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Teaching and Math Methodology
Instruction
Introduction to Teaching
Challenges
 Teaching
involves many challenges, 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" (http://www.las.iastate.edu/diversity/definition.shtml).
Learners and teachers themselves bring to the learning
environment a host of variables, such as beliefs, attitudes, perceptions,
self-efficacy, motivation, learning styles, habits of mind, cultural influences
and demographics (e.g., male/female, sexual orientation, ethnicity,
ability/disability, socio-economic status, religion/spirituality, etc.).
It is certainly helpful for teachers to be aware of their personal biases,
beliefs, and attitudes, as those influence interactions with learners.
Often teachers come
into the profession with a conviction that they really will help learners and
are prepared to do so, only to shortly change that conviction to a hope to
help learners. According to Jane Pollock (2007), teachers need to make
that hope a certainty by adhering to a Big Four approach, which means
-
Using precise terminology to describe what students will
learn
-
Undertaking purposeful instructional planning and delivery
-
Employing purposeful assessment
-
Applying deliberate assessment and feedback strategies (p.
7).
The Big Four is just the tip of the iceberg. Just to be
considered proficient, the National Board for Professional Teaching Standards
includes that teachers have "a broad grounding in the liberal arts and sciences;
knowledge of the subjects to be taught, of the skills to be developed, and of
the curricular arrangements and materials that organize and embody that content;
knowledge of general and subject-specific methods for teaching and for
evaluating student learning; knowledge of students and human development; skills
in effectively teaching students from racially, ethnically, and
socioeconomically diverse backgrounds; and the skills, capacities and
dispositions to employ such knowledge wisely in the interest of students"
(NBPTS, 2002, p. 2, sec: What Teachers Should Know, para. 2).
Teachers also accept responsibility for student
success, develop communities of respect, and help students become partners in
their own success. They take on three instructional roles: direct
instructor, facilitator, and coach (Tomlinson & McTighe, 2006). Their
professional responsibilities include reflecting on their own teaching,
maintaining accurate records, communicating with families, participating in
professional communities, growing professionally in content and pedagogical
skills, and showing professionalism with their own integrity and ethical conduct
(Danielson, 2007, ch. 1).
Thus, teachers are challenged to know and communicate subject
matter; to design curriculum, instruction, and assessments; to be knowledgeable about diverse
student populations, to be knowledgeable about effective uses of data and
technology, to conduct action research to improve their practice, to
implement existing research, and to be learner-centered in their approach.
On top of all this is the need to continually grow in the profession, maintain sanity, minimize stress,
learn from mistakes, and let us not forget--prepare students for
standardized testing.
So where does one begin? Resources on the current page will
assist you with your knowledge of students and instructional practices. CT4ME's section on
Professional Development includes a variety of resources to assist you with
becoming more knowledgeable about the mathematics content you teach. Our section
on Standardized Test Preparation
provides solid advice and resources. Technology Integration
will assist you
with strategies for incorporating technology into your instruction, including
designing your classroom web site, and incorporating multimedia into math
projects. You can learn more about scientifically based research and action
research at our Research Corner.
Teacher Goal Setting
Gaining confidence in teaching takes time, and requires goal
setting, reflection, dialogue and collaboration among colleagues. Goals
might revolve around adequate planning and classroom management, effective
discipline and motivation strategies; subject matter standards and benchmarks,
curriculum frameworks; and developing a range of instructional and assessment strategies.
Goal 1: Adequate Planning and Classroom Management, Discipline, Motivation
Classroom management plays a significant role in effective
teaching and ultimate achievement of learners. It's more than organizing
the physical space for student safety and easy access to materials. It's
more than deciding how you will manage classroom procedures, instructional
groups and student behavior. It means creating a classroom environment of
respect and rapport, and a culture for learning (Danielson, 2007, ch. 1). Marzano, Marzano, and
Pickering (2003) conducted a meta-analysis of 100 reports on this issue,
addressing four general components of effective classroom management: rules and
procedures, disciplinary interventions, teacher-student relationships, and
mental set. This latter refers to an ability to remain emotionally
objective and businesslike and "to identify and quickly act on potential
behavioral problems" (p. 75). They found "on the average, students in classes
where effective management techniques are employed have achievement scores that
are 20 percentile points higher than students in classes where effective
management techniques are not employed" (p. 10).
