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Math Methodology:
Instruction Essay: Introduction to Teaching Challenges

 

Math Methodology: Instruction Zone

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 standards-based classroom four instructional strategies are key:

  • Inquiry and problem solving
  • Collaborative learning
  • Assessment embedded in instruction
  • Higher order questioning

Math Methodology is a three part series on instruction, assessment, and curriculum. Sections contains relevant essays and resources:

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Teaching and Math Methodology: Instruction

 

Introduction to Teaching Challenges

 

"A student is not a container you have to fill, but a torch you have to light up." --Albert Einstein

Teaching Challenges

Various definitions of teaching over time have been proposed and certainly a working definition of the term is needed, if one is to discuss all the challenges involved in teaching.  According to James Hiebert and Douglas Grouws (2007), "Teaching consists of classroom interactions among teachers and students around content directed toward facilitating students’ achievement of learning goals" (p. 372).  Further, this definition notes a two-way process and makes teaching "largely under the control of the teacher" (p. 377).

Consider the complexities of teaching as viewed by the National Board for Professional Teaching Standards (NBPTS):

The fundamental requirements for proficient teaching are relatively clear: 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, What Teachers Should Know section, para. 2).

The NBPTS makes it clear that:

This enumeration suggests the broad base for expertise in teaching but conceals the complexities, uncertainties and dilemmas of the work. The formal knowledge teachers rely on accumulates steadily, yet provides insufficient guidance in many situations. Teaching ultimately requires judgment, improvisation, and conversation about means and ends. Human qualities, expert knowledge and skill, and professional commitment together compose excellence in this craft. (NBPTS, 2002, p. 2, What Teachers Should Know section, para. 3).

In Bob Sullo's view (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" (Introduction section).  What is striking is that CT4ME includes discussion of many of those innovations throughout this site.

Enhancing Professional Practice: A Framework for Teaching by Danielson

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).  Danielson's work on Enhancing Professional Practice: A Framework for Teaching is grounded in constructivist theory.

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.html). Queensborough Community College also noted that 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.  However, it is also important to note Hiebert and Grouws (2007) who stated, "Characteristics of teachers surely can influence their teaching, but these characteristics do not determine their teaching. Teachers with different characteristics can teach in essentially the same way and vice versa" (p. 377).

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) in Improving Student Learning One Teacher at a Time, teachers need to make that hope a certainty by adhering to a Big Four approach, which means

The Big Four is just the tip of the iceberg.  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).

The fundamental challenges of the teaching profession are also well-articulated in Connecticut's 2010 Common Core of Teaching, which includes "six domains and 46 indicators that identify the foundational skills and competencies that pertain to all teachers, regardless of the subject matter, field or age group they teach" (p. 2).  They are useful for teacher preparation programs, beginning teachers, and experienced teachers.  For example, among professional responsibilities, the Connecticut Department of Education (2010) indicated "Continually engaging in reflection, self-evaluation and professional development to enhance their understandings of content, pedagogical skills, resources and the impact of their actions on student learning" (p. 10).  Collaboration and proactive communication with colleagues, administrators, students, and families are featured elements, as are understanding the legal rights of individuals with disabilities, and the role that race, gender, and culture might have on professional interactions with students, families, and colleagues, and ethical uses of technology.

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" (Introduction section).

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" (Preface section, 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 and how to enhance your teaching skills. 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 Education Research section.

 

What are the Dimensions of Learning?

Dimensions of Learning (Marzano & Pickering, 1997) is an instructional framework based on five types of thinking, called the dimensions of learning, that are considered essential to student learning and academic performance:

  • Positive attitudes and perceptions about learning

  • Acquiring and integrating knowledge

  • Extending and refining knowledge

  • Using knowledge meaningfully

  • Productive habits of mind

See how Prince George's County Public Schools applies dimensions of learning among its instructional strategies:  http://www.pgcps.pg.k12.md.us/~elc/dolref.html

Source: Marzano, R., & Pickering, D. (1997). Dimensions of learning teacher's manual (2nd ed.). Alexandria, VA: ASCD.  Retrieved from http://www.ascd.org/ASCD/pdf/siteASCD/publications/books/Dimensions-of-Learning-Teachers-Manual-2nd-edition.pdf  

What are "Habits of Mind"?

