He did so poorly in his introductory psychology class that his professor insisted that he pursue another major. Determined to succeed, Sternberg earned a BA summa cum laude, and was elected to Phi Beta Kappa, gaining honors and exceptional distinction in psychology. Sternberg continued his academic career. What Can I Do With a Degree in Psychology? Each year, approximately 73,000 to 75,000 students. Phrase, “You'll never get a job with a psychology degree.” It is true that you cannot be a licensed psychologist. Career paths in psychology: Where your degree can take you. Washington, DC: American Psychological.

Book Descriptions: Online Reading And Free Download Career Paths In Psychology Where Your Degree Can Take You 7th Printing 2004 Paperback Book That Written By Donald R. Release On 2000-03 By Energy Institute Press, Career Paths In Psychology Where Your Degree Can Take You 7th Printing 2004 Paperback Book Is One Of The Best Arts & Photography Book!

How it works: • 1. Register a free 1 month Trial Account. Download as many books as you like (Personal use) • 3. Cancel the membership at any time if not satisfied.

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 evidence-based 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, ). 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. Learn more about selecting teacher resource apps. 'Teacher resource apps are designed to assist teachers in completing common tasks (e.g., taking attendance, communicating with parents, monitoring student learning and behavior, etc.)' (Cherner, Lee, Fegely, Santaniello, 2016, p. The market is flooded with such apps that teachers can use to manage their classrooms and tasks.

Quality varies, however, and selecting one based on a star-rating system might not be sufficient. So, how do you select one? HOT: See This Rubric Cherner, T., Lee, C-Y., Fegely, A., & Santaniello, L. A detailed rubric for assessing the quality of teacher resource apps.

Journal of Information Technology Education: Innovations in Practice, 15,117-143. Retrieved from Cherner and his colleagues developed their Evaluation Rubric for Teacher Resource Apps (see Appendix A) to help teachers assess strengths and weaknesses of resource apps they might wish to use.

As discussed in their 2016 publication, the dimensions fall within three domains: Efficiency, Functionality, and Design. • 'Six dimensions are used to evaluate an app’s efficiency that include (A1) Productivity, (A2) Frequency, (A3) Guidance, (A4) Relevance, (A5) Credibility, and (A6) Differentiation' (p. • 'To measure an app’s functionality, the following dimensions are used: (B1) Multipurpose, (B2) Collaboration & Communication, (B3) Ability to Save Progress, (B4) Modifications, (B5) Platform Integration, and (B6) Security' (p. • The dimensions to evaluate the design of the app 'include (C1) Navigation, (C2) Ease of Use, (C3) Customization, (C4) Aesthetics, (C5) Screen Design, (C6) Information Presentation, (C7) Media Integration, and (C8) Free of Distractors' (p.

Goal 2: Know and apply standards, benchmarks, and curriculum frameworks. 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. Are at the forefront, as many states have adopted them. Full House Plans Free Download. 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 & Operations-Fractions 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 As educators examine the Common Core Standards for Mathematical Practice, it becomes clear that the standards 'reflect the view that learning is a social process, implicitly calling for teaching practices that leverage the power of group work and collaborative learning' (Charles A. Dana Center at The University of Texas at Austin and the Collaborative for Academic, Social, and Emotional Learning (CASEL), 2016, p. The eight mathematical practice standards embed core SEL competencies, which CASEL identified: 'self-awareness, self-management, social awareness, relationship skills, and responsible decision making' (p. Consider the first standard of mathematical practice (SMP 1).

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

Improving Instruction The following delves into theory and research; learning styles, multiple intelligences and thinking styles; and differentiated instruction and the educator's ideology. Theory and Research Every teacher should have some knowledge on how students learn and be able to connect research to what they do in the classroom. In the, the Deans for Impact (2015) provide a valuable summary of cognitive science research on how learning takes place. In it you'll find cognitive principles and practical implications for the classroom related to six key questions on how students understand new ideas, learn and retain new information, and solve problems; how learning transfers to new situations; what motivates students to learn; and common misconceptions about how students think and learn (About section). Likewise, the Centre for Education Statistics and Evaluation (2017) in New South Wales, Australia elaborates on research that teachers really need to understand about cognitive load theory: what it is, how the human brain learns, the evidence base for the theory, and implications for teaching. For example, when teaching, you'll learn about the effect of using worked examples with novices and learners who gain expertise, the effect of redundancy (unnecessary information might actually lead to instructional failure), the negative effect of split-attention (processing multiple separate sources of information simultaneously in order to understand the material), and the benefit of using supporting visual and auditory modalities. In their review of over 200 studies in, 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: • teachers’ content knowledge, including their ability to understand how students think about a subject and identify common misconceptions (strong evidence); • quality of instruction, which includes using strategies like effective questioning and the use of assessment (strong evidence); • classroom climate (moderate evidence); • classroom management (moderate evidence); • teacher beliefs (some evidence); and • professional behaviors (some evidence). 2-3) Their review also delved into some common practices that can be harmful to learning and have no grounding in research. Examples included: • lavish praise; • allowing learners to discover key ideas for themselves; • ability grouping; • encouraging re-reading and highlighting to memorize key ideas; • addressing issues of confidence and low aspirations before you try to teach content; • presenting information to learners in their preferred learning style; and • ensuring learners are always active, rather than listening passively, if you want them to remember. 22-24) 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.

