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King-Fitch, Catherine C.Assist Students in Improving Their Math Skills.Module M-5 of Category M--Assisting Students inImproving Their Basic Skills. Professional TeacherEducation Module Series.Ohio State Univ., Columbus. National Center forResear in Vocational Education.Depart& of Education, Washington, DC.ISBN-0-89A6-176-08568p.; For related documents, see ED 249 373, CE 040497-498 and CE 040 594-597.American Association for Vocational InstructionalMaterials, 120 Driftmier Engineering Center,University of Georgia, Athens, GA 30602.Guides Classroom Use Materials (For Learner(051)

MFO1 /PCO3 Plus Postage.*Basic Skills; Case Studies; Classroom Techniques;*Competency Based Teacher Education; IndividualizedTnstruction; Job Skills; Learning Activities;.earning Modules; Mathematics Instruction;*Mathematics Skills; Postsecondary Education;Secondary Education; Skill Development; TeacherEvaluation; *Teaching Methods; Teaching Skills;Vocational Education; *Vocational EducationTeachers

ABSTRACTThis module, one in a series of performance-based

teacher education learning packages, focuses on a specific skill thatvocational educators need in order to integrate the teaching andreinforcement of basic skills into their regular vocationalinstruction. The purpose of the module is to give educatorscompetency in assisting students in improving their math skills. Itprovides techniques for (1) assessing students' math skills inrelation to the math requirements for the occupational area, (2)assessing one's own recciiness to assist students with these skills,and (3) working with students to improve math skills. The teacheralso gains skill in identifying specific kinds of errors studentscommonly make and in helping students to improve skills in thesespecific areas. Introductory material provides terminal and enablingobjectives, a list of resources, and general information. The mainportion of the module includes three learning experiences based onthe enabling objectives. Each learning experience presents learningactivities with information sheets, samples, checklists, and casestudies. Optional activities are provided. Completion of these threelearning experiences should lead to achievement of the terminalobjective presented in the fourth and final learning experience. Thelatter provides for a teacher performance assessment by a resourceperson. An assessment form is included. (YLB)

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Learning ExperienceOVERVIEW

After coMpleling the required reading, devise a method of assessing stu-,dente proficlenoy'levels in the math skills required for a selected unit of in-struction in your own occupational specialty. ,

Activity Ybu will be' reading the Enformatian sheet, Improving Basic Math Skills inVdcatiPna1 Edu r I

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You may wish to refresh your knowledge and skill in basic mathematics byreviewing a math textbook such as the following: Boyce at al., Mathematicsfor Technical and ttocational Schools. Or you may wish to review a basicInlithernatics textbook used-in your school orcoltege.

You may wish to consult a math specialist to obtain information and re-sources for assisting students in improving their basic math skills, given themath requirements for your occupational specialty

You will be identifying the math skiffs required for achieving the student per-formance objectives in a selected unit of instruction in your occupational spe-cialty.

You will be evaluating your competency in identifying required men skills bycompleting the Math Skins Checklist, pp. 19-20.°

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Some of your students may lack the bait meth sldlls requkid fdr entry intotheir chosen occeapations. For information on how you can help them to ire-prove their basic Math skills, read thd following irOsxmagoriatreet. _

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"IMPROVING BASIC MATH SKILLS IN VOCATIONALEDUCATION

How often do use math from day to d$y?

an average day with no, math skies. Nim wouldn't beable to decide whether youhact_enough_buy gas, or coffee, or lunch. srto couldn't count your

_change, pay a bill, follow a recipe, estimatea length_or distance -balance

Carried to the extreme, you would have to have ajob in which you never had to count, read or writenwnbers, recognize a number. as being larger or!knitter than another, add, subtract, multiply, divide,measure, and so on. Pretty farfetched? Most adults,you might say, can count and compare numbers.Anyone knows that 10 *lamer than_Z_dgbt-?-

Ww jaw 932 and 1/2? Which is larger? Or .50 and1.00? Or 314 and .75?

ltxr may have students in your class who lacksome of the math sifts that are considered to be

bark. -ff the skills they lack-are needed on the job krthe occupational area for which they are preparing,your students may be in trouble.

Even if they could get by on the job without beingable to work math problems In conventional ways,they might never get a chime to try. Many employ-ers requke job applicants to pass an employmenttestincluding math problemsbefore they willeven Interview them. An applicant who can't multiply37 40 may never gets chance to show thetheorshe can run a machine, make a sale, or type a busi-ness letter with ease.

Your RoleNW may feel that you shouldn't have to be re-

sponsible for teaching basic math. After all, you'retrying to teach your students to operate machines,make sales, type letters, or gain other job skills.

Isn't it the math teacher's job to teach basic math?Isn't it your job to prepare students for employment?'Rue enough, but if your students need to be able touse basic math in order to enter the occupation, you-must be involved to some extent in developing theirmath skills.

Does this mean that you should everythingand teach remedial math? No, of cou not. Nbuprobably have neither the time nor the training to doso. But if your students need to improve their basicmath skills fn order to perform on the job, there areways you can help them in the course of your regu-lar instruction.

As a vocational teacher, you may even have anadvantage when it comes to ad&rm_little "painless"

--math instrUcticif to your lessorisiwry studentswho have done poorly in math are afraid of it. It mayseem compficated turd utterly "academic"an ob-stacle in school and Irrelevant in adultly,od.

never rteed to know a numerator from a de-nominator after I leave this place," bey may say. Or,"The square of the hypotenuse won't put money Inmy pocket; why bother with kr

Such atOludes often result from a series of earlyfailures. Having failed at mastering math in the earlygradesfor whatever reasonsstudents maycome to think that they just can't do it. They thenavoid taking any more math, or even any coursesthat include math. By avoiding math, they get furtherand further behind.

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you include all these math skills as part of your reg-ular instruction? It may be helpful to keep severalpoints in mind

First, vie are talking about improving math skills,not teaching remedial math. You are not going totake on the responsibility of teaching your studentseve ylhinQwin be helping their to improve in specific areas

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where their skills are defident.

Second, students in your occupational area may

S need all the skills Setae One of the first stepsyou will need to takeis k.they do need and in which skills individual studentsneed help.

Finally, help is available. Math specialists in yourinstitution can help you by providing advice, texts,rekrences, assessment devices, math exercises,games, and other activities to use with your stu-dents. If you have students with special learningproblems, special education teachers or learningdisabildy specialists may be aval to help you.

10:: 77; a st ts mathskills are so poor that you feel it is beyond your ca-pacity- to help the student. In those situations, itwould probably be best to arrange for remedial in-struction by a math specialist and to provide sup-portive activities in your own class.

Assistinwstudents. in iniproving their basic mathskills.bas-two-phaseiv-preparatlen

you make the necessary assessments) and actuallyworidtig with the students. Lets explore some..general straisiglet-Kkican use-in the two phases ofthis process.

9

re to Assist lour StudentsBefore you can begin helping your students to im-

prove their math skills, you will need to prepare forthe task by assessing the situation and your readi-ness to meet ft. Mau will need to assess the follow-ing:

What math skiffs students wlH need as they pur-sue their occupational goalsStudents' competency levels in relation to therequired skills',bur own readiness to teach those skillsThe adequacy of your instiuctional materials formeeting students' needs

Let's look at each of these assessments separately.

Meese the math requirements. As part of yourinstructional planning, you will probably have usedoccupational analyses or competency profiles toidentify the minimum competencies required for en-try into occupations in your vocational-technicalarea. lbu will also have identified the long-rangegoals of the students enrolled ir your program.From these two b9dies of information, you will havederived the oompelencies to, be covered in your in-struction and the Roaming objectives for individualstudents,

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If you do choose to use formalized testing, the fol-. 10.4 suggestions can help you tc minimize prob.Mises such as these:

Keep tests short. It a student is having troublewith one or two problems, why give the studentten problems of the same type? If you havemany different skills to assess, it may be betterto give several short tests at different times thanto administer one long testBe sure readkig levels, vocabulary and testingmethods match your students' capabilities. Beaware of the reading (acuities and other spe--dal needs of individual students. Try alternativeapproaches that meet their needs, such as giv-ing word problems orally.2Make sample problems realistic and relevant toyour program and students' occupational inter-ests.Try to make the testing situation a positive, non-threatening experience. Explain the purposeand procedures of the test clearly Review vo-cabulary if necessary. Explain the relationshipof math abilities to the rest of the instructionalProgram.Afterward, review the reset:. c;f the test witheach student. In order to pi ipcint specific diffi-culties, go over each probleiT: ins student gotwrong, and have the student explain the pro-cess he or she used to solve each problem.Show students how to-self-evaluate. Let themparticipate in deciding where they need addi-tional help. Taking responsibility for this deci-sion may help them to approach the subjectconstructively.

lbu may choose to limit formal testing and to takean Informal vproach. Such an approach might, infact, give you inforqpation that is more useful for yourpurposes than a more formal approach. By focusingon the ways nath is really used on the job, you candevise informal situates to assess students' abili-ties to perform the related math.

For example, you might decide to have studentsdo individual shop projects. You could ask them tofigure out the exact materials (type, size, amount)they will need for their projects and to come to youwith their orders. By reviewingwith the studentstheir orders and their methods of arriving at them,you could assess their abilities in a given set of mathskills. From this and similar situations, you could de-termine student needs in relation to basic math andderive a set of learning objectives for each student.

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2. To an skill in selecting testing methods Mat are appropriate for stu-dents' special needs. you may wish to refer to Module 1-0, Assess theProgress of Exceptional Students.

11

Ammo your own skills. For you to make mathsimple and clear to your student" it must be simpleand clear to you. But it would not be uncommon,when stepping outside an occupational specialtyand kV° err area like math (which we tend to takepretty much for granted at the basic level), for youto find yourself saying something like, 1 can do it,but I can'freally explain just how I do ft."

!bit will need to assess your knowledge and coLi.petency in the math skills you have identified. Thisshould include the following:

'bur understanding of the underlying conceptslbw ability to perform the mathematical opera-tionsbur ability to help your students iearn those

operations

YOu can test yourself informally by workingsample problems and trying to explain how youworked them. It may also be helpful to review a ba-sic math text to refresh your memory or refine yourskills. Again, a math specialist can serve as a valu-able resource in your self-assessment and in help-ing you brush up wherever necessary.

Assess existing materials. With the requiredmath skills and the sk....rents' present skill levels inmind, you will need to review your instructional ma-terials. In determining whether they are adequatefor improving basic math skills, you should considerthe following questions:

Do they cover math skill development ade-quately, or do they assume basic math profi-ciency and go on to more complex math skills?Do they present basic math in ways that areappropriate for your students' learning needs?(For example, do they use communicationchannefiauditory, visual, or kinestheticthrough which individual students learn best?Are reading and complexity levels apprnpriate?Are the organization and pace suited to the stu-dents' attention spans and ability to focus?)Are explanations clear and simple?Do the materials provide enough opportunityfor practice?Are the practice problems relevant to your stu-dents' interests? Are they relevant to your pro-gram?Is the, material presented in a way that is ap-pealing and nonthreatening?

Based on your ...,sessment, you may need toadapt your materials or to locate other materials tosupplement or replace them. For example, youmight revise existing materials to include more de-tail, to simplify explanations, to reduce the readinglevel, or to make sample problems more relevant toyour students' interests.

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Say, for instance, that you have given a problemand are calling on individual students: "John, what'syour answer?" "Amy, what answer do you have? Asanswers are given, you could write ail of them onthe board, randomly, right or wrong. Then you couldwork the problem on ilk: board, ignore all the wronganswers, and reinforce the right one. No one in thissituation is penalized for being wrong, and everyonesees how your answer is obtained.

A third approach to creating a positive atmo-sphere is to encourage students to take respon-sibility for improving their own math skills. That is,you want to encourage them to make a consciousdecision that they need to improve their *Ms andthat they will work toward that improvement. The fi-nal responsibility for learning must be theirs. NWcan help them improve by proilang instruction, ma-*dais, and assistance, but only they can do thelearning and the practic 3 that are required for skilldevelopment.

To make this kind of commitment, the studentsmust be motivated. They need to understand justhow important math is to them. Each student in yourprogram has an occupational goal. It may be helpfulto discuss with your students specific ways in whichmath relates to their goals. For example, you mightreview some of the job duties related to a student'soccupational goal, pose hypothetical on-the-job sit-uations, and have the student consider how well he*or she would get along with his/her present levels ofmath skill.

Some students may be better motivated byseeing the importance of math for outside interests(e.g., understanding batting averages, figuring ma-terials needed for a hobby, even playing poker) orfor getting along as a consumer. For example, youcould say to students something like the following:"You're on your own, you've got a job, and you'remaking money. You can finally buy that color televi-sion you've boob wanting. Are they going to comeand take it away in three months because you didn'tunderstand the interest rate for the installment pur-chase and can't make the payments?"

For other students, such topics as earnings, pay-checks, overtime, bonuses, and getting ahead maybe the key. Knowing your students' interests willhelp you determine how to motivate them to takeresponsibility for improving their math oldie.

Students are Elso more likely to take respOnsibil-ity for math skills improvement if they take part inthe assessment and decision-making process. Stu-dents who evaluate their own performance and de-termine where they need help are much more likelyto take the help seriously and to work towardachievement.

Nbu could, for example, Involve a student in re-viewing the math skills analysis for the chosen oc-cupational goal, taking a skill test for the requiredmath, assessing his or her own performance, andidentifying weak areas. The student would thenneed to decide whether he or she wants to work toimprove in the weak areas in order to pursue thelong-range goal. If so, the sti:dent could then takepart in setting his/her own learning objectives.

A student who participates in this kind of processis likely to feel a greater sense of responsibility forachieving the objectives than a student who issimply told that he or she needs to improve. And theatmosphere is going to be much more positive whenstudents are working toward chosen goals.

Another way to create and sustain a positive at-mosphere is to build on success. This can help toincrease students' confidence in themselves. Theassessments that you have done will show not onlywhat the students can't do, but. what they can do.This Is a good place to start. You can focus on thestudents' present skills by giving. them problems youknow they can handle. Then you can reinforce their

mathsuccess

and point out practical applications for thethey Weedy know

For mple, if you have a student who can addmultigife-digit whole numbers fairly well, you mightpoint out how this skill applies to balancing a check-

/ book, totaling a sales check, and so on. Or youmight choose to give the student tasks such asthese on which to demonstrate his or her ability.

1315

As your students' confidence grows from realizingwhat they can do, you can gradually work up to newskills. Of course, in order to build on success, stu-dents need to experience success with each newskill. As you introduce new material, you should belooking for ways to foster success.

One strategy is to prepare the students properlyfor each new skill. For some students, reviewingnew vocabulary will help them to understand a newconcept. Relating new ideas to known concepts isalso very helpful.

For example, 100 yards might he meaningless toa student until he or she relates it to the length of afootball field. Staying with the sports analogy for amoment, a student who isn't grasping the conceptof division might already understand the concept of"hatf the distance to the goal line" without havingrecognized it as division.

It is also important to avoid overwhelming stu-dents with problems they can't handle. It can bevery discouraging for a student who is strugglingwith a new concept to see thirty problems on asheetof paper. It would be far better to let the student tryone or two problems. Then, when the student isready, you can give more problems for reinforce-ment.

Finally, an important part of creating a positive at-mosphere is to make learning enjoyable. For moststudents having trouble with math, math has neverbeen fun. But if you are corr.uent your students cansucceed, if the environment is-nonthreateninf0f--your students want to learn, and if they are experi-encing some successif all these corrditions aremet, then improving theic math skills at least will notbe a negative experience. \bu can make it evenmore positive by choosing learning activities thatare enjoyable for the students.

Games, brainteasers, tricks, and shortcuts forsolving certain kinds of problems can be more funand, in some cases, more instructivethan workinga page of problems. A math teacher can probablyhelp you locate a variety of enjoyable activities thatyou can tie in with your instruction.

Knowledge of your field will probably give you ma-terial you can use to create hypothetical job situa-tions involving math. It can be both fun and inStruc-five, for example, for students to figure out whatwent wrong, mathematically, to cause the person inthe hypothetical situation to arrive at a silly outcome.

Another way of making math enjoyable is to keepstudents' interest high by relating math to things inwhich you know they have an interesthobbies,sports, home-life, career goals, and so on.

When you really get the students involved, youcan rake advantage of their interest level by stop-ping the activity midstreamwhile interest is stillhighand picking it up later. In the meantime, tofitheayamight even spend some time thinking aboutyou were working on.

you area competency-based vocational program, you willfind it OW natural to individualize instruction formath skills improvement In the same way that youhave done sr, for the rest of your program. If you arein a more traditionally organized program that em-ploys primarily grow instruction, you still may find ithelpfuleven necessaryto individualize the mathportions of your instruction.

We have discussed the importance of assessingoccupational math requirements and students' re-lated abilities in order to set learning objectives. Wehave also noted the need for a positive atmosphereto enhance students' commitment to achieving theeobjectives. Naturally, you will also want to plan yourinstruction so that it enables each individual studentto meet his/her objectives. In aodition, you may findthat individualization permits you to meet students'needs with the least interruption to your ongoingprogram.