 The Brain- a gif used as metaphor for "Habits of Mind"According to Art Costa and Bena Kallick (n.d.), "A "Habit of Mind” means having a disposition toward behaving intelligently when confronted with problems, the answers to which are not immediately known" (para. 2).  Such habits include, but are not limited to the following:

  1. Persisting
  2. Managing impulsivity
  3. Listening with understanding and empathy
  4. Thinking flexibly
  5. Thinking about thinking (metacognition)
  6. Striving for accuracy
  7. Questioning and posing problems
  8. Applying past knowledge to new situations
  1. Thinking and communicating with clarity and precision
  2. Gathering data through all senses
  3. Creating, imagining, innovating
  4. Responding with wonderment and awe
  5. Taking responsible risks
  6. Finding humor
  7. Thinking interdependently
  8. Remaining open to continuous learning

Source: Costa, A. L., & Kallick, B. (n.d.). Describing 16 habits of mind.  Retrieved from http://www.habits-of-mind.net/

Visit the Habits of Mind Blog from the Education Development Center's Mathematical Practice Institute.  This blog began January 2014 and is intended to explore mathematical habits of mind or ways of thinking about mathematics in K-16.  Such habits are "also recognizable today in the Standards for Mathematical Practice."  The goal is to provide a vehicle for exploring ideas "to bring serious mathematics to all learners" (Welcome post, January 15, 2014).

What Mind-sets Drive Teacher Effectiveness?

Art Costa, Robert Garmston, and Diane Zimmerman (2012) defined five states of mind that "create a growth mind-set that is a potent force for fostering collective excellence and influencing, motivating, and inspiring our intellectual capacities."  They include the drive for efficacy, the drive for consciousness (reflection on one's actions and those of others), the drive for flexibility, the drive for craftsmanship, and the drive for interdependence.  Effective teachers demonstrate those dispositions.

Source: Costa, A., Garmston, R., & Zimmerman, D. (2012, November 8). What mind-sets drive teacher effectiveness? ASCD Express, 8(3).  Retrieved from http://www.ascd.org/ascd-express/vol8/803-costa.aspx

What impact does a growth mindset have on learners?

According to Dr. Carol Dweck (2006), author of Mindset: The New Psychology of Success, "When students and educators have a growth mindset, they understand that intelligence can be developed. Students focus on improvement instead of worrying about how smart they are. They work hard to learn more and get smarter."  Through years of research, she and colleagues have found that "students who learn this mindset show greater motivation in school, better grades, and higher test scores" (Mindset Works, The Science section).

Visit Mindset Works for resources and how to promote mindset in your school.

 

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Teacher Goal Setting

 

Gaining confidence in teaching takes time, and requires goal setting, reflection, dialogue and collaboration among colleagues. Three goals are essential for success: 1. adequate planning and effective classroom management, effective discipline and knowledge of motivation strategies; 2.  knowledge of subject matter standards and benchmarks, curriculum frameworks; and 3.  developing a range of instructional and assessment strategies and test preparation methods.

Goal 1: Acquire adequate planning and classroom management skills and effective discipline and motivation strategies.

What's classroom management all about?  Classroom management plays a significant role in effective teaching and ultimate achievement of learners.  Per Kratochwill (n.d., Introduction section), "Although there is no agreed-upon definition of classroom management, the framework offered by Evertson and Weinstein (2006) represents a current and widely accepted view. According to Evertson and Weinstein, classroom management has two distinct purposes: “It not only seeks to establish and sustain an orderly environment so students can engage in meaningful academic learning, it also aims to enhance student social and moral growth” (p. 4).  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).

How does the classroom environment affect motivation?  In such an environment just described, 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)

Classroom Management that Works by MarzanoWhat are components of effective classroom management?  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). 

