AGT Contoh Kajian Maths Achievement

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Making the Connection between Literacy and Mathematics Achievement

Asha Jitendra, Ph.D.

Professor of Special Education

Lehigh University

E-mail: [email protected]

Middle Grades Literacy Forum

National Governors Association (NGA)

Phoenix, AZ

mailto:[email protected]

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Background• Media reports about student’s mathematics

performance include inadequate teaching, poorly designed curricula, and low test scores [National Research Council (National Research Council, NRC, 2001].

• Mathematics instruction has not received as much in-depth study and analysis as reading even though mathematics competence is emphasized by researchers and in several national reports.

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Mathematics and Reading

“All young Americans must learn to think mathematically, and they must think mathematically to learn” (NRC, 2001, p.1).

All students must learn to read and read to learn.

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Reading and Mathematics: Similarities

1. Competence in reading/mathematics is important in determining children’s later educational and occupational prospects.

2. Understanding the common features of reading/mathematics development is as important as understanding the special characteristics of learning in each domain.

3. Learning to read or developing mathematical proficiency rests on a foundation of concepts and skills.

National Research Council (2001, pp. 17-18)

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Reading and Mathematics: Differences1. International comparisons

suggest that U.S. schools have been relatively successful in developing skilled reading, with improvements in both instruction and achievement.

2. Students having reading difficulties get a variety of intervention programs designed to address their reading problems.

3. After a certain point, reading (i.e., phonological system) requires little explicit instruction.

1. International comparisons indicate that by eighth grade, the mathematics performance of U.S. children is well below that of other industrialized countries.

2. There is very little early targeted enrichment in mathematics to help students overcome special difficulties.

3. Students continue to require explicit instruction in mathematics.

National Research Council (2001, pp. 17-18)

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Reading and Mathematics: Differences4. Reading uses a core set of

representations. (e.g., the English alphabetic writing system).

5. Variations in children’s reading skill are associated with large differences in reading experience.

6. Schools generally have reading specialists.

7. Many school districts ensure that adequate reading instruction is given in the elementary grades.

4. Mathematics has many types and levels of representation.

5. Similar data are not available for mathematics, but differences in the amount of time spent on mathematics are likely to be less than reading.

6. Schools generally lack mathematics specialists.

7. Mathematics instruction has yet to receive similar attention as reading instruction in early grades.

National Research Council (2001, p. 17-18)

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Developing Students’ Mathematical Competence: Background

The Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, NCTM, 2000) –What students should know - number and

operations, measurement, algebra, geometry, data analysis and probability

–What students should be able to do with the content - problem solve, reason, communicate, connect, represent

Note: Skill in basic arithmetic is no longer a sufficient mathematics background for most adults.

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Emphasize Grades 5-8

• Mathematics builds on prior knowledge and many students have not learned the primary grade content in depth

• High school initiatives are important, but continuing to build proficiency in the middle grades is critical

• Children have to learn new and more advanced content in middle grades.

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Reading and Mathematics: Critical Elements

1. Phonemic awareness

2. Decoding3. Fluency4. Vocabulary5. Comprehension

1. Conceptual understanding

2. Procedural fluency3. Strategic competence4. Adaptive reasoning5. Productive disposition

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Mathematical Proficiency

1. Conceptual understanding – comprehension of mathematical concepts, operations, and relations

2. Procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

3. Strategic competence – ability to formulate, represent, and solve mathematical problems

4. Adaptive reasoning – capacity for logical thought, reflection, explanation, and justification

5. Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

NRC (2001, p. 5)

PROCESS STANDARDS CONTENT STANDARDS 1. Problem

Solving 2. Reasoning and

Proof 3. Communication 4. Connections 5. Representation

1. Number & Operations

2. Algebra

3. Geometry

4. Measurement

5. Data Analysis & Probability

Mathematical ProficiencyConceptual Understanding

Procedural FluencyStrategic CompetenceAdaptive Reasoning

Productive Disposition

Standards and Mathematical Proficiency

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Recommendations Regarding Mathematical Proficiency

• Integrated and balanced development of all five strands of mathematical proficiency should guide the teaching and learning of school mathematics.

