NCLB mandated that states and districts adopt programs and policies supported by scientifically based research. The Every Student Succeeds Act of 2015 recognizes other forms of research in that this law "requires that states and districts use evidencebased interventions to support school improvement" (Dynarski, 2015, para. 1; 114th Congress, 2015). Drawing upon research and an extensive collection of evidence from multiple sources, the Common Core State Standards (2010) were developed to reflect the knowledge and skills that young people need for success in college and careers. Those standards impact teachers in several ways, including to guide them "toward curricula and teaching strategies that will give students a deep understanding of the subject and the skills they need to apply their knowledge" (Common Core State Standards Initiative, FAQ section). For many the standards require changes in how mathematics is taught, thus they will influence instructional strategies that educators use. In a standardsbased classroom four instructional strategies are key:
Math Methodology is a three part series on instruction, assessment, and curriculum. Sections contains relevant essays and resources:
Part 1: Math Methodology: Instruction
The Instruction Essay (Page 1 of 3) on this page contains the following subsections:
The Instruction Essay (Page 2 of 3):
The Instruction Essay (Page 3 of 3) addresses the needs of students with math difficulties and contains the following subsections:
Math Methodology Instruction Resources also includes resources for special needs students (e.g., hearing and visually impaired, learning disabilities, English language learners).
Part 2: The Role of Assessment, essay and resources
Part 3: Curriculum: Content and Mapping and resources
"A student is not a container you have to fill, but a torch you have to light up." Albert Einstein
"[E]xcellence in teaching is the single most powerful influence on achievement." John Hattie (2003, p. 4)
When it comes to student achievement, teachers matter. Your role is to positively affect achievement in all you do. If you become a teacher, you'll soon learn, however, that there are multiple sources of variance in predicting achievement. Per John Hattie (2003), "It is what students bring to the table that predicts achievement more than any other variable," accounting for about 50% of that variance (p. 1). This is followed by the influence of teachers, who account for about 30% of that variance. "It is what teachers know, do, and care about which is very powerful in this learning equation" (p. 2). The home, schools (including their principals), and peer effects account for the remaining major sources of variance, about 510% each. But, it is excellence in teachers, not just teachers, that makes a difference (Hattie, 2003).
But, what is teaching? 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 twoway 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 subjectspecific 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 quartercentury 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, scopeandsequence 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.
The road to incorporating those best practices in teaching is filled with challenges, as learning to teach is a complex neverending 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, selfefficacy, motivation, learning styles, habits of mind, cultural influences and demographics (e.g., male/female, sexual orientation, ethnicity, ability/disability, socioeconomic 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 wellarticulated 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, selfevaluation 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 learnercentered 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 forgetprepare 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).
Years of experience does not equate to being an expert, however. From a review of literature and a synthesis of over 500,000 studies on differences between experience and expertise in teaching, John Hattie (2003) and colleagues identified five major dimensions illustrating the expertise of excellent teachers and then elaborated on 16 attributes within those. Expert teachers:
 can identify essential representations of their subject,
 can guide learning through classroom interactions,
 can monitor learning and provide feedback,
 can attend to affective attributes, and
 can influence student outcomes. (p. 5)
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.
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
Source: Marzano, R., & Pickering, D. (1997). Dimensions of learning teacher's manual (2nd ed.). Alexandria, VA: ASCD. Retrieved from http://www.ascd.org/Publications/Books/Overview/DimensionsofLearningTeachersManual2ndEdition.aspx
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:


Source: Costa, A. L., & Kallick, B. (n.d.). Describing 16 habits of mind. Retrieved from http://www.artcostacentre.com/html/habits.htm
Visit the Habits of Mind Blog from the Education Development Center's Mathematical Practice Institute (MPI). This blog began January 2014 and is intended to explore mathematical habits of mind or ways of thinking about mathematics in K16. 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). The MPI developed a fivepart webinar series for Exploring the Standards for Mathematical Practice, which is designed for middle and high school educators. The series provides useful tips to "help you engage all students in enjoyable, challenging learning that enables them to "puzzle through problems" and excel."
