NCLB mandated that states and districts adopt programs and policies supported by scientifically based research. Drawing upon research and an extensive collection of evidence from multiple sources, the Common Core State Standards were developed to reflect the knowledge and skills that young people need for success in college and careers. Those standards impact teachers in several ways, including to guide them "toward curricula and teaching strategies that will give students a deep understanding of the subject and the skills they need to apply their knowledge" (Common Core State Standards Initiative, FAQ section). For many the standards require changes in how mathematics is taught, thus they will influence instructional strategies that educators use. In a standardsbased classroom four instructional strategies are key:
Math Methodology is a three part series on instruction, assessment, and curriculum. Sections contains relevant essays and resources:
Part 1: Math Methodology: Instruction
The Instruction Essay (Page 1 of 3) contains the following subsections:
The Instruction Essay (Page 2 of 3) contains the following subsections:
The Instruction Essay (Page 3 of 3) addresses the needs of students with math difficulties and contains the following subsections:
Math Methodology Instruction Resources also includes resources for special needs students (e.g., hearing and visually impaired, learning disabilities, English language learners).
Part 2: The Role of Assessment, essay and resources
Part 3: Curriculum: Content and Mapping and resources
Although learning for understanding is unique to an individual, teachers can enhance the process of learning with their own knowledge of how people learn. The National Research Council (2005) stated three fundamental and wellestablished principles of learning, which all teachers should understand and incorporate in their teaching:
What does it mean to be mathematically literate and proficient?
According to the National Research Council (2012), "Deeper learning is the process through which a person becomes capable of taking what was learned in one situation and applying it to new situations – in other words, learning for “transfer.” Through deeper learning, students develop expertise in a particular discipline or subject area" (p. 1). As mathematics educators, we want our learners ultimately to be mathematically literate and proficient in mathematics. To achieve this, educators will need to focus on deeper learning and learning for understanding.
Volker Ulm (2011) noted that mathematical literacy involves several competencies:
Developing proficiency, as the National Research Council (2001) pointed out, embodies "expertise, competence, knowledge, and facility in mathematics" and the term mathematical proficiency entails what is "necessary for anyone to learn mathematics successfully" (p. 116). It has five interwoven and interdependent strands:
conceptual understanding—comprehension of mathematical concepts, operations, and relations
procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
strategic competence—ability to formulate, represent, and solve mathematical problems
adaptive reasoning—capacity for logical thought, reflection, explanation, and justification
productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. (National Research Council, 2001, p. 116)
Becoming mathematically literate and proficient are ongoing processes. Writing in IAEpedia, David Morsund and Dick Ricketts (2010) noted that becoming proficient is a matter of developing math maturity, which certainly varies among students and which involves how well they learn and understand the math, how well they can apply their knowledge and skills in a variety of mathrelated problemsolving situations, and in their long term retention.
While teachers have a role to play in helping students to develop understanding, students also have a role to play in the process, which cannot be overlooked. They must have intrinsic motivation, as in Eric Booth's (2013) words: "Learning can be transformed into understanding only with intrinsic motivation. Learners must make an internal shift; they must choose to invest themselves to truly learn and understand" (p. 23). This kind of motivation involves fulfilling their need for creative engagement, which is where the teacher's role in the design of instruction and corresponding assessments comes into play.
But why focus on understanding?
Understanding "typically entails being able to explain processes and relationships, apply conceptual knowledge to new situations, and interpret ideas in intended ways" (Zwiers, O'Hara, & Pritchard, 2014, p. 210). In mathematics, understanding goes beyond just getting the right answer to a problem. Students develop many misconceptions over time, which must be addressed in instruction. Consider the following examples. 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." Teachers Magazine with the help of Tim Coulson of the National Numeracy Strategy in England provided 10 such Maths misconceptions (2006) and suggestions to correct the situation. That article sets the tone for the need to teach mathematics right the first time with a focus on understanding.
In Wayne Snyder's (2013) view, "Without eliciting the mathematical thinking that lies behind the answer, there is no way to tell where understanding breaks down. Only by providing the structure, environment, and opportunity for students to share the reasoning behind their answers can we dislodge misconceptions" (para. 7). Further, reasoning is a cornerstone to understanding:
Eliciting students' thinking is not just about determining their misconceptions; it is also about understanding and encouraging their correct mathematical reasoning. ... Mathematical reasoning is a required part of the new paradigm of mathematics education, underscored and made explicit in the Common Core Standards for Mathematical Practice. Reasoning does not take the place of mathematical processes; rather, it strengthens the understanding, retention, and application of processes. Teaching for mathematical reasoning involves allowing and requiring students to express and to share their reasoning; listening to their explanations; and responding, guiding, and celebrating their mathematical thinking. (para. 910)
A focus on understanding is among six key instructional shifts for implementing the Common Core State Standards (2010). Certainly, understanding is an element of proficiency and literacy. While fluency is among those shifts with students being "expected to have speed and accuracy with simple calculations," for deep understanding, teachers will be expected to "teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures." Further, teachers will need to ensure that students "demonstrate deep conceptual understanding of core math concepts by applying them to new situations. as well as writing and speaking about their understanding" (EngageNY, 2011).
This is not to negate the role of some memorization in mathematics. Morsund and Ricketts (2010) also noted, "It is well recognized that some rote memory learning is quite important in math education. However, most of this rote learning suffers from a lack of long term retention and from the learner’s inability to transfer this learning to new, challenging problem situations both within the discipline of math and to mathrelated problem situations outside the discipline of math. Thus, math education (as well as education in other disciplines) has moved in the direction of placing much more emphasis on learning for understanding. There is substantial emphasis on learning some “big ideas” that will last a lifetime" (section 1.1: Math Maturity, para. 12).
What does literacy look like in the mathematics classroom?
A central strategy for developing mathematical literacy is "enabling students to find their own independent approaches to learning" (Ulm, 2011, p. 5). According to the Ohio Department of Education (2012), there are multiple ways for developing literacy in the mathematics classroom:
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. 23). "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 A^{2}+ B^{2}. The following visual helped clarify (A+B) (A+B) = A^{2} + 2AB + B^{2}