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 best-practice examples they might need., 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 (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. 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 and. 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. 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. Are you new to teaching? Consider these four tips to help improve your math instruction.

Small changes in math instruction can help students to make sense of mathematics and empower them as mathematicians. In her work with novice teachers, Corey Drake (2016) emphasizes the following strategies, which are easily managed within the classroom, and meaningful to students: • Ask students “why at least once every day.

Why did that strategy work? Why does that strategy make sense? Why would this work for all numbers?

• Instead of looking only for whether a student’s answer was right or wrong, focus on what was right in the student’s work. Then build on what the student did understand in your next discussion and next task.

• Use your textbook as a tool. Find meaningful tasks in the materials — or tasks that could be meaningful and accessible for students with small changes in numbers or contexts. • Provide at least one opportunity each day for students to solve and explain problems mentally (without pencils, paper, calculators, or computers). This promotes students’ sensemaking, creativity and, most importantly, their sense that they are mathematicians.

(para. 5) Learning Styles, Multiple Intelligences, Thinking Styles Many students experience math anxiety. Much of this stems from a one style fits all approach to teaching. Traditionally, approaches to teaching mathematics have focused on linguistic and logical teaching methods, with a limited range of teaching strategies.

Some students learn best, however, when surrounded by movement and sound, others need to work with their peers, some need demonstrations and applications that show connections of mathematics to other areas (e.g., music, sports, architecture, art), and others prefer to work alone, silently, while reading from a text. All of this is reflected in, which has found its way into schools (Moran, Kornhaber, & Gardner, 2006; Smith, 2002), along with the concept of learning styles. (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. 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. 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. 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. 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' (, 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: includes his inventory and more information on learning styles.

• (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 is a 44-question on-line instrument with automatic scoring on the Web that was developed by Richard Felder and Barbara Soloman of North Carolina State University.

This model assesses learning preferences on four dimensions (active/reflective, sensing/intuitive, visual/verbal, and sequential/global). As an alternative to determining learning styles, a personal and 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, 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). Caution: 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 Dembo and Howard (2007, p. Rather, Dembo and Howard indicated, 'The best practices approach to instruction can help students become more successful learners' (p. 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. Cognitive scientists Pashler, McDaniel, Rohrer, and 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. 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. This position is further confirmed by Willingham, Hughes, and Dobolyi (2015) who concluded in their scientific investigation into the status of learning theories: 'Learning styles theories have not panned out, and it is our responsibility to ensure that students know that' (p. 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. Tommy Emmanuel Dare To Be Different Download. 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) Additional MI and LS Resources 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: • Visit Howard Gardner's website to learn more about him, his theory, and publications: Visit the: his Official Authoritative Site of Multiple Intelligence S. • Multiple Intelligences Institute: is committed to understanding and application of this theory in educational settings from pre-school through adult education. • Tapping into Multiple Intelligences, a workshop from Concept to Classroom at Thirteen Ed Online: • Walter McKenzie's Surfaquarium: 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. 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: 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: 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.

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, ',' 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 include varieties of expertise that mathematics educators should strive to develop in students at all levels: • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision.

• Look for and make use of structure. • Look for and express regularity in repeated reasoning. 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. Tomlinson's website,, will enhance your knowledge of differentiated instruction. She also clarifies myths and misconceptions about differentiation in an ASCD podcast,. 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 'on-the-fly.' Teacher-led group instruction is only one model of instruction.

So, teachers also need to know about additional resources beyond what's in the textbook or available in print format that can be used to help learners. For example, students can also learn from each other in collaborative groups, or from virtual instructors in online settings, or by working alone using software or apps.

They can get different perspectives on a topic from viewing podcasts. 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 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. 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. 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). 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.

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 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 has multiple. Curriculum Associates, Inc.

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. 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 (1953). ASCD devoted its to the theme 'The Challenge of Individual Difference,' which is available online. In the lead article,, Carleton Washburne presented a short history of reform efforts aimed at making education more individualized. Differentiation in the Mathematics Classroom In Creating a Differentiated Mathematics Classroom, Richard Strong, Ed Thomas, Matthew Perini, and Harvey Silver (2004) indicated that student differences in learning mathematics tend to cluster into four mathematical learning styles: • Mastery style--tend to work step-by-step • Understanding style--search for patterns, categories, reasons • Interpersonal style--tend to learn through conversation, personal relationship, and association • Self-Expressive style--tend to visualize and create images and pursue multiple strategies. Students can work in all four styles, but tend to develop strengths in one or two of the styles.

Each of these styles tends toward one of four dimensions of mathematical learning: computation, explanation, application, or problem solving. 'If teachers incorporate all four styles into a math unit, they will build in computation skills (Mastery), explanations and proofs (Understanding), collaboration and real-world application (Interpersonal), and nonroutine problem solving (Self-Expressive)' (p. 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.

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. 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.