In terms of planning and classroom management, certainly new teachers would benefit
from the wisdom of their more experienced colleagues. They and mentors can
serve as resources for initial concerns such as "setting up the classroom and
preparing for the first weeks of school, covering the required curriculum
without falling behind or losing student interest, grading fairly, dealing with
parents, and maintaining personal sanity" (Mandel, 2006, p. 67).
Experienced teachers
are better able to integrate and draw connections between current, past, and
future learning and relate their content to other curricular areas. They tend to
be able to better use such classroom management skills as voice, gestures,
reading student facial expressions and body language, and proximity. They can
see the big picture--in planning they can anticipate problems and a need for
alternative plans and adjust their practice accordingly. They also know their students' needs and evaluate their
lessons according to students' learning growth--that is they measure
effectiveness of a lesson beyond meeting the broad objective of the day.
Plus, they are knowledgeable about school and community resources that can
benefit students. They understand the culture of the school, and have
amassed strategies to effectively engage parents in collaborative activities.
They understand how to motivate students and maintain their interest even in the
face of temporary failure (NBPTS, 2002).
In the classroom, Marzano et al. (2003) say that it is important
to involve students in the design of classroom rules and procedures.
Although rules will vary, they should be specific and generally address
expectations for behavior, beginning and ending the day or period, procedures
for transitioning from one activity to the next, interruptions, materials and
equipment, group work, seatwork and teacher-led activities (p. 26).
To help build your confidence in teaching, understanding student
behavior and learning styles, and classroom management, you might experience the potential of digital games-based learning at
simSchool, a simulation program for
educators. You can practice
your teaching skills and get immediate feedback on how your selected strategies
affect student learning. Plus, as in a real classroom, the simStudents
will react to user-selected task design, as well as your teacher moves. Curriculum Associates, Inc., has
a
free mini-course on classroom discipline strategies,
accompanied by audio, which provides tips and strategies that teachers might use
in classroom settings. In four lessons you will learn about setting
expectations, procedures, rules, and consequences; reasons for and managing
disruptive behavior, including how to avoid power struggles with students; how
to document incidences of misbehavior objectively; and strategies for positive
parent conferences on discipline issues and follow-up. There is also a
free mini-course on
motivating students to learn.
You Can Handle Them All, a Web site
on discipline help for teachers and parents, lists over 100 behaviors (e.g.,
arrogant, class clown, cheater, disorganized, overly aggressive, whiner), the
affect of each, actions to take to change the behavior, and mistakes in dealing
with the behavior.
Goal 2: Know Standards, Curriculum Frameworks, Instructional Strategies
All teachers must include goals to become familiar with teacher standards, subject matter standards and benchmark
indicators at the state and national levels. CT4ME provides this
information in our section on Standards. These frameworks specify
standards that students should achieve, but do not specify the curriculum and
teaching methods to be used. For this, teachers need to examine the
district curriculum for how their schools and teachers aligned standards with
content to be taught. They need to examine scope and sequence,
instructional materials, implementation strategies, and any suggested
pedagogical methods. All teachers should consider the role of active or
constructivist learning, as opposed to use of the lecture method. Active student
involvement reinforces learning. This is not to say, however, that
teachers ought not to ever tell students anything directly.
Pekin Public Schools District #108 in Illinois (ILS) is
exemplary in its effort to provide curriculum frameworks and a set of math
instructional strategies commonly supported by rigorous research in schools and
classrooms. Educators, parents, and students can access
ILS math benchmarks
by grade level (K-8). Each benchmark contains teacher clarifications
and additional statements for parents and students that can serve as a checklist
of what students should know and be able to do. The district's
Best Instructional Practices in Mathematics is based on meta-analyses of
research from documents published by the Association for Supervision and
Curriculum Development, Mid-continent Research for Education and Learning, North
Central Regional Educational Laboratory, and various state agencies such as the
Illinois Department of Education. Their best practices address curriculum,
teaching/learning experiences, problem solving and critical thinking,
accommodating diversity, attitudes, parental involvement, and assessment.
Goal 3: Investigate Assessment Methods and Test
Preparation
Another goal for teachers is to investigate assessment methods
and how they might be incorporated into lesson plans. Assessing student
understanding and designing instruction to meet learners' needs are challenging
tasks. Certainly formative assessment plays a major role, and its
importance might be overlooked in our zeal to prepare students for mandated
accountability tests. See Part 2 of this essay for more on the role of
assessment. Test preparation is a reality and Curriculum Associates, Inc., has
a
free-mini course on test preparation strategies
to introduce you to some of
the research behind test preparation and factors that affect test performance.