Bob Sornson (2010) provided tips, which fall within those four components, for creating a positive classroom environment that will enhance student achievement:

  1. Build connections daily.  This includes such things as personal greetings, asking about students' lives, sharing your own stories.
  2. Use consistent procedures and routines.
  3. Respond quickly to misbehavior.
  4. Notice specific positive behaviors.  Mention those to the student(s).
  5. Use instructional design for success.  Pay attention to students' level of readiness and set high expectations.
  6. Neutralize arguments.
  7. Sometimes delay consequences when dealing with a problem.
  8. Develop an empathetic classroom culture.  This includes students for each other and from the teacher.
  9. Build connections with families.  Send home good news, too, not just the bad.
  10. Use enforceable statements.  These pertain to managing classroom events.
  11. Offer choices.
  12. Teach problem-solving skills. (Sornson, 2010)

In terms of Sornson's tip on offering choices, Bryan Goodwin (2010) reminded educators that while research suggests that students be given choices, the number of choices should be limited.  Less experienced students might be offered fewer choices, perhaps just two, while more advanced students might gradually be given from three to five options.  When learners have too many choices, they might spend too much time making the choice and be less satisfied in that choice at the expense of completing work with quality.  There is also risk that their motivation to do a good job might decrease, if they've spent their mental energy making the choice and then worry if it was the right one.

And in terms of setting high expectations for all, Robert Marzano (2010) reminded educators that this is easier said than done.  It's the "teachers' behaviors toward students [that] are much more important than their expectations," as students "make inferences on the basis of these behaviors" (pp. 82-83).  Students become easily aware of differences, as "teachers tend to make less eye contact, smile less, make less physical contact, and engage in less playful or light dialogue" with low-expectancy students.  They also pose fewer and less challenging questions to them, and delve into their answers less deeply and reward them for less vigorous responses" (p. 83).  The key to overcome this is for teachers to be aware of their own behaviors: identify students as early as possible for whom they have low expectations, identify their similarities and differential treatment of them, and then set out to change and treat low-expectancy and high-expectancy students the same.

Teachers can demonstrate positive behaviors to influence students' perceptions, as those perceptions have a great deal to do with effective instruction.  Echoing Sornson (2010), Marzano (2011) stated that developing positive perceptions involves teachers showing interest in students' lives, advocating for students with such actions as the appearance of wanting students to do well and providing assistance to that end, never giving up on students, and acting friendly.  "With good relationships in place, all other instructional strategies seem to work better" (p. 82).

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).

How do you plan for better classroom management? 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 the following in which readers will note some of the tips Sornson (2010) also provided:

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.

Goal 1 Resources

To help build your confidence in teaching, understanding student behavior and learning styles, and classroom management, consider the following resources:

THE Classroom Management Book by Wong and Wong

 

 

Goal 2: Know and apply standards, curriculum frameworks, and a variety of 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.

Findings from research are particularly relevant for pedagogical methods that have been successful with learners, particularly in terms of reducing math anxiety.  Math anxiety has been identified as a common reason for lack of success in mathematics.  Blazer's (2011) review of strategies that teachers can use to help reduce students' anxiety, thus leading to better achievement for all, revealed:

 

How-to-Help-Students-Meet-the-CCSS-for-Math-Infographic

 

 

Goal 3: Develop a range of instructional and assessment methods and test preparation methods.

Linda Gojak (2012), former NCTM President, noted that "Over the last three decades a variety of instructional strategies have been introduced with a goal of increasing student achievement in mathematics. Such strategies include individualized instruction, cooperative learning, direct instruction, inquiry, scaffolding, computer-assisted instruction, and problem solving" with the flipped classroom being a recent addition to the list (para. 1).  Thus, another goal for teachers is to investigate instructional and assessment methods and how they might be incorporated appropriately 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.

 

Learn more about math pedagogy from math educators around the world.

