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Teaching and Math
Methodology
Content and Curriculum Mapping
Peter Oliva (2005) defines curriculum as “a
plan or program for all the experiences that the learner encounters under the
direction of the school. In practice, the curriculum consists of a number of
plans, in written form and of varying scope, that delineate the desired learning
experiences. The curriculum, therefore, may be a unit, a course, a sequence of
courses, the school’s entire program of studies—and may take place outside of
class or school when directed by the personnel of the school” (p. 7).
Students need both quality instruction and quality curriculum. Typical instruction in schools suffers from the "twin sins" of
"activity-focused teaching and coverage-focused teaching." In both cases, "there
are no explicit big ideas guiding the teaching and no plan for ensuring the
learning" (Wiggins & McTighe, 2005, p. 3). The Center for Applied Special Technology (CAST) advocates curriculum based on
universal design for learning (UDL) principles, which stem from research into
how individuals learn and process information through the recognition,
strategic, and affective networks of the brain. According to CAST, a
universally-designed curriculum offers multiple means of representation,
engagement, and expression. "UDL provides a blueprint for creating flexible
goals, methods, materials, and assessments that accommodate learner differences"
(CAST, n.d., What is Universal Design for Learning?, para. 3). CAST also
provides a UDL Curriculum Self-Check
for checking your curriculum for those goals, methods, materials, and
assessments. Wiggins and McTighe provide the Understanding by Design (UBD) framework for the associated
curriculum mapping.
As national associations, states, and provinces have established
content standards for what students should know and be able to do, educators are
faced with decisions on how to develop a curriculum to best address those
standards. In many cases, there are just too many standards
identified, which might lead to a surface covering of many without the depth
required for true understanding. Because of so many standards, some
educators rely on textbooks to specify content they teach. In doing so,
one might find topic area gaps in instruction. Some educators might leave
out areas they don't feel comfortable teaching, such as statistics, probability,
and data analysis in mathematics. Their own knowledge might have a gap. In other
cases, teaching to the text might lead to too many content-overlaps for
students. The order of
content presented in a textbook should not specify the curriculum. "Finding the
best textbooks for standards-based teaching and learning is possible only after
the district has determined the grade levels and sequence in which critical
standards are to be taught" (O'Shea, 2005, p. 37). Fortunately, the National Council of
Teachers of Mathematics has released its
Curriculum Focal Points,
which outlines the most important math topics for each grade level
Pre-kindergarten through Grade 8.
The implication is that adequate planning is essential and an
ongoing process for effective teaching. Identifying the learning
progression toward mastery of any topic is not easy. James Popham (2007)
points out that "with few exceptions, there is no single, universally accepted
and absolutely correct learning progression underlying any given high-level
curricular aim" (p. 83). Task analyses by different educators will yield
different progressions; however, the important point is "any carefully conceived
learning progression is more likely to benefit students than teachers'
off-the-cuff decision making" (p. 83).
Long-range and Short-range Curriculum Maps
A successful mapping program will help to ensure "measurable
improvement in student performance in targeted areas" and a "process
for ongoing curriculum and assessment review" for schools and districts (Jacobs,
2004, p.
2). A long-range map serves as a horizontal and vertical alignment
mechanism for operational curriculum within a district or school. All
teachers benefit from short-range unit curriculum maps and year-long curriculum
maps because the curriculum map is like a blueprint for aligning content and
skills to be taught, and assessments (Jacobs, 2004).
The essence of curriculum mapping is that it is "an approach to
curriculum and instruction designed to engage students in inquiry, promote
transfer of learning, provide a conceptual framework for helping students make
sense of discrete facts and skills, and uncover the big ideas of content"
(Wiggins & McTighe, 2005, p. 4). Maps can help educators identify
gaps in instruction, places where repetitions occur, and places where content
might be integrated across subject areas. Maps help educators to decide
what should stay and what should be cut from instructional units to best address
essential standards. They can assist with pacing and differentiating
instruction. If you are not familiar with curriculum mapping, Heidi Jacobs provides
resources and sample maps at her Web site,
Curriculum
Designers. North Central Regional Educational Laboratory describes the
step-by-step process in
Curriculum
Mapping: A Process for Continuous Quality Improvement (2003).
The curriculum we teach should revolve around enduring
understandings that we wish all learners to have about mathematics and any other
disciplines. In the words of Carol Ann Tomlinson and Jay McTighe (2006), this will help
"uncover" the content standards deemed essential. Understanding by Design (UBD)
is a model of curriculum development that focuses on what to teach and how, and
the assessment evidence to collect. It is the companion to differentiated
instruction. McTighe and Grant Wiggins (2004) defined the
model, which is often called “Backwards Design,” as a three stage process in which
alignment is a key word. The model is useful for both short range and long
range planning.