A 
B 
A 


B 


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
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5
x 3
1
x 3
1
x 3
1
5
2
0
2
5
2
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1
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What are the avenues to understanding?
What goes on in the classroom on a daily basis and over the course of a unit of instruction is key to processing information for understanding. Robert Marzano (2009) identified five avenues to understanding: chunking information into small bites, scaffolding, interacting, pacing, and monitoring. Of those, scaffolding is key to the entire process, as it involves the content of those chunks and their presentation in a logical order. After presenting a chunk of reasonable length, it is important for teachers to pause and allow students to interact with each other. A high rate of interaction among learners is a necessary component for understanding. Monitoring enables teachers to determine if a chunk has been understood before moving on. Pacing, how fast or slow to move through chunks, is not easily predetermined. It depends on being able to read students' understanding and engagement with the content.
Within the classroom, how teaching is organized also matters. Spacing out learning over time with review and quizzing helps learners retain information over the course of the school year and beyond. According to research, such spacing and exposure to concepts and facts should occur on at least two occasions, separated by several weeks or months. Students will learn more when teachers alternate their demonstration of a worked problem with a similar problem that students do for practice. This helps students to learn problem solving strategies, enables them to transfer those strategies more easily, and to solve problems faster. Student learning is improved if teachers connect abstract ideas and concrete contexts via stories, simulations, handson activities, visual representations, realworld problem solving, and so on. Teachers can also enhance learning by using higher order questioning and providing opportunities for students to develop explanations. This ranges from creating units of study that provoke questionasking and discussion to simply having students explain their thinking after solving a problem (Pashler, Bain, Bottge, et al., 2007).
A Note on RealWorld Contexts
According to the National Mathematics Advisory Panel (2008):
The use of “realworld” contexts to introduce mathematical ideas has been advocated, with the term “real world” being used in varied ways. A synthesis of findings from a small number of highquality studies indicates that if mathematical ideas are taught using “realworld” contexts, then students’ performance on assessments involving similar “realworld” problems is improved. However, performance on assessments more focused on other aspects of mathematics learning, such as computation, simple word problems, and equation solving, is not improved. (p. xxiii)
Marzano, Pickering, and Pollock (2001) included nine researchbased 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 "should not be expected to work equally well in all situations." They are grouped together into three categories for strategies that provide evidence of learning, strategies that help students acquire and integrate learning, and strategies that help students practice, review, and apply learning, as suggested by Pitler, Hubbell, Kuhn, and Malenoski (2007) in Using Technology With Classroom Instruction That Works:
Strategies that provide evidence of learning:
Setting objectives and providing feedbackset 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.
Reinforcing effort and providing recognitionyou 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.
Strategies that help students acquire and integrate learning:
Cues, questions, and advance organizersthese should be highly analytical, should focus on what is important, and are most effective when used before a learning experience.
Nonlinguistic representationincorporate words and images using symbols to show relationships; use physical models and physical movement to represent information
Summarizing and note takingprovide guidelines for creating a summary; give time to students to review and revise notes; use a consistent format when note taking
Cooperative learningconsider common experiences or interests; vary group sizes and objectives. Core components include positive interdependence, group processing, appropriate use of social skills, facetoface interaction, and individual and group accountability.
Note: Reinforcing effort from the first category also fits into this category to help students.
Strategies that help students practice, review, and apply learning:
Identifying similarities and differencesgraphic forms, such as Venn diagrams or charts, are useful
Homework and practicevary 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
Generating and testing hypothesesa deductive (e.g. predict what might happen if ...) , rather than an inductive, approach works best.
Learn more about how to teach for understanding and mathematical literacy.
Tools for Understanding is 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.
Each month you can freely download an issue in the series Towards New Teaching in Mathematics from SINUS International (Germany). These are in English and great for middle and high school. Issues 18 address:
Read Strengthening the Student Toolbox: Study Strategies to Boost Learning by John Dunlosky (2013, Fall) in American Educator. In Dunlosky's view, "teaching students how to learn is as important as teaching them content, because acquiring both the right learning strategies and background knowledge is important—if not essential—for promoting lifelong learning" (pp. 1213).
Laney Sammons (2011) offers assistance with this endeavor in Building Mathematical Comprehension: Using Literacy Strategies to Make Meaning. She links reading comprehension strategies and research to mathematics instruction. You'll find comprehension strategies for mathematics, strategies for recognizing and understanding vocabulary, the role of schema theory and strategies for making math connections, good questioning strategies, the importance of visualization of mathematical ideas, and how to help learners enhance their mathematical understanding via inferences and predictions. There are strategies for teaching learners how to determine mathematical importance, to synthesize for meaning and for monitoring and repairing comprehension. Sammons also considers the components of a guided math classroom. "This resource is aligned to the interdisciplinary themes from the Partnership for 21st Century Skills and supports the Common Core State Standards."
So what can you do to put research into practice?
Educators should have one goal in mind in everything they do: achievement of learners, which includes their ability to transfer knowledge to new situations. To this end, researchbased instructional strategies focusing on deep learning should be used, as suggested by the National Research Council (2012):
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). Such "accountability includes actions of adults, not merely the scores of students" (p. 83). Among those actions of adults is to assist students with gaining proficiency in a range of their own academic learning skills and behaviors. Writing in relation to the new Common Core State Standards, David Conley (2011) emphasized:
These behaviors include goal setting; study skills, both individually and in groups; selfreflection and the ability to gauge the quality of one's work; persistence with difficult tasks; a belief that effort trumps aptitude; and timemanagement skills. These behaviors may not be tested directly on common assessments, but without them, students are unlikely to be able to undertake complex learning tasks or take control of their own learning. (p. 20)
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, studentfaculty 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
Consider learning some basic facts about the brain and the geography of thinking.
Visit The National Institute of Neurological Disorders and Stroke for an introduction to the brain and how it works.
See the table of Thinking and Learning Characteristics of Young People with suggested teaching strategies, presented at PUMUS, the online journal of practical uses of math and science. The table is subdivided into sections for grades K2, 35, and 68.
Teaching critical thinking is very hard to do, but there are strategies consistent with research to help learners acquire the ability to think critically. According to Daniel Willingham (2007), a professor of cognitive psychology, "the mental activities that are typically called critical thinking are actually a subset of three types of thinking: reasoning, making judgments and decisions, and problem solving" (p. 11). Studies have revealed that:
First, critical thinking (as well as scientific thinking and other domainbased thinking) is not a skill. There is not a set of critical thinking skills that can be acquired and deployed regardless of context. Second, there are metacognitive strategies that, once learned, make critical thinking more likely. Third, the ability to think critically (to actually do what the metacognitive strategies call for) depends on domain knowledge and practice. (p. 17)
Rupert Wegerif (2002) noted, “[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” (p. 3). In agreement, Willingham (2007) added that when learners "don't have much subject matter knowledge, introducing a concept by drawing on student experiences can help" (p. 18). Further, "Learners need to know what the thinking skills are that they are learning and these need to be explicitly modeled, drawn out and reapplied in different contexts. The evidence also suggests that collaborative learning improves the effectiveness of most activities" (Wegerif, 2002, p. 3).
Not only must the strategies be made explicit, but practice is an essential element. Willingham (2007) suggested:
The first time (or several times) the concept is introduced, explain it with at least two different examples (possibly examples based on students’ experiences ...), label it so as to identify it as a strategy that can be applied in various contexts, and show how it applies to the course content at hand. In future instances, try naming the appropriate critical thinking strategy to see if students remember it and can figure out how it applies to the material under discussion. With still more practice, students may see which strategy applies without a cue from you. (p. 18)
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) elaborated on each of those:
Informationprocessing 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. 45)
Donald Treffinger (2008) distinguished between creating thinking and critical thinking, stating that effective problem solvers need both, as they are actually complementary. The former is used to generate options and the latter to focus thinking. Each form of thinking has associated guidelines and tools, illustrated in the following table.
Guidelines and Tools for Creative vs. Critical Thinking 