Specific strategies for math and other content areas are included. CT4ME
has an entire section devoted to standardized test preparation.
Improving Instruction
Attention to theory, research, learning styles, thinking styles,
multiple intelligences, differentiated instruction and the educator's ideology play a role in improving
instruction.
Theory and Research
In Improving Mathematics Instruction,
James Stigler and James Hiebert (2004) indicate 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.
Teachers who view classroom instruction from other countries, which was gathered
in the Trends in International Mathematics and Science (TIMSS) 1999 video study, might learn an alternative methodology that holds
promise to improve math instruction in the U.S. Details and videos are
available at http://www.lessonlab.com.
Japanese Lesson Study is growing in the U.S. as a result of the TIMSS study
(O'Shea, 2005).
The process involves teachers working together to develop, observe, analyze, and
revise lessons and focuses on preparing students to think better mathematically
through more effective lessons. For more on the work of TIMSS, see
http://nces.ed.gov/timss/ and
http://timss.bc.edu/.
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.
Mike Schmoker (2006) states that "the most well-established elements of good
instruction [include]: being clear and explicit about what is to be learned and
assessed; using assessments to evaluate a lesson's effectiveness and making
constructive adjustments on the basis of results; conducting a check for
understanding at certain points in a lesson; having kids read for higher-order
purposes and write regularly; and clearly explicating and carefully teaching the
criteria by which student work will be scored or evaluated" (p. 25). In
mathematics classrooms, teachers might tend to ignore writing about the
discipline; however, to develop complex knowledge, "students need opportunities
to read, reason, investigate, speak, and write about the overarching concepts
within that discipline" (McConachie et al., 2006, p. 8).
Learning Styles, Multiple Intelligences, Thinking Styles
 As
a mathematics teacher, you are aware that many students experience math anxiety.
Much of this stems from a one style fits all approach to teaching.
Traditionally, approaches to teaching mathematics have focused on linguistic and
logical teaching methods, with a limited range of teaching strategies.
Some students learn best, however, when surrounded by movement and sound, others
need to work with their peers, some need demonstrations and applications that
show connections of mathematics to other areas (e.g., music, sports,
architecture, art), and others prefer to work alone, silently, while reading
from a text.
All of this is reflected in
Howard
Gardner's Theory of Multiple Intelligences, which has found its way into
schools (Moran, Kornhaber, & Gardner, 2006; Smith, 2002), along with its relevance for determining learning styles.
Moran et al. (2006) indicate that the theory proposes viewing intelligence in
terms of nine cognitive capacities, rather than a single general intelligence.
Thus, a profile consists of strengths and weaknesses among "linguistic,
logical-mathematical, musical, spatial, bodily-kinesthetic, naturalistic,
interpersonal, intrapersonal, and (at least provisionally) existential" (p. 23).
Overall, the theory has been misunderstood in application.
The multiple intelligences approach does not require a
teacher to design a lesson in nine different ways to that all students can
access the material...In ideal multiple intelligences instruction, rich
experiences and collaboration provide a context for students to become aware
of their own intelligence profiles, to develop self-regulation, and to
participate more actively in their own learning. (p. 27)
Educators should be aware that multiple intelligences
(MI) and learning styles (LS) are not interchangeable terms. According to
Barbara Prashnig (2005), "LS can be defined as the way human beings prefer to
concentrate on, store, and remember new and/or difficult information. MI is a
theoretical framework for defining/understanding/assessing/developing people's
different intelligence factors" (p. 8). Consider LS as "explaining
information 'INPUT' capabilities" and MI "more at the 'OUTPUT' function of
information intake, knowledge, skills, and 'talent'--mathematical, musical,
linguistic" and so on (p. 9).
Knowledge of students' learning styles assists teachers in
developing lessons that appeal to all learners. However, determining a
student's learning style cannot be done strictly by observation. Various
models and inventories have been designed to determine a learning style, which
does not remain fixed over time. Therein lies the problem of relying on
inventories, as their validity and reliability might be in question (Dembo &
Howard, 2007), and they differ. The following are among those inventories:
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The Dunn and Dunn Model includes "environmental, emotional,
sociological, physiological, and cognitive processing preferences" (International
Learning Styles Network, sec: About Learning Styles).