Art Costa, Robert Garmston, and Diane Zimmerman (2012) defined five states of mind that "create a growth mindset 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 mindsets drive teacher effectiveness? ASCD Express, 8(3). Retrieved from http://www.ascd.org/ascdexpress/vol8/803costa.aspx
According to 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 website, The Science section para. 1).
Dweck is also author of Mindsets and Math/Science Achievement (2008), an article prepared for the Carnegie Corporation of New YorkInstitute for Advanced Study. She provided evidence that mindsets "can predict math/science achievement over time" and "can contribute to math/science achievement discrepancies for women and minorities." Further, "interventions that change mindsets can boost achievement and reduce achievement discrepancies; and educators play a key role in shaping students’ mindsets" (p. 2).
Visit the following resources for supporting growth mindsets:
Mindset Works for resources, teaching strategies, and how to promote mindset in your school.
The Mindset Kit, produced by PERT (Project for Education Research That Scales) at Stanford University is a "free, online resource designed to help educators and parents learn about adaptive academic mindsets, learn practices that promote these mindsets, and download activity ideas" (PERT description).
With Math I Can is an Amazon initiative supported by several leading organizations (e.g., NCTM, ASCD, PERTS, Common Sense Education, the Teaching Channel, Edutopia, and more). The goal is to change students' attitudes about math. You'll find several resources on mindset for the classroom, district, and the home.
Youcubed at Stanford University promotes a growth mindset with teaching ideas, a searchable database of tasks, and videos (e.g., number talks, brain science, enquiry based learning, etc.).
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.
What's classroom management all about? Classroom management plays a significant role in effective teaching and ultimate achievement of learners. It occurs at two levelswhat can be seen in the classroom during actual contact with learners, and what goes on "behindthescenes" outside of the classroom. The latter would typically include planning and followup activities that teachers do in connection with classroom management.
Per Kratochwill (n.d., Introduction section), "Although there is no agreedupon 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).
In terms of enhancing a culture of learning, technology can also assist teachers in planning and managing tasks they typically do. For example, teachers can use online calendars and todo lists. Rather than making multiple copies of class handouts, they can post a single copy to a class website. They can use online grade books and grading tools. Technology enables them to provide individualized feedback to electronic documents/assignments submitted by their learners.
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 neverending 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)
What are components of effective classroom management? Marzano, Marzano, and Pickering (2003) conducted a metaanalysis of 100 reports on this issue, addressing four general components of effective classroom management: rules and procedures, disciplinary interventions, teacherstudent 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:
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. 8283). 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 lowexpectancy 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 lowexpectancy and highexpectancy 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. 5255).
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 picturein 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 growththat 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 teacherled 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 socialemotional 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. 5657). 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:
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:
Assertive Discipline: Positive Behavior Management for Today's Classroom (4th Ed.) by Lee Canter (2010). This awardwinning book "outlines a threestep approach for positive behavior management by creating a classroom discipline plan that includes: rules that students must follow at all times; positive support that students will receive consistently for following the rules; and corrective actions that the teacher will use consistently when students choose not to follow the rules" (Description section).
Better Classroom Management from Day One is featured in ASCD Express, August 1, 2013, volume 8, issue 22. This issue contains several resources in text and video formats.
Classroom Management: Teachers Modules by Tom Kratochwill is hosted by the American Psychological Association. Following an introduction, you'll find dos and don'ts, why classroom management works, for whom it works, FAQs, developmental differences, and additional resources.
THE Classroom Management Book by Harry Wong and Rosemary Wong (2014) offers 50 procedures that can be used as is or adapted to fit in any classroom management plan at any grade level. The authors define classroom management and how it differs from discipline. You'll find procedures for before and on the first day of school, procedures for students and teachers, procedures for the classroom, for instruction (e.g., discussion, group work, notetaking, reading a text, test taking, for students correcting work, cultivating social skills), for the special needs classroom, and more.
Collaborative for Academic, Social, and Emotional Learning (CASEL) features information on this topic and programs based on social and emotional learning: What is SEL?, SEL in action, SEL in policy, SEL research.
