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ED 252 739
AUTHORTITLE
INSTITUTION
SPONS AGENCYREPORT NOPUB DATENOTE
AVAILA3LE FROM
PUB TYPE
EDRS PRICEDESCRI?TORS
DOCUMENT RESUME
CE 040 596
0
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)
-
--11111110
AssistStudents
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ath Skills
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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
OptionalActivity
2
10
OptionalIL ActivityNo 3
41%
Activity
4
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. _
MI
"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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IligtivaistiteMany. meth skis depend on anundemNinding of now
ineaSureinerds on Marant scales relate to sad) other
(0.4,,maitic.and_
-Faacnsitriddecimals, feet and
yards,-----ounearandTiotimts);
Organization of dateStudents successmay Often depend on their
understanding differ-
- ent-tvays -numerical dais- are displayedarsPhe. wales,
schedules, tables, charts);1Lable to oolled,( gatilie and
interpret.....dets; and developing the confidence not to
beintimidated by sheer inataini nurnOenk
For some areas, students need tobe obis 10 set up and solve
equations and to
**Wiens to
Specific Strategies
to use geometric principles and fonnulasangles, area, volume,
perimeter,
soon.
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
theYTeed 61,1014i about basicmadiAbu
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,
11
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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.
..
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
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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:
14
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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.
11..111.`
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?
at is I ;
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
Computing blidgetsYou canirsesiroWernsspch as these to
demonstrateo..-the-job problem solving. garnple 1 shaws
hcnir"--problem-solving questions and related math can beapplied to
a classroom problem (in this case, buyingfabric for draperies).
1 16 18
-
SAMPLE 1
11) APPLIED PROBLEM SOLVING
Figuring Drapery Fabric
Whet do) need to find aid?How much fabric should I buy?
What do I already know?
3 identical windows, sizes as shown
Finished length: 7 in.Header allowance: 3 in. + 3 in. = 6
in.Horn allowance: 3 In. + 3 in. 6 in.Total width: face of rod + 12
In.Fabric width: '44 in.
What other In orUnfinished drapery length per windowNumber of
panels per window
How can I use available kdormadon to learn more?
WM,1-54" ROD -01
Unfinished length = finished length+ header allowance+ hem
allowance
Panels per window - total width x 2
Om... OW. OW .01 0.41,
FABRIC
4
HEM
72,+ 6 , "
8 4" urifinisha:1 lencifh
fabric width 54 66 F>erx 4 4, 132. wteldow
-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