Friendly reminder Gif

Mathagogy includes several two-minute videos from math educators around the world who are sharing how they approach teaching various topics.  For example, teachers have uploaded how they introduce sine and cosine graphs, teach inquiry, algebraic literacy, prime numbers, proportions, probability, proof, and how they teach using Cuisenaire rods or using one question lessons.

 

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Improving Instruction

Math teacher at blackboard GIF

Attention to theory and research; learning styles, multiple intelligences and thinking styles; and 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.timssvideo.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

Learning styles: see, hear, doMany 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 the concept of learning styles.

Moran et al. (2006) indicated that multiple intelligences 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)

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).

Indeed, Howard Gardner has stated that multiple intelligences are not learning styles.  In Gardner's view, a style or learning style "is a hypothesis of how an individual approaches the range of materials."  There are two basic problems.  First, "the notion of 'learning styles' is itself not coherent. Those who use this term do not define the criteria for a style, nor where styles come from, how they are recognized/assessed/exploited."  Second, "When researchers have tried to identify learning styles, teach consistently with those styles, and examine outcomes, there is not persuasive evidence that the learning style analysis produces more effective outcomes than a 'one size fits all approach.' " Labeling a style (e.g., visual or auditory learner) might actually be unhelpful or inconceived.  Putting a label on it does not mean the "style" fits all learning scenarios (Gardner, in Strauss, 2013). 

Knowledge of how students learn best 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.  Labeling a "style" poses an additional problem in that a style does not remain fixed over time. Therein lies the main concern 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:

As an alternative to determining learning styles, a personal Thinking Styles Inventory and The Multiple Intelligence Inventory based on Gardner's work will also benefit teaching and learning. 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 although 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).

Ultimately, although we might continue to argue over terminology and labels associated with learning, Gardner provided the following lessons for educators:

  1. Individualize your teaching as much as possible. Instead of “one size fits all,” learn as much as you can about each student, and teach each person in ways that they find comfortable and learn effectively. Of course this is easier to accomplish with smaller classes. But ‘apps’ make it possible to individualize for everyone.
  2. Pluralize your teaching. Teach important materials in several ways, not just one (e.g. through stories, works of art, diagrams, role play). In this way you can reach students who learn in different ways. Also, by presenting materials in various ways, you convey what it means to understand something well. If you can only teach in one way, your own understanding is likely to be thin.
  3. Drop the term "styles."  It will confuse others and it won't help either you or your students.  (Gardner, in Strauss, 2013)

 

Additional MI and LS Resources

For more on multiple intelligences and learning styles, consult the following:

 

Video and audio clips support multiple intelligences and varied learning preferences and disabilities.

Friendly reminder GifUsing video and audio to support multiple intelligences and varied learning preferences and disabilities is one of the strategies noted by Tomlinson and McTighe's (2006) to support differentiated instruction.  Here's a sampling of video sites for your consideration in support of their recommendation:

HOT: Knowmia is a repository of free tutorial videos featuring teachers explaining various concepts, which the site moderators have gathered from the web.  There are multiple subjects and topics for K-12.  Mathematics features pre-algebra, algebra, linear algebra, developmental math, geometry, pre-calculus, calculus, trigonometry, statistics & probability.  The site also has additional teacher resources.

HOT: SchoolTube contains numerous videos on mathematics in their category of Academics and Education, which would help learners review concepts presented in class and in some cases offer a different instruction perspective. "SchoolTube provides students and educators a safe, world class, and FREE media sharing website that is nationally endorsed by premier education associations."

HOT: Teacher Tube provides "an online community for sharing instructional videos. ... It is a site to provide anytime, anywhere professional development with teachers teaching teachers. As well, it is a site where teachers can post videos designed for students to view in order to learn a concept or skill."

For additional video resources at this site, see Math Resources: Integrating Podcasts, Vodcasts and Whiteboards into Teaching and Learning.

What do teachers say about using video in instruction?