Curriculum Associates has a free minicourse 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 followup. There is also a free minicourse on motivating students to learn.
Inclusive Teaching Strategies from the Center for Research on Learning and Teaching at the University of Michigan.
Learn NC from the University of North Carolina School of Education includes best practices on classroom management for professional development. Resources range from designing your classroom to getting organized to building a classroom culture via effective daytoday management.
The National Education Association has a series of articles among its tools on classroom management.
simSchool is a simulation program for educators. You can experience the potential of digital gamesbased learning as you practice your teaching skills. You'll get immediate feedback on how your selected strategies affect student learning. Plus, as in a real classroom, the simStudents will react to userselected task design, as well as your teacher moves.
The Teacher's Guide has a wealth of information on classroom management with strategies, lessons, videos, discussion groups, books, and printouts.
TeachLivE provides an opportunity for preservice and inservice educators to develop and hone their instruction and management skills using the safe environment of a virtual middleschool classroom, complete with props, whiteboards, and students.
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.
25 Incredible Grading Apps That No Teacher Should Live Without
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. Common Core Math Standards are at the forefront, as many states have adopted them. 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.
Domains within the Common Core Math Standards  
K  1  2  3  4  5  6  7  8  HS 
Counting & Cardinality  
Number & Operations in Base 10  Ratios & Proportional Relationships  Number & Quantity  
Number & OperationsFractions  The Number System  Algebra  
Operations & Algebraic Thinking  Expressions & Equations  Algebra  
Operations & Algebraic Thinking  Functions  Functions  
Geometry  Geometry  
Measurement & Data  Statistics & Probability  Statistics & Probability  
Source: Common Core State Standards (2010). Mathematics Standards. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers. Retrieved from http://www.corestandards.org/Math 
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, computerassisted 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.
All teachers should consider the role of active or constructivist learning, as opposed to using only a lecture method. Active student involvement reinforces learning. This is not to minimize the role of direct instruction, however. Per Robert Coe, Cesare Aloisi, Steve Higgins, and Lee Elliot Major (2014, "...if you want students to remember something you have to get them to think about it. This might be achieved by being ‘active’ or ‘passive’ " (p. 24).
The IRIS Center at Vanderbilt University provides details on three evidencebased strategies for teaching mathematics: explicit or direct instruction lesson, peer tutoring, and cooperative learning. John Malouff (n.d.) grouped and explained over 50 problemsolving strategies, many of which can be applied to mathematics problem solving. Strategies can help understand the problem, simplify the task, determine the cause of a problem, involve external aids to help identify problem solutions, use logic to help identify possible solutions. Strategies can also identify a possible solution to serve as a starting point to solve a problem, or determine which possible solution is best. Strategies can employ geometric thinking, help you to function optimally while problem solving, and help solve multiple problems. George Polya's ProblemSolving Techniques contain details of his four principles that have become a classic for math problemsolving: understand the problem, devise a plan, carry out the plan, and look back.
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 paperandpencil exams, such as those made up of True/False, shortanswer, or multiplechoice 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 selfreport 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 gradegiving, such as when a teacher periodically projects multiplechoice 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 freemini 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.
Mathagogy includes several twominute 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.
The following delves into theory and research; learning styles, multiple intelligences and thinking styles; and differentiated instruction and the educator's ideology.
In their review of over 200 studies in What makes great teaching?, Robert Coe, Cesare Aloisi, Steve Higgins, and Lee Elliot Major (2014) identified elements of teaching with the strongest evidence of improving student achievement. In order of strength, those factors included:
Their review also delved into some common practices that can be harmful to learning and have no grounding in research. Examples included:
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.
There is much to be learned about improving instruction by examining initiatives within the U.S. that provide educators with the bestpractice 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 tools for educators, problems of the month, classroom videos, Common Core resources, and performance assessment tasks. 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/.
Effective lessons incorporate bestpractice. 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 teacherapprentice approach, authentic experiences, and reflective assessment. Further, Mike Schmoker (2006) stated that "the most wellestablished 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 higherorder 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.