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David Kolb's Learning Styles Inventory categorizes in four
dimensions (converger, diverger, assimilator, or accommodator) based on the
degrees to which one possesses "concrete experience abilities, reflective
observation abilities, abstract conceptualization abilities and active
experimentation abilities" (Smith, 2001, sec: David Kolb on Learning
Styles).
-
VARK
(Visual, Aural, Read/write, and Kinesthetic) is only part of a learning
style, according to developer Neil Fleming who states "VARK is about one
preference -our preference for taking in, and putting out information in a
learning context"; "VARK is structured specifically to improve learning and
teaching." The VARK questionnaire (just 16 short questions) is
available online.
-
Memletics Learning Styles Inventory posted at
Learning-styles-online.com
is a free online inventory (70 questions) with graphical feedback to
determine your dominant and secondary styles: visual, social, physical,
aural, verbal, solitary, and logical.
-
The
Index of Learning Styles is a 44-question on-line instrument with
automatic scoring on the Web that was developed by Richard Felder and
Barbara Soloman of North Carolina State University. This model
assesses learning preferences on four dimensions (active/reflective,
sensing/intuitive, visual/verbal, and sequential/global).
As an alternative to determining learning styles, Sternberg-Wagner's
Thinking Styles Inventory
and The
Multiple Intelligence Inventory based on Gardner's work will also benefit
teaching and learning, and are brought to you by the Learning Disabilities
Resource Community of Canada. Students with learning disabilities or
attention-deficit-disorder can find practical tips on how to make your learning
style work for you at LdPride.net, which also contains more information on multiple intelligences.
With so many inventories available, teachers might wonder how
their teaching can accommodate so many styles. Li-fang Zhang and Robert
Sternberg (2005) indicate, however, that teachers need only to attend to "five
basic dimensions of preferences underlying intellectual styles: high degrees of
structure versus low degrees of structure, cognitive simplicity versus cognitive
complexity, conformity versus nonconformity, authority versus autonomy, and
group versus individual. Furthermore, [they] believe that good teaching treats
the two polar terms of each dimension as the two ends of a continuum and
provides a balanced amount of challenge and support along each dimension" (p.
43).
Readers should also be aware that while determining
learning styles might have great appeal, "The bottom line is that there is no consistent evidence that matching instruction to
students' learning styles
improves concentration, memory, self-confidence, grades, or reduces anxiety,"
according to Myron Dembo and Keith Howard (2007, p. 106). Rather, Dembo
and Howard indicate, "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).
Differentiated Instruction and Ideology
Learning to teach in a flexible manner that responds to the
unique needs of learners is a
challenge. Often total lessons or the pace of individual lessons need to
be adjusted "on-the-fly." So, teachers also need to know about additional
resources beyond what's in the textbook that can be used to help learners. Carol Ann Tomlinson and Jay McTighe (2006) promote
differentiated instruction, which is primarily an instructional design model
that focuses on "whom we teach, where we teach, and how we teach" (p. 3).
How one teaches is based on one's ideological perspective. According to
David Ferrero (2006), educators are divided by traditionalism and innovation,
but teaching that leads to achievement gains when one embraces standardized
testing does not mean that educators have to choose between one or the other.
There is a concept of "innovative traditionalism" that is student-centered, yet
has been shown to improve standardized achievement test scores. This has
been accomplished in two Chicago-area high schools by "a combination of test
prep, classical content, and collaboratively developed thematic projects
grounded in controversy and designed to cultivate student voice and civic
engagement" (p. 11). The following table (Ferrero, 2006, p. 11)
illustrates the essential differences in education's ideological divide, which
can be bridged.
|
Education's Ideological Divide |
| Traditional |
Innovative |
| Standardized tests |
Authentic assessment |
| Basic skills |
Higher-order thinking |
| Ability grouping |
Heterogeneous grouping |
| Essays/research papers |
Hands-on projects |
| Subject-matter disciplines |
Interdisciplinary integration |
| Chronology/history |
Thematic integration |
| Breadth |
Depth |
| Academic mastery |
Cultivation of individual talents |
| Eurocentrism |
Multiculturalism |
| Canonical curriculum |
Inclusive curriculum |
| Top-down curriculum |
Teacher autonomy/creativity |
| Required content |
Student interest |
|
Source: Ferrero, D. (2006). Having it all. Educational Leadership, 63(8), 11. |
The goals of differentiated instruction and innovative
traditionalism are to ensure effective learning for all. Best practice
learning adheres to 13 principles. Best practice is student-centered,
experiential, holistic, authentic, expressive, reflective, social,
collaborative, democratic, cognitive, developmental, constructivist, challenging
with choices and students taking responsibility for their learning (Zemelman,
Daniels, & Hyde, 1998, as cited in Wilcox & Wojnar, 2000).