Creative 
Critical 

Guidelines  Defer judgment, seek quantity, encourage all possibilities, look for new combinations that might be stronger than any of their parts.  Use affirmative judgment as opposed to being critical, be deliberateconsider the purpose of focusing, consider novelty and not only what has worked in past, stay on course. 
Tools  Brainstorming  Hits and Hot Spotsselecting promising options and grouping in meaningful ways 
ForceFittingforcing a relationship between two seemingly unrelated ideas  ALoUacryonym for what to consider when refining and developing options: A  Advantages, L  Limitations, o  ways to overcome limitations, U  Unique features  
Attribute Listing  PCA or Paired Comparison Analysisused to rank options or set priorities  
SCAMPERacronym for how to apply checklist of action words to look for new possibilities: S  Substitute, C  Combine, A  Adapt, M  Magnify or Minify, P  Put to other uses, E  Eliminate, R  Reverse or Rearrange)  Sequence: SMLsequence short, medium, or longterm actions  
Morphological Matrixidentify key parameters of task) 
Create Evaluation Matrix consider all options and possibilities 

Adapted from Treffinger, D. (2008, Summer). Preparing creative and critical thinkers [online]. Educational Leadership, 65(10). Retrieved from http://www.ascd.org/publications/educational_leadership/summer08/vol65/num10/toc.aspx 
Research scientists Derek Cabrera and Laura Colosi (Wheeler, 2010) identified yet another approach to teaching thinking skills, the DSRP method, that is tied to four universal patterns that structure knowledge:
DSRP focuses on making teachers and students more metacognitive and can be used in any standardsbased curriculum. Cabrera and Colosi believe the system works because it is so simple.
In How to Teach Thinking Skills Within the Common Core, James Bellanca, Robin Fogarty, and Brian Pete (2012) identified three essential thinking skills for explicit teaching within each of seven student proficiencies. Proficiencies and related skills are critical thinking (analyze, evaluate, problem solve), creative thinking (generate, associate, hypothesize), complex thinking (clarify, interpret, determine), comprehensive thinking (understand, infer, compare/contrast), collaborative thinking (explain, develop, decide), communicative thinking (reason, connect, represent), and cognitive transfer (synthesize, generalize, apply). Such skills are explicitly stated within the CCSS or are implicit in the language of the standards. Each chapter includes an explicit teaching lesson, a classroom content lesson, a CCSS performance task lesson, and reflection questions. Online and print resources and reproducibles are also included.
Incorporate Writing and Journaling in Math
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. Reeves (2006) noted, "The most effective writing is nonfictiondescription, 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.
Principles and Standards for School Mathematics (NCTM, 2000) call for students to communicate about mathematics. Writing across the grades preK12 is encouraged and should enable all students to
Port Angeles School District (WA) emphasizes writing in math, as illustrated by their Math Practice Problems for the Washington assessments. Problems by grade level (K8 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) stated 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 reallife 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 alreadycompleted math activities.
To illustrate Burns' (2004) ideas, Marian Small (2010) suggested providing parallel tasks to learners as a way to differentiate math instruction. Students might choose between two problems, which differ in difficulty. However, regardless of choice, teachers might pose a set of common questions for all students to answer. Such questions focus on common elements. For example, one question might be a reflection on the estimation of the answer to the problem itself before calculating the exact answer. Another might ask students to explain why a particular operation(s) is needed to solve it, or what would happen if one number was changed, or how mental math might be used, or to explain the exact strategy actually used to solve the problem (p. 32). Students might then write answers to such questions in a journal.
Among Burns' (2004) other 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 might consult the following. You'll also find resources for products to assist with writing in math:
Improve Questioning and Dialogue
The Common Core Standards (2010) for Mathematical Practice include that students "Construct viable arguments and critique the reasoning of others" (Standard 3). In addressing this standard through questioning and dialogue, teachers facilitate interactive participation to promote their students' conceptual understanding and problem solving abilities. As students communicate with others and present their ideas, the discourse process can also help them to "Attend to precision" as they "try to use clear definitions in discussion with others and in their own reasoning" (Standard 6). To improve questioning and dialogue, both the teacher role and the students' role should be considered.
Teacher Role
Mathematical discussions can involve concepts, procedures, explanations, and various representations, which lead to a key principle that a discussion about mathematics should achieve a goal. As not all discussions are conducted the same way, each requires planning to achieve a particular goal. For example, a discussion might be openended, if the goal was to generate as many ideas as possible on a topic (e.g. strategies for solving a particular problem, such as 25 x 18 mentally). Or, a discussion might be more targeted on a particular idea. It is the teacher's role to determine the goal, plan for how the discussion will be conducted, present the mathematical ideas, and orient learners to how they will share ideas with each other. It is also the teacher's role to ensure that all learners believe their ideas are valued and that all learners contribute to achieving the goal (Kazemi & Hintz, 2014, Ch. 1).
Participating in a mathematical community through discourse is as much a part of learning mathematics as a conceptual understanding of the mathematics itself. As students learn to make and test conjectures, question, agree, or disagree about problems, they are learning the essence of what it means to do mathematics. If all students are to be engaged, teachers must foster classroom discourse by providing a welcoming community, establishing norms, using supporting motivational discourse, and pressing for conceptual understanding. (Stein, 2007, p. 288)
The process of building a community begins with what the teacher says and the way teachers pose questions, as this affects the richness of a discussion. According to Paul and Elder (1997), "The oldest, and still the most powerful, teaching tactic for fostering critical thinking is Socratic teaching. In Socratic teaching we focus on giving students questions, not answers." Mastering the process of Socratic questioning is highly disciplined:
The Socratic questioner acts as the logical equivalent of the inner critical voice which the mind develops when it develops critical thinking abilities. The contributions from the members of the class are like so many thoughts in the mind. All of the thoughts must be dealt with and they must be dealt with carefully and fairly. By following up all answers with further questions, and by selecting questions which advance the discussion, the Socratic questioner forces the class to think in a disciplined, intellectually responsible manner, while yet continually aiding the students by posing facilitating questions. (Paul & Elder, 1997)
Paul and Elder (1997) noted multiple dimensions for questioning and dialogue:
We can question goals and purposes. We can probe into the nature of the question, problem, or issue that is on the floor. We can inquire into whether or not we have relevant data and information. We can consider alternative interpretations of the data and information. We can analyze key concepts and ideas. We can question assumptions being made. We can ask students to trace out the implications and consequences of what they are saying. We can consider alternative points of view. (Paul & Elder, 1997)
In Intentional Talk: How to Structure and Lead Productive Mathematical Discussions Kazeemi and Hintz (2014) characterized other possible targeted discussion structures for mathematics, as noted below.
Examples of Structures and Goals of Targeted Mathematics Discussions  
Targeted Discussion Structure  Goal 
Compare and Connect  to compare similarities and differences among strategies 
Why? Let's Justify  to generate justifications for why a particular mathematical strategy works 
What's Best and Why?  to determine a best (most efficient) solution strategy in a particular circumstance 
Define and Clarify  to define and discuss appropriate ways to use mathematical models, tools, vocabulary, or notation 
Troubleshoot and Revise  to reason through which strategy produces a correct solution or figure out where a strategy when awry 
Adapted from Kazemi, E., & Hintz, A. (2014). Intentional talk: How to structure and lead productive mathematical discussions, p. 3. Portland, ME: Stenhouse Publishers. 
However, to promote thinking and understanding for all learners, the effective questioner also needs to "draw as many students as possible into the discussion," and "periodically summarize what has and what has not been dealt with and/or resolved" (Paul & Elder, 1997). Unfortunately, this does not always occur in classrooms. Too often, math teachers tend to look for one right answer, which leads to one of the biggest problems in the art of questioningteachers do not have an appropriate waittime between posing the question and getting the answer. Students need time to process the question and reflect on it before answering. When there is insufficient time given, teachers tend to answer their own question, or will call on students who they are relatively certain will have that answer. Thus, the whole class is not involved.
That discourse was 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). NCTM provides a twopart collection of tips: Asking Good Questions and Promoting Discourse.
In his Questioning Toolkit, Jamie McKenzie listed 17 types of questions and elaborated 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 (McKenzie, 1997). Unfortunately, too often teachers unskilled in the art of questioning will pose questions that involve "only simple processes like recognition, rote memory, or selective recall to formulate an answer." Such cognitivememory questions are at the lowest level of Gallagher and Ascher's Questioning Taxonomy: cognitivememory, convergent, divergent, and evaluative questions (Vogler, 2008, Gallagher and Ascher's Questioning Taxonomy section).
As in the online learning environment, the richest discussions will come from higher order openended questions (i.e., divergent or evaluative questions), as opposed to centering or closedended questions (i.e., cognitivememory or convergent questions), and then probing followup questions (Muilenburg & Berge, 2000). Open ended questions also better involve the whole class and thus enable teachers to better differentiate instruction. Marian Small (2010) suggested four strategies on how one might do this in the mathematics classroom:
Teachers should use answers to a question to help formulate the next question, enabling questions to build upon each other. Kenneth Vogler (2008) suggested how sequencing and patterns can be accomplished:
Extending and Liftinginvolves asking a series of questions (extending) at the same cognitive level, then asking a question at the next higher level (lifting).
Circular Pathask an initial question (this one perhaps was not answered) followed by a series of questions leading back to the first one.
Same Pathall questions are asked at the same level, typically at a lower level (e.g., a series of "what is ..." questions).
Narrow to Broadlowerlevel, specific questions are followed by higherlevel, general questions.
Broad to Narrowlowerlevel, general questions are followed by higherlevel, specific questions.
A Backbone of Questions with Relevant Digressionsthe series of questions relate to the topic of discussion, rather than focus on a particular cognitive level.
Student Role
