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Are you confused by terms that educators use? The ASCD Lexicon of Learning might be what you need.

 

 

 

Math Methodology

Part 1: Instruction Essay (Page 1 of 3)
Introduction to Teaching Challenges

 

 

Teaching and Math Methodology

Instruction

 

Introduction to Teaching Challenges

According to Bob Sullo, (2009) teaching over the past quarter-century has become more professional due to the emergence of a number of "best practices" that have significantly affected curriculum and instruction. "A sampling of innovations includes differentiated instruction, Understanding by Design, the emergence of state standards, the development of curriculum frameworks, scope-and-sequence charts that inform teachers of what to teach and when to teach it, the expanded use of technology in education, active literacy, curriculum mapping, and the proliferation of professional learning communities. Formative assessment informs instruction like never before" (sec: Introduction).  What is striking is that CT4ME includes discussion of many of those innovations throughout this site.

However, the road to incorporating those best practices in teaching is filled with challenges, as learning to teach is a complex never-ending process.  One might organize that process into four major domains: planning and preparation, the classroom environment, instruction, and professional responsibilities (Danielson, 2007). Regardless of level of experience, teachers always are challenged with how to motivate learners, particularly when you consider the extent of diversity encountered in many schools in the United States. Such diversity involves "not only ways of being but ways of knowing" and "knowing how to relate to those qualities and conditions that are different from our own and outside the groups to which we belong, yet are present in other individuals and groups"  (Queensborough Community College (NY), Definition for Diversity: http://www.qcc.cuny.edu/diversity/definition.asp). Learners and teachers themselves bring to the learning environment a host of variables, such as beliefs, attitudes, perceptions, self-efficacy, motivation, learning styles, habits of mind, cultural influences and demographics (e.g., male/female, sexual orientation, ethnicity, ability/disability, socio-economic status, religion/spirituality, etc.). It is certainly helpful for teachers to be aware of their personal biases, beliefs, and attitudes, as those influence interactions with learners.

Often teachers come into the profession with a conviction that they really will help learners and are prepared to do so, only to shortly change that conviction to a hope to help learners. According to Jane Pollock (2007), teachers need to make that hope a certainty by adhering to a Big Four approach, which means

• Using precise terminology to describe what students will learn
• Undertaking purposeful instructional planning and delivery
• Employing purposeful assessment
• Applying deliberate assessment and feedback strategies (p. 7).

The Big Four is just the tip of the iceberg. Just to be considered proficient, the National Board for Professional Teaching Standards includes that teachers have "a broad grounding in the liberal arts and sciences; knowledge of the subjects to be taught, of the skills to be developed, and of the curricular arrangements and materials that organize and embody that content; knowledge of general and subject-specific methods for teaching and for evaluating student learning; knowledge of students and human development; skills in effectively teaching students from racially, ethnically, and socioeconomically diverse backgrounds; and the skills, capacities and dispositions to employ such knowledge wisely in the interest of students" (NBPTS, 2002, p. 2, sec: What Teachers Should Know, para. 2).

Teachers also accept responsibility for student success, develop communities of respect, and help students become partners in their own success. They take on three instructional roles: direct instructor, facilitator, and coach (Tomlinson & McTighe, 2006). Their professional responsibilities include reflecting on their own teaching, maintaining accurate records, communicating with families, participating in professional communities, growing professionally in content and pedagogical skills, and showing professionalism with their own integrity and ethical conduct (Danielson, 2007, ch. 1).

Thus, teachers are challenged to know and communicate subject matter; to design curriculum, instruction, and assessments; to be knowledgeable about diverse student populations, to be knowledgeable about effective uses of data and technology, to conduct action research to improve their practice, to implement existing research, and to be learner-centered in their approach. On top of all this is the need to continually grow in the profession, maintain sanity, minimize stress, learn from mistakes, and let us not forget--prepare students for standardized testing.

With all this in mind, Carole Frederick Steele (2009) would add that teachers need to be adept at improvising, interpreting events in progress, testing hypotheses, demonstrating respect, showing passion for teaching and learning, and helping students understand complexity.  Fortunately, she reminded us that "No teacher is likely to excel at every aspect of teaching....What experts attend to and ignore is markedly different from what beginners notice. The growth continuum ranges from initial ignorance (unaware) to comprehension (aware) to competent application (capable) to great expertise (inspired)," paralleling Bloom's taxonomy.  "Lack of awareness occurs before Bloom's categories. The awareness stage is a fair match for Bloom's stage of knowledge and understanding. Teachers at the capable stage use application and analysis well. Educators who reach the inspired stage have become skilled at synthesis and evaluation in regard to their thinking about teaching and learning" (sec: Introduction).

W. James Popham (2009) summed up the nature of teaching in the 21st century.  He stated, "once we strip away its external complexities, teaching boils down to teachers' deciding what they want their students to learn, planning how to promote that learning, implementing those plans, and then determining if the plans worked" (sec: Preface, para. 7).   So where does one begin? Resources on the current page will assist you with your knowledge of students and instructional practices. CT4ME's section on Professional Development includes a variety of resources to assist you with becoming more knowledgeable about the mathematics content you teach. Our section on Standardized Test Preparation provides solid advice and resources. Technology Integration will assist you with strategies for incorporating technology into your instruction, including designing your classroom web site, and incorporating multimedia into math projects. You can learn more about scientifically based research and action research at our Research Corner.

