Math Methodology

Part 1: Instruction Essay (Page 2 of 3)
Learning for Understanding

Teaching and Math Methodology

Instruction

Teaching Mathematics Right the First Time: Learning for Understanding

Have you ever heard students say (or have you as the teacher said), "To multiply by 10, just add a zero after the number"?  Or, "The product of two numbers is always bigger than either one"?  How about, "The number with the most digits is the biggest"?  These are examples of math myths, but they and other misconceptions can be overcome. Teachers Magazine with the help of Tim Coulson, who leads the National Numeracy Strategy in England, provides 10 such Maths misconceptions (2006) and suggestions to correct the situation.  This article sets the tone for the need to teach mathematics right the first time with a focus on understanding.

Carpenter, Blanton, Cobb, Franke, Kaput, and McClain (2004) proposed that "there are four related forms of mental activity from which mathematical and scientific understanding emerges: (a) constructing relationships, (b) extending and applying mathematical and scientific knowledge, (c) justifying and explaining generalizations and procedures, and (d) developing a sense of identity related to taking responsibility for making sense of mathematical and scientific knowledge" (pp. 2-3).  "Placing students' reasoning at the center of instructional decision making... represents a fundamental challenge to core educational practice" (p. 14).

According to Steve Leinwand and Steve Fleishman (2004), since the 1980s research results "consistently point to the importance of using relational practices for teaching mathematics" (p. 88).  Such practices involve explaining, reasoning, and relying on multiple representations that help students develop their own understanding of content.  Unfortunately, much instruction begins with instrumental practices involving memorizing and routinely applying procedures and formulas.  "In existing research, students who learn rules before they learn concepts tend to score lower than do students who learn concepts first" (p. 88).

Examples:

The importance of addressing misconceptions using relational practices and multiple representations was made clear when a teacher recently voiced concern about being unable to convince a beginning algebra student that (A + B)² was not A² + B².  The following visual helped clarify (A+B) (A+B) = A² + 2AB + B²

This same discussion brought up a comparison to using such a visual for understanding the typical multiplication algorithm in which students have been taught to "leave off the zeroes and move each successive row of digits when multiplying left one place."  Students often have no idea as to why they are doing that.  Consider the multiplication problem 31 x 25 and how the distributive property plays a role in the algorithm:

	20	5
30
1

The visual suggests that 31 x 25 = (30 + 1)(20 + 5) = (30 x 20) + (30 x 5)+ (1 x 20) + (1 x 5) and that there will be four values (600 + 150 + 20 + 5) to add together after the products are found.  As addition can be done in any order, the above might make the transition to the traditional vertical presentation of the algorithm easier to understand, as in the following illustration:

	2	5			2	5			2	5
	x 3	1			x 3	1			x 3	1
		5
	2	0			2	5			2	5
1	5	0
6	0	0		7	5	0		7	5
7	7	5		7	7	5		7	7	5

 

Robert Marzano, Debra Pickering, and Jane Pollock (2001) included 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:

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

2. 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

3. 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.

4. 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

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

6. 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.

7. 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

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

9. 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."

So what can you do to put research into practice?  

Educators should have one goal in mind in everything they do: achievement of learners.  According to Douglas Reeves (2006), "Schools that have improved achievement and closed the equity gap engage in holistic accountability, extensive nonfiction writing, frequent common assessments, decisive and immediate interventions, and constructive use of data" (p. 90).  Reeves stated that "accountability includes actions of adults, not merely the scores of students" (p. 83).  "The most effective writing is nonfiction--description, analysis, and persuasion with evidence" (p. 85).  Writing includes "editing, collaborative scoring, constructive teacher feedback, and rewriting" (p. 84) in all subject areas, including math. 

Assessments are not just summative, but also formative occurring at least quarterly or more with immediate feedback.  Beyond a score, feedback contains detailed item and cluster analysis, and is used to inform future instruction.  While individual class teachers might not be able to change student schedules to provide double classes in math or literacy for students in need, they can provide such interventions as homework supervision, break down projects into incremental steps, provide time management strategies, project management strategies, study skills, and help with reading the textbook, all of which are among immediate and decisive intervention strategies.  An analysis of data in a constructive manner would reveal effective professional practices and lead to discussion on how they might be replicated (Reeves, 2006).