Likewise, students have a role in discourse. The art of questioning can be introduced to them as earlier as Kindergarten. 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 Right Question Institute (RQI) developed The Question Formulation Technique, which is a strategy for how to teach learners to question. RQI has provided several resources to learn the process and facilitate the technique.
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 openended 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 can play in increasing dialogue about mathematics. These might take the form of wikis, podcasts, blogs, or voting options (known as clickers) that often come with interactive whiteboards. 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 pedagogical potential of wikis, see Wiki Pedagogy posted at WikiBooks. 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). Marzano (2009) noted that whiteboard voting technologies "allow students to electronically cast their vote regarding the correct answer to a question. Their responses are immediately displayed on a pie chart or bar graph, enabling teacher and students to discuss the different perceptions of the correct answer" (p. 87).
For more on podcasts and blogs for learning, read articles by Patricia Deubel (2007): Podcasts: Where's the learning? and Moderating and ethics for the classroom instructional blog.
Improve Feedback
Another key to successful instruction is effective feedback and reinforcement. However, strictly speaking, feedback is not advice, praise, grades or evaluation, as "none of these provide the descriptive information that students need" about their efforts to reach a goal (Wiggins, 2012, p. 11). According to Susan Brookhart (2008) in How to Give Effective Feedback to Your Students, feedback strategies can vary in timing (when given and how often), amount, mode (oral, written, visual/demonstration), and audience (individual, group/class) (p. 5). It's "always adaptive. It always depends on something else. Feedback is based on the learning target, the particular assignment, the particular student, and the characteristics of a given piece of work. Feedback also depends on the depth of the teacher's understanding of the topic and of how students learn it" (p. 112).
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, Self Assessment by Pupils section).
Jan Chappuis (2012) provided the following five characteristics of effective feedback:
Everyone makes mistakes. That is, sometimes we do things that are uncharacteristic of work we might have done in past and which we might be able to correct ourselves through greater attention. So, in providing corrective feedback, we should focus on true errors, rather than pointing out all mistakes. True errors "occur because of a lack of knowledge" and fall into four broad categories," according to Douglas Fisher and Nancy Frey (2012):
David Nicol and Debra MacfarlaneDick (n.d.) provided additional principles of good feedback, which are drawn from their formative assessment model and review of research literature:
Use Varied Instructional Approaches
Putting research into practice involves teaching for understanding by using a variety of instructional approaches. James Hiebert and Douglas Grouws (2009) stated that "conceptual understandingthe construction of meaningful relationships among mathematical facts, procedures, and ideas; and skill efficiencythe rapid, smooth, and accurate execution of mathematical procedures" are "central to mathematics learning and have often competed for attention" (p. 10). While teachers might wrestle with selecting effective instructional methods for increasing learning, an important point to remember is that "particular methods are not, in general, effective or ineffective. Instructional methods are effective for something" (p. 10). The key is to "balance these two approaches, with a heavier emphasis on conceptual understanding" (p. 11).
Teachers also need to remember that varying instructional approaches is part of differentiated instruction. The Rochester Institute of Technology (2009) noted 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.
Teachers might question if their approach should be more teachercentered or more studentdirected. The National Mathematics Advisory Panel (2008) noted, "Highquality research does not support the exclusive use of either approach" (p. 45). The terms themselves are not uniquely defined with "teacherdirected instruction ranging from highly scripted direct instruction approaches to interactive lecture styles, and with studentcentered instruction ranging from students having primary responsibility for their own mathematics learning to highly structured cooperative groups" (p. 45). Ball, FerriniMundy, Kilpatrick, Milgram, Schmid, and Schaar (2005) expressed:
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. (Areas of Agreement section)
Teachers should exercise caution if students are to use a discovery approach to learning. Discovery learning is a form of partially guided instruction. Partially guided instruction is known by other names, including "problembased learning, inquiry learning, experiential learning, and constructivist learning" (Clark, Kirschner, & Sweller, 2012, p. 7).
According to Alfieri, Brooks, Aldrich, and Tenenbaum (2011), a review of literature would suggest that "discovery learning occurs whenever the learner is not provided with the target information or conceptual understanding and must find it independently and with only the provided materials" (p. 3). The extent that assistance is provided would depend on the difficulty students might have in discovering target information. Findings in their 2011 metaanalysis of 580 comparisons of discovery learning (unassisted and assisted) and direct instruction suggested that generally "unassisted discovery does not benefit learners, whereas feedback, worked examples, scaffolding, and elicited explanations do" (p. 1). Thus, Alfieri et al. indicated the following implications for teaching:
