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Math Methodology

Part 1: Instruction Resources

 

 

 

Math Methodology Instruction Resources

Instructional resources here are grouped into sections: for all learners, for special needs learners, the instructional environment and behaviors.

For All Learners

 

Adding It Up: Helping Children Learn Mathematics (2001) is a longer book (over 400 pages), available online, by the Mathematics Learning Study Committee and editors Jeremy Kilpatrick, Jane Swafford, and Bradford Findell of the Center for Education at the National Research Council.  Its focus is school mathematics from pre-kindergarten to eighth grade. Much of this report attends to the learning and teaching of number, but authors emphasize that their perspective is considerably broader than just computation.  Authors say, "Three kinds of knowledge are crucial for teaching school mathematics: knowledge of mathematics, knowledge of students, and knowledge of instructional practices" (p. 370).

The Art of Teaching Resources from TeacherVision.com.

21st Century Problem Solving, by H. McAllister of the University of Hawaii, is devoted to providing teachers and students with training in how to solve problems.  The methodology described is useful for all students from grade school to grad school and beyond.  Begin with an understanding of why students can't solve word problems reliably.  Learn to use the 3 R's of problem solving, Request-Response-Result, and a verification of the result.  The examples and theory given at this site deal specifically with subjects that use mathematics for problem solving, including algebra, physics, and chemistry.  The problem-solving principles apply equally well to problems which require non-mathematical tools or a mixture of mathematical and non-mathematical tools.

Centre for Innovation in Mathematics Teaching (CIMT) was established in 1986 at the University of Exeter in the UK and is now housed at the University of Plymouth.  Content is focused on research and curriculum development for mathematics teaching and learning.  K-12 educators will be particularly interested in the CIMT curriculum resources, such as "pupil texts, lesson plans, classroom resources, assessment materials and on-line interactive resources."  There is also a section for math misconceptions.  Don't miss the project, The Mathematics Enhancement Programme, which offers school curriculum and teaching material (posted online) for mathematics for all grade levels with lesson plans, copymasters, and tests.  Read articles from the International Journal for Mathematics Teaching and Learning.

Classroom Instruction that Works (2001) by Robert Marzano, Debra Pickering, and Jane Pollock includes nine research-based instructional strategies that have a high probability of enhancing student achievement for all students in all subject areas at all grade levels:

• Identifying similarities and differences--graphic forms, such as Venn diagrams or charts, are useful

• Summarizing and note taking--provide guidelines for creating a summary; give time to students to review and revise notes; use a consistent format when note taking

• Reinforcing effort and providing recognition--you might have students keep a weekly log of efforts and achievements with periodic reflections of those.  They might even mathematically analyze their data.  Find ways to personalize recognition, such as giving individualized awards for accomplishments.

• Homework and practice--vary homework by grade level; keep parent involvement to a minimum; provide feedback on all homework; establish a homework policy; be sure students know the purpose of the homework

• Nonlinguistic representation--incorporate words and images using symbols to show relationships; use physical models and physical movement to represent information

• Cooperative learning--consider common experiences or interests; vary group sizes and objectives.  Core components include positive interdependence, group processing, appropriate use of social skills, face-to-face interaction, and individual and group accountability.

• Setting objectives and providing feedback--set a unit goal and help students personalize that goal; use contracts to outline specific goals students should attain and grade they will receive if they meet those goals; use rubrics to help with feedback; provide timely, specific, and corrective feedback; consider letting students lead some feedback sessions

• Generating and testing hypotheses--a deductive (e.g. predict what might happen if ...) , rather than an inductive, approach works best.

• Cues, questions, and advance organizers--these should be highly analytical, should focus on what is important, and are most effective when used before a learning experience.

The authors caution, however, that instructional strategies are only tools and "they should not be expected to work equally well in all situations."

Columbia Education Center Resources for Schools and Teachers contains resources for mathematics and science teachers including Internet based elementary and secondary lesson plans , the Third International Mathematics and Science Study, technology resources, curriculum projects, Federal Government math/science organizations, and more.

Concrete and Virtual Manipulatives Research: The George Mason University Mathematics Education Center focuses on the study of concrete and virtual manipulatives.  You will find a list of published articles and abstracts on this topic.

