This section on curriculum is part 3 of the Math Methodology series on instruction, assessment, and curriculum design.
The short essay that follows, Content and Curriculum Mapping, is part 3 of the essay, Teaching and Math Methodology. It includes:
Part 1: Math Methodology: Instruction
Part 2: The Role of Assessment and resources
Part 3: Curriculum: Content and Mapping
Curriculum Mapping (Page 1 of 2) on this page contains the following subsections:
Peter Oliva (2005) defined 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).
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. Cynthia Curry of Maine's Learning Technology Initiative has an excellent presentation in iTunes on UDL and Accessibility. The goal of this presentation is to help ensure "access to the curriculum for all students" and includes examining "tools, resources, and strategies for supporting all students, including students with disabilities and students from diverse cultural and linguistic backgrounds."
As national associations, states, and provinces have established content standards for what students should know and be able to do, educators have been faced with decisions on how to develop a curriculum to best address those standards. In many cases, there have been just too many standards identified, which has 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).
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) pointed 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).
A number of documents have been developed to assist districts with curriculum and implementing standards. The National Council of Teachers of Mathematics developed Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (NCTM, 2006), which outlines the most important math topics for each grade level Pre-kindergarten through Grade 8. Focus in High School Mathematics: Reasoning and Sense Making (NCTM, 2009) addresses mathematics education in high school by providing reasoning opportunities in five areas: numbers and measurements, algebraic symbols, functions, geometry, and statistics and probability. Of particular significance is the eBook, Making it Happen: A Guide to Interpreting and Implementing Common Core State Standards for Mathematics (NCTM, 2011).
As the majority of states and the District of Columbia have adopted the Common Core State Standards (2010), change in on the horizon in terms of curriculum for mathematics. According to David Conley (2011):
To reach the new levels envisioned in the common standards and assessments, students must actively participate in their own learning. Curriculum that includes interesting problems, investigations, debates, simulations, games, Socratic questioning, presentations, projects, and other forms of learning that demand engagement will help maximize retention of key content and concepts. (p. 20)
Educators should not wait to incorporate Conley's recommendation. Note: Educators might also appreciate the Common Core Video Series posted at EngageNY.org. Although meant for New York educators, this 15-part series will provide valuable insights for all who have adopted the Common Core State Standards. Educators and administrators will "learn step-by-step how to implement the Common Core for ELA/Literacy and Math in their schools and classrooms. You’ll also gain a deeper understanding of the rationale behind the Common Core and what it will mean for students ..." (EngageNY, Common Core Video Series, August 1, 2011, para. 1).
Students need both quality instruction and quality curriculum. However, typical instruction in schools has suffered 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" (McTighe & Wiggins, 2005, p. 3). McTighe and Wiggins provide the Understanding by Design (UBD) framework for curriculum mapping.
A successful mapping program as described by Heidi Jacobs (2004) in Getting Results with Curriculum Mapping 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 (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" (McTighe & Wiggins, 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. The Rubicon International Podcast Channel will help you keep up on the latest developments in education, technology, and curriculum mapping. Among podcasts are Why Map?, The Road to Mapping Quality, and Curriculum Mapping: Prologue and Setting the Stage, which are episodes 1, 3, and 9, respectively.
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). Jay McTighe (2013) provided seven characteristics of an essential question in Essential Questions: Opening Doors to Student Understanding. An essential question is open-ended, thought-provoking and intellectually stimulating, calls for higher-order thinking and "cannot be effectively answered by recall alone." It "points toward important, transferable ideas within (and sometimes across) disciplines," raises additional questions, requires support and justification, and "can and should be revisited again and again" (p. 3).
What students know and will be able to do are closely tied together, but are not the same. Jane Pollock (2007) distinguished 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" (McTighe & Wiggins, 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 McTighe and Wiggins (2005) proposed at least one task be developed this way for assessing understanding in a major unit or course (p. 158). Representative stems follow, as selected from McTighe and Wiggins (2004, p. 172):
G, Goal: Your goal is to
R, Role: Your job is ...