Stage 1,
Desired Results, includes writing goals linked to
state and national standards, identifying enduring understandings (the "Big
Ideas") framed as full sentences, writing essential questions tied to those
understandings, and identifying what students will know and be able to do
(skills).
Enduring understandings
cannot be read in a book. They are abstract and require “uncovering.”
The understandings need to complete the
stem, "Students will understand that..." The essential questions need to be
provocative and engaging enough to serve as a hook for students; good essential
questions will lead to retention and transfer. Foundational
knowledge and skills need to be a comprehensive list of facts and skills to
underpin the unit (McTighe & Wiggins, 2004).
What students know and will be able to do are
closely tied together, but are not the same. Jane Pollock (2007)
distinguishes between declarative (content mastery) and procedural (skill
mastery) knowledge. "In a curriculum document, the statements of
declarative knowledge (facts, concepts, generalizations and principles) are
identified by the words understands or knows" (p. 35) that "serve
as placeholders for active verbs, which translate into activities and
experiences that help students organize declarative knowledge." For
procedural knowledge, a statement of student learning would begin with "a verb
that describes the steps that need to be practiced to attain automaticity such
as add, compose, sing, draw, or graph" (p. 36). The latter requires
extensive repetition and practice.
In terms of differentiated instruction, the
established goal (content standard), understandings and essential questions
should not be differentiated. Knowledge and skills may be differentiated
(Tomlinson & McTighe, 2006, p. 36).
Stage 2,
Assessment Evidence, includes development of performance tasks and providing
other evidence of learning. "A performance ability lies at the heart of
understanding" and is linked to a real work task that an adult might typically
do. It is the "evidence of being able to transfer what we know" (Wiggins & McTighe, 2005, p. 7). Performance tasks can be constructed by completing
stem statements associated with the GRASPS model.
Not every task needs to be formed using GRASPS,
although Wiggins and McTighe (2005) propose at least one task for assessing
understand in a major unit or course be developed this way (p. 158). Representative stems follow,
as selected from McTighe and Wiggins (2004, p. 172):
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G, Goal: Your
goal is to…
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R, Role: Your
job is ...
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A, Audience:
Your target audience is ...
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S, Situation:
The challenge involves dealing with…
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P, Product,
Performance, and Purpose: You need to develop…so that…
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S, Standards
and Criteria for Success: Your product must meet the following standards: …
Performance assessments should involve
meaningful, authentic, and engaging tasks. Rubrics, which are criterion-based
scoring tools, should be included by which you and learners can assess their
products or performances. They should have point values attached to assessment
criteria, with the traits specified from greatest to least strengths. For
example, a four-point scale (high to low) might include criteria for exceeding
expectations, meeting expectations, almost meeting expectations, and not meeting
expectations. Care should be taken that the rubric is easy to use. Other
evidence of learning should include opportunities for student
self-assessment and self-adjustment based on feedback.
In terms of differentiated instruction,
performance tasks and other types of assessment evidence may be
differentiated. Response modes might have been orally, visually, or in
writing. However, key criteria for evaluating should not be
differentiated, as they are linked to content goals (Tomlinson & McTighe, 2006,
p. 35).
Stage 3,
Learning Plan, involves planning learning activities and an action plan
that engages learners. Learning
activities should be organized and well-sequenced. They should align with
enduring understandings, essential questions, and standards.
The action plan follows a WHERETO model. Strategies suggested in Tomlinson
and McTighe (2006, pp. 120-126) follow steps in the model. The learning
plan should be differentiated (p. 36):
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W indicates that you are helping learners to know
where the unit is headed and what is expected from them. You are also
determining what their prior knowledge is. Strategies: Provide rubrics
with examples from prior student work tied to different levels of the
rubric.
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H stands for the need to hook the learners and hold
their interest. Strategies: Hooks might take the form of "provocative
essential questions, counterintuitive phenomena, controversial issues,
authentic problems and challenges, emotional encounters, and humor"
(Tomlinson & McTighe, 2006, p. 123).
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E is equipping learners to succeed, enabling them
to experience key ideas and explore issues. Strategies: Provide a
balance of constructivist learning experiences, structured activities, and
direct instruction.
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R is providing opportunities for learners to
rethink and revise their work and understandings. Strategies: Rethinking
and revision might be encouraged by "playing the devil's advocate,
presenting new information, conducting debates, establishing peer-response
groups, and requiring regular self-assessment" (Tomlinson & McTighe, 2006,
p. 124).
-
E, again, allows students to evaluate their work
and set future goals. Strategies: Provide regular opportunities for
students to develop metacognitive skills of self-evaluation,
self-regulation, and reflection.
