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Teaching Strategies for “Algebra for All”

James R. Choike

ducators and policymakers alike recognize thatalgebra is an important gatekeeper course, not onlyfor college preparation but also for preparation forthe world of work. To prepare students for futuresuccess, many school districts and state legislaturesnow make algebra a graduation requirement for allhigh school students. Of course, the burden forpreparing all students in algebra falls on the shoul-ders of the classroom teacher. Whether an algebra-for-all initiative is school-, district-, or state-based,a teacher faces the difficult challenge of teaching ahigh-standards course to a classroom of studentswhose beginning knowledge may range from farbelow to far above the course prerequisites.

Since 1990 I have been actively involved in theCollege Board’s algebra-and-geometry-for-allschool-reform project called Equity 2000. Equity2000 is an educational reform initiative for gradesK–12 that is based on the Curriculum and Evalua-tion Standards for School Mathematics (NCTM1989). It uses mathematics—algebra and geometryby the tenth grade—as a lever to enhance academicachievement for all students, especially minoritystudents, thereby increasing the likelihood thatthey will enroll in and complete college. My workfor Equity 2000 consists of serving as chair of theEquity 2000 National Mathematics Technical Assis-tance Committee and as an in-service presenter atEquity 2000 professional development institutes formathematics teachers. This article shares somelessons that I have learned from my Equity 2000experiences.

The article presents classroom strategies thathelp me in my own teaching at the university leveland in in-service presentations. These strategieshave evolved over time. They have been the resultof a gradual personal professional developmentstemming from the challenges that I have faced inbeing asked by the College Board to lead Equity2000 teachers by offering insights and strategies tohelp them reach all students. They have also beenformed from working with teachers and students inEquity 2000. In this leadership role, I have workedtoward preparing algebra-for-all exemplar class-room materials for Equity 2000 in-service presenta-

tions to teachers and have demonstrated theselessons for teachers by modeling their use and ped-agogy with actual classes of students that are typi-cal of those that they are likely to face within theirdistricts.

For example, in 1992, I teamed with J. T. Sutcliffe,a mathematics teacher at Saint Mark’s School ofTexas, to create and demonstrate five algebra-for-alllessons with a group of thirty-two eighth gradersfrom Prince George’s County, Maryland. Subse-quently, in 1993, I teamed with Rheta Rubensteinof Schoolcraft College, in Livonia, Michigan, in cre-ating and demonstrating lessons in geometry withthirty tenth-grade students in an Equity 2000 Sum-mer Math Institute in Milwaukee.

Each algebra-for-all teaching strategy includes abrief explanation of the nature and scope of thesuggested strategy. Although each one has generalapplication to teaching mathematics, the strategiesare presented in the setting of teaching algebra.

Emphasize conceptual understanding, that is, emphasize “big ideas”A daunting task that teachers face throughout theschool year is “getting through the book.” The tableof contents of most algebra textbooks compartmen-talizes algebra into many chapters and many top-ics within each chapter. But algebra essentiallyinvolves just a few conceptual themes, or “bigideas.” Focusing instruction on a few big ideasaccomplishes two important instructional goals:(1) it helps all students understand how a topicthat they are currently learning connects withmaterial previously learned; and (2) it gives a con-ceptual framework for teaching a high-standardscourse without being bound to follow the textbookone section after another. Selected big ideas areillustrated throughout the rest of this article.

Jim Choike, choike@math.okstate.edu, teaches at OklahomaState University, Stillwater, OK 74078. He is interested inpedagogy, curriculum development, technology, and profes-sional development for teachers in algebra, geometry,Advanced Placement calculus, and collegiate mathematics.

E

MATHEMATICS TEACHER

Focusinstruction

on a few big ideas

Copyright © 2000 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

Vol. 93, No. 7 • October 2000 557557557

Eliminate numbers as distracters to understanding new conceptsConsider the following problem.

A department store has a Surround Sound/CDplayer system on sale for a down payment of$13.80 and a monthly payment of $22.69. a) Write an algebraic expression that allows you

to compute the amount paid after any pay-ment number.

b) How much will you have paid for the CD sys-tem after the twenty-fourth payment?

This problem has a lot to offer algebra students.Unfortunately, the reality-based amounts of $13.80for the down payment and $22.69 for the monthlypayment may be more than some students’ existingskills can handle. Such numbers as 13.80 and 22.69can actually distract students from understandingsimple concepts that underlie the solution of aproblem. As a preliminary step to solving the prob-lem, eliminate the decimals. For example, have stu-dents solve a similar problem in which the downpayment is $5 and the monthly payment is $10.With these “friendly” numbers, more students arelikely to build a table of cost in terms of paymentnumber and to recognize the pattern in this tableboth in words and as an algebraic expression, asrequested in part (a) of the problem. This prelimi-nary problem also offers students insights abouthow they can solve the original problem.

Ensure entry into a problem or activityIn other words, questions of clarification from yourstudents should always be permitted. Consider thefollowing problem, which appeared as the firstproblem in Addison-Wesley Mathematics: Grade 6(Eicholz et al. 1991) in a section entitled “UsingWhole Number Division to Solve Problems.”

A golden eagle takes about 56 hours to hatch. Itmay take the chick 3 times as long to break outof the shell as to peck the first hole. How longdoes it take the chick to peck the first hole?

This problem includes three ambiguous elementsthat are not inherently mathematical, that confusestudents, and that prevent them from getting start-ed on, that is, entering, the problem solution. Theseelements are as follows:

• Eagle versus chick: students may be unclearabout the relationship between the golden eagleand a baby chick. If the problem statement hadused the words baby eagle instead of chick, thisrelationship would then be clear to students.

• The use of the words about, as in “about 56 hoursto hatch,” and may, in “may take the chick,”introduce an element of uncertainty to some stu-

dents about the numbers to use in this problem.Compare these wordings with the wording in thequestion statement of the problem, which verydefinitely asks, “How long does it take the chickto peck the first hole?”

• The meaning of hatching is somewhat ambigu-ous. Does hatching begin when the chick startsto peck the hole (this interpretation leads to anequation x + 3x = 56) or when the first sign of acrack appears to an outside observer (this inter-pretation leads to the equation 3x = 56)?

The golden-eagle problem is an example of a prob-lem in which a student’s inability to solve it mayhave no direct connection with his or her level ofmathematical knowledge. Both the strong studentand the weak student may have trouble with thisproblem. Providing entry to a mathematical activi-ty or a problem is a strategy that ensures that allstudents understand the setting of a problem.

Emphasize multiple representation: words, tables of data, graphs, and symbolsMultiple representation is a content “big idea” ofalgebra. Students should be taught the value ofrepresenting mathematics verbally in words,numerically in tables, visually in graphs, and alge-braically in symbols and should learn how thesevarious representational forms of mathematics areconnected.

Multiple representations also help reach the goalof algebra for all. A teaching strategy that connectsthe various forms of multiple representations todescribe mathematics is an effective strategy forreaching out to students with different learningstyles. In this sense, multiple representation pro-vides an algebra-for-all pedagogy strategy in theclassroom.

Multiple representation is also a problem-solvingstrategy. Using the full variety and power of all theforms of describing and presenting mathematics isan effective means for gaining insight into, andunderstanding of, a problem situation. In thissense, multiple representation is a problem-solvingprocess.

As an example of the power of multiple represen-tation, consider the following problem:

List the ways to change a fifty-dollar bill intofive-dollar and/or twenty-dollar bills.

A straightforward, numerical trial-and-errorapproach by students yields the following threeways to change a fifty-dollar bill: (1) two 5s and two20s, (2) six 5s and one 20, and (3) ten 5s and zero20s.

By placing the algebraic equation 5x + 20y = 50on the chalkboard, then asking students to explainthe meaning of the variables x and y in this equa-

Clarificationquestionsshouldalways beencouraged

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tion and the relationship of the equation to theoriginal problem, the teacher can enrich students’understanding of the concept of variables.

Next, by incorporating a coordinate graph to rep-resent their numerical solutions, students can seethat all their numerical solutions lie on the sameline; that 5x + 20y = 50 describes all points that lieon this line; and that a point on this line, namely,(–2, 3), gives a surprising but quite reasonablemethod of making change for a fifty-dollar bill. Seefigure 1.

tree. In this simple setting, students can investi-gate, through multiple representations, patterns inwords, tables, symbols, and graphs. By having stu-dents link these representations with one another,they experience a rich lesson in many of the bigideas of algebra.

When the class is familiar with this setting, useit again but with an additional twist: give the snaila nine-foot head start on the turtle. Students canthen investigate how the various multiple represen-tations have changed. This added condition raisesthe question of which critter reaches the elm treefirst. The question can be solved using tables, set-ting up and solving an algebraic equation, or locat-ing an intersection point of graphs. This multiple-representation problem-solving approach extendsand deepens students’ understanding of importantconcepts of algebra. In addition, the class timeneeded to cover these higher-level topics is reduced,since students are already familiar with the setting.

An additional variation on this setting is to startthe turtle walking from the oak and the snailcrawling from the elm toward each other. Again,students can investigate how the various multiplerepresentations have changed as they attempt tosolve where and when the critters meet. This twistintroduces students to decreasing distance relativeto the oak tree, negative rate of change, setting upand solving an algebraic equation, and intersectinggraphs as a solution to an equation.

Finally, be alert for situations that are popularwith students; such situations are often related tothe latest school fads. Embedding a mathematicalactivity within a contextual setting that is linked toa school craze, fad, or even a school event is aneffective way to engage students’ interest in mathe-matics and its role in their lives.

Do not begin the year by remediatingWith each new class, every teacher faces studentswhose mathematical knowledge is widely divergent.How can a teacher carry out her or his responsibili-ty to effectively teach all students so that all reachhigh levels of learning by the end of the year? Thisperennial question has no simple answer.

However, planning a remediation effort for thefirst two to three weeks of the school year is fatedto fail for several reasons. How can you decide whatto review before you really know the deficienciesthat your students have? Some may need a reviewof basic skills; but others may be deficient in suchother prerequisites as graphing skills, proportionalreasoning, the concept of a fraction, the concept of avariable, and number sense. Also, spending two orthree weeks on review and remediation wastesvaluable class time, and at the end of that time,students who had deficiencies in prerequisiteknowledge continue to have them.

x

y 5x + 20y = 50

(10, 0)

(6, 1)(2, 2)(–6, 4)

(–2, 3)

Fig. 1A graphical representation ofthe “making change” problem

Students should be invited to describe in words,in writing or orally in class, a scenario for changinga fifty-dollar bill that matches the coordinate pair(–2, 3). Notice how a problem that started out havinga simple numerical-only solution has been enrichedby the application of multiple representation.

Revisit rich or popular settingsUsing or revisiting a familiar setting is a powerfulalgebra-for-all teaching strategy. In general, a newsetting of a word problem can intimidate students.Students may not understand the meanings ofsome words, or situations may not be related totheir everyday experiences or their culture. Whennew problem settings are presented to students, bealert for any signs that indicate that barriers arepreventing students from entering or engagingmathematically with the problem. After class timehas been invested to familiarize students with aproblem setting, use the setting again, especially ifit can be modified to enrich and extend their under-standing of mathematics.

As an illustration, consider the following setting.A turtle walks at five feet per minute, and a snailcrawls at three feet per minute. The turtle and thesnail start from an oak tree and head toward anelm tree that is located thirty feet from the oak

Revisit a setting

familiar tostudents

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Instead of starting the school year with remedia-tion activities, begin immediately with the curricu-lum that defines the course. Indeed, some studentsin your classroom will have deficiencies. But use thepreviously described “friendly” numbers that havebeen chosen to be within reach of their skill levels toinvolve them in activities that emphasize the bigideas of algebra and algebraic thinking. Then, asyou introduce students to aspects of the curriculum,fold into your lessons any review of prerequisitesthat individual students may need. Thus, the strate-gy is to tailor remediation to the needs of your stu-dents without sacrificing your curriculum objectives.

Another tip for helping students in need ofremediation is to use the first five to ten minutes ofeach class—a time when not much can be accom-plished because of such activities as studentschanging classes, roll taking, and other classroom-management duties—to place a warm-up problemor activity on the chalkboard. Choose these activi-ties carefully for their ability to incorporate neededreview work in an interesting and conceptual way.As an example, consider the following as a warm-up problem.

The value of the area of the square in figure 2is 1. What are the values of the areas of each ofthe pieces of the square?

any area value can be assigned to any piece of thissquare to construct a new problem that is based onthis figure and the concept of fraction.

Involve students in guided explorations; uselearning-by-discovery teaching strategiesTo capture the essence of algebra, I like to definealgebra as follows:

Algebra is the process of organizing the arith-metic needed to find an answer to a questioninvolving quantities that are not yet known.

According to this definition, algebra is a natural ex-tension of arithmetic. Thinking of algebra as a cur-riculum strand for grades K–12 makes sense. Alge-bra is not just a list of topics in a textbook. Algebrais a process that is called algebraic thinking. Noticealso that everyone needs algebra, because as peoplegrow older, their quantitative questions becomemore complex by involving quantities that are notyet known, whether or not they have taken algebrain school. The reader is invited to compare this defi-nition with three others that were showcased in theJuly/August 1997 issue of the NCTM NewsBulletin.

Guided explorations or learning-by-discoveryteaching strategies actively involve students in thisprocess of organizing their arithmetic to findanswers to questions. For example, when the previ-ously described turtle-and-snail setting is present-ed as a guided-exploration lesson, students areinvolved in algebraic thinking and investigating arich panorama of big ideas of algebra: arithmetic-sequence patterns; such multiple representationsas data in tables, in graphs, in symbols, and inwords; rate of change; slope; y-intercept; linearfunctions; evaluating functions; setting up and solv-ing linear equations; and solving simultaneousequations in such multirepresentational ways astables, graphs, and symbols. These approaches aremore likely to keep students engaged and hence, toencourage them to construct a personal under-standing of the mathematics.

Learn to recognize correct thinking in students even when it may be incomplete orlacking in closureIn other words, be a good listener and be as alert toyour students as you are to what you are teaching.A vignette about a mathematics teacher, whom wewill call Mr. Relentless, helps illustrate this algebra-for-all strategy. This vignette is based on a realclassroom event.

Mr. Relentless gave his class the following prob-lem on the big idea of proportionality: “Frank’scar travels twenty-five miles in thirty minutes.What is the car’s average speed?”

Fig. 2A square with area 1

A BC

D

E F

G

HI

JK

L

Although problems similar to the one above havebeen in the literature for a long time, they exempli-fy problems that are very conceptual and rich incontent. The problem deals with proportionality,that is, the concept of a fraction, and with suchelements of geometry as shapes and area. It alsopermits a variety of solution approaches that canlead to rich class discussions. Finally, this warm-upactivity has tremendous potential for follow-upactivities. For example, suppose that the area of thetriangle labeled D is 1. What is the value of thearea of each of the other pieces? As a matter of fact,

Be a good listener

�

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The answer that Mr. Relentless expected tothis question was fifty miles per hour.

In class, he called on Kisha to give her answerto his question. Kisha responded, “The averagespeed of Frank’s car is 5/6 mile per minute.” Mr.Relentless brushed Kisha’s answer aside as heasked whether anyone else in class could givethe correct answer to this problem.

Did Kisha understand proportionality andrate of change?

Be a teacher who listens carefully and who val-ues diversity of thinking. Be a teacher who looks forunderstanding of the big ideas rather than just ananswer that conforms exactly to an answer key.

Mold lessons, whenever possible, around the interests of individual studentsThe following situation actually occurred in aninth-grade algebra class in a high school in aneastern urban inner city. Students in this ninth-grade class were presented with the following performance-based guided-exploration task.

A farmer had 24 yards of fencing. What are thedimensions of a rectangular pen that gives hissheep maximum grazing area?

For this task, the students were equipped withgraphing calculators and were assigned to work ingroups of three.

As an entry activity, the teacher had studentsuse coordinate grid paper to draw sample rectangu-lar pens that met the conditions in the problemstatement. The students were enthusiasticallyinvolved in this problem and were busily drawingsample rectangular pens on grid paper and noticingthat the pens could be constructed in many ways.All students were engaged, that is, except for oneyoung lady who had quietly detached herself fromher group. She preferred, instead, to focus on hergrooming. She fluffed her hair and checked her eyeshadow and her nail polish. When asked why shewas not interested in the problem, she commentedrespectfully, yet honestly, that she could not careless about sheep in a pen. As an inner-city youth,she was stating that the problem setting had norelationship to her world.

Could this problem be changed so that it wouldbe relevant to this young lady? Faced with thisquestion, the teacher discovered that the studentliked flowers, in particular, red roses. Homing in onthis volunteered interest, the teacher restated theproblem for Maria as follows:

Maria has 24 yards of fencing, which she can useto make a rectangular garden for growing redroses. Each rosebush needs space to grow. A one-yard-by-one-yard square gives a rosebush suffi-cient growing room. What dimensions will allow

her to plant the maximum number of rosebushesin this rectangular garden?

With this new setting, tailored to Maria’s inter-est in roses, Maria smiled delightedly at theteacher and became involved in the activity.

Establish a safe classroom environmentA safe classroom environment is one in which ateacher establishes that—

• everyone in class, including the teacher, is partof a community of learners;

• taking risks on the part of students, by askingquestions, volunteering answers, and so on, isencouraged and respected;

• different solution methods and different ways ofthinking about problems and mathematics arevalued and give the teacher instructional oppor-tunities; and

• collaboration with classmates is valued.

CONCLUSIONIn closing, consider for a moment the followingproblem.

Mr. Relentless, a mathematics teacher, is plan-ning his algebra course for the school year. Hehas 180 days to cover the new textbook. The bookis 983 pages long. How many pages, on the aver-age, must he cover each class day to complete thecourse by the end of the year?

An answer to this problem that sums up what itmeans to be an algebra-for-all teacher is that algebra-for-all teachers teach students, not pages.

BIBLIOGRAPHYEicholz, Robert E., Phares G. O’Daffer, Randall I.

Charles, Sharon L. Young, Carne S. Barnett, andCharles R. Fleenor. Addison-Wesley Mathematics:Grade 6. Menlo Park, Calif.: Addison-Wesley Pub-lishing Co., 1991.

National Council of Teachers of Mathematics (NCTM).Curriculum and Evaluation Standards for SchoolMathematics. Reston, Va.: NCTM, 1989.

———. Principles and Standards for School Mathe-matics. Reston, Va.: NCTM, 2000.

“What Is Algebra.” NCTM News Bulletin 34(July/August 1997): 5. ¿

Algebra-for-all teachers

teach students, not pages

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