly, the improvements made by PM students maintained better than those of DI students over six-week (Experiment 1) and eight-week (Experiment 2) follow-up periods.
This study examined the effectiveness of innovative curriculum-based measurement (CBM) classwide decision-making structures within general education mathematics instruction, with and without recommendations for how to incorporate CBM feedback into instructional planning. Forty general educators, each of whom had at least one student with an identified learning disability for math instruction, were randomly assigned to three groups: CBM with classwide reports that summarized information and provided instructional recommendations, CBM with reports but without recommendations, and contrast (no CBM). Results indicated that only the CBM teachers who received instructional recommendations designed better instructional programs and effected greater achievement for their students.
Technical Report
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Increased attention is being paid to students who demonstrate difficulty in learning and applying mathematics concepts. The purpose of this special series was to address issues related to students and mathematics learning disabilities (LD). We identify Response to Intervention (RtI) as it relates to early mathematics instruction and a multi-tiered service delivery system. Further, because RtI has focused primarily on young children and the prevention of LD, we present information about older students who have been identified as having mathematics LD and provide strategies for helping them access the general education curriculum. Six papers on various mathematics topics, grade levels, and service delivery will be provided in this special series. Authors report findings on research efforts and offer implications for practice.
The purpose of this study was to examine the effects of strategy instruction on the mathematical problem solving of 3rdgrade students with learning disabilities. Participants were 24 students whose teachers were randomly assigned to 4 conditions: (a) control, (b) acquisition, (c) low-dose acquisition plus transfer, or (d) full-dose acquisition plus transfer. During the 3-week study, students in each experimental group received instruction on a 6-step procedure for solving word problems that required finding half of a number. Across Groups A, B, and C, treatment comprised explicit instruction with heavy use of worked examples and practice with a higher achieving classmate. Analyses of variances were conducted on improvement between pre- and posttreatment measures in terms of number of problems solved correctly and amount of work showing the steps taught in the treatment. For problems solved correctly, statistically significant improvement favored the full-dose acquisition plus transfer group over the control and over the low-dose acquisition plus transfer groups. For amount of work, significant differences favored the low-dose acquisition plus transfer and full-dose acquisition plus transfer groups over the control group. Student and teacher attitudes about the instructional strategy and working with a partner were positive. Mathematical strategy instruction and pairing students for instruction is discussed with respect to directions for practice and future research.