Although direct teaching is better than unassisted discovery, providing learners with worked examples or timely feedback is preferable. ... Furthermore, [their metaanalysis suggested] teaching practices should employ scaffolded tasks that have support in place as learners attempt to reach some objective, and/or activities that require learners to explain their own ideas. The benefits of feedback, worked examples, scaffolding, and elicited explanation can be understood to be part of a more general need for learners to be redirected, to some extent, when they are misconstructing. Feedback, scaffolding, and elicited explanations do so in more obvious ways through an interaction with the instructor, but worked examples help lead learners through problem sets in their entireties and perhaps help to promote accurate constructions as a result. (p. 12)
Richard Clark, Paul Kirschner, and John Sweller (2012) further put to rest the debate on the use of partially guided instruction. After a half century of such advocacy, "Evidence from controlled, experimental studies (a.k.a. "gold standard") almost uniformly supports full and explicit instructional guidance" (p. 11). Elaborating, they revealed:
Decades of research clearly demonstrate that for novices (comprising virtually all students), direct, explicit instruction is more effective and more efficient than partial guidance. So, when teaching new content and skills to novices, teachers are more effective when they provide explicit guidance accompanied by practice and feedback, not when they require students to discover many aspects of what they must learn. ... this does not mean direct, expository instruction all day every day. Small group and independent problems and projects can be effectivenot as vehicles for making discovery, but as a means of practicing recently learned content and skills. ... Teachers providing explicit instructional guidance fully explain the concepts and skills that students are required to learn. Guidance can be provided through a variety of media, such as lectures, modeling, videos, computerbased presentations, and realistic demonstrations. It can also include class discussions and activities. (p. 6)
Uncover BIG ideas. Using instructional approaches such as "problembased 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 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 selfassess (McTighe & Seif, 2002). 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 ageappropriate levels and sequenced so that one leads to the next. Students need to understand key vocabulary associated with those questions.
Emphasize vocabulary development. The emphasis on vocabulary development is particularly important for learning mathematics with understanding. "Contentarea concepts, thinking skills, and literacy all depend on students' abilities to use complex language" (Zwiers, O'Hara, & Pritchard, 2014, p. 6).
Key vocabulary must be explicitly taught, and reinforced by posting symbols with definitions and examples to clarify meaning. Marzano and Simms (2013) provided a sixstep method for vocabulary instruction::
Granite School District in Salt Lake City, Utah, includes math vocabulary organized by grade level (K7 and secondary including for the Common Core standards, and dual immersion) and resources for teaching vocabulary development. East Moline School District 37 (IL) also includes Common Core mathematics terms by K8 grade level and high school by math strand at its website.
English language learners also benefit from materials presented in their native language, where possible. 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). 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:
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:
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:
Note: For examples on how to use openended problemsolving that enables learners to develop their own approaches, read Volker Ulm's (2011) Teaching mathematics  Opening up individual paths to learning.
Need more ideas for instructional strategies?
Visit the Teaching Channel for highquality, free videos on effective teaching practices, inspiring lesson ideas, and the Common Core State Standards.
Consider using whiteboard technology to improve the quality of your lessons.
Steven Ross and Deborah Lowther (2009) noted several valuable features for improving lesson quality when using interactive whiteboards:
Further, when interactive response systems (known as clickers) are used, teachers can pose questions to students, enabling them to get immediate feedback with answers "instantly aggregated and graphically displayed" (Ross & Lowther, 2009, p. 21). This is the kind of feedback enabling timely review of lessons and studentcentered community learning.
For whiteboard resources at this site, see Math Resources: Integrating Podcasts, Vodcasts and Whiteboards into Teaching and Learning.
Teach Reading the Math Text
Zwiers, O'Hara, & Pritchard (2014) reminded educators that the Common Core Standards include that students should read and understand "complex texts" independently and proficiently. From their work on Common Core Standards in Diverse Classrooms:
A complex text can be any written, visual, audio, or multimedia message that conveys information or ideas for learning purposes. Complexity varies, of course. A text can be complex for some students and not complex for others. More often than not, gradelevel texts in school are complex for academic English learners and others who have not been exposed to the wide ranges of ways that authors of school texts use text structure, and vocabulary to communicate their messages. (p. 63)
Hence, students should 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 discussed 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 nonnumeric 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 provided reading strategies for math texts.
Taking notes when reading a math textbook will also help learners become active readers and will help them to comprehend the content and raise questions for further discussion. Lent (2012) suggested twocolumn notes with prompts to help learners think about what they are reading. In the first column students respond to prompts appropriate to the reading selection. In the second column, students elaborate on their response or answer the question provided.
TwoColumn Notes and Prompt Ideas  
What is important?  What is trivial? 
The most important sentence in this section is:  Because 
This paragraph  Could be summarized as 
The big idea is  The details are 
The problem is asking me to  The most important words in this problem are 