In planning how you will assist students in improv-ing their math skills, you may find it helpful to keepthe following guidelines in mind:

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Instruction should contribute directly to eachstudent's lorxj-range occupational goals and

1110 , immediate objectives for math skills improve-ment. The students' learning objectives will diobtate what you teach. Whether you use group orindividual !nstruction for specific portions of thecontent will depend to a large extent on howmany students need help in a given area.Learning activities should be varied and shouldpermit each student to begin at the appropriatelevel and proceed at his or her own rate.Instructional strategies should be consistentwith students' individual needs with regard tolearning style and preferred grouping.For example, one student might respond to aone-to-one explanation of a skill that IS givinghim or her troublesay, multiplying fractions.Another might need practice in multiplying frac-tions to refine his or her skill. Perhaps learningactivity packages, a game, a simulation, or re-peated practice on supervised laboratory taskswould help this student.Evaluation procedures should allow each stu-dent to be evaluated when ready. Opportunitiesfor self-evaluation should be provided to en-courage in students an ongoing sense of re-sponsibility for their own progress.Since checking one's answers is a good prac-tice in math, methods for doing so should be

110 taught with each computation skill. Self-evaluation can be used as a natural outgrowth

Students' vocational-technical interests mayserve as the basis for individualiied instruction.If you capture their interest with something that"hits home," you may well accomplish thatsometimes-difficult task of motivating individu-als.For example, is a student in a hospitality man-agement program dreaming of unning a fishinglodge one day? By having that student plan thefacilities and staffing for such a lodge, the in-structor could give him/her a lot of practice onbasic math skills.Personal interestsleisure activities, home in-terests, hobbies, and so onmay also be goodsources of math activities. For example, if oneof your students is a displaced homemaker withsmall children, you could come up with manyproblems related to family budgeting. For a stu-dent who puts a lot of time Into working on cars,you could devise problems related to costs ofparts, speed, RPMs, horsepower, fuel costs,

1 and so on.4

10 4. To gain additional skill in individualizing instruction, you may wish torefer to Module C-18, individualize Instruction.

-Roach math in the context of occupational skilldevelopment. As a vocational-techical instructor,your main task is to prepare students for entry intotheir chosen occupations. If your students need toimprove their math skills for job entry, you will needto look for ways to incorporate math into other areasof your instruction.

As we have already discussed, weaving math intoyour ongoing instruction will benefit both you andyour students. For you, it will reduce the amount ofextra time required beyond your regular vocational-technical subject matter. For your students, math fora specific purpose will have more relevance thanmath as a separate subject. It is likely, therefore, tobe more interesting, and it may be easier to under-stand as well.

As part of your math skills analysis for the occu-pation, you will have identified tasks in which mathis used. These tasks are the most likely context forteaching math. Your own situation will dictate howyou incorporate the math, but let's Iodic at an ex-ample of how it can be done.

Imagine that students in a clothing program aregoing to learn about altering patterns, and the in-structor knows that one student is having troublewith fractions. Without taking advantage of this con-text, the instructor could decide just to p'.ug in a re-medial unit on the related math before getting intopattern alterations with that student. (Using this ap-proach, chances are that the instructor would eitherturn off or scare off the student.)

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On the other sand, the instructor might devise away to work the math Into the unit on alterations.Here's an example. Beginning with the importanceof proper fit, the instructor might show pictures ofwomen wearing improperly fitted clothing. He or shemight then describe a set of measurements that donot correspond to any given pattern size and askquestions such as the following: How would this per-son look in a size 12? How would you make a size12 fit her properly?

Then, as the instructor explains how to make spe-cific adjustments (for example, evenly reducing thepattern 2 inches in the hips), he or she could givethe struggling student extra help with fractions.

You can see that the difference between the twoapproaches is that, in the second, the additionalmath help was given as a part of the occupationalskill being learned, not as a separate math unit. Assuch, it probably would have had more meaning andpurpose for the student.

Use visual and tactile means to reinforce mathconcepts. A major stumbling block for some stu-dents is their inability to relate mathematical con-cepts to things they already know Math by nature issymbolic. To really grasp the underlying meaning,students have to be able to "see" the symbolism.And they need to see that mathematical facts aretrue no matter what objects they are applied tothat the sum of 8 and 4 is 12, whether cans orinches are being added.

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clear. One is to use physical objects for demonstrat-ing math concepts and to have students manipulatethose objects. Hands-on experience is one of themost effective means of learning math concepts.

Say, for example, that a carpentry instructor istalking about dividing a board into equal lengths.Demonstrating this concept by dividing an actualboard into 6 equal lengths may be much clearerthan dividing 12 by 6 on a chalkboard. Having stu-dents do the measuring and cutting would be evenmore effective. You can probably think of many otherexamples in your own areausing money, cups offlour, containers of water, lengths of yard goods,quantities or nails, or whatever objects are appro-priate.

Visual demonstrations using .suchAigis as count-ers, pie charts, graphs, Or 'folded pap*" are alsohelpful ways for clarifying math concepts. The morethe students manipulate these materials them-selves, the better their chances of understandingthe concepts.

Another visual technique, which we have alreadytalked about, is to relate mathematical ideas to con-cepts the students already know For example, howlong is 15 millimeters? Some students would drawa blank. But, compare it to 16nvn film, and studentswho have handed movie film will probably get arough idea.

Endless examples could be given for visual tech-niques. Nbu can probably think of many that natu-rally occur in your instruction.

Provide practical math activities. ltu are al-ready aware of some of the disadvantages of usingtypical word problems to provide practice in problemsolving in the vocational-t classroom. Formany reasons, they tend to be an Ineffective testofproblem-solving skills and often seem irrelevant tothe vocational-technical skills the students arelearning.

However, your class or lab is-a natural source ofreal problems to be solved through math. Situationsthat arise on the job or la. daily living can also besimulated in class to give students practice in askingthemselves the right questions:

What do I need to find out?What do I already know that can help me findthe answer?What more do I need to know in order to findthe answer?How can I use what I already know to find outmore?

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The student who can ask the right questions is morelikely to be able to solve problems on the job usingother math skits (computation, algebra, and so on).

You can probably identify many problems thatarise in your vocational-technical area that can besolved mathematically:

Increasing or decreasing formulas or recipesPlanning amounts of materials needed for spe-cific shop projectsPredicting results of changes in speed, velocity,temperature, or other variablesPredicting ;nventory needs

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-tegB told width 137,'.What Is the solution ?.

Amount of fabric needed

84X 325Zx 37 5 "

unfinished lengthx number of panels per windowx number of windows

'1% inches36 inches per yard

As students participate in solving these kinds ofproblems. they become better able to use their skillsindependently. This is important: it will allow them tosaLvR.problems that ariseJater on the sob.

Problems related to personal interests and dailyliving can also provide the same kind of practiceFor example

Planning a schedule to get specific things donein a given amount of timeReccncilinq a hank statement with the checkbook register

17

84x 3

2.1 yards ofneFMb

34,'36

Figuring interest on an installment purchaseFiguring total. tip. and tax on a salescheck andsplitting the check among several people

If time permits. you 'night haVe each student bringsuch a problem to be solved by the class The classcan work together to set up the problem. ideetilyavailable and needed int( yrmation and COrnpute theanswer

OptionalActivity

2%111 1110

If you are ;- In improving your own skills and knowledge of basicmathematics, may wish to review a basic mathematics textbook used inyour school or

Another option IS to review poignant sections of the following supplementaryreference: Boy0 at al., Mathematics for Technical and lbcational Schools.This text is written in simple and Is intended for people who will beworldng with . . and machinety. 1 through 4 deal with commonfractions, . ..: . and ratios and proportions. Also addressed are ge-ometry, grwhs, ; ..s. . instruments, and a variety of industrial applica-tions of math. problems' are vocationally oriented.

If you desire ad4tional help in preparing to assist your students in improvingtheir basic math! skins, you may wish to consult a math specialist. Such aspecialist may be able to provide the following kinds of assistance:

Sample ass sment devices or assistance in devising your own meansofAssistance In evaluating instructional materials for basic math skills iftl-provementAdvice and resources for adapting or supplementing existing materialsSources of basic math materials and activities related to your contentarea

Nbu should be to locate a math specialist who can help you within themath of your own. school or college.

-

Assurse-thist y4 tire teaching in a vocational-technical program in your ownoccupational specialty. Identify a unit of instruction in your program that re-

for students to achieve the student performance objectives in the selectedunit of instruction. Be sure to include not only broad areas of math skill (e.g.,computation), but also specific skills (e.g., addition of whole numbers).

After you have identified the math skills required, for students to achieve thestudent performance objectives in a selected unit of instruction, use the MathSkills Checklist, pp. 19-20, to evaluate your work.

O

MATH SKILLS CHECKLIST

Ottectione: Place an X In the NO, PARTIAL, or FULL box to indicate that *meeach of the following performance components was not accomplished, par-tially accomplished, or fully accomplished. If, because of special circum- Doestances, a performance component was not applicable, place an X in theNIA box. Resource Paton11111=1MOIIIIMMIlmi111111...1.

--In Identifying the requited math Skills, you:I. reviewed the student performance objectives included in the unit of

instruction

2. identified broad areas of math skill required for achieving the objec-tives, including the following, as appropriate:a. quantification

-^

b. computation

c. measurement

d. estimation

e. problem solving

f. comprehension of equivalents

g. organization oislata_.

LEVEL OF PERFORMANCE

h. algebra

i. geometry

. 3. identified specific math skills needed for achieving the objectives, in-duding, as appropriate, the ability to:a. read and write numbers, count, and order numbers

b. add, subtract, multiply, and divide whole numbers

c. add, subtract, multiply, and divide fractions and mixed numbers

d. add, subtract, multiply, and divide decimals

e. make,. use, and report measurements (e.g., of time, temperature,or other units pertinent to the specific objectives to be achieved)

f. estimate measurements and quantities

g. use problem-solving techniques in solving on-the-job problems

19 21

AP 4it

Ell El

gip / 4st sh. relate measurements on thfterent scales (e.g., metric and English,fractions and decknals, feet and rude, or other scales used in the .selected unit of instruction) El

i. apply algebraic and geometric principies to practical, on-the-jobProb0m,

Level of Performance: All items must receive FULL or N/A responses. If any Item re vies._. a_Isha or_PARTIAL response, review the material in the information sheet Improving Basic Math Skills in VocationalEducation, pp. 7-17, or check with your resource person if necessary.

E._%46

[te:i7back

For the unit of instruction you selected, develop In writing an activity forassessing students' levels of proficiency in the math skills required to achievethe student performance objectives in that unit. Your activity may includewritten tests and/or other classroom or lab activities through which you canassess proficiency in the identified skills.

After you have developed your assessment activity, use the Assessment Ac-tivity Checklist, pp. 21-22, to evaluate your work.

2022

1111111.1..-

ASSESSMENT ACTIVITY CHECKLIST

Dereetions: Place an X in the NO, PARTIAL or FULL box to indicate thateach of the following performance wmponents was not accomplished, par-tally accomplished, or fully accomplished. If, because of special circum-stoves, a performance component was not mplicable, place an X in theNIA box.

Nave

Date

RIPOOVICO FOAM

%bur assessment actIvfty1. includes one or more of the following assessment devices:

a. informal situations devised to assess students' ability to performthe required skills

b. existing tests covering the required skills Elc. tests specialty developed to assess the required skills

2. covers all the math skills identified within the selected instructionalunit El

3. includes activities/test items that focus on ways the identified mathsums are actually used on the job

4. includes activities/test items that require students to apply problem-. solving tectmiques D5. includes problems that are realistic and relevant to:

a. the selected unit of instruction

b. students' vocational interests

LEVEL OF PERFORMANCE

6. provides opportunities for self-evaluation

7. Includes one or more of the following strategies for making the situa-tion as nonthreatening as possble:a. presenting the activity/test in the context of regular activities

b. explaining the relationship of math to other program act -Mee

c. using just one or two problems to test each math skill

d. using several short activities/tests rather than one long one

e. ensuring that reading levels, vocabulary, and testing methodsmatch studerts' capabikbes C]

f. explaining assessment purposes and procedures In advance 0g. reviewing vocabulary, if needed

h. reviewing assessment results with each student E]

2123

Level of Perforinencec AN Items must nrwive FULL or tiliA responses. If any Item receives a NO orPARTIAL response, review the material In the inlormatIon sheet, Improving Basic Math Skills In socsitiOnalEducation, pp...7-17, or check with your resource person If necessary

Lear mg xperience dlOVERVIEW

1113ta may wish to review vo0ationally oriented student end teed* *dilateclas4 to imPrale sPecitic math Ortils.sitctlt 45 the Matheinalits LearningActivity Pack4ges Produced by Me Imitate

I sti4YMtuni Coneortiuth (ACC) . r "

23

Activity

int:111

Once you know Oat basic math skiff your students need and where theyneed help, there are a viviety rof techniques you can 'use to he them im-prove in specific skills. By and large, you can do this within the context ofyour regular vocationol-technical content. For information on helping stu-dents 10 knprove specific math lids; road the following information attest

IMPROVING SPECIFIC MATH SKILLSLet us say that, through your assessment activi-

ties, you- have iderdffied-math-skills-that-your adents need to Improve. Stu have assessed yourown math skills and are premed to work with thestudents individually and ki a 'positive atmosphere.How can you do this in the cordext of your ongoingvocational-technlcal program?

First, you will need to identify exactly where eachstudent is having trouble so that you can give eachstudent just the kind of help he or she needs. Thereare several things you can then do to help studentsimprove specific math skiffs:

Pinpoint the Difficulty

Pin poird the difficulty.Use simple explanations, visual aids, and-ma--

activffies to explain mathematicalconcepts.Nabrk on specific problem areas and build to thelarger *M.Provide practice activities.

Lets take a closer look at each of these tasks inrelation to specific math skffis.

Nbur initial assessment is usually confined to thebroader areas of math with which your. students

. . Ora 11E.. tali a A

found initially that one of his/her marketing and dis-tributive education students was having difficultymaking extensions. The student needs this skiff forfiguring discounts, quantity prices, and the tax incompleting an invoice. In the instruckwis assess-ment, he/she found that the student was havingtrouble multiplying fractions and decimals.

But the instructor stiff needs more information.Does the student need the basics of multiplication?Are multiplication tables the problem? Or is it some-thing like carrying numbers or pining decimalpoints?

One way to pinpoint the problem more speefficanyis to use diagnostic tests. Nbu may be able to ob-tain such tests from a math specialist. They are de-signed so that the students pattern of right andwrong answers shows exactly what past Of the op-eration the student is doing incorrectly.

Another way to pinpoint the difficulty is to develOpa checklist of the component skills, or subtasks,with which to `zero in on the problem area. Nbu candevelop such checklists by listing the math applica-tionsfrom simple to COMplex__---Amedil your pro-gram and the subtasks that make up those applica-tions. It may also be helPful to review a basic mathtext to identify subtasks. Samples 2 trough 4 illus-trate checklists for selected math applications.

24

The Waked in malhatnatloil problemsolving be less easily defined In general, theywill' the basic problem-solving steps (deter-mining i you need to *id out, what you akeadyknow i is useftd, what more you needs know tosolve i problem, and so on). But the specific sub-tasks will vary because they will be closely tifid tothe panlcular problem'. being solved. Sample 5shows subtaske for mathematically solving agiven occupational problem.

26

..M11

SAMPLE 2

CHECKLIS

Asking Whole Okiebers

The student:1. demonstrates Imowledge of the basic addition facts ,

2. adds a single column of whole numbers on pepper

3. adds whole numbers mentally

4. adds multiple-dIgit whole numbers:-a. fines up numbers correctly

b. starts with right-hand column, vrorics left

c. adds two digits in the column at a time, gets a sum, and adds the nextnumber to it

d. writes total sum for column directly below column

0. if sum has two digits, writes only the second digit under the column andcarries the firel-diglt to-the next-column

f. adds "carried over digits when totaring the column

g. ireeats process. for each column

h. for final (left-hand) colunin, writes both digits of sum to complete thefinswer

"

11,111::. :4; r1.0 00, .1- 40$4,00. rr,r,

0

*

:AO "r=x- "r-'

i.

You can develop checklists such as those shownin samples 2 through 5 for any math competenciesthat you identify in your occupational area. With themath operations broken down into subtasks in thismanner, it should be easy to pinpoint what skills andsubtasks are giving students trouble.

Another way to pinpoint difficulties is to have stu-dents demonstrate how they work problems. Ifyou have a student who is having trouble with divi-sion, for example, it can be very enlightening just to

. watch the student work a few division problems. Ifyou discovered the student's difficulty initiallythrough a test, you. could simply follow up by havingthe student show you the process he or she used in

solving those.division problems. With your checklistof division subtasks in hand or in mind, it should befairly easy to pinpoint where the student needs toimprove.

One student might need help with the basic num-ber facts. Another might need to review the wholeconcept and process of division. Still another maysimply be getting hung up on a particular subtask,such as bringing down numbers or inserting zeros.It would be pointless to give each of these studentsa detailed explanation of how division works. By pin-pointing the trouble, you can tailor your instructionto each student's needs.

SAMPLE 3

alsixePliate irteaSUring Inatrionent (e.g., ruler, tape, micrometer) forthe lent0 to be rreasuredand*o of accuracy needed 1=1

'PlaCee ir:istiUment

3. reacis.lbe Measurement correctly D4. if riecgssarY, *vet* measurement to more practical units (usually by mulflPIY-

ire or dividing).

S. identifies units of measure In the reported measurement II El

Ws No N/A

SAMPLE 4

MATH CHECKLIST

The stildant:f: eSei60-01.i4titaarkialicablelitenis 0 El 02. lindikorre0 (taller 'catunin on cove wide of table

3. *ids cerrect cents range on 00ser mkt

4. reef* tax amount whoa columns intersect

5. enters tax properly on salsa check El

awm.a..

Yes No N/A

26

I

L

ha I I

OM/art pctdan'or

inwhich1,isedetennitm the number i#'niiiniiP 011'0

t determines the 00 at tha .Cke,vides $10.74 by 6 and (Worn. that the coripwr-vdeiave "1

5. determines the price per serving bi dividing the price ofnumber of servings n the package -

Explain Mathematical Concepts

You are likely to find that some students really dolack basic understanding in one or more areas ofmath. Perhaps a student can't handle division be-cause he or she has never really understood the re-lationship between multiplication and division. Sucha student will need some basic groundwork beforehe or she will be able to handle the division orocessvery well.

You will have to decide whether you can providethe help the student needs or whether you shouldrefer the student to a specialist for more intensiveinstruction. If you do refer the student, you will prob-ably want to plan some supportive activities to usein your own program.

It you determine that you can assist the student,you need to plan learnirj activities that include thefollowing:

Simple, clear explanations and basic rulesEgalanations Gr. applications related to programcontentDemonstraticns cf step-by-step procedures

a Visible or tangible teacrd ig aidsManipulative activities

Sample 6 shows an excerpt from a handout incor-porating several of these strategies, which might beused in an automotive program to help explain amath concept. The handout could be used in com-bination with an oral explanation, a chalkboard dem-onstration, and the use of real objects.

Manipulating objects is another effective way fora student to learn underlying math concepts. Youcan probably think of many hands-on activities ap-propriate to your own program that could be used toteach math concepts. For example:

Adding coins. nails, or the inch markings on aruler

Subtracting money pipe lengths. or patient in-put outputMultiplying ingredients, tax, or stockDividing a board length, a pie. or a template intoequal partsMeasuring wire, floor space. or paperFolding paper or fabric into equal parts or geo-metric shapes

9ne point you will need to consider in teachingmath basics is how much your students really needto know for occupational competency. Take, for example, definitions or labels. Your students moy notreally need to know the terms multiplier, nult plicand, and product,

On the other hand, you may decide that usingthose terms makes it easier for you to teach the pro-cess of multiplication. In that case, you would prob-ably explain the terms so that the students wouldknow what you are talking about. But you wouldprobably riot require them to know the terms in orderto demonstrate competency in multiplication.

27

SAMPLE 6

Or, you**, add 4 + 4 and 906 Cara

,16

But .Y041***0:17 c4.7.43 paidi tiotdo court*,,The *Or. !ha riOn*Tai tOeld 0000 **1 1

eritaert itor**4. you oilocation,

Sad

Procedures:

Now let's look at some sample palms. ...

28

Work on Specific Problem Areas

le have talked about starting at the beginningthe basic math conceptsfor students who have areal gap in their mathemadcal knowledge. For manystudents, howevm this would be both unnecessaryand ineffective. A student who is turned off by mathmay just stop listening when you start talidng aboutfundamentals. Besides, the trouble spot may not bethat basic.

dent Is is having problems. Imagine that a student is-showing you how she solves an addition problemand you see that she isn't (=Tying numbers cor-recdy. That's a good place to start. If you work withher on carrying numbers, you may find that heroverall skiff th addition improves significantly. Bybudding competency in the subtasks, the studentbuilds competency in the larger skill.

Basic Number Facts as a Problem AreaNia ad know that you can't do math unless you

know basic number facts (4 + 5 = 9, 7 x 7 = 49,and so on). The number facts for addition and sub-traction usually are not a big problem because theyonly involve combinations from 0 to 10. Multiplica-tion facts, of course, are another matter; they seem

111 be astudemajon r stumbling block for most math-

deficient ts.

This presents a dilemma for a teacher trying tohelp students improve their math skills. Do you workwith the students to help them learn the numberfacts? Or do you help them find shortcutsways toget around learning the tables? Or do you perhapstry to do both?

It may be unrealistic to expect some students whoare having trouble with math to memorize multipli-cation_ facts at this stage of the game. It may havebeen the insistence that they must learn their tablesthat created the banter in the first place.

Then how do you help these students? The an-swer lies partly in the occupational math require-ments. If the worker is going to-have to recall num-ber facts quickly in order to do on-the-spotcalculations, then there may be no way aroundlearning the tables. In other job situations, usingshortcuts and aids may be an acceptable alterna-tive.

For example, you might suggest that your stu-dents refer to a multiplication number chart (seesample 7) as they do their math. Most students willquickly learn some of the easier combinations, suchas the 2s and the 5s. By using the chart, they areapt to team other combinations to which they referoften. Number charts can also be used for additionand subtraction, as shown in samples 8 and 9.

-might suggest that the-studentscount ontheir fingers, make lines on paper, or construct anumber line (see sample 10) if they need to.

Hand-held oak:Waters are another option. Youmay argue that a person Is better off having thefacts handyin his or herown headthem depend-ing on a calculator. This is probably true, but if thevalculator mattes the difference between being ableto work math problems and not being able to (andespecially if calculators can be used on the job),then the calculator can be .a valuable tool. In fact,research has shown that the use of calculatorshelps, rather than hinders, learning.

Besides, student use of calculators is increasingdramatically. As calculators have become smallerand less expensive, many more students havegained access to them. Some even have them builtinto their wristwatches.

So, you are likely to have students who are goingto wet calculators no matter what you decide abouttheir appropriateness. But students shoUldn't de-pend totally on a calculator, because things can gowrong. A student may push the wrong button or notpush hard enough. A student may push two buttonsat once or the same button twice. Or the battery maybe weak;

Ability to estimate is the key to recognizingwhether the calculators answer is correct. (If a stu- .dent expects an answer to be somewhere around4,000 and gets an answer around 4,000,000, itshould be clear to him or her that something iswrong.) And in order to estimate, students need tokr,:..w basic number facts.

Consequently, whether or not caiculators andcharts can'be used on the job, knowledge of thenumber facts is still valuable. Learning those factsthrough memorization, flash cards, and other rotemethods is effective with some students.

SAMPLE 7

.

,

, ,,

.

,,,..

. 4. 8

...._

12 AS

...'','' '

5 10 15 20

il.

, i',' (%\griV 'f ,

1

i.r.

,

SAMPLE 8

ADDITION CHART

Directions: To add two numbers, find one number along the top edge and the other along the left edge.Run your fingers along the two rows until they meet.

Example: 8 4- 4

Find 8 on the top edge.Find 4 on the left edge.The "8" row (going down) and the "4" row (going across) meet at 12. The answer Is 12.

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10 11

2 3 4 5 6 7 8 9 10 11 12

3 4 5 6 7 8 9 10 11 12 13

4 5 6 7 8 9 10 11 12 13 14

5 6 7 8 9 10 11 12 13 14 15

6 7 8 9 10 11 12 13 14 15 16

7 8 9 10 11 12 13 14 15 16 17

8 9 10 11 12 13 14 15 16 17 18

9 10 11 12 13 14 15 16 17 18 19

10 11 12 13 14 15 16 17 18 19 20 I

31

s

SAMPLE 9

subtract two postiNe ni /item. jowlOw minuend along the top edge and find the subtrahendRun your *Vero along Mt, rows until they meet.

4 0-theltinuortd; 4 l the Stibtrealfmd)

32

SAMPLE 10

NUMBER LINES

Number lines are two sr irate, movable strips of paper (rulers can also be used) 'may 00 trOVed inrelation to each other to add or subtract.

Example: 4 + 6

Place two number lines on the table,. one above** other.Find 4 on the first line.Place the front end (the zero point) olthe SecOnd number line (tiredly bed! the.4/orithe:firetFind 6 on the second line and read the rhanber aboie' it on the fihtt Threanewer

To subtract, reverse the process.

Fiowever rote methods may repel other studentsai id actually prevent Mem from learning. F-or thesestudents using memory aids (charts and calculatorsi (1Uy riot seem to them like a tedious learningexrcise Thus inev may ref am at least some of themost commonly used combinations, so that theyeveniwijy !efe1 to the aids ie6s As students learn

11011:1t)ei t0Ctt, tinled ;IS recorded on ci cnartto ,,how prr)we'r,.-; May shIrienis 101,11n?tit

t3evurld !ricit),()(;,, few hints ul.tytn,1!

ad(ar:in and multiplication combinations are the:i,-,Linte in reverse Students need to know Mat if7 fi escapes them they can turn it aroundmaybe ;) 7 is e;isier Likewise / J is the ;drileat.; 5

YOU can SSO suggest that students move up of(10Wn 0 notch or) ttle!r mental t.0t)!et.; when theystock F Or exampR'. it 8 G is hard they need toknow that they can try 8 to and then add anothert3 The 1'ick with both ot theic strati:idles is to' sti,

#fOM what they do k'IcrA

33 ,

SAMPLE 11

GROUPING NUMBERS IN AN ADDITION PROBLEM

SAMPLE PROBLEM: 475362563621

+19

SOLUTION:

1. GROUP NUMBERS

415= 5

65

10 = 32

1 - 1

2. ADD RIGHT COLUMN AND CARRY

3. ADD LEFT COLUMN

15

10

1

+ 3)

29 4 = 294

10

212

(3) 10

carry 4

35

Subtracting the top number from the bottomnumberThe sluclent subtracted 2 from 3, in-stead of borrowing and than subtracting 3 from12.

.52030.347

5113Neglecting to bonowThis student didn'tborrow to make it possible to subtract 7 from10. Instead, the student tried td subtract 7 from0 (impossible) and dedded the answer must be0. The same mistake was made in dying to sub-tract 3 from 2.

52001341

5020Forgetting to decrease the number boo-rowed fromThis time the student borrowedand correctly subtracted 7 from 10 to get 3. Butthen, instead of decreasing the 8 and subtract-ing 4 from 5, the student subtracted 4 from 6.The same mistake was repeated In borrowingfrom the 5.

5260347

5923Neglecting to borrow from more than oneplace valueTM; student should have bor-rowed from three place values in order to sub-tract 3 from 10, 0 from 9, 0 from 9, and 6 from6. Instead, the student borrowed incorrectlyfrom one place value.

441,0006J00.3

43, Ocl

It may he your students to review the concept ofplace valuers and the basic ruin of subtraction (e.g.;shop work from fight b let be sum the top num-ber Is bigger; and remember that subge np 0 kma number does not change the number).skating the procedures for borrowing is aim Woot-ton In addition, you should encourage studenhe todo the fallowing:

Wirt% in the borrowed and decreased valrs asthey work, to lessen chance of WOE

44$3414'113

Chock an answers by adding the answer andthe bottom number to get the top nimbler.

344+44113

5260Estimate answers before they begin, to catchany gross errors in subtraction. For example, inthis problem, 597 is about 800 and 308 isabout 300. Thus, the answer should be about300.

5'3'4308

Multiplication of whole numbers. The most fre-quent problems In multiplicadon are in carryingre-membering which number is carried end what to dowith it when mariplying the next set of numbers. Be-low are some typical errors made in multiplicationproblems.

Not cartykigIn \ this problem, the studentmultiplied 3 x 5 add got 15. instead of writingthe 5 and carrying the 1, the student wrote 15.The some mistake was made when multkriying2 x 5.

36

65

1815

13,

38

Adding the canted number Wore multiply-ingThe student correctly ,mulNplied 3 x 5(15), tool* the 5, and corned the 1. However,instead of licit multiplying 3 x 6 (18) and thenadding the carried number (18 + 1 = 19), thestudent added the carried number first(6 + 1 7), then multiplied (3 x 7 = 21). Thesame mistake was made when multiplying2 x 65.

165x 25

2151

EMultiplying the carried number Msteed ofadding ItThis student correctly multiplied3 x 5 (15), wrote the 5, and carried the 1. How-ever, the student then multiplied the cantednumber (3 x 1= 3) before multiplying 3 x 6(18). The same mistake was made in multiply-ing 2 x 65.

X Zb. 1935122013,035

Forgetting which number was awnedInthis problem, the student correctly multiplied3 x 5 (15), wrote the 5, and carded the 1. Butin the next step, the student reversed numbersand multiplied the carried minter first(3 x 1 = 3) and then added the 8 to get 9, in-stead of calculating correctly: 3 x 6 =18 + 1 = 19. The same mistake was made inmultiplying 2 x 65.

ex5X 23

4580895

-\ 37

Forgetting to Indent the second set of num-bers Me the productThis student did al themultiplications =molly However, Instead ofbeginning the second product under the 2.1n thenumber 23, the student aligned It with the 3.This gives the wrong place values to the sec-ond product and a wry wrong anew of 325imaged of the correct 1495.

1 5130326-

Agate, reviewing place values and the proceduresfor carrying numbers may be helpful to your stu-dents. They should be encouraged to do the follow-ing:

Kite carried numbers above the next cokimn.Check answers by reversing the problem andmultiPlYkig

For multiplying mentaNy, it may be helpful for stu-dents to separate the factors kilo combinations thatare easier to won with or to multiply by rounded offnumbers and then to adjust the answer. Sample 12Illustrates these two processes.

Division of whole numbers. In division,dents make more varied mistakes becausealso Wolves multiplication and subtraction. Any'tufty that students are having with thesemath operations will also show up in division.following are examples of some typical errors in di-vision problems.

Not keeping numbers in correct pieces Intti this problem, the student placed the 6

rectly above the 4 of the dividend ofabove the 0. This caused the studentan extra 0 M the quotient, and it that40.6 = 60.

39

SAMPLE 12

SKATING F TORS AND ROUNDING FACTORS

$91141ne *WI12 -k Ptt otli to 45.

.00' 96 am 646

RfauttidniFactors12 48 12 ),1/4....i...inititurt2

6,00 24 to 5

Failing to bring down a zero to maintainplace valuesAfter correctly dividing(7 3 2. with a remainder of 1), the studentshould have brought down the 0 and divided 3into, 10. By overlooking the 0 and bringing downthe 9. the student lost the place value repre-sented by the 0.

1 coq19

Failing to place a zero iv the c' atient- Thisstudent correctly divided (105 52 2, with aremainder of 1) and brought down the 6. At thatpoint, he she should have tried to divide 16 by52 and entered a 0 above the 6, then broughtdown the 2. By neglecting the zero, the studentgave the final 3 in the quotient the wrong placevalue.

52 j10562-704

102-150

38

Choosing the wrong quotientThe studentincorrectly divided (40 6 5), leaving a re-mainder larger than the divisor, From there, thesolution went from bad to worse. The studentdivided 106 by 6 (17) and wrote this two-digitnumber in the quotient.

1_517_)406-30

10(0

Subtracting incorrectly In this problem. thestudent incorrectly subtracted (7 6 2)This type of error can be easily missed by stu-dents, because it may not interfere with com-pleting the rest of the problem

11..

Multiplying incerredlyThe student kw-redly multiplied in this problem (6 x 8 = 38).As with subkaction _errors, this type of mistakeoften goes undetected If the student is able tocontinue with the problem:

6140626,

Not recognizing too large a remainder--Here the student chose the wrong quotientwhen dividThg 46 by 6, leavkig a remainderlarger than the divisor.

0/44116-36

4(0-3610

A review of place values and careful demonstra-tion and explanation, of the division process may behelpful to your students. They should also be en-couraged to do the following:

Check all remainders against the divisor to besure a high-enough quotient was used.Check subtractions and multiplications.Check final answers by multiplying the quotientby the divisor, then adding the remainder to getthe dividend.

Addition and subtraction of fractions andmixed number*. Difficulty with adding and sub-tracting fractions and mixed lumbers often has to

. do with denominators (e.g., converting to fractionswith common denominators and converting to im-proper fractions). Students who have trouble with

.....,-L-multiplicatfori ire also apt to run into *rouble whenconverting fractions. The following a ,,,me typicalenucs.

39

Not converting to fractions with commondenominakins before computingk1 thisproblem, the student tried to subtract fractionswith different denominators. This then led thestudent to subtract the denominato rs.

8 5 3Not leaving denominators constant whencomputing Here the student added the de-nominators, getting an anew of 7A6 instead of7/43.

Computing new numerators incorrectlyInconverting 2% to an improper fraction, the stu-dent did not add in the old numerator (2). Thestudent should have computed as follows tofind the new numerator: 5 x 2 = 10 + 2 = 12.

2,*

5 5Not converting mixed numbers to Improperfractions before cbmputingIn this problem,the student converted the fractions within themixed numbers to fractions with common de-nominators (% = "112 and % =1%4. But be-cause the problem still contains mixed num-bers, the student will probably get into troublewhen trying to subtract %2 10/12.

3 11-25-1=

t

41

,4g7.7.7.r.nznr-sn. 74c-:

Not reducing final answers to proper frac-tions or snored numbersTo complete thisproblem, the student should have reduced 47/6to a mixed number 7%.

3* +41 =IQ 4.32ff+.22.40111.

Multiplying incorrectlyLack of skill in mul-*kaftan got this student into trouble whenconverting % to a fraction with a denomina-tor of 6. The student incorrectly multiplied3 x 9 = 18, getting the wrong numerator.

i 1,,IQ_ ±3 2

For most fraction cllfficulties, it w be helpful toreview comparative sizes of fractions. Studentsneed to have an idea of the relative sues of com-mon fractions: that 1/2 is larger than Vs; that .% islarger than 1/4; and so on. 141ti can demonstratethese relationships using charts, circle graphs, othervisual aids, or objects pertinent to your occupationalspecialty.

Depending on the area of difficulty, you may wishto review such concepts and procedures as the fol-lowing:

Basic concepts and procedures related to de-nominators (the number of parts into which youare splitting the whole)Converting to common denominatorsConverting to Improper fractionsReducing fractionsThe_rule that the value of a fraction remains thesame if both numbers are multiplied by thesame-number

ft is usually helpful to begin with problems thathave like denominators and then toproceed to prob-lems with different denominators after the first fOehas been mastered.

Multlpftcation of fractions and mixed num-bers. Multiplying fractions is a simple process if thestudent knows the multiplication tables. Aside fromerrors caused by lack of skill in multipkation, mosterrors relate to conversion of mixed numbers andimproper fractions. Difficulties with division will showup when converting answers to proper fractions.The following am some typical mistakes.

Multiplying IncorrectlyIn this problem, thestudent incorrectly muftiplied (4 x 8 36).

X-2- x214 8

Not converting mixed numbers to improperfractions beton) computingHere the ski-dent should have converted 1% to 1% beforemultiplying. Instead, the student multiplied thefractions and simply transferred the whole num-bar to the answer.

X

Not reducing final answers to proper frac-tionsIn this problem, the final answer shouldhave been reduced to Wm

14-'xCo

15

Aside from working on multiplication tables anddivision procedures as needed, studeMs may needto review a few principles of working with fractions:

In multiplying fractions, multiply both the nu-merators and the denominators.To convert a mixed number to an improper frac-tion, multiply the whole number by the denomi-nator, and add the old numerator to find the newnumerator.To reduce an improper fraction, divide the nu-merator by the denominator.

4240

Division of fractions and mixed numbers. Stu-_ dent division mistakes we often caused by simplemultiplication errors ,a failure to find invert theset The following are examples.

Nat inverting before multiplying In thisproblem, the student forgot to invert %.

.101

÷Multiplying Incorrectly Here the student in-verted the divisor but then multiplied incorrectly(3 x 5 = 18).

5 2.4

x ji8 8

Reviewing the need to invert the divisor and theprocesses for multiplying fractions may be helpful toyour students. Again, working on multPlicationtables may be important for some students.

Addition and subtraction of decimals. Any stu-dent who works with money or financial recordsneeds to be able to work with decimals (also calleddecimal fractions). Aside from ordinary addition andsubtraction procedures, most difficulties with deci-mals relate to placement of the decimal point and todealing with zeros. Below are typical errors in deci-mal problems.

Falling to line up decimal pointsBy align-ing the decimals incorrectly, the student gave17 a value of only .17 anti 4.5 a value of only.45.

2o17.

A c82)

I

41

Failing to extend numbers with zeros insubtraction problems ----In this problem, thestudent should have extended 4. to 4.00. Thismould have made It possible to borrow and thento subtract 6 from 10, 1 from 9, and 3 from 3---for a correct answer of .84.

4.

1.16Adding or subtracting incorrectlyHere thestudent subtracted (7 - 2 = 5) instead of add-ing (7 + 2 = 9).

G.2 7.

C9.55When having trouble In this area, students need

to remember the following:

A whole number has a decimal point after it.All decimal points must be lined up in both ad-dition and subtraction problems.In the answer, the decimal point is placed di-rectly beneath the decimal points in the prob-lem.

Multiplication of decimals. Again, the mainproblem (other than not knowing the multiplicationtables) is placing the decimal point in the answer.Following are typical errors in multiplying decimals.

Failing to count decimal places In areswersln this problem, the student multipliedcorrectly but placed the decimal point in the an-swer below the decimal point in the multiplier.He/she should have counted the decimalplates in both the multiplier and the multipli-:And (three places) and placed the decimalpoint to the left of the 4 in the answer

4.13x826

2005

43

2 147.6

Multiplying IncorrectlyHere the studentforgot to wry when multiplying 5 by 4.13.

.4. 13

826205

Aside from working on the multiplication tables,students need to remember to count the total num-ber of decimal places in the two factors and then tocount the same number of places In the answer tolocate the decimal point correctly.

Division of decimals. The major stumblingblocks in the division of decimals we difficulty withdivision of whole numbers and placement of thedecimal point. The following we two typical errorsthat students tend to make:

Not moving decimal pointsIn this problem,the student foiled to make the divisor (.15) awhole number (15) by moving the decimal pointtwo places to the right and also felled to makethat same change in the dividend (30.37).If done correctly, the problem would havebeen stated and solved as follows:3037 ÷ 15 = 202 (remainder 7).

24. 0

.15 .30.37

307

Not placing the *militia point in the quo-tientHere the student correctly moved thedecimal points in the divisor and dividend. Buthe/she :alled to place a decimal point in thequotient. The correct anew would be 27.8, not278.

Students need W. remember the following basicrules of dividing decimals:

Move the decimal point in the divisor &lithe wayto the right to make It a whole number.Move the decimal point in the dividend thesame number of pieces.Place the decimal point in the quotient directlyabove the decimal point In the dividend.

It may also be helpful to explain that movement ofthe decimal points in the divisor and dividend rep-resents multiplying them both by the same number(e.g., by 10 or by 100). If stunts are having troublewith division, multiplication, or subtraction in work-ing the problem, they probably need to work on thetables.

Computation is not the only area of math thatgives students trouble. They may also have mob-!ems In the other basic math Witsmeasurement,estimation, equivalents, problem solving, and so on.The Wowing are studegies for helping studentswho need to Improve their sidle in these areas.

Measurement, estimation, and equivalents. Asa vocational-technical teacher, you have an advan-tage in teaching measurement and estimation sidleand comprehension of equivalents. These sidits arebest taught In relation to their practical uses on thejob. The specific measuring devices used and theways they are used will depend on the occupationalarea.

Measurement usually requires other fundwnentalskills as wen, such as reading and writing numbers,counting, ordering numbers, and working with frac-tions.

An widerstandThg of equivalents Is also involvedin many measuring tasks. For example, a person Inthe construction trades will need to measure in feet,inches, and fractions of Inches and to understandsuch equivalents as 12 inches = 1 foot.

A person In commercial foods will have to mea-sure ingredients by teaspoons, tablespoons, cups,and fractions thereof, and In metric quantities (e.g.,grams and liters). He or she also needs to under-stand how one measuring unit rbiates to another hifact, in many occupations, workers are increasinglyconverting to the metric system. Students who leaveyour program familiar with metric equivalents WI bemuch better prepared for future trends in their oc-cupations. ,

actuat-en-theiebappliaatione-mabe-an-effective way of improving students' skills, not onlyIn measurement and comprehension of equivalents,but also in the underlying computation sidlis. Whatbetter way is there to learn fractions, for example,than by manipulating them in taking accurate mea-surements?

42

44

Some students may find it helpful to memorizethe equivalents most commonly used In the partic-ular occupational area. Doing so reduces theamount of computation they will need to do in thefuture and makes it easier to do mental figuring.

For example, a student in a textiles programshould know the decimal equivalents of 1/8, 3/4, %,7/10, 1/4, Va, and 3/4 in order to figure yardage costsmore easily. The following are examples of howfraction and decimal equivalents can be inter-changed for easy mental calculation. Notice that thetechnique of splitting factors, shown in sample 12,is used in both these examples.

24 yards at $2.5024 yards at 21/2 =-

33/4 yards at $2.00

3.75x2=24x2 = 48 3x2= 6.00

24 1/2 - 12 ,75 x 2 1.5048 + 12 $60 6.00 + 1.50 = $7.50

A helpful rule of thumb for students to rememberin converting from one measurement unit to anotheris this: multiply to get more units of a smaller mea-sure; divide to get fewer units of a larger measure.

Estimation is a skill that comes primarily frompractice. You can devise a variety of games andexercises relevant to your vocational-technical areathat will give students practice In estimating. For ex-ample, you might line up several containers and ask

' stuckints to estimate how many ounces each willhold. Then you could have them actually pour waterinto them to check their accuracy The same kind ofactivity could be done using board lengths, numberof feet or yards from one point to another, and soon.

