Ultimately, most teachers accumulate a great deal of resource materials. It is not unusual for teachers to collect exemplary lesson plans, activities, classic problems, assignment ideas, useful handouts, and a wealth of other teaching resources from books, journals, the Internet, professional conferences, and colleagues. But although collecting these ideas is relatively simple, organizing them in a useful manner is not always so easy. We often hear teachers saying, “I know I have it somewhere because I took notes on that topic during a conference one time, but I have no idea where I put it.” Therefore, it is important—particularly for teachers who are new to the profession—to establish a use-ful method for collecting and organizing resource materials. After all, what good is an exemplary hands-on activity if you can’t fi nd it on the day that you want to use it in class? Here are four practical suggestions for organizing a resource fi le:
Obtain a box of fi le folders. Each time that you fi nd a good problem, activity, or lesson, place a master copy in one folder and title the folder with a de-scriptor, such as “Grains of Rice” or “Orange Grove,” on the label. Then organize the fi le folders in alphabetical order by content area. For example, you could place the grains-of-rice problem under Measurement because it involves weighing a sample of rice to estimate the total weight of the rice.
You could fi le the orange grove problem under Algebra because the problem involves patterning and writing an equation that represents the problem. It is important, however, that you use a fi ling scheme that works for you. The grains-of-rice problem, for example, could just as easily be fi led under Alge-bra because the solution can include an analysis of exponential functions.
Consequently, some teachers fi le all of the problems and activities alphabeti-cally. However you choose to organize the resources, it is important to place only one idea in each fi le folder. Otherwise, you may spend a great deal of time rummaging through a fi le to fi nd one teaching idea that has been mixed in with a dozen others. And the process becomes simple: Each time that you encounter an idea that you think will be useful, give it a title on a fi le folder, place the idea in the folder, and fi le it in an appropriate location.
Obtain several three-ring binders and a plastic storage tub or a small fi le cabinet. Label each binder for a particular content area, such as the concep-tual categories in the Common Core State Standards —number and quan-tity, algebra, functions, modeling, geometry, and statistics and probability.
Then, each time that you locate a useful teaching idea, three-hole-punch the pages, and place the idea in the appropriate binder. If you store all of the binders in a plastic tub or fi le cabinet, you will always know where to look for the teaching ideas. As new ideas are added, you may choose to change the organization scheme, and this is very easy when using binders and three-hole-punched pages. For example, if you decide to create a new binder on problem solving that includes some ideas you have already collected, you
only need to label a new binder and either move some activities from their former location to the new one or make a second copy of the activity and place one copy in each binder.
Obtain a set of index cards and an index card fi le box. Each time that you discover a new teaching idea, write the idea title at the top of the card, a short description of the idea on the card, and fi le it in a fi le box by content topic (or by chapter of the textbook if your class uses a particular one over a period of time). Then keep one master fi le of all of the teaching ideas on pa-per in alphabetical order, titled at the top to match the titles on the index cards. Whenever you are going to teach a lesson on a particular topic, you can quickly thumb through the short descriptions on the index cards to lo-cate a useful idea. Then go to the master fi le of ideas and pull it out for du-plication or use in your class. It is often quicker and easier to locate a problem or lesson by looking through a small box of index cards than by fl ipping through binders fi lled with papers.
Scan your favorite lesson ideas (or save them in PDF format) and put them on a fl ash drive or burn them onto a CD-ROM. The fl ash drive or CD-ROM can serve as an electronic fi le from which lessons and activity pages can be readily retrieved or printed. With a fl ash drive, each time a new idea is lo-cated, it can simply be saved and added to the drive. It is also easy to run a search for a particular lesson or key word when the resource fi le is saved in an electronic format. As an alternative, some teachers store their teaching ideas and Web links to lessons at free or subscription Internet sites, such as LiveText.com. Information stored at a Web site can be readily searched and downloaded when needed for a lesson. Updating the ideas at such a site is easy, and the lessons can be retrieved from any computer with Internet ac-cess, without having to physically take a CD or fl ash drive along to school.
Whatever method you choose—including, perhaps, some other scheme that is not listed here but makes sense to you—it is important to seek out exemplary prob-lems, activities, and lessons from resource books, journals, the Internet, and other sources and to organize them for easy reference. These resources will facilitate your ability to meet the mathematical goals and objectives set forth by the school district for your grade level.