Some might not appreciate the true essence of cooperative
learning. Learners are responsible for not just their own learning, but
the learning of others. Shared learning leads to success for all, as each
member of a learning group has a specific role to play in reaching a common
goal. Successful groups include positive interdependence--if one fails,
the entire group is affected. There is both individual and group
accountability; although some work might be completed individually, some must be
accomplished by group interactions. Typical
cooperative learning
strategies include think-pair-share, the three-step interview, the jigsaw,
and numbered heads.
Techniques might
include focused listing to brainstorm or examine concepts and descriptions,
structured problem solving, one-minute papers, paired annotations, guided
reciprocal peer questioning, and send-a-problem.
According to P. 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) addresses four ways to differentiate instruction:
content (requires pre-testing to determine the depth and complexity of the
knowledge base that learners will explore), process (leads to a variety of activities and
strategies to help students gain knowledge), product (complexity varies in ways
for assessing learning), and manipulating the environment or
accommodating learning styles. Fairness is a key concept to emphasize with
learners, who will recognize that not everyone will work on the same thing at
the same time. They need to appreciate that not everyone has the same
needs. Likewise, Hall (2002) presents a visual for the Learning Cycle and Decision Factors Used in Planning
and Implementing
Differentiated Instruction.
Hall also provides a number of links to learn more about this topic.
In Creating a Differentiated Mathematics Classroom, Richard Strong, Ed Thomas, Matthew Perini, and Harvey Silver
(2004) indicate that student differences in learning mathematics tend to cluster
into four mathematical learning styles:
-
Mastery style--tend to work step-by-step
-
Understanding style--search for patterns, categories,
reasons
-
Interpersonal style--tend to learn through conversation,
personal relationship, and association
-
Self-Expressive style--tend to visualize and create images
and pursue multiple strategies.
 Students can work in all four styles, but tend to develop
strengths in one or two of the styles. Each of these styles tends toward
one of four dimensions of mathematical learning: computation, explanation,
application, or problem solving. "If teachers incorporate all four styles
into a math unit, they will build in computation skills (Mastery), explanations
and proofs (Understanding), collaboration and real-world application
(Interpersonal), and nonroutine problem solving (Self-Expressive)" (p. 74).
From an instructional styles perspective, Silver, Strong, and
Perini (2007) note that teachers who use mastery strategies focus on increasing
students' abilities to remember and summarize. "They motivate by providing
a clear sequence, speedy feedback, and a strong sense of expanding competence
and measurable success." When focusing on interpersonal strategies,
teachers use "teams, partnerships, and coaching" to help students better relate
to the curriculum and each other. Understanding strategies help students
to reason and use evidence and logic. Teachers "motivate by arousing
curiosity using mysteries, problems, clues, and opportunities to analyze and
debate." Self-expressive strategies highlight students' imagination and
creativity. Teachers employ "imagery, metaphor, pattern, and what ifs to
motivate students' drive toward individuality and originality." Finally,
it's possible to use all four styles at the same time to achieve a balanced
approach to learning (sec: Part One: Introduction, Figure B).
The implication for mathematics instruction is
that "any sufficiently important mathematics topic requires students to
learn the topic in four dimensions: procedurally, conceptually, contextually,
and investigatively" (Strong et al., 2004, p. 75). Even taking that
approach, we are challenged
to help students overcome misconceptions.
Example:
The importance of addressing these four dimensions was made
very clear in a recent query I had from an individual [let's call him Mac]
seeking help
for
a learner in the 5th grade
who was struggling to multiply decimal numbers. The learner had incorrectly calculated: 0.032 * 0.16
=0.0512. But why? Apparently the learner was taught an algorithm, but used it incorrectly.
Let's examine
the problem that arises in understanding if teaching is done only
procedurally.