Educators in all instructional settings who put research into practice should apply "The Seven Principles of Good Practice in Undergraduate Education."  Such practice emphasizes "active learning, time management, student-faculty contact, prompt feedback, high expectations, diverse learning styles, and cooperation among students" (Garon, 2000, para. 1).  However, to reach an entire class, educators need to create an opportunity for full participation and cooperation among students. 

Putting research into practice also involves building a community of learners who can dialogue effectively about mathematics, and "do" mathematics.  Much depends on the teacher's ability to assist learners with developing thinking skills, which includes incorporating writing and journaling in math classes as a way to demonstrate thinking, and their ability to question, provide feedback, use varied instructional approaches, assist learners with reading math texts and doing homework, and use tools and manipulatives, all of which help concept development. Elaboration of those follows.

Embed Thinking Skills within the Curriculum

Thinking skills can be taught, but “[t]he emerging consensus, supported by some research evidence, is that the best way to teach thinking skills is not as a separate subject but through ‘infusing’ thinking skills into the teaching of content areas”  (Wegerif, 2002, p. 3).    In support of this, Rupert Wegerif indicates that "Learners need to know what the thinking skills are that they are learning and these need to be explicitly modeled, drawn out and re-applied in different contexts. The evidence also suggests that collaborative learning improves the effectiveness of most activities" (p. 3).

So what are valued thinking skills that might be embedded within a curriculum?  Among those are information processing skills, reasoning skills, enquiry skills, creating thinking skills, and evaluation skills.  Wegerif (2002) elaborates on each of those:

Information-processing skills: These enable pupils to locate and collect relevant information, to sort, classify, sequence, compare and contrast, and to analyze part/whole relationships.

Reasoning skills: These enable pupils to give reasons for opinions and actions, to draw inferences and make deductions, to use precise language to explain what they think, and to make judgments and decisions informed by reasons or evidence.

Enquiry skills: These enable pupils to ask relevant questions, to pose and define problems, to plan what to do and how to research, to predict outcomes and anticipate consequences, and to test conclusions and improve ideas.

Creative thinking skills: These enable pupils to generate and extend ideas, to suggest hypotheses, to apply imagination, and to look for alternative innovative outcomes.

Evaluation skills: These enable pupils to evaluate information, to judge the value of what they read, hear and do, to develop criteria for judging the value of their own and others’ work or ideas, and to have confidence in their judgments. (pp. 4-5).

Incorporate Writing and Journaling in Math

Principles and Standards for School Mathematics (NCTM, 2000) call for students to communicate about mathematics.  Writing across the grades preK-12 is encouraged and should enable all students to--

• organize and consolidate their mathematical thinking though communication;
• communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
• analyze and evaluate the mathematical thinking and strategies of others;
• use the language of mathematics to express mathematical ideas precisely.

As in other curricular areas, writing and journaling in math class helps students to organize and clarify their thoughts and to reflect on their understanding of concepts.  See:  Port Angeles School District, Washington, Sample Math Questions for the Washington Assessment of Student Learning (WASL) assessments.  This district really emphasizes writing in math.  Problems by grade level (K-8 and High School) presented in the web site are recommended for student use to communicate (in written form) understanding of math content.  The series of problems are grouped by number sense, measurement, geometry, algebraic sense, probability and statistics, logic, and problem solving strategies. 

Students also need to learn how to revise their writing.  Strategies include using graphic organizers to plan writing exercises, writing on every other line so that there is room for revision, and then rereading a response to see if it makes sense and responds to the topic of the exercise.  See for example: 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.

Marilyn Burns (2004) states that writing assignments fall into four categories: keeping journals, solving math problems, explaining concepts and ideas, and writing about learning processes.  Teachers might provide initial statements, prompts, and guidelines for topics of the day for when students write to a journal.  Students might write about their reasoning and problem solving process as they solve math problems.  They might comment on why their solution makes sense mathematically and as a real-life solution.  When explaining a concept or idea, students might also provide an example.  Some writing might include commentary about the general nature of the learning activity, such as what they liked the most or least about a learning unit, or their reactions to working alone or in a group.  They might show their creative side to develop a game or learning activity, or compose directions for others on how to do one of their own already-completed math activities.

Among Burns' (2004) strategies to incorporate writing in math is to have students discuss their ideas before writing, post useful vocabulary on a class chart, and use students' writing in subsequent instruction.  Posting vocabulary reminds students to use the language of math to express their ideas.  Above all students should know that writing supports their learning and helps you to assess their progress.  They should share their writing in pairs or small groups so that they can get alternative viewpoints or bring to light conflicting understanding.  This latter provides a stringboard for further discussion.

Individuals interested in learning more about how to use writing and journaling in math classes should consult Tools for Understanding, a resource guide for extending mathematical understanding in secondary schools.  This project at the University of Puget Sound was funded by a grant from the US Department of Education, Office of Special Education Programs.

Improve Questioning and Dialogue

Effective questioning and dialogue promote thinking and understanding.  That discourse is among NCTM's (1991) Professional Standards for Teaching Mathematics. Teachers orchestrate discourse by "posing questions and tasks that elicit, engage, and challenge each student's thinking" (Standard 2).  The art of questioning involves knowing when to listen, when to ask students to clarify and justify their ideas, when to take ideas that students present and pursue those in depth, and when and how to convert ideas into math notation.  Teachers must decide when to add their own input, when to let students struggle with difficulties, and monitor and encourage participation (Standard 2)  They enhance discourse with tasks that employ computers, calculators, and other technology; concrete materials used as models; pictures, diagrams, tables, and graphs; invented and conventional terms and symbols; metaphors, analogies, and stories; written hypotheses, explanations and arguments; and oral presentations and dramatizations (Standard 4).

Likewise, the student has a role in discourse.  They, too, must listen, initiate questions and problems, and respond to others; use a variety of tools to explore examples and counterexamples; convince themselves and others of the representations, solutions, conjectures, and answers.  They must rely on evidence and argument to determine validity (NCTM, 1991, Teaching Standard 3).

The art of questioning can be introduced to students as earlier as Kindergarten, and it is the way teachers pose questions that affects the richness of a discussion.  One of the biggest problems in the art of questioning is that teachers do not have an appropriate wait-time between posing the question and getting the answer.  Students need time to process the question and reflect on it before answering.  Too often when there is insufficient time given, the teacher tends to answer his/her own question, or will call on students who they are relatively certain will have that answer.  Thus, the whole class is not involved.

As in the online learning environment, the richest discussions will come from higher order open-ended questions, as opposed to centering or closed-ended questions, and then probing follow-up questions (Muilenburg & Berge, 2000).   In his Questioning Toolkit, Jamie McKenzie (1997) lists 17 types of questions and elaborates on their role in addressing the essential questions related to a unit of study.  Among those are organizing, elaborating, divergent, subsidiary, probing, clarification, strategic, sorting/sifting, hypothetical, planning, unanswerable, and irrelevant. 

New teachers, and some of us veterans, might have difficulty in getting students to discuss mathematics in class.  You will find helpful suggestions for discussion in How to Get Students to Talk in Class from Stanford University's Center for Teaching and Learning.  Among those are to decentralize responses to you as teacher by encouraging learners to direct them specifically to others in the class, share discussion authority with student facilitators, ask open-ended questions, give students time to think and perhaps brainstorm answers to questions with a classmate, be encouraging to those who take risks to answer even if the answer was incorrect, use strategic body language, take notes on student responses to help summarize views later or keep discussion moving, and use active learning strategies.

Consider also the role that new technology tools, such as podcasts, blogs, or wikis, can play in increasing dialogue about mathematics.  Students might use their classroom wiki to create their own textbook with group understandings of various topics, or for collaborative problem solving, projects, applications of math in everyday life, and so on.  They might create podcasts in which they vocalize understandings individually or as a group to share with others.  For more on the pedagogic value of podcasts and wikis, see Wiki Pedagogy by Renée Fountain.  Blogs would be useful for monitoring individual contributions of learners in discussion on a variety of topics.  Their commentaries are revealed in reverse chronological order (i.e., the most recent is listed first).

Improve Feedback

Another key to successful instruction is effective feedback and reinforcement.  Feedback should be clearly understood, timely, immediately useable by students, consistent, comprehensive, supportive, and valued (Garon, 2000).  "When anyone is trying to learn, feedback about the effort has three elements: recognition of the desired goal, evidence about present position, and some understanding of a way to close the gap between the two" (Sadler, in Black & Wiliam, 1998, sec: Self Assessment by Pupils).

David Nicol and Debra Macfarlane-Dick (n.d.) provide examples of good practice strategies related to each of the following principles of good feedback, which are drawn from their formative assessment model and review of research literature:

1. Facilitates the development of self-assessment (reflection) in learning.

2. Encourages teacher and peer dialogue around learning.

3. Helps clarify what good performance is (goals, criteria, expected standards).

4.  Provides opportunities to close the gap between current and desired performance.

5. Delivers high quality information to students about their learning.

6. Encourages positive motivational beliefs and self-esteem .

7. Provides information to teachers that can be used to help shape the teaching. (p. 3)

Use Varied Instructional Approaches

Putting research into practice involves teaching for understanding by using a variety of instructional approaches.  While teachers might question if their approach should be more teacher-centered or more student-directed, the National Mathematics Advisory Panel (2008) noted, "High-quality research does not support the exclusive use of either approach" (p. 45).  The terms themselves are not uniquely defined with "teacher-directed instruction ranging from highly scripted direct instruction approaches to interactive lecture styles, and with student-centered instruction ranging from students having primary responsibility for their own mathematics learning to highly structured cooperative groups" (p. 45).

The Rochester Institute of Technology (2008) notes how a mix of strategies might benefit visual, auditory, and kinesthetic learners.  Visual learners appreciate lessons with graphics, illustrations, and demonstrations.  Auditory learners might learn best from lectures and discussions.  Kinesthetic learners process new information best when it can be touched or manipulated; thus, for this group of learners, written assignments, note taking, examination of objects, and participation in activities are valued strategies to consider.

According to Ball et al. (2005):

Students can learn effectively via a mixture of direct instruction, structured investigation, and open exploration. Decisions about what is better taught through direct instruction and what might be better taught by structuring explorations for students should be made on the basis of the particular mathematics, the goals for learning, and the students' present skills and knowledge. For example, mathematical conventions and definitions should not be taught by pure discovery. Correct mathematical understanding and conclusions are the responsibility of the teacher. Making good decisions about the appropriate pedagogy to use depends on teachers having solid knowledge of the subject. (section: Areas of Agreement)
 

Using instructional approaches such as "problem-based learning, scientific experimentation, historical investigation, Socratic seminar, research projects, problem solving, concept attainment, simulations, debates, and producing authentic products and performances" (Tomlinson & McTighe, 2006, p. 110) will help you to uncover the BIG ideas related to content that lie below the surface of acquiring basic skills and facts. 

When teaching for understanding, a unit or course design incorporates instruction and assessment that reflects six facets of understanding.  Students are provided opportunities to explain, interpret, apply, shift perspective, empathize, and self-assess (McTighe & Seif, n.d.).  Framing the essential or BIG questions in a unit is an important skill for educators to acquire, as these questions offer the organizing focus for a unit. Tomlinson and McTighe (2006) suggested two to five essential questions per unit, which are written at age-appropriate levels and sequenced so that one leads to the next.  Students need to understand key vocabulary associated with those questions.

The emphasis on vocabulary development is particularly important for learning mathematics with understanding, especially for students for whom English is a second language.  Imagine their possible confusion upon encountering homophones like "pi/pie, plane/plain, rows/rose, sine/sign, sum/some" (Bereskin, Dalrymple, Ingalls, et al., 2005, p. 3). Key vocabulary must be explicitly taught, and reinforced by posting symbols with definitions and examples to clarify meaning. Such learners also benefit from materials presented in their native language, where possible.  In TIPS for English Language Learners in Mathematics, Bereskin, Dalrymple, Ingalls, and others from the Ontario (CA) Ministry of Education and their Partnership of School Boards proposed the following types of mathematical activities that help to develop both mathematics and language skills:

• Computing

• Recalling facts

• Manipulating

• Using manipulatives and technology

• Exploring

• Hypothesizing

• Inferring/concluding

• Revising/revisiting/reviewing/reflecting

• Making convincing arguments, explanations, and justifications

• Using mathematical language, symbols, forms, and conventions

• Explaining

• Integrating narrative and mathematical forms

• Interpreting mathematical instructions, charts, drawings, graphs

• Representing a situation mathematically

• Selecting and sequencing procedures (p. 3).

In discussing essential principles of effective math instruction for all learners, including learners with disabilities and those at risk of school failure, Karen Smith and Carol Geller (2004) said common attributes that have been identified as positively affecting student learning include:

• Teaching concepts

• Connecting learners' prior knowledge and the new concept.

• Presenting information from a problem solving perspective that is relevant to the learner (i.e., authentic).

• Demonstrating word problems at concrete (e.g., with manipulatives), pictorial, and abstract levels.

• Providing examples and non-examples of the concept.

• Scripting or visually representing the concept with necessary steps to solve the problem.  This promotes mastery of the concept at the abstract level.

• Providing guided practice with feedback.

• Completing an error analysis of a learner's work, as well as having learners verbally describe their strategy and generate their own problem to illustrate the concept.  This helps to evaluate learner mastery of the concept and to determine the next step in instruction.  The practice also promotes retention.

Notice that Smith and Geller (2004) also noted the importance of feedback.  In support of the above attributes, Leinwand and Fleishman (2004) suggested the following to teach for meaning:

• Pose open-ended questions: Why do you think that? Explain your reasoning.

• Make explicit connections and incorporate pictures, concrete materials, and role playing in instruction so that students have alternative ways for developing understanding.

• Avoid instruction focused on teaching a single correct approach to problem solving.

Teach Reading the Math Text

Students must be taught how  to read a math textbook.  Most students, in my experience, have never learned how, and rely greatly on explanations from their teachers and jump right in to doing their homework problems without reading the text.  According to Mariana Haynes (2007), "The research is clear that when teachers across content areas help students use reading comprehension strategies (such as summarizing, generating questions, and using semantic and graphic organizers), student learning improves substantially. Studies show that explicitly teaching these strategies requires students to actively process information and connect new learning with prior concepts and experiences" (p. 4).

Reading a math text is different from reading texts in other subject areas.  Diana Metsisto (2005), who discusses this issue in depth in Reading in the Mathematics Classroom, stated that math texts contain a greater number of concepts per sentence and paragraph than in texts for other subjects.  Reading is complicated by the use of numeric and non-numeric symbols, specialized vocabulary, graphics which must be understood, page layouts that are different from other texts, and topic sentences that often occur at the end of paragraphs instead of at the beginning.  The text is often written above the reading level of the intended learner.  Some small words when used in a math problem make a big difference in students' understanding of a problem and how it is solved.  Metsisto provides reading strategies for math texts.  Cynthia Arem (Pima Community College) also provided a concise list of tips on Reading a Math Textbook, which can easily be shared with students. 

Provide Homework Assistance

The issue of assigning homework is controversial in terms of its purpose, what to assign, the amount of time needed to complete it, parental involvement, its actual affect on learning and achievement, and impact on family life and other valuable activities that occur outside of school hours.  To help ensure that homework is completed and appropriate, consider the following research-based homework guidelines provided by Robert Marzano and Debra Pickering (2007, p. 78):

• Assign purposeful homework.  Legitimate purposes for homework include introducing new content, practicing a skill or process that students can do independently but not fluently, elaborating on information that has been addressed in class to deepen students' knowledge, and providing opportunities to explore topics of their own interest.

• [E]nsure that homework is at the appropriate level of difficulty.  Students should be able to complete homework assignments independently with relative high success rates, but they should still find the assignments challenging enough to be interesting.

• Involve parents in appropriate ways (for example, as a sounding board to help students summarize what they learned from the homework) without requiring parents to act as teachers or to police students' homework completion.

• Carefully monitor the amount of homework assigned so that it is appropriate to students' age levels and does not take too much time away from other home activities.  (p. 78).

A rule of thumb for homework might be that "all daily homework assignments combined should take about as long to complete as 10 minutes multiplied by the students' grade level" and "when required reading is included as a type of homework, the 10-minute rule might be increased to 15 minutes" (Cooper, 2007, cited in Marzano & Pickering, 2007, p. 77).  Other tips for getting homework done are in Helping Your Students with Homework, a 1998 booklet based on educational research from the U.S. Department of Education. 

Use Tools and Manipulatives

Students' thinking and understanding will be enhanced by their use of a variety of tools, such as graphic organizers, thinking maps, calculators, computers, and manipulatives. 

Graphic organizers help learners to visually organize and interrelate information.  According to Judy Willis (2006), "Graphic organizers are a creative alternative to rote memorization"; they "coincide with the brain's style of patterning" and promote this patterning "because material is presented in ways that stimulate students' brains to create meaningful and relevant connections to previously stored memories" (Ch. 1, sec: Graphic Organizers).  Students can generate their own graphic organizer using the following sample instructions, adapted from Willis (2006, ch. 1):

Graphic organizers come in many forms and might be classified as sequential, relating to a single concept, or multiple concepts.  They are often used in brainstorming. Common forms include continuum scales, cycles of events, spider maps, Venn diagrams, compare/contrast matrices, and network tree diagrams.  The Enhance Learning with Technology Web site contains numerous resources on graphic organizers.  For example, a Venn diagram (two or more overlapping circles) could be used to compare and contrast sets, such as in a study of least common multiple and greatest common factor, or classifying geometric shapes.  A tree diagram is useful for determining outcomes in a study of probability of events, permutations and combinations.

As another example, Metsisto (2005) suggested the Frayer Model  and Semantic Feature Analysis Grid.  The Frayer Model is used for vocabulary building and is a chart with four quadrants which can hold a definition, some characteristics/facts, examples, and non-examples of the word or concept.  The word or concept might be placed at the center of the chart.  In Think Literacy: Mathematics Approaches for Grades  7-12, the Ontario Association for Mathematics Education (2004) further elaborates on reading, writing and oral communication strategies and provides a thorough discussion of the Frayer Model. 

The Semantic Feature Analysis Grid is a matrix or chart to help students to organize common features and to compare and contrast concepts.  Spreadsheets are useful to design these kinds of charts.  In Knowledge Maps: Tools for Building Structure in Mathematics, Astrid Brinkmann (2005, October 25) discusses the rules for developing mind maps and concept maps and illustrates how they are used to graphically link ideas and concepts in a well-structured form.

Thinking maps are closely aligned to graphic organizers; however, in the words of David Hyerle, they are "a LANGUAGE of interdependent graphic primitives....teachers and student thrive within the dynamism of eight integrated tools based on thinking patterns. (a simple analogy may be made to complexity of 8 parts of speech and how they are relatively meaningless in isolation, and convey complexity when used together... this also leads to deep, authentic assessment" (personal communication, Oct. 6, 2007).  Thinking maps are open-ended, allow students to draw on their own experience, and help them to identify, "organize, synthesize, and communicate patterns of information by using a common visual language.  They enable students to explore multiple perspectives and to develop metacognitive strategies for planning, monitoring, and reflecting" (Lipton & Hyerle, n.d., p. 6).  The eight maps are discussed and illustrated with student examples at mapthemind.com.  Lipton and Hyerle also described them, which I have adapted for the following table:

Overall, Harold Wenglinsky (2004) concluded that "teaching that emphasizes higher-order thinking skills, project based learning, opportunities to solve problems that have multiple solutions, and such hands-on techniques as using manipulatives were all associated with higher performance on the mathematics" National Assessment of Educational Progress among 4th and 8th graders (p. 33).  Using such practices to teach for meaning promotes high performance for students at all grade levels.  CT4ME has an entire section devoted to Math Manipulatives. 

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References

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