A, Audience: Your target audience is ...
S, Situation: The challenge involves dealing with…
P, Product, Performance, and Purpose: You need to develop…so that…
S, Standards and Criteria for Success: Your product must meet the following standards: …
Performance assessments should involve meaningful, authentic, and engaging tasks. Such tasks "are better suited to assess more complex concepts and 21st century skills, such as mathematical reasoning, scientific investigations, issues analysis, creative problem solving, oral communications, and technology applications" and should "include both content specific and interdisciplinary performances." Teachers should implement them "as part of the curriculum at designated time periods during the school year" (McTighe & Wiggins, 2011, p. 17).
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 (McTighe & Wiggins, 2004).
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).
Results from performance tasks and rubrics might be placed in a Student Standards Folder, as part of a "systematic collection of assessment evidence related to Core Standards and other important educational goals" (McTighe & Wiggins, 2011, p. 17).
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):
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.
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).
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.
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. ASCD has a collection of resources to learn more about UBD.
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|
Order of events or operations
Source: Guskey (2005, p. 34)
Jane Pollock (2007) discussed 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):
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).
Access prior knowledge (APK). Strategies include non-linguistic representations, advance organizers, and cooperative learning, for example.
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.
Apply thinking skills or real-world situation (APP). Pollock indicated, "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.
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.
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.
Become Knowledgeable about Curriculum Terminology
The following resources provide core vocabulary associated with curriculum:
Think about adopting textbooks that include multimedia.
Multimedia provides the multiple means of representation, engagement, and expression to customize learning.
A Guide to Curriculum Mapping: Planning, Implementing, and Sustaining the Process by Janet Hale (2007) is a step-by-step guide, which examines the stages of contemplating, planning, and implementing curriculum mapping initiatives that can improve student learning and create sustainable, systemic change. Heidi Jacobs contributed the foreword.
Curriculum Designers: Heidi Jacobs provides resources, including mapping software, and sample maps at her Web site for curriculum mapping.
Galileo Educational Network: Creating Essential Questions by Pat Clifford and Sharon Friesen, who explain the key components.
PBS TeacherLine has two courses on curriculum mapping, noted in their online catalog, based on the work of Heidi Jacobs.
Roadmap to Success: A Curriculum Mapping Primer from Glencoe Teaching Today.
Atlas Curriculum Mapping is web-based from Rubicon International.
BrainHoney from Agilix Labs offers free accounts to individuals and teachers to set up classes and manage content for those--an online learning system. There's an online gradebook, student/parent portals, and curriculum mapping support. The software comes with pre-loaded state education standards. After selecting those for your state and content area, use a drag and drop feature to move those selected into modules you've identified for learning, add your activities and assessments. Outcomes will help with individualized learning and preparing for state standardized testing.
Curriculum Mapper is part of CCStudio, and is a web-based mapping system developed by Collaborative Learning.
TeacherEase provides standards-based curriculum mapping on the web. State standards are pre-loaded, but districts can add their own standards, too. Product has other features to help teachers manage their instruction, including lesson planning, gradebook, report cards, and parent access.
Montgomery Public Schools (MD) course pages for math contain expectations, essential questions, enduring understandings, indicators, and vocabulary, illustrating stage 1.
CtCurriculum.org from the Connecticut Department of Education has a series of performance tasks for multiple subject areas. Great for ideas for stage 2 in UBD.
Big Ideas e-journal from Authentic Education is a primary site for UbD stages. Resources include UbD Staples: What is a big idea?; What is transfer?; resources for Essential Questions; Habits of Mind; and What is an Essential Question? Grant Wiggins is President of Authentic Education.
Common Core's Curriculum Maps in Mathematics (P-5) provided at K-12 Blueprint include a 180 day plan for each grade level.
EngageNY Year Long Draft Curricular Maps in ELA and Mathematics for the Common Core standards.