Conclusion
The teaching and learning process begins with a very specifi c set of statements about what a student should feel, know, and be able to do at each grade level in a Pre-K–12 program. Although national standards and state models provide a framework for teaching, a local course of study is generally written to prescribe the details of what students should be exploring each year they are in school. The course of study is essen-tially the contract between the school and the community in that it provides direction for the instructional process. Objectives within the course of study document vary from simple knowledge-level items to conceptual and higher-order application situa-tions. The wording of these objectives can also suggest the type of teaching methods that are expected in a district, including the use of hands-on materials and technology.
After the NCTM published Curriculum and Evaluation Standards for School Mathematics in 1989, funding from the National Science Foundation brought about the writing of several reform mathematics curricula at both the secondary
and middle school levels. These materials place the student at the center of the les-sons, using inquiry and a constructivist approach to teaching as described in Chap-ter 3 . Teaching units in these programs are rooted in real-life problems so that students can explore mathematical concepts in the context of problem solving. The materials emphasize the connections between content subjects and the use of tech-nology in problem solving. Details on these programs can be found by accessing the COMPASS and Show-Me Center Web sites or by contacting the publishers directly using information at the end of this chapter.
But, although the active, hands-on engagement of students in the learning process makes sense on paper, this type of teaching becomes possible only when the teacher has access to lessons and activities that support a more constructivist approach. And although a textbook can be a valuable tool for providing direction and serving as a source for problems, often many additional ideas are found in resource books and on the Internet. Ideally, the teacher will use the textbook as a general guide for instruc-tion but will supplement the book with a multitude of ideas from other sources.
With the release of Principles and Standards for School Mathematics in 2000, NCTM followed up by producing a series of resource books entitled Navigations, which provide educators with lessons and ideas that shed additional light on the meaning of the standards. Together with NCTM’s secondary and middle school journals and Web site, these resources are available to assist teachers in implement-ing the standards in their classrooms. The thoughtful collection and organization of ideas from all of these resources are important skills for an effective mathemat-ics teacher. The art of teaching is all about sharing ideas with one another so that we can use the experiences and successes of others in our own planning.
This chapter concludes our discussion of the mathematics curriculum. Chapter 4 explored the NCTM Principles and Standards for School Mathematics and the Com-mon Core State Standards, the use of state-level curriculum models, and the notion of a core curriculum. In this chapter, we discussed the issues of writing courses of study;
the use of goals and objectives in curriculum planning; and the selection, use, and organization of supplementary resource materials. In the four chapters that make up Unit 3, we turn our attention to the art of teaching mathematics. In Chapter 6 , we discuss how daily lesson plans and long-term unit plans are prepared to meet the goals and objectives set forth in a course of study, and Chapter 7 deals with organiza-tion of the classroom and the role of the mathematics teacher. In Chapters 8 and 9 , we look at some of the issues involved in teaching specifi c content areas, including number sense, algebra, geometry, statistics and probability, and discrete mathematics, in the secondary and middle school settings.
For of an outcome referring to attitudes or feelings that should be displayed by a student after experiencing a les-son, series of lessons, course, or mathematics program.
Bloom’s Taxonomy: Developed in 1956, this hierarchy describes six levels of increasing complexity of cogni-tion (thinking), which include knowledge, comprehen-sion, application, analysis, synthesis, and evaluation.
Other revisions of this taxonomy have been published since then. These levels should be considered when
writing a course of study as well as when designing classroom lessons and assessments.
Cognitive Objective: A cognitive objective is a statement of an outcome referring to skills and concepts the stu-dent should understand after experiencing a lesson, series of lessons, course, or mathematics program.
Cognitive objectives are often subcategorized as knowl-edge and skill, concept, and application-level outcomes.
Course of Study: A course of study is a document that prescribes the curriculum, by grade level, for a state,
county, or individual school district. It includes a dis-trict philosophy, overarching goals, a list of objectives for each grade level, and pupil performance objectives for mastery-level outcomes.
Curriculum Mapping: Curriculum mapping is the pro-cess of scheduling the sequence and timing for teaching major topics throughout a school year. The idea is to keep all teachers of a course or grade level to follow the same syllabus and pacing schedule.
District Philosophy: A district philosophy is a broad statement of beliefs held by educators in that system.
The philosophy should provide the underlying founda-tion on which specifi c objectives are written.