 

 

Teacher Goal Setting

Gaining confidence in teaching takes time, and requires goal setting, reflection, dialogue and collaboration among colleagues. Goals might revolve around adequate planning and classroom management, effective discipline and motivation strategies; subject matter standards and benchmarks, curriculum frameworks; and developing a range of instructional and assessment strategies.

Goal 1: Adequate Planning and Classroom Management, Discipline, Motivation

Classroom management plays a significant role in effective teaching and ultimate achievement of learners. It's more than organizing the physical space for student safety and easy access to materials. It's more than deciding how you will manage classroom procedures, instructional groups and student behavior. It means creating a classroom environment of respect and rapport, and a culture for learning (Danielson, 2007, ch. 1).  In such an environment, Sullo (2009) indicated that there is no fear factor, which some teachers themselves invoke just by their tone and what they say in reaction to learners' deeds and actions.  Sometimes teachers are not even aware of the affect their sarcasm and negativism might have on motivation.  In a culture of success, the teacher's message should be "This is important.  You can do it.  I won't give up on you" (ch. 2, Getting Started section).  Teachers know how to internally motivate learners, rather than relying on external motivators as coercions and rewards/punishments that do not work for the majority of learners.

In fact, students are internally driven by the needs built into their genetic code, and they behave in a never-ending quest to satisfy the universal needs to connect, be powerful, make choices, and have fun in a safe, secure environment. Our success as teachers is largely determined by how effective we are at creating learning environments where students can meet their needs by immersing themselves in the academic tasks we provide. (Sullo, 2009, Ch. 3, Basic Needs section)

Marzano, Marzano, and Pickering (2003) conducted a meta-analysis of 100 reports on this issue, addressing four general components of effective classroom management: rules and procedures, disciplinary interventions, teacher-student relationships, and mental set. This latter refers to an ability to remain emotionally objective and businesslike and "to identify and quickly act on potential behavioral problems" (p. 75). They found "on the average, students in classes where effective management techniques are employed have achievement scores that are 20 percentile points higher than students in classes where effective management techniques are not employed" (p. 10). 

Diversity plays a significant role in classroom management.  Disabilities and cultural differences impact behavioral differences.  It important to know the nature of a disability.  For example, an autistic child might require consistency in his/her schedule as disruptions in routine might trigger inappropriate behaviors. In responding to students with disabilities, some learners might need individualized plans for behavior management.  Ideas might be to develop a behavior progress monitoring form with categories such as "Brought supplies, Worked productively, Was respectful of others" for various time frames (e.g., periods in a school day) or to develop a behavioral contract.  In terms of cultural differences, teachers and all learners in a class should be aware of each others' interaction styles.  What is acceptable in one culture might not be in another.  For example, there are cultural differences in what is acceptable in speaking to others (e.g., one at a time, and loud voice), levels of physical activity and verbal discourse needed with thinking and learning, attitudes about sharing and respecting physical space, authority figures, what constitutes an authority figure and the manner in which deference is shown to authority figures (Voltz, Sims, & Nelson, 2010, pp. 52-55).

In terms of planning and classroom management, certainly new teachers would benefit from the wisdom of their more experienced colleagues. They and mentors can serve as resources for initial concerns such as "setting up the classroom and preparing for the first weeks of school, covering the required curriculum without falling behind or losing student interest, grading fairly, dealing with parents, and maintaining personal sanity" (Mandel, 2006, p. 67).

Experienced teachers are better able to integrate and draw connections between current, past, and future learning and relate their content to other curricular areas. They tend to be able to better use such classroom management skills as voice, gestures, reading student facial expressions and body language, and proximity. They can see the big picture--in planning they can anticipate problems and a need for alternative plans and adjust their practice accordingly. They also know their students' needs and evaluate their lessons according to students' learning growth--that is they measure effectiveness of a lesson beyond meeting the broad objective of the day. Plus, they are knowledgeable about school and community resources that can benefit students. They understand the culture of the school, and have amassed strategies to effectively engage parents in collaborative activities. They understand how to motivate students and maintain their interest even in the face of temporary failure (NBPTS, 2002).

In the classroom, Marzano et al. (2003) said that it is important to involve students in the design of classroom rules and procedures. Although rules will vary, they should be specific and generally address expectations for behavior, beginning and ending the day or period, procedures for transitioning from one activity to the next, interruptions, materials and equipment, group work, seatwork and teacher-led activities (p. 26).