Teachers have several reasons for using television and video in their instruction, according to results of a 2009 national online survey of 1,418 full time pre-K and K-12 teachers on their use of media and technology.  The study, "Digitally Inclined," was conducted by Grunwald Associates for PBS.  Teachers believed television and video reinforces and expands on content they are teaching (87%), helps them respond to a variety of learning styles (76%), increases student motivation (74%), changes the pace of classroom instruction (66%), enables them to demonstrate content they can't show any other way (57%), enables them to introduce other learning activities (51%), and helps them teach current events and breaking news (38%).  Further, teachers perceived benefits to instruction.  They agreed that using television and video stimulates discussion (58%), helps them be more effective (49%) and creative (44%).  They agreed that students prefer television and video over other types of instructional resources or content (48%), and it stimulates student creativity (36%). (PBS & Grunwald Associates LLC, 2009, p. 8)

 

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, Strangman, & Meyer, 2003, 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).  Tomlinson's website, DifferentiationCentral, will enhance your knowledge of differentiated instruction.

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.

Education's Ideological Divide

Traditional Innovative
Standardized tests Authentic assessment
Basic skills Higher-order thinking
Ability grouping Heterogeneous grouping
Essays/research papers Hands-on projects
Subject-matter disciplines Interdisciplinary integration
Chronology/history Thematic integration
Breadth Depth
Academic mastery Cultivation of individual talents
Eurocentrism Multiculturalism
Canonical curriculum Inclusive curriculum
Top-down curriculum Teacher autonomy/creativity
Required content Student interest

Source: Ferrero, D. (2006). Having it all. Educational Leadership, 63(8), 11.

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, Strangman, and Meyer (2003) presented a graphic organizer called the Learning Cycle and Decision Factors Used in Planning and Implementing Differentiated Instruction and also provided a number of links to learn more about this topic. ASCD also has multiple resources on differentiated instruction.

 

Learn more about the history of differentiated instruction.

Friendly reminder Gif

The concept of differentiated instruction is not new.  Historically it has been discussed in other terms related to addressing individual differences in instruction.

Read the ASCD Express My Back Pages: A Brief History of Differentiated Instruction (1953).  ASCD devoted its entire December 1953 issue of Educational Leadership to the theme "The Challenge of Individual Difference," which is available online.  In the lead article, Adjusting the Program to the Child, Carleton Washburne presented a short history of reform efforts aimed at making education more individualized.  What a find.

 

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:

Misconception when adding decimalsStudents 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.

Decimal multiplication linked to fraction multiplication

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

Laughing Out Loud gifPlay video gifFor a humorous take on the importance of developing conceptual understanding, watch the YouTube Video, Ma and Pa Kettle Math--they prove to you that 25 divided by 5 is 14!

Play video gifSometimes personal strategies for problem solving work better than algorithms. Students who use them demonstrate conceptual understanding, as the YouTube Video, Algorythm and Personal Strategy, illustrates.

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, Everyday Mathematics includes algorithms for grades 2-6 and their animations, located in free resources 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) indicated, "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" (Areas of Agreement section).

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:

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

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.

Guiding Questions

With so many strategies, Linda Gojak (2012), an NCTM President, offered the following questions about process, which can be helpful when deciding how to structure and present a lesson:

  • Is this instructional approach appropriate for the grade level of students at this time?
  • Can I adapt this strategy so that my lesson incorporates the NCTM Process Standards and encourages students to make sense of the mathematics?
  • Does this lesson build from a rich mathematical task?
  • What questions can I ask students that will encourage them to think more deeply about the mathematics that I want them to understand?
  • How can I encourage rich discussions with and among students as they develop understanding and apply the mathematical ideas in a variety of contexts?
  • Will my instruction help students to reason and make sense of the mathematics in the lesson?
  • In what ways do I anticipate students will represent their thinking about the mathematics?
  • How does the mathematics in this lesson connect to previous concepts as well as future concepts? (para. 5)

 

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Bloom's Taxonomy and Levels of Understanding

 

There is more than one kind of learning.  In 1956 Benjamin Bloom identified three domains of learning: cognitive, affective, and psychomotor.  The cognitive domain focus on knowledge or mental skills; the affective on the growth of feelings, emotions, attitudes; and the psychomotor on manual or physical skills (Clark, 2013).  Within Revised Bloom's TaxonomyBloom's Taxonomy of Learning Domains, the cognitive domain contains six levels: knowledge, comprehension, application, analysis, synthesis and evaluation. The taxonomy was later revised in the mid-90's and now has levels of remembering, understanding, applying, analyzing, evaluating, and creating (Clark, 2013; Forehand, 2005). You can learn more on the Revised Bloom's Taxonomy at the Educational Origami web site.