Critical Thinking: Glazer, E. (2001). Using Web-Based Resources to Promote Critical Thinking in High School Mathematics. Wellington, New Zealand, NZAMT Annual Conference, July 3-6, 2001 Available: http://math.unipa.it/~grim/AGlazer79-84.PDF

The Critical Thinking Community comprises The Center for Critical Thinking and Moral Critique and the Foundation For Critical Thinking.  "The work of the Foundation is to integrate the Center’s research and theoretical developments, and to create events and resources designed to help educators improve their instruction" (Mission).  Resources are numerous at this site, particularly articles that define critical thinking and elaborate on the dimensions of critical thought.  See Sample Teaching Strategies for K-12 Teachers and For Students, for example.  Articles are relevant for teaching critical thinking within the mathematics classroom.

Everyday Mathematics algorithms for grades 2-6 and their animations from the publisher are free and are also good for family use.  A series of Everyday Mathematics Algorithms posted at Mary Bryan Elementary School (Indianapolis, IN) includes demonstration videos of teachers explaining how to use them.

Helping Children Learn Mathematics (2002) is an short book (52 pages), available online, by the Mathematics Learning Study Committee and editors J. Kilpatrick and J. Swafford of the Center for Education at the National Research Council.  The authors stress that mathematics proficiency involves five intertwined strands: understanding mathematics, computing fluency, applying concepts to solve problems, reasoning logically, and engaging with mathematics, seeing it as sensible, useful, and doable.  Mathematics in grades K-8 should be taught for an hour a day.  Time should be divided so that all of the strands receive adequate attention--the key being that instructional materials should integrate the five strands. All students can and should be proficient in mathematics.

Houghton Mifflin Math Education Place has teaching models and tools (glossary, online manipulatives, graphic organizers, games) associated with each of the chapters in their K-6 texts to help you teach concepts--excellent resource for methodology.

IMAGES: Improving Measurement and Geometry in Elementary Schools contains a discussion of cognitive and developmental issues related to designing and teaching an instructional unit to elementary students.  Teaching strategies and assessment issues are addressed with how-to's, lesson plans and activities:

Strategies

• Have high expectations for all students
• Base practice on educational research
• Integrate content areas
• Incorporate standards
• Collaborate with others
• Promote cooperative learning
• Use technology as a tool
• Use inquiry-based learning
• Promote mathematical reasoning and problem solving
• Use hands-on activities to model topics
• Integrate assessment and instruction
• Reflect on teaching and learning
• Include strong academic content
• Cluster concepts

Assessments

• Observation, questioning, interviewing
• Performance tasks, self-assessment and peer-assessment
• Work samples, portfolios, writings
• Teacher-designed written tests, achievement tests

Inside Mathematics, which grew out of the Noyce Foundation's Silicon Valley Mathematics Initiative, is exemplary as "a professional resource for educators passionate about improving students' mathematics learning and performance. This site features classroom examples of innovative teaching methods and insights into student learning, tools for mathematics instruction [organized by grade level and subject] that teachers can use immediately, and video tours of the ideas and materials on the site" (Welcome section).  The Ohio Department of Education developed a Correlation of Inside Mathematics Tasks to CCSS (Common Core State Standards) as of June 2010.

Instructional Methods Information by Dr. Bob Kizlik includes advantages, disadvantages, and required preparation related to using direct teaching, cooperative learning, lecture, lecture with discussion, a panel of experts, brainstorming, video tapes/slides, discussion, small group discussion, role playing, worksheets/surveys, guest speakers, and values clarification.

Instructional Strategies from Prince George's County Public Schools (MD) lists a variety of instructional strategies and research-based best practices, including performance-based instruction, portfolios, questioning, dimensions of learning, and cooperative learning.

Instructional Strategies Online from Saskatoon Public Schools (CA) includes five categories of instructional strategies and explanation of these five categories: direct instruction, indirect instruction, experiential learning, independent study, and interactive instruction.  There are additional sections for instructional skills, and instructional methods by strategy.

Instructional Tools Related to Mathematics from Special Connections at the University of Kansas.  Tools relate to building initial mathematical understanding and then extending it, building math proficiency (e.g., via instructional games), evaluating student needs and making effective math decisions.