In fact this involvement of students in the design of classroom rules and procedures is one of the practices of the Responsive Classroom approach to improving interactions with children in elementary school.  Some of the practices would also apply well with upper grades.  The approach is based on the premise that children learn best when they have both academic and social-emotional skills and is the result of research and thinking from child development and constructivist educators (e.g., Piaget, Gesell, Montessori, Dewey, Erikson, and Vygotsky).  As a general approach to teaching, educators might consider:

• Morning Meeting—gathering as a whole class each morning to greet one another, share news, and warm up for the day ahead
• Rule Creation—helping students create classroom rules to ensure an environment that allows all class members to meet their learning goals
• Interactive Modeling—teaching children to notice and internalize expected behaviors through a unique modeling technique
• Positive Teacher Language—using words and tone as a tool to promote children’s active learning, sense of community, and self-discipline
• Logical Consequences—responding to misbehavior in a way that allows children to fix and learn from their mistakes while preserving their dignity
• Guided Discovery—introducing classroom materials using a format that encourages independence, creativity, and responsibility
• Academic Choice—increasing student learning by allowing students teacher-structured choices in their work
• Classroom Organization—setting up the physical room in ways that encourage students’ independence, cooperation, and productivity
• Working with Families—creating avenues for hearing parents’ insights and helping them understand the school’s teaching approaches
• Collaborative Problem Solving—using conferencing, role playing, and other strategies to resolve problems with students.  (Responsive Classroom web site, 2010, About section: FAQ)

In summary, when inappropriate behaviors occur, apply the ABC method.  Determine the antecedents of the behavior, how the behavior happens, and consequences for the student after the inappropriate behavior occurs.  You might be able to help the learner substitute a more appropriate behavior as a result of your assessment (Voltz, Sims, Nelson, 2010, pp. 56-57).  And, if your teaching is not going as well as you'd like and is affecting the success of your learners, Elizabeth Breaux (2009) might suggest to consider if you've done any of the following:

1. Are you assuming students already know your expectations?
2. Do you attempt to control students too much?
3. Do you engage in power struggles?
4. Do you take student behavior personally?
5. Are you foregoing the real-life connection in what you teach?
6. Do you teach only to the whole group?
7. Are you allowing for unstructured time?
8. Have you failed to organize?
9. Have you been unprofessional?
10. Have you lowered your expectations?
11. Have you treated students unfairly?
12. Have you neglected priorities?
13. Have you been inconsistent?
14. Have you planned haphazardly?
15. Have you misused administration?
16. Do you delay feedback?
17. Have you discouraged participation?
18. Have you sacrificed the parent-teacher alliance?
19. Do you believe in the façade you've created?
20. Have you refused to acknowledge your mistakes?

If you answer is yes to any of the above mistakes, it might be time to make changes in your teaching.

To help build your confidence in teaching, understanding student behavior and learning styles, and classroom management, you might experience the potential of digital games-based learning at simSchool, a simulation program for educators. You can practice your teaching skills and get immediate feedback on how your selected strategies affect student learning. Plus, as in a real classroom, the simStudents will react to user-selected task design, as well as your teacher moves. Curriculum Associates has a free mini-course on classroom discipline strategies, accompanied by audio, which provides tips and strategies that teachers might use in classroom settings. In four lessons you will learn about setting expectations, procedures, rules, and consequences; reasons for and managing disruptive behavior, including how to avoid power struggles with students; how to document incidences of misbehavior objectively; and strategies for positive parent conferences on discipline issues and follow-up. There is also a free mini-course on motivating students to learn. You Can Handle Them All, a Web site on discipline help for teachers and parents, lists over 100 behaviors (e.g., arrogant, class clown, cheater, disorganized, overly aggressive, whiner), the affect of each, actions to take to change the behavior, and mistakes in dealing with the behavior.

Goal 2: Know Standards, Curriculum Frameworks, Instructional Strategies

All teachers must include goals to become familiar with teacher standards, subject matter standards and benchmark indicators at the state and national levels. CT4ME provides this information in our section on Standards. These frameworks specify standards that students should achieve, but do not specify the curriculum and teaching methods to be used. For this, teachers need to examine the district curriculum for how their schools and teachers aligned standards with content to be taught. They need to examine scope and sequence, instructional materials, implementation strategies, and any suggested pedagogical methods. All teachers should consider the role of active or constructivist learning, as opposed to use of the lecture method. Active student involvement reinforces learning. This is not to say, however, that teachers ought not to ever tell students anything directly.

Pekin Public Schools District #108 in Illinois (ILS) is exemplary in its effort to provide curriculum frameworks and a set of math instructional strategies commonly supported by rigorous research in schools and classrooms. Educators, parents, and students can access ILS math benchmarks by grade level (K-8). Each benchmark contains teacher clarifications and additional statements for parents and students that can serve as a checklist of what students should know and be able to do. The district's Best Instructional Practices in Mathematics is based on meta-analyses of research from documents published by the ASCD, Mid-continent Research for Education and Learning, North Central Regional Educational Laboratory, and various state agencies such as the Illinois Department of Education. Their best practices address curriculum, teaching/learning experiences, problem solving and critical thinking, accommodating diversity, attitudes, parental involvement, and assessment.