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.

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.

Join a Discussion Group or Participate in a Blog

Online Discussion Group It's easy for teachers to discuss topics in education and math methodology with colleagues around the world.

  • HOT:  Microsoft's Partners in Learning Network enables educators from all over the world to communicate, collaborate and exchange ideas and methods on how best to incorporate technology into the classroom.
  • TThe Math Forum's Epigone discussion archives include mathematics and math education-related newsgroups, mailing lists, and Web-based discussions. For example, join K12.ed.math, math-learn, math-teach, math.teaching.styles, or math.teaching.technology.
  • You might also select from the many newsgroups and listservs at The Math Archives, hosted by the University of Tennessee at Knoxville. A listserv is an automatic emailing list to which people subscribe.

Use Web 2.0 tools for blogging:

  • Blog at Classroom 2.0
  • HOT: Participate in the blog at Let's Play Math! This blog is about learning, teaching, and just playing around with K-12 mathematics.
  • HOT: MathNotations is a blog by Dave Marain, a math educator with considerable experience.  He said, "Look for fully developed math investigations that are more than one inch deep, math challenges, Problems of the Day and standardized test practice. The emphasis will always be on developing conceptual understanding in mathematics. There will also be dialogue on issues in mathematics education with a focus on standards, assessment, and pedagogy primarily at the 7-12 level through AP Calculus."

 

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References

Ball, D. L., Ferrini-Mundy, J., Kilpatrick, J., Milgram, R. J., Schmid, W., & Schaar, R. (2005). Reaching for common ground in K-12 mathematics education.  Retrieved from http://www.ed-msp.net/public_documents/document/Reaching%20for%20Common%20Ground%20in%20K-12%20Mathematics%20Education.pdf

Blazer, C. (2011). Strategies for reducing math anxiety. Information Capsule IC1102.  Miami, FL: Miami Dade County Public Schools, Research Services, Office of Assessment, Research, and Data Analysis.  Retrieved from http://drs.dadeschools.net/InformationCapsules/IC1102.pdf

Breaux, E. (2009). How the best teachers avoid the 20 most common teaching mistakes. Larchmont, NY: Eye on Education.  Retrieved from http://www.eyeoneducation.com/

Clark, D. (2013). Bloom's taxonomy of learning domains.  Retrieved from http://www.nwlink.com/~donclark/hrd/bloom.html

Common Core State Standards (2010). Mathematics, Introduction, Standards for Mathematical Practice. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers.  Retrieved from http://www.corestandards.org/Math/Practice

Connecticut State Department of Education. (2010).  2010 Common core of teaching: Foundational skills. Hartford, CT: Author. Retrieved from http://www.sde.ct.gov/sde/cwp/view.asp?a=2618&q=320862

Danielson, C. (2007). Enhancing professional practice: A framework for teaching (2nd ed.). Alexandria, VA: ASCD.

Dembo, M., & Howard, K. (2007). Advice about the use of learning styles: A major myth in education. Journal of College Reading and Learning, 37(2), 101-109.

Ferrero, D. (2006). Having it all. Educational Leadership, 63(8), 8-14.

Forehand, M. (2005). Bloom's taxonomy: Original and revised. In M. Orey (Ed.), Emerging perspectives on learning, teaching, and technology. Retrieved from http://www.coe.uga.edu/epltt/bloom.htm

Goodwin, B. (2010). Choice is a matter of degree. Educational Leadership, 68(1), 80-81.

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Instruction--ResourcesAssessment and Curriculum: Content and Mapping