Iowa Education Agency 267: Curriculum, Instruction, Assessment contains specific sections on teaching strategies, thinking skills, and mathematics.  For developing thinking skills in mathematics, see http://www.aea267.k12.ia.us/math/index.php?page=math_thinking_skills

LINKS Learning is an "E-learning site, [which] has involved Washington's Office of Superintendent of Public Instruction and partner organizations including the Technology Based Learning and Research Project at Arizona State University; the Washington school districts involved in the Washington Alliance for Better Schools; Carkhuff Thinking Systems; Limelight Technologies, Inc.; Fairfax County, Virginia schools; and Educational Service District 113 in Olympia, Washington." The Teacher Section for Math is designed to provide information and resources to assist teachers and others in understanding and providing high quality instruction for all students.  You will find:

• a general overview of the five math content areas
• content and curriculum maps
• videos on math curriculum, instruction and assessment; and
• a number of classroom resources.

The math concept maps in Curriculum Planning are superb with examples of the key concepts for each of the five math content areas. When you click on the bolded concepts, you will see a definition and, in some cases, examples.

Mathematics Education: The Math Forum from Swarthmore College contains information on topics as assessment, block scheduling, calculus reform, collaborative learning, curriculum development, grouping and tracking, interdisciplinary math, math education reform, mathematicians, NCTM standards, and the WWW and pedagogy.  See math resources for people with disabilities.   Read about teaching issues/strategies and technology in math education.

Mathematics Models, by Paul Griffith, illustrates how teachers can use mathematical modeling as a methodology in a  4th-8th grade curriculum.  Each page on this site includes a concept and basic models that accompany a lesson from Math Alive!  (formerly known as the Visual Mathematics), which is a hands-on, concept-building curriculum that emphasizes exploration and discussion. The curriculum has been published by the Math Learning Center at Portland (Oregon) State University since about 1989.  Teachers, parents, and students will benefit from the models that illustrate concepts.  This is a very complete site to accompany your lessons.

The Most Common Errors in Undergraduate Mathematics have been articulated by E. Schechter of Vanderbilt University's Math Department, along with the likely causes of those errors, and their remedies.  Dr. Schechter presents errors in communication, algebra errors, confusion with notation, errors in reasoning, unwarranted generalizations, and common calculus errors.  Any teacher of upper level secondary mathematics will also benefit from this presentation, which should be shared with students--they make the mistakes after all.  Pointing out these common errors as part of your teaching methodology might help students to become more successful in learning algebra and calculus.

Northwest Regional Educational Laboratory Math Problem Solving Model "helps educators meet the challenges of teaching and assessing open-ended problem solving. The model includes a scoring guide for problem solving, open-ended tasks, and examples of student work for practice in scoring."  The design of the model was informed by Robert McIntosh and Denise Jarrett's (2000) TEACHING MATHEMATICAL PROBLEM SOLVING: IMPLEMENTING THE VISION, which is a review of recent research and literature on the essential traits and processes of teaching and learning mathematics through open-ended problem solving.

Promoting Reading Strategies for Developmental Mathematics Textbooks by Anne E. Campbell, Ann Schlumberger, and Lou Ann Pate of Pima Community College presents three reading and study strategies designed to facilitate student comprehension of and learning from developmental mathematics textbooks.  The discussion includes a preview, predict, read, and review reading strategy; concept cards; and a Question Answer Relationship technique.  For example, concepts cards can include definitions, characteristics, examples, and nonexamples. Common kinds of concept cards in math include: (a) strategy cards for solving problems; (b) fact cards that include rules, laws, or theorems; and (c) cards for symbols and specialized vocabulary.

Supporting the Math Classroom through Literacy Development posted at Knowledge Loom: http://www.knowledgeloom.org .  Strategies that support literacy development and understanding of math concepts include "Think Alouds, graphic organizers, word problems, brainteasers, math journals, Inquiry Models, Quick Writes, Word Walls, concept maps, flow charts, computer or graphic programming, creation of texts, Socratic Questioning, and WebQuests."  The Northeast and Islands Regional Educational Laboratory at Brown University, Center for Resource Management, presents a series of articles illustrating how those strategies can be used in the high school mathematics classroom.

Teaching and Learning Elementary Math from Patricia Stohr-Hunt at the University of Richmond has numerous resources for preK-6, including instruction, assessment, pedagogy and content by NCTM strands.

Teaching Math to Young Children by Rick Garlikov is "one of a series of Web pages to help students understand math, and to help parents teach their children math -- especially to help children have a good foundation."  Some other essays in this series include:

• "The Concept and Teaching of Place-Value"
• "The Socratic Method: Teaching by Asking Instead of by Telling"
• "A Supplemental Introduction to Algebra" and a separate webpage "The Way Algebra Works"
• "Understanding 'Rate' Word Problems"

Universal Design for Learning Lesson Builder from the Center for Applied Special Technology (CAST).  Learn more about universal design for learning, explore model lesson plans designed using UDL principles, then create, save and edit your own UDL lesson plans.  You will create an account for this latter, but it is free.

 

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For Special Needs Learners

Access Center contains a number of information briefs on strategies for teaching K-8 math to all learners, and specifically for those with mild learning disabilities.  For example, read about using mnemonic instruction, peer tutoring, computer-assisted instruction, the Concrete-Representational-Abstract Instructional Approach, direct/explicit instruction, learner accommodations and instructional modifications for students with learning disabilities, math differentiation, problem-solving strategies, graphic organizers, and more.  Presentations such as Research Based Math Interventions are included.  The Access Center is a national technical assistance (TA) center funded by the Office of Special Education Programs.

Center on Instruction: Mathematics offers materials and resources "to build educators’ knowledge of instruction for students with low achievement in mathematics, improve professional development models for math teachers, and build teachers’ skills in monitoring student growth toward important math outcomes."  The Center on Instruction maintains a collection of scientifically based research for reading, math, science, special education and English language learners,  and exemplars of best practice.  It is one of five content centers serving as resources for the 16 regional U.S. Department of Education Comprehensive Centers.

Clearinghouse On Mathematics, Engineering, Technology, and Science (COMETS). http://www.rit.edu/~comets. National Science Foundation Grant HRD-9550468.  Dr. Harry Lang of COMETS says, "Much of the information on this website has been developed from 20 years of teaching a methods course to prepare teachers for effective instruction of deaf students in science and math." K-12 teachers will appreciate the section on teaching tips and preparing a lesson for deaf students.  Educators will appreciate the classroom teaching tips, lesson plans, links to science/math projects for deaf students, and bibliography of science/math education for deaf students. Additional resources are provided for pre-service teachers, college faculty, high school and college students, parents, administrators, and support personnel.

Citing work from Marsha C. Stevens and Madeline Hunter, Dr. Lang elaborates on the essential components of a lesson, which include:

• Objective
• Purpose
• Materials
• Anticipatory Set
• Procedures

• Input
• Modeling
• Check for Understanding

• Guided Practice
• Closure
• Independent Practice

Dyscalculia.org addresses teaching and learning strategies for learners with dyscalculia (math LDs) and dyslexia (reading LDs).  A diagnostic is available (fee attached), along with special education resources.

Dyscalculia: Symptoms, strategies to help students learn mathematics, and technology suggestions.  Page is by Ed Keller of West Virginia University.

Education for All: The Report of the Expert Panel on Literacy and Numeracy Instruction for Students With Special Education Needs, Kindergarten to Grade 6 (2005) was financially supported by the Ontario Ministry of Education in Canada.  The report recommends practices, based on research, "to improve and reinforce effective instruction of reading, writing, oral communication, and mathematics to students from Kindergarten to Grade 6 who have special education needs" (p. 1).  However, practices discussed can enhance the learning of all students in mathematics and literacy.  Chapters address Universal Design and Differentiated Instruction, Assessment and Evaluation, Developing Learning Profiles, Professional Learning Communities, Research to Practice: What Works for Both Literacy and Numeracy; Effective Instructional Approaches for Numeracy, Effective Instructional Approaches for Literacy, Organization and Management, Computer-based Assistive Technology, and Professional Development.

Inclusion in Mathematics Education for Students with Disabilities addresses proven strategies that work with students with disabilities.  "Eight general types of disabilities are presented across four Mathematics teaching methods, viz., teacher presentation, recitation, reading, and discussion (24 subsets). Over 600 teaching strategies are presented" (section: About this site).  Strategies are noted for attention deficit disorder, learning disabilities, behavior disorders, intellectual disorders, communication disorders, motor/orthopedic disorders, hearing impairments, and vision impairments.  Software and special hardware possibilities are also included.  Site is associated with West Virginia University and Ed Keller.

Learning Disabilities Online contains a series of articles with strategies for  teaching mathematics to students with learning disabilities, but those strategies are suitable for use with all students.   Read how to modify mathematics instruction to promote success and understanding in the areas of mathematical readiness, computation, and problem-solving; how use cooperative learning in the math classroom, and how to break math difficulties down into different types.  Another article discusses techniques that have been demonstrated to be effective with secondary students who have learning disabilities in mathematics.

Learning Toolbox from the James Madison University Special Education Program contains tools and resources to enable students with learning difficulties to become better learners.  Sections are devoted to secondary learners, teachers of middle and secondary students, and parents.  Academic areas and strategies include organization, test taking, study skills, notetaking, reading, writing, math, and advanced thinking.  Advanced thinking addresses organizing information sequentially, comparing and contrasting ideas, understanding categories, determining cause and effect relationships, and problem solving.

Mathematics Education: The Math Forum from Swarthmore College also contains math resources for people with disabilities.  

Math VIDS! "MathVIDS is an interactive website for teachers who are teaching math to students who are having difficulty learning mathematics. The development of MathVIDS was sponsored through funding by the Virginia Department of Education" (sec: Introduction). The site contains Foundational Information, Instructional Strategies, and Teaching Plans. Foundational Information is devoted to Math Disabilities and Metacognitive Strategies. The Instructional Strategies section describes and models fifteen, research-based effective math instructional strategies for students with learning difficulties. The Teaching Plans section provides detailed instructional plans and selected video models for teaching specific K-5 math concepts/SOL.

National Center on Accessing the General Curriculum established by CAST (Center for Applied Special Technology) provides "a vision of how new curricula, teaching practices, and policies can be woven together to create practical approaches for improved access to the general curriculum by students with disabilities" (http://www.cast.org/policy/ncac/). The publications on curriculum (e.g., effective classroom practices and curriculum enhancements), teacher practices, and policies are available on the Web or for download.

Special Connections, according to the Web site, "is a Project of National Significance (CFDA #84.325N) funded through the federal Office of Special Education Programs (OSEP) and coordinated through the University of Kansas. The ultimate goal of the project is to provide educators, both classroom teachers and university faculty, with tools and resources that support students with special needs in general education settings and in accessing the general education curriculum in meaningful ways. Four main areas of focus include Instruction, Assessment, Behavior Plans, and Collaboration." Tools for mathematics address building initial mathematical understanding, extending that understanding, building proficiency through games, evaluating student needs and making effective mathematics instructional decisions.

Teaching Math to Visually Impaired Students: S. Osterhaus of the Texas School for the Blind and Visually Impaired offers strategies and resources for teaching mathematics to visually impaired students.  She provides links to math education and Nemeth code, tactile math graphics, calculators, current research in math for VI students and more.

TIPS for English Language Learners in Mathematics (2005) from the Ontario (CA) Ministry of Education includes grade-level support materials for those working with English language learners in Grades 7 to 10. "This resource contains models for adjustment of instruction for pairs, small groups, or the entire class and ways of differentiating instruction and assessment for English language learners so they can achieve literacy and mathematics goals. These adjustments in student groupings, teaching strategies, timing, and materials are based on recent research of ways to support English language learners" (Introduction, p. 1).

 

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Instructional Environment and Behaviors

Center for Effective Collaboration and Practice has several miniwebs in relation to preventing behavior problems and appropriate interventions: Functional Behavior Assessment, Prevention Strategies that Work, Prevention and Early Intervention, and more.  Several issues (e.g., cultural competence, school safety) are included with extensive resources.

Committee for Children addresses programs for prevention of bullying and violence, and includes a personal safety curriculum.

You Can Handle Them All, a Web site on discipline help for teachers and parents, lists over 100 behaviors (e.g., arrogant, class clown, cheater, disorganized, overly aggressive, whiner), the affect of each, actions to take to change the behavior, and mistakes in dealing with the behavior.

 

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The Amazon widget below shows books using the search phrase: mathematics teaching.  You can also use the widget to search with other key words. Suggestions include:

• math for elementary school children
• math and middle school
• math and secondary
• mathematics English language learner
• math disability
• dyscalculia
• math and algebra [NOTE: you can substitute the word "algebra" for a topic of your choice.]
• math inclusion
• differentiating instruction

 

 

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See other Math Methodology pages:

Instruction--Essay,  Assessment and Curriculum: Content and Mapping