Goal 3: Investigate Assessment Methods and Test Preparation

Another goal for teachers is to investigate assessment methods and how they might be incorporated into lesson plans. Assessing student understanding and designing instruction to meet learners' needs are challenging tasks. Popham (2009) noted that assessment is a broad term:

Assessment includes both traditional paper-and-pencil exams, such as those made up of True/False, short-answer, or multiple-choice items, and a much larger collection of procedures that teachers can use to get a fix on their students' status, including the use of portfolios to document students' evolving skills and the use of anonymous self-report inventories to measure students' attitudes or interests. Assessments also include the variety of informal techniques a teacher might use to check on the status of students' skills for the purpose of guiding instruction rather than for grade-giving, such as when a teacher periodically projects multiple-choice questions on a screen during a lesson and asks students, "on the count of three," to hold up one of four prepared index cards showing the letter of what each student believes is the correct answer. (Popham, 2009, Preface section, para. 6)

Thus, formative assessment plays a major role, and its importance should not be overlooked in our zeal to prepare students for mandated accountability tests. See Part 2 of this essay for more on the role of assessment. Test preparation is a reality and Curriculum Associates, Inc., has a free-mini course on test preparation strategies to introduce you to some of the research behind test preparation and factors that affect test performance. Specific strategies for math and other content areas are included. CT4ME has an entire section devoted to standardized test preparation.

Improving Instruction

Attention to theory, research, learning styles, thinking styles, multiple intelligences, differentiated instruction and the educator's ideology play a role in improving instruction.

Theory and Research

In Improving Mathematics Instruction, James Stigler and James Hiebert (2004) indicated that teachers need theories, empirical research, and alternative images of what implementation of problem solving strategies looks like. U.S. teachers need assistance with making connections problems. As they might never have seen what it looks like to implement such problems effectively, they tend to turn making connections problems into procedural exercises.

Effective lessons incorporate best-practice. According to Daniels and Bizar (1998, as cited in Wilcox & Wojnar, 2000), there are six methods that matter in a " best practice classroom." These are integrative units, small group activities, representing to learn through multiple ways of investigating, remembering, and applying information; a classroom workshop teacher-apprentice approach, authentic experiences, and reflective assessment.  Further, Mike Schmoker (2006) stated that "the most well-established elements of good instruction [include]: being clear and explicit about what is to be learned and assessed; using assessments to evaluate a lesson's effectiveness and making constructive adjustments on the basis of results; conducting a check for understanding at certain points in a lesson; having kids read for higher-order purposes and write regularly; and clearly explicating and carefully teaching the criteria by which student work will be scored or evaluated" (p. 25). In mathematics classrooms, teachers might tend to ignore writing about the discipline; however, to develop complex knowledge, "students need opportunities to read, reason, investigate, speak, and write about the overarching concepts within that discipline" (McConachie et al., 2006, p. 8).  These perspectives illustrate that teaching mathematics is not easy.

There is much to be learned about improving instruction by examining initiatives within the U.S. that provide educators with the best-practice examples they might need. Inside Mathematics, which grew out of the Noyce Foundation's Silicon Valley Mathematics Initiative, is exemplary as "a professional resource for educators passionate about improving students' mathematics learning and performance. This site features classroom examples of innovative teaching methods and insights into student learning, tools for mathematics instruction that teachers can use immediately, and video tours of the ideas and materials on the site" (Welcome section).  The Ohio Department of Education developed a Correlation of Inside Mathematics Tasks to CCSS (Common Core State Standards) as of June 2010.

Teachers can also improve instruction by examining what takes place in other countries.  For example, the Trends in International Mathematics and Science (TIMSS) 1999 video study examined an alternative methodology that holds promise to improve math instruction in the U.S. Details and videos are available at http://www.lessonlab.com. Japanese Lesson Study is growing in the U.S. as a result of the TIMSS study (O'Shea, 2005). The process involves teachers working together to develop, observe, analyze, and revise lessons and focuses on preparing students to think better mathematically through more effective lessons. For more on the work of TIMSS, see http://nces.ed.gov/timss/ and http://timss.bc.edu/.

Learning Styles, Multiple Intelligences, Thinking Styles

Many students experience math anxiety. Much of this stems from a one style fits all approach to teaching. Traditionally, approaches to teaching mathematics have focused on linguistic and logical teaching methods, with a limited range of teaching strategies. Some students learn best, however, when surrounded by movement and sound, others need to work with their peers, some need demonstrations and applications that show connections of mathematics to other areas (e.g., music, sports, architecture, art), and others prefer to work alone, silently, while reading from a text.

All of this is reflected in Howard Gardner's Theory of Multiple Intelligences, which has found its way into schools (Moran, Kornhaber, & Gardner, 2006; Smith, 2002), along with its relevance for determining learning styles. Moran et al. (2006) indicated that the theory proposes viewing intelligence in terms of nine cognitive capacities, rather than a single general intelligence. Thus, a profile consists of strengths and weaknesses among "linguistic, logical-mathematical, musical, spatial, bodily-kinesthetic, naturalistic, interpersonal, intrapersonal, and (at least provisionally) existential" (p. 23). Overall, the theory has been misunderstood in application.

The multiple intelligences approach does not require a teacher to design a lesson in nine different ways to that all students can access the material...In ideal multiple intelligences instruction, rich experiences and collaboration provide a context for students to become aware of their own intelligence profiles, to develop self-regulation, and to participate more actively in their own learning. (p. 27)

Educators should be aware that multiple intelligences (MI) and learning styles (LS) are not interchangeable terms. According to Barbara Prashnig (2005), "LS can be defined as the way human beings prefer to concentrate on, store, and remember new and/or difficult information. MI is a theoretical framework for defining/understanding/assessing/developing people's different intelligence factors" (p. 8). Consider LS as "explaining information 'INPUT' capabilities" and MI "more at the 'OUTPUT' function of information intake, knowledge, skills, and 'talent'--mathematical, musical, linguistic" and so on (p. 9).

For more on multiple intelligences, consult the following:

• Multiple Intelligences: http://lth3.k12.il.us/rhampton/mi/mi.html Roger Hampton also elaborates on each intelligence, provides lesson ideas, and technology considerations.

• Tapping into Multiple Intelligences, a workshop from Concept to Classroom at Thirteen Ed Online: http://www.thirteen.org/edonline/concept2class/mi/index.html

Knowledge of students' learning styles assists teachers in developing lessons that appeal to all learners. However, determining a student's learning style cannot be done strictly by observation. Various models and inventories have been designed to determine a learning style, which does not remain fixed over time. Therein lies the problem of relying on inventories, as their validity and reliability might be in question (Dembo & Howard, 2007), and they differ. The following are among those inventories:

• The Dunn and Dunn Model includes "environmental, emotional, sociological, physiological, and cognitive processing preferences" (International Learning Styles Network, sec: About Learning Styles).

• David Kolb's Learning Styles Inventory categorizes in four dimensions (converger, diverger, assimilator, or accommodator) based on the degrees to which one possesses "concrete experience abilities, reflective observation abilities, abstract conceptualization abilities and active experimentation abilities" (Smith, 2001, sec: David Kolb on Learning Styles).

• VARK (Visual, Aural, Read/write, and Kinesthetic) is only part of a learning style, according to developer Neil Fleming who states "VARK is about one preference -our preference for taking in, and putting out information in a learning context"; "VARK is structured specifically to improve learning and teaching." The VARK questionnaire (just 16 short questions) is available online.

• Memletics Learning Styles Inventory posted at Learning-styles-online.com is a free online inventory (70 questions) with graphical feedback to determine your dominant and secondary styles: visual, social, physical, aural, verbal, solitary, and logical.

• The Index of Learning Styles is a 44-question on-line instrument with automatic scoring on the Web that was developed by Richard Felder and Barbara Soloman of North Carolina State University. This model assesses learning preferences on four dimensions (active/reflective, sensing/intuitive, visual/verbal, and sequential/global).

As an alternative to determining learning styles, Sternberg-Wagner's Thinking Styles Inventory and The Multiple Intelligence Inventory based on Gardner's work will also benefit teaching and learning, and are brought to you by the Learning Disabilities Resource Community of Canada. Students with learning disabilities or attention-deficit-disorder can find practical tips on how to make your learning style work for you at LdPride.net, which also contains more information on multiple intelligences.

With so many inventories available, teachers might wonder how their teaching can accommodate so many styles. Li-fang Zhang and Robert Sternberg (2005) indicated, however, that teachers need only to attend to "five basic dimensions of preferences underlying intellectual styles: high degrees of structure versus low degrees of structure, cognitive simplicity versus cognitive complexity, conformity versus nonconformity, authority versus autonomy, and group versus individual. Furthermore, [they] believe that good teaching treats the two polar terms of each dimension as the two ends of a continuum and provides a balanced amount of challenge and support along each dimension" (p. 43).

Readers should also be aware that while determining learning styles might have great appeal, "The bottom line is that there is no consistent evidence that matching instruction to students' learning styles improves concentration, memory, self-confidence, grades, or reduces anxiety," according to Myron Dembo and Keith Howard (2007, p. 106). Rather, Dembo and Howard indicated, "The best practices approach to instruction can help students become more successful learners" (p. 107). Such instruction incorporates "Educational research [that] supports the teaching of learning strategies...; systematically designed instruction that contains scaffolding features...; and tailoring instruction for different levels of prior knowledge" (p. 107).  Further, cognitive scientists Hal Pashler, Mark McDaniel, Doug Rohrer, and Robert Bjork (2009) supported this position and stated, "Although the literature on learning styles is enormous, very few studies have even used an experimental methodology capable of testing the validity of learning styles applied to education. Moreover, of those that did use an appropriate method, several found results that flatly contradict the popular meshing hypothesis" (p. 105). They concluded "at present, there is no adequate evidence base to justify incorporating learning-styles assessments into general educational practice" (p. 105) and "widespread use of learning-style measures in educational settings is unwise and a wasteful use of limited resources. ... If classification of students' learning styles has practical utility, it remains to be demonstrated" (p. 117).

Differentiated Instruction and Ideology

Common Core State Standards (2010) for Mathematical Practice include varieties of expertise that mathematics educators should strive to develop in students at all levels:

1. Make Sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

Acquiring this expertise will require that educators play greater attention to the need for differentiated instruction.  "Differentiated instruction is a process to approach teaching and learning for students of differing abilities in the same class. The intent of differentiating instruction is to maximize each student’s growth and individual success by meeting each student where he or she is, and assisting in the learning process." Educators who differentiate instruction strive to "recognize students varying background knowledge, readiness, language, preferences in learning, interests; and to react responsively" (Hall, 2002, Definition section).  However, the process is challenging for educators to teach in a flexible manner that responds to the unique needs of learners. Often total lessons or the pace of individual lessons need to be adjusted "on-the-fly." So, teachers also need to know about additional resources beyond what's in the textbook that can be used to help learners.

As promoters of differentiated instruction, Carol Ann Tomlinson and Jay McTighe (2006) indicated that it is primarily an instructional design model that focuses on "whom we teach, where we teach, and how we teach" (p. 3).  How one teaches is based on one's ideological perspective. According to David Ferrero (2006), educators are divided by traditionalism and innovation, but teaching that leads to achievement gains when one embraces standardized testing does not mean that educators have to choose between one or the other. There is a concept of "innovative traditionalism" that is student-centered, yet has been shown to improve standardized achievement test scores. This has been accomplished in two Chicago-area high schools by "a combination of test prep, classical content, and collaboratively developed thematic projects grounded in controversy and designed to cultivate student voice and civic engagement" (p. 11). The following table (Ferrero, 2006, p. 11) illustrates the essential differences in education's ideological divide, which can be bridged.

The goals of differentiated instruction and innovative traditionalism are to ensure effective learning for all. Best practice learning adheres to 13 principles. Best practice is student-centered, experiential, holistic, authentic, expressive, reflective, social, collaborative, democratic, cognitive, developmental, constructivist, challenging with choices and students taking responsibility for their learning (Zemelman, Daniels, & Hyde, 1998, as cited in Wilcox & Wojnar, 2000).

Some might not appreciate the true essence of cooperative learning. Learners are responsible for not just their own learning, but the learning of others. Shared learning leads to success for all, as each member of a learning group has a specific role to play in reaching a common goal. Successful groups include positive interdependence--if one fails, the entire group is affected. There is both individual and group accountability; although some work might be completed individually, some must be accomplished by group interactions. Typical cooperative learning strategies include think-pair-share, the three-step interview, the jigsaw, and numbered heads. Techniques might include focused listing to brainstorm or examine concepts and descriptions, structured problem solving, one-minute papers, paired annotations, guided reciprocal peer questioning, and send-a-problem.

According to Theroux (2004), a teacher in Alberta (CA),

"Differentiating instruction means creating multiple paths so that students of different abilities, interest or learning needs experience equally appropriate ways to absorb, use, develop and present concepts as a part of the daily learning process. It allows students to take greater responsibility and ownership for their own learning, and provides opportunities for peer teaching and cooperative learning" (para. 2).

Theroux (2004) addressed four ways to differentiate instruction: content (requires pre-testing to determine the depth and complexity of the knowledge base that learners will explore), process (leads to a variety of activities and strategies to help students gain knowledge), product (complexity varies in ways for assessing learning), and manipulating the environment or accommodating learning styles. Fairness is a key concept to emphasize with learners, who will recognize that not everyone will work on the same thing at the same time. They need to appreciate that not everyone has the same needs. Likewise, Hall (2002) presented a visual for the Learning Cycle and Decision Factors Used in Planning and Implementing Differentiated Instruction. Hall also provided a number of links to learn more about this topic.

 

 

In Creating a Differentiated Mathematics Classroom, Richard Strong, Ed Thomas, Matthew Perini, and Harvey Silver (2004) indicated that student differences in learning mathematics tend to cluster into four mathematical learning styles:

• Mastery style--tend to work step-by-step

• Understanding style--search for patterns, categories, reasons

• Interpersonal style--tend to learn through conversation, personal relationship, and association

• Self-Expressive style--tend to visualize and create images and pursue multiple strategies.

Students can work in all four styles, but tend to develop strengths in one or two of the styles. Each of these styles tends toward one of four dimensions of mathematical learning: computation, explanation, application, or problem solving. "If teachers incorporate all four styles into a math unit, they will build in computation skills (Mastery), explanations and proofs (Understanding), collaboration and real-world application (Interpersonal), and nonroutine problem solving (Self-Expressive)" (p. 74).