In investigating Web resources for Mac on this concept,
several sites indicated using the algorithm with instructions to multiply
the digits as whole numbers (here 32*16 = 512), then count up the number of
decimal places indicated in the problem (here 5) and then to use that number
of places in the final answer. If extra zeroes are needed (here 2),
place them before the digits in the whole number answer. This kind of
wording, which I purposely made less than mathematically precise, is what a
5th grader might typically remember from only an algorithm. Notice
that the learner's answer (0.0512) did have five digits (places used
incorrectly) and two zeroes preceding 512. The answer should have been
0.00512.
-
Conceptually, the learner might have missed a connection
to prior learning on fractions, or the link was not made and reinforced
in instruction. Writing the problem in its equivalent fraction
form, using knowledge of converting decimals to fractions and vice
versa, and decimal notation and place value might eventually have helped
the learner to understand the short cut presented in the algorithm.
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 Math--they prove to you that 25 divided by 5 is 14! |
The example above noted teaching mathematical procedures using
algorithms. Algorithms play an important role in mathematics, as they
address step-by-step procedures yielding a single answer. The difficulty
arises, as the above example indicates, if the algorithm is taught without
linking it to conceptual, contextual, and investigative understanding.
Ball, Ferrini-Mundy, Kilpatrick, Milgram,
Schmid, and Schaar (2005) indicate, "Fluent
use and understanding ought to be developed concurrently." Algorithms not
only play a role in gaining whole number computation fluency, but play a role in
such examples as "constructing the bisector of an angle; solving two linear
equations in two unknowns; calculating the square root of a number by a
succession of dividing and averaging" (section: Areas of Agreement).
According to Strong et al. (2004), testing practices should also aim to
measure knowledge in all four dimensions. Teachers should be aware that
texts and their accompanying tests, however, tend to emphasize only the mastery
and understanding styles of learning. To differentiate instruction, teachers can:
-
Rotate strategies to appeal to students' dominant learning
style and challenge them to work in their less preferred styles.
Consider strategies such as using manipulatives, observing demonstrations,
sketching out a math situation, reading, having students compare their work
with a partner, or solving complex problems in a team.
-
Use flexible grouping.
-
Personalize/individualize learning for struggling students
or for those needing an extra challenge.
Among strategies for implementing differentiated instruction,
Tomlinson and McTighe (2006) suggest that teachers consider:
-
Compacting--giving students credit for what they already know;
-
Negotiated delay of due dates and times for tasks;
-
Varied homework;
-
Bookmarked Web sites on key topics in languages other than
English to support English language learners;
-
Video and audio clips to support multiple intelligences and
varied learning styles and disabilities;
-
Flexible grouping, "expert" groups, and interspersing lecture
with group discussions;
-
Guided peer review;
-
Teaching with part-to-whole and whole-to-part emphasis;
-
Tiered assignments--used when all students need to know the
same skill or concept;
-
Learning contracts;
-
Independent study/projects;
-
WebQuests and Web inquiries;
-
Learning centers--primarily used in elementary grades; and
-
Adjusting questions to accommodate levels in
Bloom's Taxonomy.
Curriculum Associates, Inc. also has a
free mini-course on
differentiated instruction. Text is
accompanied by audio. Handouts, supplementary readings, and short video
clips of teachers explaining the use of a particular strategy in their
classrooms are included. A broadband connection is recommended. The
four lessons address principles of differentiated instruction, the role of
formal and informal assessment in identifying student needs, strategies used in
differentiated instruction, and guidelines for managing a differentiated
classroom.
Back to top
Bloom's Taxonomy and Levels of Understanding
 Bloom's
Taxonomy contains six levels: knowledge, comprehension, application,
analysis, synthesis and evaluation. The taxonomy, which was revised in 2001,
now has levels of remembering, understanding, applying, analyzing, evaluating,
and creating (Forehand, 2005). Students should work at all levels of the
taxonomy. It should not be viewed as a ladder, however, nor as a framework
for differentiated instruction (Tomlinson & McTighe, 2006, pp. 119-120).
The taxonomy is helpful for breaking down state
standards into meaningful components as teachers plan their instruction.
Planning for instruction will be elaborated upon in Part 3 of this essay on
content and curriculum mapping.
Charles White (2007) provides a
closer look at how Bloom’s Taxonomy provides levels of understanding to guide
teaching and assessing knowledge. Teaching for each level has different
instructional strategies and testing techniques.