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 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, logicalmathematical, musical, spatial, bodilykinesthetic, 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 selfregulation, 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:
The Dunn and Dunn Model includes "environmental, emotional, sociological, physiological, and cognitive processing preferences" (International Learning Styles Network, About Us section).
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, David Kolb on Learning Styles section). Note: David Kolb's website: Experienced Based Learning Systems, Inc includes his inventory and more information 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.
The Index of Learning Styles is a 44question online 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, 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 attentiondeficitdisorder 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. Lifang 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, selfconfidence, 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 learningstyles assessments into general educational practice" (p. 105) and "widespread use of learningstyle 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:
 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.
 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.
 Drop the term "styles." It will confuse others and it won't help either you or your students. (Gardner, in Strauss, 2013)
For more on multiple intelligences and learning styles, consult the following:
Big Thinkers: Howard Gardner on Multiple Intelligences  listen to Gardner in this short video posted at Edutopia.org: http://www.edutopia.org/multipleintelligenceshowardgardnervideo
Visit Howard Gardner's website to learn more about him, his theory, and publications: http://howardgardner.com/biography/ Visit the MI Oasis: his Official Authoritative Site of Multiple IntelligenceS.
Multiple Intelligences Institute: http://www.miinstitute.info/ is committed to understanding and application of this theory in educational settings from preschool through adult education.
Tapping into Multiple Intelligences, a workshop from Concept to Classroom at Thirteen Ed Online: http://www.thirteen.org/edonline/concept2class/mi/index.html
Walter McKenzie's Surfaquarium: http://surfaquarium.com/MI/ has content devoted to Multiple Intelligences in Education (e.g., an overview of MI, media and software selection, MI and instruction, templates, etc.).
Institute for Learning Styles Research: http://learningstyles.org/
Video and audio clips support multiple intelligences and varied learning preferences and disabilities.
Using 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 K12. Mathematics features prealgebra, algebra, linear algebra, developmental math, geometry, precalculus, 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 preK and K12 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)
Common Core State Standards (2010) for Mathematical Practice include varieties of expertise that mathematics educators should strive to develop in students at all levels:
Acquiring this expertise will require that educators play greater attention to 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). 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.
The process of differentiation is challenging for educators, as it requires developing skills 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 "onthefly." So, teachers also need to know about additional resources beyond what's in the textbook that can be used to help learners.
A second challenge for success lies in the teacher's ideological perspective, as this latter affects how one teaches. According to David Ferrero (2006), educators are divided by traditionalism and innovation. However, teaching that leads to achievement gains (e.g., via standardized testing) does not mean that educators have to choose between one or the other. There is a concept of "innovative traditionalism" that is studentcentered, yet has been shown to improve standardized achievement test scores. This has been accomplished in two Chicagoarea 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  Higherorder thinking 
Ability grouping  Heterogeneous grouping 
Essays/research papers  Handson projects 
Subjectmatter disciplines  Interdisciplinary integration 
Chronology/history  Thematic integration 
Breadth  Depth 
Academic mastery  Cultivation of individual talents 
Eurocentrism  Multiculturalism 
Canonical curriculum  Inclusive curriculum 
Topdown 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 studentcentered, 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).
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 pretesting 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 within their work, which they 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.
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:
Mastery styletend to work stepbystep
Understanding stylesearch for patterns, categories, reasons
Interpersonal styletend to learn through conversation, personal relationship, and association
SelfExpressive styletend 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 realworld application (Interpersonal), and nonroutine problem solving (SelfExpressive)" (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." Selfexpressive 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://staff.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.
For a humorous take on the importance of developing conceptual understanding, watch the YouTube Video, Ma and Pa Kettle Maththey prove to you that 25 divided by 5 is 14!
Sometimes 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 stepbystep procedures yielding a single answer. For example, Everyday Mathematics includes algorithms for grades 26 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, FerriniMundy, 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:
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:
Compactinggiving 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 parttowhole and wholetopart emphasis;
Tiered assignmentsused when all students need to know the same skill or concept;
Learning contracts;
Independent study/projects;
WebQuests and Web inquiries;
Learning centersprimarily used in elementary grades; and
Adjusting questions to accommodate levels in Bloom's Taxonomy.