• Teachers’ professional development should be high quality, sustained, and systematically designed.

• The coordination of curriculum, instructional materials, assessment, instruction, professional development, and school organization around the development of mathematical proficiency should drive school improvement efforts.

NRC, 2001

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Recommendations Regarding Mathematical Proficiency

• Efforts to improve students’ mathematics learning should be informed by scientific evidence.

• Additional research is needed on the nature, development, and assessment of mathematical proficiency.

NRC, 2001

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What Does Research Say About Effective Practices For Students With Mathematics Difficulties?

1. Curricular approaches and instructional strategies

2. Providing feedback to teachers

3. Providing feedback to students

4. Providing feedback to parents

5. Grouping practices: Peer-mediated instruction

Baker, Gersten, & Lee, 2002; Gersten, Chard, & Baker, 2006; Kunsch, Jitendra, & Sood, 2006.

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Curricular approaches and Instructional Strategies• Explicit strategy instruction (i.e., using principles of

effective instruction) and instruction that promotes self-regulated learning are effective.

• Using physical models or visual representations to solve problems is beneficial.

Providing Feedback to Teachers• Providing teachers with data as well as suggestions to

improve teaching practices is more effective than just providing teachers with data about student performance.Baker, Gersten, & Lee, 2002; Gersten, Chard, & Baker, 2006; Kunsch, Jitendra,

& Sood, 2006.

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Organizing Principles for Designing Curriculum and Instruction

1. Big ideas

2. Conspicuous strategies

3. Scaffolding

4. Strategic integration

5. Judicious review

Kame’enui, Carnine, Dixon, Simmons, & Coyne (2002)

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Big Ideas

Mathematical concepts, principles, or strategies (heuristics) that:– form the basis for further mathematical

learning.– map to the content standards outlined by each

state.– are sufficiently powerful to allow for broad

application

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Proportional Reasoning: Big Ideas

1. A ratio is a comparison of any two quantities.

2. Proportions involve multiplicative rather than additive comparisons.

3. Proportional thinking is developed through activities involving comparing and determining the equivalence of ratios and solving proportions in a wide variety of problem-based contexts and situations without recourse to rules or formulas.

Van de Walle (2004)

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Proportional Reasoning: Mathematics Content Connections

1. Fractions - equivalent fractions and equivalent ratios are both found through a multiplicative process.

2. Algebra - concerns a study of change and, hence, rates of change (ratios) are important. Slope itself is a rate of change.

3. Similarity - When two figures are the same shape but different sizes (i.e., similar), they constitute a visual example of a proportion.

4. Data graphs - e.g., a relative frequency histogram (visual part-to-whole ratios) and a box and whisker plot that shows distribution of data along a number line.

5. Probability - is a ratio that compares the number of outcomes in an event to the total possible outcomes.

Van de Walle (2007)

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Providing Feedback to Students• Providing students with elaborative feedback as well as

feedback on their effort is effective.• Providing feedback linked to goal setting showed that

goal setting in itself is not sufficient to promote mathematics competence.

Providing Feedback to Parents• Providing parents with clear, specific feedback on their

children’s mathematics success is beneficial.


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Grouping Practices: Peer-mediated instruction• Peer-mediated instruction (PMI) is more effective for

students at risk for mathematics disabilities than students with disabilities. PMI is more beneficial for elementary-aged participants than middle and high school students, and is more effective in teaching mathematics computation than higher order skills.

• Cross age tutoring can be beneficial only when tutors are well-trained.


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Implications For Practice and Policy

What are the implications of the Standards and research in mathematics for:

– Curriculum development?

– Teaching?

contexts

The Instructional Triangle

Cohen and Ball, 1999, 200

Teacher

MathematicsStudents

contexts

contexts

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Textbooks as the Curriculum

Textbooks “serve as critical vehicles for knowledge acquisition in school” and can “replace teacher talk as the primary source of information” (Garner, 1992, p. 53).

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Findings from Research on Content Analysis Studies

• Although US mathematics textbooks have improved in their adherence to principles of effective instruction, they do not meet the Standards needed to move students to higher levels of achievement (e.g., Jitendra, Deatline-Buchman, & Sczesniak, 2005; Jitendra, Griffin, Deatline-Buchman, DiPipi, Sczesniak, Sokol, & Xin, 2005).

• A comparison of reform-based (i.e., Everyday Mathematics) and traditional mathematics textbooks suggests that they differ in the development of critical concepts and adherence to principles of effective instruction (Sood & Jitendra, 2006).

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Implications• Publishers - continue to improve the quality of educational

materials by considering: – Curriculum Focal Points developed by NCTM for each grade

level (pre-K-8)– principles of effective instruction

• Professional development to use the materials– provide strong connections between understanding of

mathematics content, knowledge of students’ thinking and errors, and systematic teaching using relevant activities and strategies.

– emphasize understanding and attending to principles of effective instruction when modifying or adapting instruction in mathematics textbooks to meet the needs of a range of students.

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The Teacher

What the teacher does in the classroom can influence student achievement.

“how the teacher presents the curriculum, what subject matter is taught, the teacher's belief systems and intentions, together with the context in which instruction occurs" (Hafner, 1993, p. 72).

Teachers’ Knowledge and Use of Mathematical

Content

Teachers’ Attention to and Work With Students

Students’ Engagement in and Use of Mathematical

Tasks

Mathematics Teaching

NRC (2001)

Effective Instruction and Student Understanding

Proficiency in teaching mathematics = effectiveness + versatility

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Findings from Research on TeachingTeachers and Content• Opportunity to learn (OTL) - the single most predictor of student

achievement– The curriculum is a potent force in OTL– Instructional time allocated to various subjects and management of time

influence OTL• Task selection and use

– Select tasks that build on students’ prior knowledge– Scaffold tasks to maintain student engagement at a high level– Allocate adequate time (neither too much nor too little)– Have students explain their solution processes and attach meaning to the

symbols they are using• Planning

– Lesson Study, for example

NRC (2001)

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Findings from Research on TeachingTeachers and Students• Teacher expectations - beliefs about what students need to

learn and are capable of learning. – Related to teacher expectations is teachers’ sense of efficacy.

• Maintaining an expectation of success1. Assigning tasks on which students can succeed2. Providing scaffolding to help students acquire and apply concepts,

skills, and abilities3. Helping students to commit to goals and assessing their progress

toward those goals• Motivation• Managing discourse• Teaching students with special needs and diverse learners (e.g.,

English language learner)• Grouping • Assessment NRC (2001)

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Assessment• Assessment of student performance serves to:

– Validate instructional practices – help identify students who are not achieving and need special

assistance so that they do not fall further behind

• The linking of assessment to instructional efforts is critical. Two recommendations concerning assessment and instruction for Title 1 students are frequent and regular assessments in classrooms to:

• monitor individual students’ performance and adapt instruction to improve their performance (NRC, 2001, p. 44)

• improve the quality and appropriateness of instruction.

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Findings from Research on Teaching

Students and Content• Students and tasks

– Mathematical tasks should support student learning - neither too difficult nor too trivial

• Practice– Role of practice

• Important to execute procedures automatically– Kinds of practice

• Practice that is directly associated with the topic of the lesson• Distributed practice• Practice on problem solving

– Homework (also a means to communicate with parents)– Manipulatives– Calculators

NRC (2001)

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Implications for Teaching

• Protect instructional time• Hold high expectations regarding student learning • Ensure high levels of engagement with varied

instructional activities• Teaching should be more than simply delivering

information. The instructional sequence should focus on:– an integrated and balanced development of students’

mathematical proficiency– Adapting instruction based on a fundamental understanding

of the nature and manifestations of learning disabilities.

• Use assessment to inform instruction

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Implications for Teacher Preparation and Professional Development

Develop specialized knowledgePrepare teachers to have a deep understanding of mathematics

1. Mathematical knowledge (e.g., knowledge of mathematical facts, concepts, procedures, and the relationship among them; knowledge of the ways that mathematical ideas can be represented; and knowledge of mathematics as a discipline)– Teachers of grades pre K-8 should have a deep

understanding of the mathematics of the school curriculum and the principles behind it.

2. Knowledge of students and how they learn mathematics.

3. Knowledge of instructional practice (e.g., curriculum, tasks and tools for teaching important mathematical ideas).

NRC (2001)

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Prepare teachers to teach mathematics to students in such grades as pre-K - 2, 3 - 5, and 6 - 8.

Mathematics specialists should be available in every elementary school.

Work togetherProvide teachers with more time for planning and conferring with each other on mathematics instruction with appropriate support and guidance.

NRC (2001)

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Capitalize on Professional Meetings

Sustain Professional Development

LEA’s should give teachers support, including stipends and released time, for sustained professional development.

Providers of professional development should know mathematics.

Organizations and agencies that fund professional development in mathematics should focus resources on multiyear, coherent programs.

NRC (2001)

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Summary: Five things to consider

1. Increase amount of direct school-based instruction.

2. Teach to the big ideas in the mathematics curriculum.

3. Prepare teachers to know the mathematics they teach and know how to design effective instruction for a range of diverse learners to move them toward higher levels of mathematics achievement.

4. Involve parents in their children’s learning.

5. Ensure that assessment information is coordinated with curricular goals to improve the quality of instruction.

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References• Baker, S., Gersten, R., & Lee, D. S. (2002). A synthesis of empirical research on

teaching mathematics to low-achieving students. Elementary School Journal, 103, 51-73.

• Garner, R. (1992). Learning from school texts. Educational Psychologist, 27, 53-63.

• Gersten, Chard, & Baker (2006) • Kunsch, C. A., Jitendra, A. K., Sood, S. (2006). The effects of peer-mediated

mathematics instruction for students with disabilities: A review of the literature. Manuscript submitted for the Special Issue on Mathematics intervention for students with Learning Problems in Learning Disabilities Research & Practice.

• National Research Council (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.) Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

• Van de Walle, J.A. (2007). Elementary and middle school mathematics (6th ed.). Longman: New York, New York.

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Jitendra ReferencesMathematics Textbook Analysis • Carnine, D., Jitendra, A. K., & Silbert, J. (1997). A descriptive analysis of mathematics

curricula materials from a pedagogical perspective: A case study of fractions. Remedial and Special Education, 18(2), 66-81.

• Jitendra, A. K., Carnine, D., & Silbert, J. (1996). A descriptive analysis of fifth grade division instruction in basal mathematics programs: Violations of pedagogy. Journal of Behavioral Education, 6(4), 381-403.

• Jitendra, A. K., Deatline-Buchman, A., & Sczesniak, E. (2005). A comparative analysis of third-grade mathematics textbooks before and after the 2000 NCTM standards. Assessment for effective Intervention, 30(2), 47-62.

• Jitendra, A. K., & Griffin, C., Deatline-Buchman, A., DiPipi, C., Sczesniak, E., Sokol, N., & Xin, Y.P. (2005). Adherence to NCTM Standards and instructional design criteria for problem-Solving in mathematics textbooks. Exceptional Children, 71(3), 319-337.

• Jitendra, A. K., Salmento, M., & Haydt, L. (1999). A case study of subtraction analysis in basal mathematics programs: Adherence to important instructional design criteria. Learning Disabilities Research & Practice, 14(2), 69-79.

• Sood, S., & Jitendra, A. K. (2006). A comparative analysis of number sense instruction in first grade traditional and reform-based mathematics textbooks. Manuscript submitted for publication.

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Jitendra ReferencesInstructional Strategies• Jitendra, A. K. (2002). Teaching students math problem-solving through

graphic representations. Teaching Exceptional Children, 34(4), 34-38.• Jitendra, A. K., DiPipi, C. M., & Grasso, E. (2001). The role of a graphic

representational technique on the mathematical problem solving performance of fourth graders: An exploratory study. Australasian Journal of Special Education, 25, 17-33.

• Jitendra, A. K., DiPipi, C. M., & Perron-Jones, N., (2002). An exploratory study of word problem-solving instruction for middle school students with learning disabilities: An emphasis on conceptual and procedural understanding. Journal of Special Education, 36(1), 23-38.

• Jitendra, A. K., Griffin, C., Haria, P., Leh, J., Adams, A., & Kaduvetoor, A. (in press). A comparison of single and multiple strategy instruction on third grade students’ mathematical problem solving. Journal of Educational Psychology.

• Jitendra, A. K., Griffin, C., McGoey, K., Gardill, C, Bhat, P., & Riley, T. (1998). Effects of mathematical problem solving by students at risk or with mild disabilities. Journal of Educational Research, 91(6), 345-356.

• Jitendra, A. K., Griffin, C., Deatline-Buchman, A., & Sczesniak, E. (2006). Understanding teaching and learning of mathematical word problem solving: Lessons learned from design experiments. Manuscript submitted to Journal of Educational Research.

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Jitendra ReferencesInstructional Strategies• Jitendra, A. K., & Hoff, K. (1996). The effects of schema-based instruction on

mathematical problem solving performance of students with learning disabilities. Journal of Learning Disabilities, 29(4), 422-431.

• Jitendra, A. K., Hoff, K., & Beck, M. (1999). Teaching middle school students with learning disabilities to solve multistep problems using a schema-based approach. Remedial and Special Education, 20(1), 50-64.

• Montague, M., & Jitendra, A. K. (Eds.) (2006). Teaching mathematics to middle school students with learning difficulties. New York: The Guilford Press.

• Xin, Y. P., & Jitendra, A. K. (2006). Teaching problem solving skills to middle school students with mathematics difficulties: Schema-based strategy instruction. In M. Montague & A. K. Jitendra (Eds.), Teaching mathematics to middle school students with learning difficulties (pp. 51-71). New York: Guilford Press.

• Xin, Y. P., Jitendra, A. K., & Deatline-Buchman, A. (2005). Effects of mathematical word-problem solving instruction on middle school students with learning problems. The Journal of Special Education, 39(3), 181-192.

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Jitendra ReferencesAssessment• Jitendra, A. K., Parker, R., & Kameenui, E. J. (1997).

Aligning the basal curriculum and assessment in elementary mathematics: The experimental development of curriculum-valid survey tests. Diagnostique, 22(2), 101-127.

• Leh, J., Jitendra, A. K., Caskie, G., Griffin, C. (in press). An evaluation of CBM mathematics word problem solving measures for monitoring third grade students’ mathematics competence. Assessment for Effective Intervention.

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Jitendra ReferencesResearch Synthesis• Jitendra, A. K., & Xin, Y. (1997). Mathematical problem

solving instruction for students with mild disabilities and students at risk for math failure: A research synthesis. The Journal of Special Education, 30(4), 412-438.

• Kunsch, C. A., Jitendra, A. K., Sood, S. (2006). The effects of peer-mediated mathematics instruction for students with disabilities: A review of the literature. Manuscript submitted for the Special Issue on Mathematics intervention for students with Learning Problems in Learning Disabilities Research & Practice.

• Xin, Y. P., & Jitendra, A. K. (1999). The effects of instruction in solving mathematical word problems for students with learning problems: A meta-analysis. The Journal of Special Education, 32(4), 207-225.

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Jitendra ReferencesOther• Jitendra, A. K. (2005). How design experiments can

inform teaching and learning: Teacher-researchers as collaborators in educational research. Learning Disabilities Research & Practice, 20(4), 213-217.

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AGT Contoh Kajian Maths Achievement