-
Knowledge: Memorization
and recitation fall within the knowledge level. Teachers might rely on a
lecture method and assigned readings. They are transmitters of knowledge.
Students remain passive and acquire familiarity with the material, take
notes, memorize, and study enough so that they can recall information at
least long enough to pass tests, which might be multiple-choice or
true/false. Such tests rely on one-right answer. “Opinions and values are
excluded from this type of testing” (p. 162). When writing, students tend to
parrot back what the teacher has said. However, the ability to recite
information that has been memorized does not mean that students know what
they are saying.
-
Comprehension: At a comprehension level, students are able to discuss
what they’ve learned in their own words rather than in the teacher’s words,
express their feelings, participate in classroom debate, and are thus taking
ownership of content and remembering it better. They would be able to
explain a graph, a calculation using a formula, or an equation (e.g., linear
regression), but not necessarily be able to implement associated tools.
At this level,
“short- and medium-length answers [in students’ own words] combined with
complex multiple-choice formats often serve as the medium of test material”
(p. 162).
-
Application: While key
words for comprehension are explaining and discussion, application involves
doing. Novices lacking understanding might only be able to apply knowledge
when given step-by-step instructions that can be used without deviation.
However, at this level, students must be able to demonstrate that they can
use concepts and theories in problem-solving. They might be given all the
information necessary to do calculations or tasks. Memory at this level is
enhanced with repetition. Testing includes unstructured problems that might
not have been encountered in the text or during a lecture, requiring
students to determine a solution method using what they have learned.
Novice students might still turn to the teacher for a correct solution.
-
Analysis: At this
level, application is taken a step further. Students must be able to take a
situation apart, diagnose its pieces, and decide for themselves what tools
(e.g., graph, calculation, formula, etc.) to apply to solve the problem at
hand. Rather than just understanding and applying individual concepts,
students understand the relationship among concepts. Case studies in
business, for example, fit this level. The level of difficulty can be
controlled for novices to experts by the number of issues presented in the
cases requiring analysis. Likewise, this process to control difficulty can
be used for any mathematics problem-solving scenario based on level of
expertise of learners. For example, at elementary levels, students are
introduced to analysis when a few extraneous facts are included in a
problem, which are not needed to solve it. At an analysis level, students
are able to appreciate that some problems do not have a unique solution and
there is more than one way to defend a position or solution method, as in a
case study.
-
Synthesis: In contrast
to analysis (i.e., taking apart), at the synthesis level students put things
back together. Given the pieces, there might be more than one way to do
this. In terms of mathematics, students might take the pieces they’ve
learned, and put them together to solve problems not yet encountered in the
actual classroom setting. Synthesis is involved when creating something
new. Advanced students might be asked to create a new theory. Synthesis is
tested via major projects, for example, which might be long term involving
creativity and application of all that students have learned on a topic.
-
Evaluation: Teachers
evaluate student work all the time, particularly exams and homework. The
difficulty in evaluation arises when judging multiple perspectives and
varied problem-solving approaches, as one must be thoroughly familiar with
content. At this level, students might be asked to problem-solve via debate,
for example. At the evaluation level, one is able “to judge the work of
others at any level of learning with regard to its accuracy, completeness,
logic, and contribution” (White, 2007, p. 161). Rubrics help teachers to
evaluate work, particularly for that involving application, analysis and
synthesis.
White (2007) presents a novel way
to test levels of understanding. He proposes writing two test questions on a
topic, allowing students to choose only one of those to answer. The first is
written for the knowledge and comprehension levels (e.g., key verbs: list,
describe), and the second is written for the higher critical thinking levels of
application, analysis, and synthesis. Points possible would be indicated for
each, so that students would recognize that only those answering the second
could be awarded maximum points toward an A+ grade. The option to choose
enables the less able student to better demonstrate what he does know and
perhaps earn a B grade, rather than risk failure because of an inability to
demonstrate critical thinking. For either question, students could fail.
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| Math Methodology Instruction Essay: Page 1 |
2 |
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NCLB mandates that states and districts adopt programs and
policies supported by scientifically based research, which will influence
instructional strategies that educators use. In a standards-based
classroom four instructional strategies are key:
The
Instruction Essay (Page 1 of 3) on this page contains the following subsections:
See
other Math Methodology pages: 