Teaching Secondary and Middle School Mathematics

Teaching Secondary and Middle School Mathematics
4e
ISBN 978-1-29204-206-0
9 781292 042060
Teaching Secondary and Middle
School Mathematics
Daniel J. Brahier
Fourth Edition
Pearson Education Limited
Edinburgh Gate
Harlow
Essex CM20 2JE
England and Associated Companies throughout the world
Visit us on the World Wide Web at: www.pearsoned.co.uk
© Pearson Education Limited 2014
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the
prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom
issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS.
All trademarks used herein are the property of their respective owners. The use of any trademark
in this text does not vest in the author or publisher any trademark ownership rights in such
trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this
book by such owners.
ISBN 10: 1-292-04206-0
ISBN 10: 1-269-37450-8
ISBN 13: 978-1-292-04206-0
ISBN 13: 978-1-269-37450-7
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Printed in the United States of America
Implementing a Course of Study
ideas, problems of the week, links to other mathematics Web sites, and even a link to
Ask Dr. Math, that allows students to leave questions to be answered by a mathematician. In fact, the future of teacher resources may very well reside with the Internet, as
more materials are being made available every day and are easy to access with the
click of a mouse. Some professional organizations have already begun to make their
journals available on-line.
Organizing a Resource File
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 useful method for collecting and organizing resource materials. After all, what good is an
exemplary hands-on activity if you can’t find it on the day that you want to use it in
class? Here are four practical suggestions for organizing a resource file:
Obtain a box of file folders. Each time that you find a good problem, activity,
or lesson, place a master copy in one folder and title the folder with a descriptor, such as “Grains of Rice” or “Orange Grove,” on the label. Then
organize the file 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 file 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 filing scheme that works for you. The
grains-of-rice problem, for example, could just as easily be filed under Algebra because the solution can include an analysis of exponential functions.
Consequently, some teachers file all of the problems and activities alphabetically. However you choose to organize the resources, it is important to place
only one idea in each file folder. Otherwise, you may spend a great deal of
time rummaging through a file to find 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 file folder,
place the idea in the folder, and file it in an appropriate location.
Obtain several three-ring binders and a plastic storage tub or a small file
cabinet. Label each binder for a particular content area, such as the conceptual categories in the Common Core State Standards—number and quantity, 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 file 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 threehole-punched pages. For example, if you decide to create a new binder on
problem solving that includes some ideas you have already collected, you
123
Implementing a Course of Study
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 file 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 file it in a file 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 file of all of the teaching ideas on paper 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 locate a useful idea. Then go to the master file of ideas and pull it out for duplication 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 flipping
through binders filled with papers.
Scan your favorite lesson ideas (or save them in PDF format) and put them
on a flash drive or burn them onto a CD-ROM. The flash drive or CD-ROM
can serve as an electronic file from which lessons and activity pages can be
readily retrieved or printed. With a flash drive, each time a new idea is located, 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 file 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 access, without having to physically take a CD or flash 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 problems, 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 specific 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 essentially 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 situations. 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
124
Implementing a Course of Study
and middle school levels. These materials place the student at the center of the lessons, using inquiry and a constructivist approach to teaching. 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 technology 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 instruction 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 implementing the standards in their classrooms. The thoughtful collection and organization
of ideas from all of these resources are important skills for an effective mathematics 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.
Glossary
Affective Objective: An affective objective is a statement
of an outcome referring to attitudes or feelings that
should be displayed by a student after experiencing a lesson, series of lessons, course, or mathematics program.
Bloom’s Taxonomy: Developed in 1956, this hierarchy
describes six levels of increasing complexity of cognition (thinking), which include knowledge, comprehension, 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 student should understand after experiencing a lesson,
series of lessons, course, or mathematics program.
Cognitive objectives are often subcategorized as knowledge 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,
125
Implementing a Course of Study
county, or individual school district. It includes a district 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 process 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 foundation on which specific objectives are written.
Goals: Goals are broad statements about what a student
should be able to accomplish as a result of participating in a district’s mathematics program. The goal statement should follow logically from the district’s
philosophy and provide a framework for more specific
grade-level objectives.
Objective: An objective is a very specific 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 mathematician, although there is evidence that the triangle existed
long before Pascal’s lifetime. Patterns in Pascal’s Triangle are numerous as one views numbers vertically,
horizontally, and diagonally. Determining binomial
distributions, finding combinations, and locating
famous number patterns, such as the triangular numbers, 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 student should be physically able to do after experiencing
a lesson, series of lessons, course, or mathematics program. An example of a psychomotor objective is, “the
student will be able to successfully do at least 10 consecutive jumping jacks,” where the emphasis is on a
physical activity required of the student. Psychomotor
objectives are common in the areas of physical education and the arts and generally not associated directly
with mathematics education.
Pupil Performance Objective (PPO) : A pupil performance objective is a specific description of what a student should be able to do at a particular grade level.
PPOs flow naturally from the objectives for a grade
level, generally reflect 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 student has mastered the outcome.
Unpacking: A process by which one specifies what a student should know or be able to do to demonstrate mastery 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-
126
ment rather than select a larger representative committee including administrators, guidance counselors,
and community members?
3. One of the problems with including affective objectives in a course of study is that it can be difficult 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 proficient at
working with square roots. Write three objectives
involving the use of square roots—one at the knowledge
Implementing a Course of Study
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 accompany each of the following content objectives if mastery 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 student will classify quadrilaterals. (d) The student will
find 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 discuss the difficulties that may have confronted you
while running the search and the practicality of using
the Internet to find teaching ideas.
9. In a small group, discuss the options for organizing a
resource file 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 differences between this curriculum and a more traditional curriculum that you may have experienced.
What are the benefits and possible drawbacks to
using the NSF-funded curricula?
bibliographic references and resources
Anderson, L. W. (Ed.), Krathwohl, D. R. (Ed.), Airasian,
P. W., Cruikshank, K. A., Mayer, R. E., Pintrich, P. R.,
Raths, J., & Wittrock, M. C. (2001). A taxonomy for
learning, teaching, and assessing: A revision of Bloom’s
taxonomy of educational objectives (Complete edition).
New York, NY: Longman.
Ball, D., & Cohen, D. (1996). Reform by the book: What
is—or what might be—the role of curriculum materials
in teacher learning and instructional reform? Educational
Researcher, 25 (9), 6–8, 14.
Bloom, B. S., & Krathwohl, D. R. (1956). Taxonomy of
educational objectives: The classification of educational
goals. Handbook I: Cognitive domain. New York: Longmans, Green.
Common Core State Standards Initiative. (2010). Common
core state standards for mathematics. Washington, DC:
National Governors Association Center for Best Practices
and Council of Chief State School Officers.
Coxford, A. F., et al. (1998). Contemporary mathematics in
context: A unified approach (Course 1, Part A). Chicago:
Everyday Learning.
Hamilton, B. (2011). No reason to fear the common core
standards. Retrieved July 1, 2011, from the World Wide
Web: http://www.insidetheschool.com/articles/noreason-to-fear-the-common-core-standards
Hart, E. W., Kenney, M. J., DeBellis, V. A., & Rosenstein,
J. G. (2008). Navigating through discrete mathematics in
grades 6–12. Reston, VA: National Council of Teachers
of Mathematics.
Kenney, M. J. (Ed.). (1991). 1991 Yearbook of the NCTM:
Discrete mathematics across the curriculum, K–12. Reston, VA: National Council of Teachers of Mathematics.
Krathwohl, D. R. (2002). A revision of Bloom’s taxonomy:
An overview. Theory into Practice 41 (4), 212–218.
Marzano, R. J. (2001). Designing a new taxonomy of educational objectives. Thousand Oaks, CA: Corwin Press.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston,
VA: National Council of Teachers of Mathematics.
Pennsylvania Department of Education. (2011). Standards
aligned system. Retrieved July 1, 2011, from the World
Wide Web: http://www.pdesas.org/module/sas/standards/anchors/unpack/#search
Rosenstein, J. G., Franzblau, D. S., & Roberts, F. S. (Eds.).
(1998). Discrete mathematics in the schools. Reston, VA:
National Council of Teachers of Mathematics.
Schwartzman, S. (1996). The words of mathematics: An
etymological dictionary of mathematical terms used in
English. Washington, DC: The Mathematical Association of America.
Stiggins, R. J., Rubel, E., & Quellmalz, E. (1988). Measuring thinking skills in the classroom. Washington, DC:
National Education Association.
Wiggins, G., & McTighe, J. (2006). Understanding by design: A framework for developing curricular development and assessment. Alexandria, VA: Association for
Supervision and Curriculum Development.
photo Credit
Daniel J. Brahier
127