From an instructional styles perspective, Silver, Strong, and Perini (2007) noted that teachers who use mastery strategies focus on increasing students' abilities to remember and summarize. "They motivate by providing a clear sequence, speedy feedback, and a strong sense of expanding competence and measurable success." When focusing on interpersonal strategies, teachers use "teams, partnerships, and coaching" to help students better relate to the curriculum and each other. Understanding strategies help students to reason and use evidence and logic. Teachers "motivate by arousing curiosity using mysteries, problems, clues, and opportunities to analyze and debate." Self-expressive strategies highlight students' imagination and creativity. Teachers employ "imagery, metaphor, pattern, and what ifs to motivate students' drive toward individuality and originality." Finally, it's possible to use all four styles at the same time to achieve a balanced approach to learning (sec: Part One: Introduction, Figure B).

The implication for mathematics instruction is that "any sufficiently important mathematics topic requires students to learn the topic in four dimensions: procedurally, conceptually, contextually, and investigatively" (Strong et al., 2004, p. 75). Even taking that approach, we are challenged to help students overcome misconceptions.

Example:

The importance of addressing these four dimensions was made very clear in a recent query I had from an individual [let's call him Mac] seeking help for a learner in the 5th grade who was struggling to multiply decimal numbers. The learner had incorrectly calculated: 0.032 * 0.16 =0.0512. But why? Apparently the learner was taught an algorithm, but used it incorrectly. Let's examine the problem that arises in understanding if teaching is done only procedurally.

In investigating Web resources for Mac on this concept, several sites indicated using the algorithm with instructions to multiply the digits as whole numbers (here 32*16 = 512), then count up the number of decimal places indicated in the problem (here 5) and then to use that number of places in the final answer. If extra zeroes are needed (here 2), place them before the digits in the whole number answer. This kind of wording, which I purposely made less than mathematically precise, is what a 5th grader might typically remember from only an algorithm. Notice that the learner's answer (0.0512) did have five digits (places used incorrectly) and two zeroes preceding 512. The answer should have been 0.00512.

• Conceptually, the learner might have missed a connection to prior learning on fractions, or the link was not made and reinforced in instruction. Writing the problem in its equivalent fraction form, using knowledge of converting decimals to fractions and vice versa, and decimal notation and place value might eventually have helped the learner to understand the short cut presented in the algorithm.

• Contextually, understanding the position of the decimal point in the final answer might have been linked to an application problem, such as buying 2.6 yards of fabric at $1.75 per yard as presented at Education Place (see: http://www.eduplace.com/math/mw/background/5/09/te_5_09_multiply_ideas.html).

• Investigatively, the learner might not have seen visual representations of the concept, such as the rectangle model for multiplication presented in the teaching model for multiplying decimals at Education Place (see: http://www.eduplace.com/math/mw/models/overview/5_13_4.html) or by Jim Reed (Argyll Centre, CA) on decimal multiplication (see: http://argyll.epsb.ca/jreed/math7/strand1/1201.htm).

In any case, the query confirmed Strong et al.'s (2004) recommendations and the need for differentiated instructional practices.

The example above noted teaching mathematical procedures using algorithms. Algorithms play an important role in mathematics, as they address step-by-step procedures yielding a single answer. For example, Mary Bryan Elementary School (Indianapolis, IN) posted a series of algorithms from Everyday Mathematics, including demonstration videos of teachers explaining how to use them. Everyday Mathematics Algorithms for grades 2-6 and their animations are also available at the publishers site: https://www.everydaymathonline.com/. The difficulty arises, as the above example indicates, if the algorithm is taught without linking it to conceptual, contextual, and investigative understanding. Ball, Ferrini-Mundy, Kilpatrick, Milgram, Schmid, and Schaar (2005) indicate, "Fluent use and understanding ought to be developed concurrently." Algorithms not only play a role in gaining whole number computation fluency, but play a role in such examples as "constructing the bisector of an angle; solving two linear equations in two unknowns; calculating the square root of a number by a succession of dividing and averaging" (section: Areas of Agreement). 

According to Strong et al. (2004), testing practices should also aim to measure knowledge in all four dimensions. Teachers should be aware that texts and their accompanying tests, however, tend to emphasize only the mastery and understanding styles of learning. To differentiate instruction, teachers can:

• Rotate strategies to appeal to students' dominant learning style and challenge them to work in their less preferred styles. Consider strategies such as using manipulatives, observing demonstrations, sketching out a math situation, reading, having students compare their work with a partner, or solving complex problems in a team.

• Use flexible grouping.

• Personalize/individualize learning for struggling students or for those needing an extra challenge.

Among strategies for implementing differentiated instruction, Tomlinson and McTighe (2006) suggested that teachers consider:

• Compacting--giving students credit for what they already know;

• Negotiated delay of due dates and times for tasks;

• Varied homework;

• Bookmarked Web sites on key topics in languages other than English to support English language learners;

• Video and audio clips to support multiple intelligences and varied learning styles and disabilities;

• Flexible grouping, "expert" groups, and interspersing lecture with group discussions;

• Guided peer review;

• Teaching with part-to-whole and whole-to-part emphasis;

• Tiered assignments--used when all students need to know the same skill or concept;

• Learning contracts;

• Independent study/projects;

• WebQuests and Web inquiries;

• Learning centers--primarily used in elementary grades; and

• Adjusting questions to accommodate levels in Bloom's Taxonomy.

