Math Methodology: Instruction Resources

Hook your students on The Beauty in Numbers in Math Wonders to Inspire Teachers and Students by Alfred S. Posamentier (2003), an ASCD publication. Chapter 1, available online, is an amazing discussion of some surprising number patterns, power relationships, beautiful number relationships, strange equalities, perfect numbers, friendly numbers (Yes, there is a definition of friendly numbers), palindromic numbers (e.g., 12321), number associations with geometric figures, Fibonacci numbers (i.e., 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,...) and more. As examples:

Note the digits of these special numbers and the consecutive exponents (p. 12)

598 = 5¹+9²+8³1,676 = 1¹+6²+7³+6⁴

A palindrome, which is a number that can be read the same in both directions, can be generated from any number by making successive additions of the number with its reversal (p. 27). This might take a few such additions.

67 + 76 = 143; 143 + 341 = 484, a palindrome

There are only five numbers (i.e., 1, 153, 370, 371, and 407) in which the number is the sum of the cubes of its digits as in 153 = 1³+5³+3³ (p. 36).

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.

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).

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.

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.

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:

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

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:

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.

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.

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.

Bullseye Gif 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.

Graphic Organizers from Enhance Learning with Technology Web site. What are they? Why use them? How to use them? The site includes numerous links on the topic, examples, and software possibilities to assist with the endeavor.

Graphic Organizers from Graphics.Org shows graphic organizers, concept mapping, and mind mapping examples related to their use: describing, comparing/contrasting, classifying, causal, sequencing, and decision making.

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:

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.

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.

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.

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:

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 Models, by P. 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.

Math VIDS! "MathVIDS is an interactive CD-ROM/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 fourteen, 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.

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.

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.

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.

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.

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.

TeacherTube.com contains numerous short video clips on math topics, which might be used to supplement classroom instruction or for topic tutorials.

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 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.

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:

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).

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.

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.

Need books on Math Methodology?

Research teaching mathematics at Questia.com, the world's largest online library.

Access the catalog of resources for the mathematics educator from the National Council of Teachers of Mathematics: http://my.nctm.org/ebusiness/ You might be particularly interested in the Navigations Series, which translates NCTM's Principles and Standards for School Mathematics into action. The books highlight major math content areas in appropriate preK-12 grade-bands: algebra, data analysis, geometry, probability, measurement, problem-solving and reasoning.

National Academies Press: http://www.nap.edu/topics.php?browse=1 has over 3,000 books online, which you can read for free. For example, math educators might be interested in How Students Learn: History, Mathematics, and Science in the Classroom (2005) by Committee on How People Learn, A Targeted Report for Teachers, Center for Studies on Behavior and Development, National Research Council.

You can also search for mathematics books at all levels at isbn.nu using search phrases such as mathematics study teaching secondary or mathematics study teaching elementary. This site will then link you to a resource for purchasing the book.

Visit Educational REALMS (Resources for Engaging Active Learners in Mathematics and Science), an organization created in 2004 after the discontinuation of the ERIC Clearinghouse for Science, Mathematics and Environmental Education. Educational REALMS hopes to pick up where ERIC/CSMEE left off by providing many of the same services and a variety of resources in science, mathematics, technology, and environmental education at all levels. Many of the public domain materials originally produced by ERIC/CSMEE are available. Lesson plans include the askERIC lesson plans and others. There are useful links to Mathematics Teaching and Learning resources and organizations, and a list of math journals.

Math Methodology

Part 1: Instruction Resources