In Good Questions: Great Ways to Differentiate Mathematics Instruction, Marian Small (2012) recognized, "It is not realistic for a teacher to try to create 30 different instructional paths for 30 students, or even 6 different paths for 6 groups of students. Because this is the perceived alternative to onesizefitsall teaching, instruction in mathematics is often not differentiated. To differentiate instruction efficiently, teachers need manageable strategies that meet the needs of most of their students at the same time" (p. 6). This can be accomplished using just two core strategies: open questions and parallel tasks, which are the focus in her book. These core strategies are addressed for grade bands preK2, 35, and 68 within each math strand: number and operations, geometry, measurement, algebra, and data analysis and probability. Per Small:
x x x x
x x x x
x x x x
Several strategies are illustrated for creating open questions:
Turning around a question. Give the answer and ask for the question. For example: 10 is a fraction of a number. What could the fraction and number be?
Asking for similarities and differences of two items: numbers, shapes, probabilities, graphs, measurements, and so on.
Replacing a number with a blank. For example, in a word problem involving addition to find the answer, students might fill in the blanks in the word problem with their own choices.
Creating a sentence: Students can create a sentence using given parameters, such a a given set of words and numbers.
Using "soft" words. Soft words add some ambiguity, yet open up the thinking process. For example, rather than asking students to create a triangle with a specific area, students might be asked to create two triangles whose areas are different, but close.
Changing the question. Teachers might take an existing question and turn it into an open question. For example, a text might have the question "A cookie has a diameter of 1.75 inches. Express the diameter as a fraction in simplest form." An associated open question might be "The diameter of a cookie is between 1 and 2 inches. Express the diameter as a fraction in two different ways." (p. 9)
Curriculum Associates, Inc. also has a free minicourse 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.
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)
Some might not appreciate the true essence of cooperative learning, a feature employed in differentiating instruction. 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 interdependenceif 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 thinkpairshare, the threestep interview, the jigsaw, and numbered heads. Techniques might include focused listing to brainstorm or examine concepts and descriptions, structured problem solving, oneminute papers, paired annotations, guided reciprocal peer questioning, and sendaproblem.
Robert Slavin (2014), director of the Center for Research and Reform in Education, discussed five strategies to make cooperative learning powerful. He stated, "It is the "learning" in cooperative learning that is too often left out. But it needn't be. Using these five strategies, teachers can get the greatest benefit possible from cooperative learning and ensure that collaboration enhances learning" (para. 3):
The IRIS Center at Vanderbilt University provided a short video of cooperative learning in action.
Visit Peeragogy.org. Peeragogy is a collection of techniques for collaborative learning and collaborative work. This handbook is a "livingdocument" created by a worldwide community of editors interested in peertopeer learning. The ideas can be applied in classrooms, in research, in business, and more.
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 Bloom'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 mid90'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. 119120). 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 multiplechoice or
true/false. Such tests rely on oneright 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.
NOTE: Memorization does not come easy for learners, but some
memorization is involved in mathematics. For example, read
The Benefits
of Memorizing Math Facts by Margaret Groves (2010).
She stated, "Quite simply, a lack of
fluency in basic math fact recall significantly hinders a child's
subsequent progress with problemsolving, algebra and higherorder
math concepts. This can have a serious impact on a child's
overall self confidence and general academic performance" (para. 1).
Get some memorization tips/techniques and learn how to
improve your short and longterm memory at
Memorization Tips.
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 mediumlength answers [in students’ own words] combined with complex multiplechoice 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 stepbystep 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 problemsolving. 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 problemsolving 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 problemsolving approaches, as one must be thoroughly familiar with content. At this level, students might be asked to problemsolve 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.
For more on Bloom's Taxonomy, see Vanderbilt University's Center for Teaching.
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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 712 level through AP Calculus."
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See other Math Methodology pages:
InstructionResources, Assessment and Curriculum: Content and Mapping