The concept  Looks like this (illustrate): 
Adapted from Lent, R. C. (2012). Overcoming textbook fatigue, p. 87. Alexandria, VA: ASCD. 
As students read a textbook, they should also have in mind questions to ask themselves. Such questions help to deepen the meaning of content, particularly when thoughts are later shared with others. Lent (2012) proposed some types:
The following are additional resources:
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 researchbased 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 10minute 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.
Classroom teachers might also make learners and their parents aware of the many homework assistance sites available on the Internet, many of which are noted at CT4ME among our Math Resources: Study Skills and Homework Help.
For more on homework, including the issue of differentiated homework, read Homework: A Math Dilemma and What to Do About It (Deubel, 2007).
Use Tools and Manipulatives
Students' thinking and understanding will be enhanced by their use of a variety of tools, including tools for visualization and analysis of mathematics such as graphic organizers, thinking maps, calculators, computers, and both concrete and virtual manipulatives. However, important variables to consider that influence effectiveness of tools and manipulatives (e.g., using graphic organizers) include such things as "grade level, point of implementation, instructional context, and ease of implementation" (Hall & Strangman, 2002, Factors Influencing Effectiveness section). CT4ME has an entire section devoted to math manipulatives, which includes use of calculators. Here I delve more into graphic organizers and thinking maps.
Learn more about the variety of visualization methods.
Graphic organizers and thinking maps are just two of the many methods for visualizing and analyzing. Ralph Lengler and Martin Eppler of visualliteracy.org have developed an entire Periodic Table of Visualization Methods from simple to complex.
Within the table (accessed via the link provided above), you can roll your mouse over each element to learn more about it. You'll find visualization methods for data, information, concepts, strategies, metaphors, and compound use of more than one method within the same scheme. Elements in the table also distinguish between those that use convergent and divergent thinking, and process and structure visualization. Chris Wallace provided a list of those methods with a picture of each.
An additional list by Chris Wallace for the Visualization Methods in the Periodic Table provides the Google images of the methods and Wikipedia links to learn more about each method.
Graphic Organizers
A graphic organizer is defined as "a visual and graphic display that depicts the relationships between facts, terms, and or ideas within a learning task. Graphic organizers are also sometimes referred to as knowledge maps, concept maps, story maps, cognitive organizers, advance organizers, or concept diagrams" (Hall & Strangman, 2002, Definition section). They are valuable as "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" (Willis, 2006, Ch. 1, Graphic Organizers section). They are often used in brainstorming and to help learners examine their conceptual understanding of new content.
Graphic organizers might be classified as sequential, relating to a single concept, or multiple concepts. In The Theory Underlying Concept Maps and How to Construct and Use Them, Joseph Novak and Alberto Cañas (2008) stated, concepts within a concept map are "usually enclosed in circles or boxes of some type, and relationships between concepts indicated by a connecting line linking two concepts. Words on the line, referred to as linking words or linking phrases, specify the relationship between the two concepts." Concept is defined as "a perceived regularity in events or objects, or records of events or objects, designated by a label. The label for most concepts is a word, although sometimes we use symbols such as + or %, and sometimes more than one word is used. Propositions are statements about some object or event in the universe, either naturally occurring or constructed. Propositions contain two or more concepts connected using linking words or phrases to form a meaningful statement. Sometimes these are called semantic units, or units of meaning" (Introduction section).
Concept maps are usually developed with in "a hierarchical fashion with the most inclusive, most general concepts at the top of the map and the more specific, less general concepts arranged hierarchically below." Crosslinks between subdomains on the concept map should be added, where possible, as these illustrate that learners understand interrelationships between subdomains in the map. Specific examples illustrating or clarifying a concept can be added to the concept map, but these would not be placed within ovals or boxes, as they are not concepts (Novak & Cañas, 2008, Introduction section). Novak and Cañas presented examples of concept maps developed with CMap Tools from the Institute for Human and Machine Cognition.
Graphic organizers come in many forms. Other common forms include continuum scales, cycles of events, spider maps, Venn diagrams, compare/contrast matrices, and network tree diagrams. 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. KWL charts are useful for investigations. Note: CT4ME includes KWL charts in our resource booklets for standardized test prep. Educators might also wish to expand the KWL chart to a KWHL chart or the ultimate KWHLAQ chart to better promote 21st century skill development. These acronyms represent the following questions:
As an example, students can generate their own graphic organizer using the following sample instructions, adapted from Willis (2006, Ch. 1):
Sample Instructions
Studentgenerated Graphic Organizer
Adapted from J. Willis, Researchbased strategies to ignite student learning, (2006, Ch. 1, Graphic Organizers section)
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 nonexamples 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 712, the Ontario Association for Mathematics Education (2004) further elaborated on reading, writing and oral communication strategies and provides a thorough discussion of the Frayer Model.
Frayer Model 