Olentangy (OH) Local School District Curriculum Maps are one page per grade level or subject. They contain essential questions, followed by standards to guide classroom instruction.
The Center for the Study of Mathematics Curriculum (CSMC) "serves the K-12 educational community by focusing scholarly inquiry on issues related to the development, implementation, and use of mathematics curriculum. Major areas of work include understanding the influence and potential of mathematics curriculum standards and textbook materials, enabling teacher learning through curriculum material investigation and implementation, and building capacity for developing, implementing, and studying the impact of mathematics curriculum" (Mission statement). You'll find three curriculum databases of interest (K-12 Mathematics Textbooks, Curriculum Research Instruments, and State Mathematics Curriculum Standards), CSMC published books and reports of research, and mathematics curriculum courses. The following is among reports:
Confrey, J., & Krupa, E. (2010). Curriculum design, development, and implementation in an era of common core state standards: Summary report of a conference. Retrieved from http://mathcurriculumcenter.org/conferences/ccss/SummaryReportCCSS
Morehead, P., & LaBeau, B. (2007, Apr. 12). Beyond curriculum mapping; Using technology to delve deeper into inquiry learning. T.H.E. Journal. Retrieved from http://thejournal.com/articles/2007/04/12/beyond-curriculum-mapping-using-technology-to-delve-deeper-into-inquiry-learning.aspx Pamela Morehead, Ph.D., and Barbara LaBeau discussed how staff at an elementary school used curriculum mapping, study groups, and an onsite technology coach as their model for professional development. Staff used a self-study process to evaluate their use of technology in the classroom and a district technology integration initiative called Project 2000.
Center for Applied Special Technology. (n.d.). What is universal design for learning? Wakefield, MA: Author. Retrieved from http://www.cast.org/research/udl/index.html
Common Core State Standards (2010). Mathematics, Introduction, Standards for Mathematical Practice. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers. Retrieved from http://www.corestandards.org/Math/Practice
Conley, D. T. (2011). Building on the common core. Educational Leadership, 68(6), 16-20.
Guskey, T. (2005). Mapping the road to proficiency. Educational Leadership, 63(3), 32-38.
Jacobs, H. H. (Ed.). (2004). Getting results with curriculum mapping. Alexandria, VA: ASCD.
McTighe, J., & Wiggins, G. (2011, Spring). Do we need an assessment overhaul? Baltimore, MD: John Hopkins University, Better: Evidenced-based Education, 16-17.
McTighe, J., & Wiggins, G. (2005). Understanding by design: Expanded 2nd edition. Alexandria, VA: ASCD.
McTighe, J., & Wiggins, G. (2004). Understanding by design: Professional development workbook. Alexandria, VA: ASCD.
National Council of Teachers of Mathematics (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: Author. Retrieved from http://www.nctm.org/store/Products/Curriculum-Focal-Points-for-Prekindergarten-through-Grade-8-Mathematics-A-Quest-for-Coherence/
National Council of Teachers of Mathematics (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: Author. Retrieved from http://www.nctm.org/store/Products/Focus-in-High-School-Mathematics--Reasoning-and-Sense-Making/
National Council of Teachers of Mathematics (2011). Making it happen: A guide to interpreting and implementing common core state standards for mathematics. Reston, VA: Author. Retrieved from http://www.nctm.org/store/Products/Making-It-Happen--Common-Core-Standards-(PDF)/
Oliva, P. F. (2005). Developing the curriculum (6th ed.). New York, NY: Allyn & Bacon.
O'Shea, M. (2005). From standards to success. Alexandria, VA: ASCD.
Pollock, J. E. (2007). Improving student learning one teacher at a time. Alexandria, VA: ASCD.
Popham, J. (2007). The lowdown on learning progressions. Educational Leadership, 64(7), 83-84.
Tomlinson, C., & McTighe, J. (2006). Integrating differentiated instruction & Understanding by Design. Alexandria, VA: ASCD.
See other Math Methodology pages: