Developmental Math: Problem Solving and Survival

Developmental Math: Problem Solving and Survival Author(s): ANN M. CHISKO Source: The Mathematics Teacher, Vol. 78, No. 8 (NOVEMBER 1985), pp. 592-596 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/27964660 . Accessed: 12/07/2014 17:42 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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Math

Developmental Problem Solving of M. By ANN

CHISKO,

University

and

courses.

These

courses

cover

pre

or beginning and inter algebra mediate topics commonly offered algebraic school levels. at the middle and secondary are re courses these At the college level, or developmental, ferred to as remedial, cur courses. A traditional mathematics arithmetic

riculum stresses

in the Since

developmental of mastery

Survival

Cincinnati, Cincinnati, OH 45236

students enroll in such college classes as calculus, mathematics level Before or and trig statistics, algebra precalculus to take need prelimi onometry, they may nary

:

mathematics

computational are skills skills. problem-solving in such a course, students often neglected ac courses without leave developmental skills for survival the necessary quiring courses. de more This article advanced that I use in the scribes some techniques to encourage the use of analyti classroom their problem and cal skills compares more a to traditional emphasis solving a approach. Although computational-skills course is college developmental prealgebra, described, many of the ideas are appropri ate for any level of mathematics. In the 1970s the mastery-learning ap was method of the teaching primary proach classes. The arithmetic and developmental into down broken skills were algebraic

units, and the theory small, manageable was that studemts could start at their level of competence, step through these units,

and arrive at the threshold of algebra, cal the postdevelopmental culus, or whatever a student this approach level was. With all the computational might indeed "learn" skills that are required for a more advanced course, but something extremely important would not be missing. Connections would

be made, analysis would not be done. on computational skill The emphasis learning was an building through mastery move to basics" of the "back outgrowth

ment and a noble attempt at remedying stu dents' lack of mathematical skills. How educators are now ever, many mathematics the need for students to under stressing stand mathematical concepts, and the em

is shifting from specific content to phasis processes of mathematical thinking. Rather

Mastery learning did not always lead to understanding concepts.

to answer students posed training we stu instead encourage questions, might dents to ask those questions that can be answered. For students to ask, and eventu reasonable ally answer, questions, they must be able to arrange and analyze infor mation. thinking is necessary. Systematic to Whimbey and Loch Indeed, according head (1982), the ability to think systemati unsuc cally is what most mathematically than

cessful students lack. of Teachers The National Council of Mathematics in An (1980), Agenda for Action: Recommendations for School Math ematics of the 1980s, lists as its first rec ommendation that problem solving be the focus of school mathematics. To stress prob lem solving in developmental classes, we first need to identify the attributes of good

problem solvers. A Russian educator, V. A. a Krutetskii (Piemonte 1982), analyzed of solvers. He group competent problem

found that in addition to high IQs and good

solvers skills, competent problem reading mathematical and comprehend concepts terms ; are aware of similarities, differences, and analogies; have the ability to estimate are flexible in switching and to generalize; methods ; and have low levels of anxiety. A

592

Mathematics

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Teacher

1979 study by Silver (Piemonte 1982) found that good problem solvers emphasize the structure of a problem rather than its de tails. Identifying some of the stages of prob

lem solving is also important. In his classic to Solve It, Polya How a (1945) provides understand the problem, four-stage model: devise a plan, carry out the plan, and look back

at the problem.

This method

involves Lesh stages. non-answer-giving (1981) claims that non-answer-giving stages are the most important for good problem solvers. Th?y assess the difficulty of the several

the time and resources problem, estimate to solve needed it, and test information of the problem. Mod verbalization through of the then follows. eling problem

A new approach emphasizes the promotion of positive attitudes, students'

encouragement involvement,

of and a

focus on survival skills in

nondevelopmental

courses.

After identifying the traits of good prob the non and lem solvers recognizing we of stages solving, problem answer-giving can devise ways to help developmental stu skills. Instead dents build problem-solving or "thought" of devising "word" problems that require answers, we can devise exer

infor cises that ask students to organize skills that might be mation and predict skills and than presenting needed. Rather in which students then posing "problems" use these skills, we can start with a prob it suggest the skills that lem and have stu to it. solve Perhaps might be needed to have dents' "getting the answer" might a level of mathematical wait for higher competency or for a computer printout, but that are the student can plan the processes needed. Problems are, after all, situations is known; for which no model of solution once a model the is developed, "getting is merely an exercise. answer" November

A New Developmental

Format

for

Mathematics

Analytical skills, lacking in many college students, can be developed through several in developmental mathematics techniques courses. The format that I use stresses three areas :developing a positive attitude toward on activeness mathematics, encouraging the part of the students, and providing sur courses vival skills for nondevelopmental that encourage the practice of problem

and skills. analytical Compu skill building is done, but the em of phasis is not so much on the mechanics solution as on the development of a plan or method of solution. solving tational

Developing Since

a positive

attitude

one

of the characteristics of good a anx solvers is low level math of problem a I have math my students complete iety, ematics attitude survey during the first or second class. Some tension can serve as a to perform well; motivator if a however, student has the debilitating of tension type with math anxiety, it will block associated learning, and efforts aimed at skill build or analytical, ing, whether computational

will be fruitless. The attitude survey asks students to respond to such statements as is creative" "Math and "Good math stu of dents do problems quickly." A discussion these attitudes, coupled with an exchange

in of both positive and negative experiences mathematics, helps students see that they are not alone in their fears. After dis cussing how attitudes affect our approach are to mathematical students situations, with of positive and examples provided statements that they might make negative a nega about mathematics. For example, tive statement would be "I can't get it; I never was good at math." A parallel posi tive statement would be "I don't know how to do this right now, but I will find out how to do it." Many students in developmental with a cer classes approach mathematics tain degree of anxiety. Since they have not been successful with it in the past, they are fearful that they will never be. These exer stu in helping cises can be instrumental how their attitudes may dents understand -593

1985

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block skills

them from acquiring mathematical in giving and them a means of and the of effects those changing reducing

attitudes. Student

involvement

After attitudes toward mathematics have been explored and students have begun to in the classroom, I attempt to participate what is most the develop single important attribute of a good student in any subject:

Students

are

encouraged

to

take control of their learning through group work

and

class

discussions.

in the Students involvement, or activeness. cannot be passive mathematics classroom will only be learned learners; mathematics it. To encourage their involve by doing we must let students articulate and ment, are communicate what and they thinking doing. We have often heard it said that one we teach learns what one teaches. When we are to forced express verbally something what we know. To express something verb the material and ally, we must organize areas. must We any clarify "fuzzy" identify

the essential components of the knowledge. We must take control of the situation. For students to understand this verbalization and control of material, they must have the of mathematics. As I precise vocabulary start each new unit, I give the students a

list of vocabulary words that are defined as we go through the unit. Students are en couraged to add any words that they do not know. Later the words are included on the unit test. When we define the terms, stu dents can see that, for example, in math ematics the word some operation means thing different than it does in everyday life. The precision and unambiguity of a math can be stressed. ematics vocabulary I also encourage students to read prob lems aloud, even ones that are not word such as evaluating problems. A problem = 4 should + for not be ap (3x 7)/5

proached visually but rather orally because the student will be forced to state what op erations need to be formed. stu As well as verbalizing the material, dents can express what they are doing in I use is to ask written form. One technique in writing what they students to describe are doing in a problem and why. Often stu dents discover that what they think is a valid method does not make sense ; this dis to the them more receptive covery makes valid methods of solution. I encourage group work and discussion in the classroom. We often work on word problems in dyads or triads. We begin early in the course with problems from Whimbey

and Lochhead's Problem Solving and Com (1980). This book illustrates how prehension solvers approach problems good problem with both mental and physical activeness. nonmathematical Many problems are pre so can students work on analytical sented, a skills without having high level of math ematical expertise. As students become ac in group work, they are tively involved more willing to question and participate during more traditional lecture classes.

in that encourages technique is to have volvement students generate their own mathematics problems. For exam ple, students can share problems found in or problems that newspapers they en stu counter in their daily lives. Having dents make up their own problems from a story can also be effective. In one story that Another

problem Any mathematical some requires degree of and analysis thought. I use

in class, a student visits a friend's to pick up some books he had stored Information about during college. and costs is in temperature, time, mileage, the story. Students terspersed throughout are asked to make up as many problems as they like and try to solve them if they can. The subsequent class discussion of these is lively; the interest in them is problems home there

594

Mathematics

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Teacher

students have created them. high because This exercise, which can be used for stu stu dents at virtually any level, encourages dents to use written, verbal, and logical skills. As we discuss the problems, specific

skills, laws, and algorithms computational are reviewed. At the same time analytical elements, such as the importance of making and knowing when sufficient assumptions data are present to solve the problems, can

For example, the student in be addressed. the story packed forty books in five boxes. asked how many books were packed When answer in each box, students generally soon questions someone eight. However, whether all the boxes and all the books were the same size. From this easily under stood example, the students can begin a pre about statistics, includ liminary discussion the of and shortcomings ing implications their use. Survival

1. Decide

what

problems and and analysis mathematical steps :

to do.

2. Doit. 3. Decide

whether

the answer

is reason

able. If the symbols "1 + 2" are written on the board and students are asked, "What do these symbols tell me to do?" more often than not the answer will be "3." Students on step 2, the "doing it" part! not surprise us that students are since concerned with "getting the answer," in our edu this aspect has been emphasized cational system. Steps 1 and 3 stress the and creative of math aspects analytical students ematics. It is difficult to convince concentrate It should

skills

Involvement is the single most important of a good student. The very en of involvement, then, is one of couragement

attribute

in the goals of the survival-skills approach survival classes. Other nondevelopmental however. skills must also be developed, for nondevelopmental Readiness classes cannot be seen as only filling a skills gap; students must also be prepared for the rea classes. In these lities of postdevelopmental to the students will be expected classes, know how to take notes, to take a variety of text, to be flex tests, to read a mathematics of solution, and to work ible in methods skills steps. The problems with multiple in classes these survival for necessary the skills of good, active problem parallel solvers.

(i.e., word problems) on the are often so contrived level developmental that they are counter and uninteresting "Problems"

skills. in developing analytical can look at a problem and deter mine the answer, then it is extremely diffi cult for the instructor to insist that stu dents go through "the steps" of solution it is "good" for them. because Instead of trying to contrive problems, I attempt to impress on my students that all productive If students

November

problems are, in a sense, word some of require degree thought. I suggest that any problem or exercise has three

Students who completed the course

which

in its new

format,

stresses problem

solving,

were

more

successful

in beginning algebra than students who completed it in the old format, a self-paced

course.

to incorporate these two steps in problem solving, since, in their perception, doing so things. To them, the "real" complicates it" part. If stu is the "doing mathematics dents are going to use calculators (and they will whether we allow them eventually in our classes or not), they must be able to do steps 1 and 3 or they will have no skills can be encouraged to ap at all. Students if home analytically proach all problems work and tests are given that require stu dents to set up problems and predict an swers that are reasonable for the question

posed. All my tests involve word problems; that test only I seldom include questions skills. computational can also use tips for doing Students -595

1985

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as part of their survival skills. Short tips, such as "start with what you can mathematics every do," "practice difficult numbers and "replace by day," " can be used when groups simpler numbers,

a grade of C new-format students achieved or better. These results, though not from a

tute variables for the numbers. Rather than to I ask students for an answer, asking decide what to do to arrive at an answer.

developmental

mathematics

I often present work with word problems. students with exercises that mix operations, that require complicated involve numbers (such as fractions), or substi computations

that this analytical Students discover step is often quite challenging yet much more crunch interesting than the dry "number ing" done in computations. skill that Another important survival the is students need ability developmental to connect and relate mathematical con can to skill this We by cepts. develop help between new op pointing out connections

studied and erations and old ones already different connections forms of between with numbers. When teaching operations use never I for fractions, quick example, tricks that help only with fractions; rather -. I carefully explain and illustrate the con then present a cepts behind the operations, to point out rational algebraic expression that the same skills that we practice with frac fractions will carry over to algebraic tions. If we point out these connections often, students will begin to look for them and will begin to develop the ability to un derstand and accept more advanced topics in future study.

study

several

spanning

en

are

years,

that the three suggest They couraging. a positive attitude areas of developing students' toward mathematics, encouraging and providing involvement and activeness, to the area be added should skills survival to prepare skill of computational building level

for higher

students

courses.

mathematics

REFERENCES in Middle Problem Solving 3 (December Problem Solving

Lesh, Richard. "Applied School Mathematics." 1981) : 1,4-6.

An of Teachers of Mathematics. Council Recommendations for School for Action: Agenda Mathematics of the 1980s. Reston, Va. : The Council, 1980.

National

Charles.

Piemonte,

in Mathematics (June

1982):

"The

How George. Polya, Princeton University

for Problem

Crusade

Education." 220-23.

Curriculum

to Solve It. 1945.

Solving

Review

20

N.J.:

Princeton,

Press,

and Jack Lochhead. Arthur, "Cognition, Journal and Math Word Problems." of De Education Remedial and (Winter velopmental 1982) :11-12.

Whimbey, Skinner,

phia

Problem :Franklin

and Comprehension. Solving Institute Press, 1980. W

1986 MATHEMATICAL

SCIENCES

Philadel

CALENDAR

from the events of mathematical chronology Falla to the Bieberbach conjecture. Babylonians and Howlers. cies quotes, cipher, Public-key informative. humor, cartoons. poems, Entertaining, : US & Can 28 pgs, 10" by 12.5". Postpd prices: in US $7.50 funds; Foreign: $11.00. M/C, Visa OK. ROME PRESS INC,Dept T, Box 31451, Raleigh, NC 27622 A

Conclusion

mathematics As a teacher of developmental for seven years, I used a modified modular, the first four during self-paced approach last three the years, I have years. During in this been using the new format described

two groups of approxi I compared students each. Of the students mately fifty with the old format, 55 percent com taught course the compared with 79 percent pleted of the students exposed to the new format. the course and Of those who completed went on to the first quarter of algebra, 75 old-format students received of the percent a C or better, whereas 94 percent of the article.

?PROFESSIONAL DATES? NCTM 64thAnnual Meeting

2-5 April 1986,Washington,

D.C.

NCTM 65thAnnual Meeting

8-11 April 1987, Anaheim, Calif..

NCTM 66thAnnual Meeting

6-9 April 1988, Chicago, III. For a listingof local and regional meetings, contact NCTM, Dept. PD, 1906 Association VA 22091, Dr., Reston, Telephone: 703-620-9840; CompuServe: 75445,1161 ;The Source: STJ228.

596

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-Mathematics

Teacher