Definition 
Characteristics Facts 

Word or Concept 

Examples 
NonExamples 
Similar to this Frayer Model, view the short ASCD video of grade 5 math teacher, Malinda Paige, using a Words in Context graphic organizer in a geometry lesson for learning vocabulary. Paige linked her lesson to realworld events. Inspiration and Kidspiration software can be used for other graphic organizers, which Rockingham County Public Schools (VA) has made available in multiple subject areas, including for math.
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. For example, Metsisto (2005) noted this grid is useful for comparing features of types of quadrilaterals.
Learn more by also reading Knowledge Maps: Tools for Building Structure in Mathematics, in which Astrid Brinkmann (2005) discussed the rules for developing mind maps and concept maps and illustrated how they are used to graphically link ideas and concepts in a wellstructured form.
The following are graphic organizer web sites to consider:
Graphic Organizers from Education Oasis include multiple types such as cause and effect, compare and contrast, vocabulary development and concept organizers, brainstorming, KWL, and more.
Graphic Organizers from Education Place include about 38 organizers. Learners can use these freely "to structure writing projects, to help in problem solving, decision making, studying, planning research and brainstorming."
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 is based on the work of Edwin Ellis, Ph.D., president of Makes Sense Strategies, and features SMARTsheets. The site also includes examples of how these graphic organizers can be used for math, literature, social studies, science, social/behavior. Register for free downloads.
The Graphic Organizer from Graphic.Org shows graphic organizers, concept mapping, and mind mapping examples related to their use: describing, comparing/contrasting, classifying, causal, sequencing, and decision making.
Thinking Maps
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, October 6, 2007). Thinking maps are openended, 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 Designs for Thinking. Lipton and Hyerle also described them, which I have adapted for the following table:
Thinking Maps 

Type 
Purpose 
Form 
Circle  helps students generate and identify information in context related to a topic written inside the inner circle; The map might be enclosed in a square for its frame of reference. 

Tree  can be used both inductively and deductively for classifying or grouping. 

Bubble  can be used for describing the characteristics, qualities or attributes of something with adjectives. Any number of connecting bubbles can extend from the center. 

Doublebubble  useful for comparing and contrasting. 

Flow  enables students to sequence and order events, directions, cycles, and so on. 

Multiflow  helps to analyze causes and effects of an event 

Brace  useful for identifying partwhole relationships of physical structures. 

Bridge  helps students to interpret analogies and investigate conceptual metaphors  
Adapted from Lipton, L., & Hyerle, D. (n.d.). I see what you mean: Using visual maps to assess student thinking, pp. 23. Thinking Foundation. Retrieved from http://www.thinkingfoundation.org/research/journal_articles/journal_articles.html 
Overall, Harold Wenglinsky (2004) concluded that "teaching that emphasizes higherorder thinking skills, project based learning, opportunities to solve problems that have multiple solutions, and such handson 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.
Need Examples?
Read the Magic of Math in which Ken Ellis (2005) described Fullerton IV Elementary School's (Roseburg, OR) nationally recognized approach to teaching math and watch the video documentary. Math is embedded throughout the curriculum. Their immersion approach has led to improved test scores. There is a focus on using precise mathematical vocabulary and problem solving in real world contexts. Instructional strategies include a mix of direct instruction, structured investigation, and open exploration. Fullerton is one of 20 Intel Schools of Distinction.
Watch the short video at Edutopia.org: Cooperative Arithmetic: How to Teach Math as a Social Activity. A teacher in Anchorage, Alaska demonstrates how he establishes a cooperative learning environment in an upperelementary math classroom.
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