Curriculum Associates, Inc. also has a free mini-course on differentiated instruction. Text is accompanied by audio. Handouts, supplementary readings, and short video clips of teachers explaining the use of a particular strategy in their classrooms are included. A broadband connection is recommended. The four lessons address principles of differentiated instruction, the role of formal and informal assessment in identifying student needs, strategies used in differentiated instruction, and guidelines for managing a differentiated classroom.

 

 

 

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Bloom's Taxonomy and Levels of Understanding

Bloom's Taxonomy contains six levels: knowledge, comprehension, application, analysis, synthesis and evaluation. The taxonomy, which was revised in 2001, now has levels of remembering, understanding, applying, analyzing, evaluating, and creating (Forehand, 2005). Students should work at all levels of the taxonomy. It should not be viewed as a ladder, however, nor as a framework for differentiated instruction (Tomlinson & McTighe, 2006, pp. 119-120). The taxonomy is helpful for breaking down state standards into meaningful components as teachers plan their instruction. Planning for instruction will be elaborated upon in Part 3 of this essay on content and curriculum mapping.

Charles White (2007) provided a closer look at how Bloom’s Taxonomy provides levels of understanding to guide teaching and assessing knowledge. Teaching for each level has different instructional strategies and testing techniques.

• Knowledge: Memorization and recitation fall within the knowledge level. Teachers might rely on a lecture method and assigned readings. They are transmitters of knowledge. Students remain passive and acquire familiarity with the material, take notes, memorize, and study enough so that they can recall information at least long enough to pass tests, which might be multiple-choice or true/false. Such tests rely on one-right answer. “Opinions and values are excluded from this type of testing” (p. 162). When writing, students tend to parrot back what the teacher has said. However, the ability to recite information that has been memorized does not mean that students know what they are saying.

• Comprehension: At a comprehension level, students are able to discuss what they’ve learned in their own words rather than in the teacher’s words, express their feelings, participate in classroom debate, and are thus taking ownership of content and remembering it better. They would be able to explain a graph, a calculation using a formula, or an equation (e.g., linear regression), but not necessarily be able to implement associated tools. At this level, “short- and medium-length answers [in students’ own words] combined with complex multiple-choice formats often serve as the medium of test material” (p. 162).

• Application: While key words for comprehension are explaining and discussion, application involves doing. Novices lacking understanding might only be able to apply knowledge when given step-by-step instructions that can be used without deviation. However, at this level, students must be able to demonstrate that they can use concepts and theories in problem-solving. They might be given all the information necessary to do calculations or tasks. Memory at this level is enhanced with repetition. Testing includes unstructured problems that might not have been encountered in the text or during a lecture, requiring students to determine a solution method using what they have learned. Novice students might still turn to the teacher for a correct solution.

• Analysis: At this level, application is taken a step further. Students must be able to take a situation apart, diagnose its pieces, and decide for themselves what tools (e.g., graph, calculation, formula, etc.) to apply to solve the problem at hand. Rather than just understanding and applying individual concepts, students understand the relationship among concepts. Case studies in business, for example, fit this level. The level of difficulty can be controlled for novices to experts by the number of issues presented in the cases requiring analysis. Likewise, this process to control difficulty can be used for any mathematics problem-solving scenario based on level of expertise of learners. For example, at elementary levels, students are introduced to analysis when a few extraneous facts are included in a problem, which are not needed to solve it. At an analysis level, students are able to appreciate that some problems do not have a unique solution and there is more than one way to defend a position or solution method, as in a case study.

• Synthesis: In contrast to analysis (i.e., taking apart), at the synthesis level students put things back together. Given the pieces, there might be more than one way to do this. In terms of mathematics, students might take the pieces they’ve learned, and put them together to solve problems not yet encountered in the actual classroom setting. Synthesis is involved when creating something new. Advanced students might be asked to create a new theory. Synthesis is tested via major projects, for example, which might be long term involving creativity and application of all that students have learned on a topic.

• Evaluation: Teachers evaluate student work all the time, particularly exams and homework. The difficulty in evaluation arises when judging multiple perspectives and varied problem-solving approaches, as one must be thoroughly familiar with content. At this level, students might be asked to problem-solve via debate, for example. At the evaluation level, one is able “to judge the work of others at any level of learning with regard to its accuracy, completeness, logic, and contribution” (White, 2007, p. 161). Rubrics help teachers to evaluate work, particularly for that involving application, analysis and synthesis.

White (2007) presented a novel way to test levels of understanding. He proposed writing two test questions on a topic, allowing students to choose only one of those to answer. The first is written for the knowledge and comprehension levels (e.g., key verbs: list, describe), and the second is written for the higher critical thinking levels of application, analysis, and synthesis. Points possible would be indicated for each, so that students would recognize that only those answering the second could be awarded maximum points toward an A+ grade. The option to choose enables the less able student to better demonstrate what he does know and perhaps earn a B grade, rather than risk failure because of an inability to demonstrate critical thinking. For either question, students could fail.

 

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References

 

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See other Math Methodology pages:

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