_"""4"1

For estimating answers to computation problems,students should be encouraged to round off num-bers for rapid estimation. They may also find it help-ful. to establish points of reference. For example,they know how big a half gallon of milk is. Is thisother container larger or smaller? Or, they know thatthis student'shi waheighllt

is about 6 feet. Now muchgher is the ?Problem solving. Problem solving is essentially

the application of judgment and computation skillsto obtain needed information. Part of the ability tosolve problems is simply knowing that you can.

As with measurement, problem solving is besttaught in relation to real, on-the-job problems. Thekinds of problems your students will be called uponto solve wil depend entirely on their vocational-technical area and occupationatgoals. Nbu can helpthem knprove their problem-solving skills by pre-senting such problems in a real contextnot aswritten problems with all the facts neatly laid out forthem.

One approach is to begin by reviewing the prob-lem-solving questions;

What do I need to find out?What do I already know that can help me findthe answer?What more do I need to know in order to findthe answer?Now can I use what I already know to find outmore?What is the solution?

vbu might then walk students throup the steps ofsolving some sample problems by asking them thekey questions and halting them provide the answers.As students gain confidence, they should becomeincreasingly able to ask and answer the questions.themselves. Eventually, they shOuld become betterable to recognize problems as they naturally occur,to intuitively sort out useful information, and to findthe answers they need.

Organization of data. Students need to be ableto set up, read, and draw conclusions from numeri-cal data in the form of charts, tables, graphs, orother graphic displais. They should realize that nu-merical data compiled in these forms can actuallybe a shortcut for themespedally if they havereading problemsbecause a lot of extra languageis eliminated.

Ilaing_able to use numerical data in these formsis partly attitudinal. That is, your students need to beconfident that they can understand the data forms ifthey use a logical approach. You can help your ski-limb Improve their skills with organized data bypresenting some of the data forms they will encoun-ter in their work and showing them how to read theforms.

43

lj

a

Provide PrecticeActivitite

No manor how - the mathematical theories areexplained and specific trouble spots ate corrected,your students will need plenty of practice to improvetheir math skiNs significantly Practice, however,does not have to consist of page after page of prob-lems. There are a variety of ways in which you canprovide practice in specific math skills within yourusual curricukun.

Class Assignments and ProjectsAmong the most important avenues for practice

are the assignments and projects that are a naturalPart of your program. NW can set up these activitiesto include the math skills in which your studentsneed practice. TIft can be done on an Individual-ized basis so that each student t work requites thetypes of math in which he or she needs to inwrove.

example, imagine that building trades stu-dents are working on blueprints. For a student whoneeds practice on working with fractions, the in-structor could assign scale and dimensions in a way

. that would emphasize computation with fractions.On The same activities, the instructor might have an-other student work with decimals to provide practicehi that

Similarly, if dietetics students are teaming to plansupply orders on the basis of a week's menus, theInstructor could arrange individual assignments toemphasize computation with whole numbers, frac-tions, decimals, percentages, or whatever skills othestudents need to practice.

Math ExercisesIf you have taught your students mathematical

shortcuts and tricks to help them improve specificmath skills, you can provide practice exercises toenable students to gain skill in their use. For ex-ample, you might provide exercises, drills, or evencontests (for speed and accuracy) in which studentsdo such tasks as the following:

Add by grouping numbersEstimate by rounding off numbersCompute by converting firbt to an easier form(e.g., fractions to decimals)Compute by using number lines or charts

In order to make math more fun and familiar tostudents, It is important that any such exercises bepresented in a positive atmosphere. Activitiesshould be designed to encourage students to In-crease their accuracy and their speed, while alsobreaking down some of the attitudinal barriers that

-students may have built concerning math.

45

A math specialist may be able to help you developmath exercises, appropriate to your eltgam, thatfocus on specific side needed by your students.

GamesMath games, if carefully chosen or constructed

and correctly integrated Into the program, can be aneffective and enjoyable way to practice math skills.They should be designed to provide practice in theSpecific skill i; that students are trying to improve,and they should be used to provide practice at theright timewhen working on those skills.

Games of chance, for example, can be con-structed to provide practice in probability and prob-lem solving, as well as computation. For example,in one such game, students have four cubes or dice:two have the numbers 0 through 5; two have thenumbers 5 through 10.

By rolling the cares one at a time, the studentstry to roll as close to a Oat of 15 as possible. Eachstudent may roll as many of the cubes as he/shewishes, may roll each cube only once, and may stopat any time. While providing practice In addition andsubtraction, the game also challenges students toconsider the probability of improving scores by se-lecting different cubes to rote

1

6. Described in Stephen S. Willoughby. Thaching Aftefleateck What isBasic? (Washington. DC: Council for Basic Education, 1901), pp. 27-28.

ltu can invent other types of gams, such as thefollowing, to provide practice in specific math

Contests of speed in the arious computation

Guessing games (guessing the volume.of con-tainers, length of distances, and soon)Brainteasers (these can be given out at the endof a class and reviewed at the beginning of thenext)

Some commercial games provide excellent praa.*ice in specific math skills. For example, suchgames as Yahtzee, backgammon, bridge, Monop-oly, Uno, and Oh-No 99 all require somemath. Nbu may feel that such games are not an ap-propriate use of class Orne. especially since the re-lationship of game playing to learning often appearsremote. However, your situation might permit stu-dent use of such games during lunch, scheduledbreaks, free periods, or even as an incentive or re-ward at certain times.

SimulationsSimulations are another way to give students

practice in specific math skills, while also develop-

IOptionalActivity

%moo

Iry occupational skins and weer avairet-Ass. Al-though creating a simulation requires a good deal ofpreparabon time, simulations have several actvan.tapes:

They can be Lsed repeatedly, often with differ.ent results.They can be designed specifically to enhanceyour vocational-technical program.They can be developed in such a way that theydo not appear childish to the students.They can be adapted to meet individual studentneeds:Roles can be deigned to estphasize specificmath skills.

Developing a. simulation calls for identifying oc-cupations related to your vocational-technical areaand developing a role, or a task to be accomplished,for each occupation. Usually simulation roles de-pend on interaction among or between role-pkwers.You can set up each role wfth facts, figures, and taskinstructions that require the use of specific math

7. To gain ski in using simulations, you may wish to refer to ModuleC-5, Employ Smulation Techniques.

To increase your awareness of available techniques and materials for im-proving specific math skills, you may wish to review student and teachermaterials such as those produced by the Interstate Distributive EducationCurriculum Consortium (IDECC). These materials include learning activitypackages for specific math skills, such as "Addition and Subtraction," "Multi-plication and Division," and "Fractions and Percents."

The Bowing case snuatIons descrke four students who need to improvespecific math skills. Read each case Corr and the question following N.Using the question as a guide, expkthl ki writing how you would help thestudent Improve his/her math skins.

CASE SITUATIONS1. Carolyn Mehaffie, a student in your program, has turned in a worksheet with the following problems on

It.

15 1-111°A.712,0671542,42

lob

2(0 I-522LZ-16157(05

2137

20076

185180

What math skills do you think Carolyn is having trouble with, and how would you help her improvethose skills?

47 49

2. Al Frized has come 61 you saying that he can't complete a class project because the dimensions don'tcome out right When he explains how he came up with his figures, you find that Al is having troubleadding and subtracting fractions.

How would you go about helping Al implove his skits in working with fractions?

3. Bijah Moore has been working on completing one forms, using self-evaluation checks to monitor his-..- own progress as he works through the unit. Elijah his come to you because his final cost figures are

on *Nom and he doesn't know why Looking over his ortisrs, you find that he is able to compute with'"" . whole numbers fairly weft but that ho does not understand placement of decimal points in decimal

computations.

How would you help Elijah !approve his skits with decline's?

4. NW have assigned a project requiring your students to (1) figure how much mate41 they will need b make an item of a given size, (2) obtain the materals, and (3) complete the pr?ject. Most of your students have begun the construction phase of the project, but three students are ilill tsylng 10 figure a out how much/Material they will need. They simply don't know how to solve the problem. How would you help these students improve their problem-solving skills?

50 52

Compare your written responses to the case situations with the model re-Feedback sponses given below bur responses need not exactly duplicate the model

responses; hottever, you should have covered the same major points.

I

MODEL RESPONSES1. Cero lyn seems to have two areas of difficulty

The first and most serious is that she has notmastered the multiplication tablesas indi-cated by such errors as 3 x 15 = 42,8 x 26 = 200, 2 x 26 = 58, 7 x 26 = 180,and 6 x 7 - 46.The second problem is that, while she appearsto have a fairly goon command of the divisionprocess, she is neglecting to bring down zerosto maintain place values.

There are several possible strategies for help-ing Carolyn to improve her multiplication skills.First, she should be encouraged to memorizethe multiplication tables, perhaps beginning withsome of the easier ones, like 2s and 5s.

Various short