-
T stands for tailoring to accommodate the diverse
needs, interests, and abilities of learners, including those with special
needs who might have individual education plans. Strategy: Provide options for
assignments with levels of difficulty associated with learners' knowledge
levels, interests, and abilities.
-
O
stands for organization to sustain engagement and the learning process.
Those who have experienced this process might conclude that it
is not so “backward” after all, in that by identifying goals and enduring
understandings at Stage 1, educators can complete a unit development aligned
with those understandings and thus avoid instruction that does not focus on
those outcomes.
Table of Specifications
 Closely tied to short range curriculum mapping is Thomas
Guskey's (2005) suggestion that teachers create a table of specifications, which
helps them to move students toward mastery of standards. This table
might also assist teachers in developing classroom assessments. According
to Guskey, teachers need to translate standards into specific classroom
experiences and ensure that classroom assessments measure that learning.
Breaking down standards into components is key. Essential questions are
"What must students learn to be proficient at this standard?" (p. 34) and "What
must students be able to do with what they learn?" (p. 35). The
development of these tables for a unit of study is closely linked to Bloom's
Taxonomy of Educational Objectives, as the specifications progress
from knowledge of basic facts and terms to the highest cognitive levels of
analysis and synthesis. The following is his general format for such a
table:
|
TABLE OF SPECIFICATIONS |
|
Knowledge of |
Translation |
Application |
Analysis & Synthesis |
|
Terms |
Facts |
Rules &
Principles |
Processes &
Procedures |
New
Vocabulary:
Words
Names
Phrases
Symbols |
Specific
Information:
Persons Events
Data Operations |
Relations
Guidelines
Organizational Cues |
Patterns
Sequences
Order of events or operations
Steps |
Identify
Describe
Recognize
Distinguish
Compute |
Use
Illustrate
Solve
Demonstrate |
Compare
Contrast
Explain
Infer
Combine
Construct
Integrate |
|
Source: Guskey (2005, p. 34) |
Daily Planning--the Teaching Schema for Master Learners Model
(TSML)
Jane Pollock (2007) discusses the TSML model for designing daily
lessons. There are six components with feedback of varying types (verbal,
non-verbal or written), voices (self-reflection, peer, and teacher), and
opportunities being a floating step incorporated throughout (p. 64):
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Set the learning goal/benchmarks or objectives (GO).
Identify the benchmark(s) as declarative or procedural, and break it down
into the daily objective(s).
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Access prior knowledge (APK). Strategies include
non-linguistic representations, advance organizers, and cooperative
learning, for example.
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Acquire new information--declarative or procedural (NI).
The difficulty is selecting of the type of strategy that helps learners
retain each type of knowledge. Lecturing is one way. However,
learners might be involved with "note-taking, using a thinking skill as a
scaffold organizer, creating a graphic organizer," questioning, and
cooperative learning (e.g., pair/sharing) (p. 71). Multimedia
presentations also help acquire new information.
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Apply thinking skills or real-world situation (APP). Pollock
indicates, "When planning for the application of declarative knowledge,
thinking skills (e.g., comparison, analysis, persuasion) can help learners
organize and reorganize facts, leading to longer retention...and how to use
the information in a constructive manner" (p. 68). Acquiring
procedural knowledge might take about 24 practices for competency.
Modeling helps students become more comfortable with applying procedures.
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Generalize or summarize back to the objective/benchmark
(GEN). Students should be involved with closure to a lesson.
Strategies that might be used during the last 5-7 minutes of a class include
"writing to a prompt, sharing aloud with a partner, summarizing using a
strategy, or briefly drawing a pictograph depicting the gist of the topic
for that lesson" (p. 69). Here's where a reflective journal might be a tool.
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Assign homework, if necessary (HW).
Both the instructional plan and the assessment plan should
address thinking skills, evidenced by verbs within Bloom's Taxonomy (knowledge,
comprehension, application, analysis, synthesis, and evaluation). Students
should know the benchmarks associated with each lesson or unit of study, although for young
learners they might need rephrasing in more kid-friendly terms, and have
resources so that they can track their own mastery of the benchmarks.
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Become Knowledgeable about Curriculum Terminology
The following resources provide core vocabulary associated with
curriculum:
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Think about adopting textbooks using multimedia.
Multimedia provides that multiple means of representation,
engagement, and expression to customize learning.
As an example,
Holt, Rinehart, Winston, Inc. has the next generation in math texts
with its
Precalculus: A Multimedia Approach and Calculus: A Multimedia
Approach. Comprehensive instructional videos are paired
with online interactives.
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This section on
curriculum is part of the Math Methodology
series on instruction, assessment, and curriculum design. 
Curriculum: Content and Mapping
See
other Math Methodology pages: 