Goals: Goals are broad statements about what a student should be able to accomplish as a result of participat-ing in a district’s mathematics program. The goal state-ment should follow logically from the district’s philosophy and provide a framework for more specifi c grade-level objectives.
Objective: An objective is a very specifi c statement that describes what a student should feel, know, or be able to do at a particular grade level. Objectives are the intended outcomes of a lesson or series of lessons.
Objectives can be subcategorized as affective, cognitive, and psychomotor.
Pascal’s Triangle: Pascal’s Triangle is a triangle made up of progressively longer rows of numbers as shown in the following illustration:
The numbers in each row are generated by adding the two numbers directly above and to the right and left of the location. Pascal’s Triangle was named in honor of Blaise Pascal, a seventeenth-century French mathemati-cian, although there is evidence that the triangle existed long before Pascal’s lifetime. Patterns in Pascal’s Tri-angle are numerous as one views numbers vertically, horizontally, and diagonally. Determining binomial distributions, finding combinations, and locating famous number patterns, such as the triangular num-bers, are only a few of the uses of this valuable tool.
Psychomotor Objective: A psychomotor objective is a statement of an outcome referring to things that a stu-dent should be physically able to do after experiencing a lesson, series of lessons, course, or mathematics pro-gram. An example of a psychomotor objective is, “the student will be able to successfully do at least 10 con-secutive jumping jacks,” where the emphasis is on a physical activity required of the student. Psychomotor objectives are common in the areas of physical educa-tion and the arts and generally not associated directly with mathematics education.
Pupil Performance Objective (PPO) : A pupil perfor-mance objective is a specifi c description of what a stu-dent should be able to do at a particular grade level.
PPOs fl ow naturally from the objectives for a grade level, generally refl ect those objectives, and are often used to write classroom or districtwide assessment items, which may take the form of problems, questions, or projects. A PPO often contains a condition under which the student should perform as well as criteria that are used to determine the degree to which a stu-dent has mastered the outcome.
Unpacking: A process by which one specifi es what a stu-dent should know or be able to do to demonstrate mas-tery of a Standard. Unpacking is often part of the backward design process.
Discussion Questions and Activities
1. Obtain a copy of the mathematics course of study for a school district near you. Examine the document, looking for the philosophy, goals, objectives, and pupil performance objectives. How effectively does the document communicate to the teacher exactly what is to be taught at each grade level?
2. Discuss the potential advantages and disadvantages of including broad representation on a course of study writing committee. Why might a school district choose to have a curriculum supervisor and a small committee of mathematics teachers write the
docu-ment rather than select a larger representative com-mittee including administrators, guidance counselors, and community members?
3. One of the problems with including affective objec-tives in a course of study is that it can be diffi cult to assess the development of dispositions. Discuss some possible alternatives that teachers have for measuring affective outcomes in a lesson or throughout a course.
4. Suppose that you want students to become profi cient at working with square roots. Write three objectives involving the use of square roots—one at the knowledge
and skill level, one at the concept level, and one at the application level.
5. Divide the class into small groups and have each group write a pupil performance objective that might accom-pany each of the following content objectives if mas-tery of the outcome is expected: (a) The student will determine the arithmetic mean of a set of numbers. (b) The student will graph a linear function. (c) The stu-dent will classify quadrilaterals. (d) The stustu-dent will fi nd the zeros of a polynomial function. Compare objectives and discuss the variety of ways in which an individual can interpret a given cognitive objective.
6. Obtain two available textbooks for a particular course or grade level. Using the questions and criteria described in this chapter, prepare a criticism of each book and a comparison that would allow an educator to select one text over the other.
7. Obtain copies of several resource books, such as the NCTM Navigations Series or other commercially available books of teaching ideas. How are the books
organized, and what features might make one resource book more desirable to the classroom teacher than another?
8. Using a computer with Internet access and a search engine such as Google, run a search for teaching ideas on the mathematical topic of your choice. Then dis-cuss the diffi culties that may have confronted you while running the search and the practicality of using the Internet to fi nd teaching ideas.
9. In a small group, discuss the options for organizing a resource fi le listed in this chapter. Which one appeals the most to you and why? What other ideas do you have for organizing resources?
10. Obtain a copy of one of the NSF-funded curriculum materials described in this chapter. Browse through the text materials and discuss the similarities and dif-ferences between this curriculum and a more tradi-tional curriculum that you may have experienced.
What are the benefi ts and possible drawbacks to using the NSF-funded curricula?