Teaching physics with math to weak math studentsMark Vondracek

Citation: The Physics Teacher 37, 32 (1999); doi: 10.1119/1.880144 View online: http://dx.doi.org/10.1119/1.880144 View Table of Contents: http://scitation.aip.org/content/aapt/journal/tpt/37/1?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in The Challenge of Teaching Introductory Physics to Premedical Students Phys. Teach. 45, 552 (2007); 10.1119/1.2809149 The MCAT physics test Phys. Teach. 38, 364 (2000); 10.1119/1.1321822 PSSC (Physical Science Study Committee) Physics Phys. Teach. 37, 294 (1999); 10.1119/1.880293 Heath Physics Phys. Teach. 37, 288 (1999); 10.1119/1.880286 Mathematical tutorials in introductory physics AIP Conf. Proc. 399, 1075 (1997); 10.1063/1.53110

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32 THE PHYSICS TEACHER Vol. 37, Jan. 1999 Teaching Physics with Math to Weak Math Students

How many physics teachers,especially those who teachgeneral, first-year high-

school physics, have been at timesfrustrated with classes that consistlargely of students who have weakmath backgrounds? My guess isprobably all. We would normallyassume that all junior and senior stu-dents have some level of competencein basic algebra, especially if thesestudents have signed up for physics.Unfortunately this is not the case,especially in urban schools.

The ChallengeWhile teaching physics in an

inner-city Chicago public highschool, it was not unusual for me tohave classes of 26 to 30 students (thatwas the room’s capacity). And withthose students it was not unusual tohave average math competencyscores, based on state standardizedtests, below the thirtieth percentile; aquarter to half of a given class maylie below the fifteenth percentile. Amajority of students may not be ableto add or subtract fractions, let alonedo algebraic manipulations. This canbe a problem for a teacher who doesnot want the course to be strictly con-ceptual. Math plays a key role inteaching physics because it allowsstudents to see how various quantitiesare related to each other in nature,and it exposes direct and indirectrelationships in a way that allows usto make predictions for a wide varietyof everyday problems.

If a physics teacher decides toinclude mathematical problem solv-ing in a class dominated by weakmath students, disaster awaits ifequations are given without explana-

tion and problems are dropped ontothe students. The students willbecome discouraged and lose all con-fidence; any hopes of turning them onto the physics will be quickly extin-guished. But does a physics teacherhave the time to (re)teach simplealgebra in order to build the skillsnecessary to solve F = ma? If otherschool districts are like Chicago’sand are going to standards-based testsin all classes and subject areas, timeis short enough to just get through thecontent you and your students will beevaluated on at the end of the year. Ifyou are in a situation like this, youmay want to try a relatively quick andsimple method that worked fairlywell in those Chicago classes. In fact,over two years the physics enroll-ment increased from six to 14 classesonce students saw they could actuallyunderstand physics and apply someof the math they had seen in otherclasses.

A SolutionThe method used is based on the

following observation: most of theequations used in an introductory,noncalculus physics class can be bro-ken down into the general form a =b/c. Things like v = d/t, a = �v/�t,density = mass/volume, slope =rise/run, and power = work/time arenaturals. But other favorites can beintroduced and taught in the sameform: a = F/m, I = V/R, v = p/m,height = potential energy / weight,and so on. Early in the year the stu-dents spend two or three days of“drill” in how to use the equation a =b/c so they become expert in solvingfor any of the three variables, andthen throughout the entire year they

keep seeing equations of the sameform and can very quickly use themin problems. Their self-confidenceincreases each time they use a newequation to solve problems involvingnew topics, and many are in factamazed at how well they begin tounderstand the physics since they nolonger worry about the math.

This method seems to work bestby starting the year off with some-thing the students can pick up quick-ly. Density, for example, can be intro-duced during the first day or two ofclass since many have at least heardof it, and some have learned about itin chemistry classes taken the yearbefore. It also works nicely becausemajor concepts such as mass and vol-ume are involved, which even weakscience and math students can identi-fy. In addition, measuring densities inthe lab is easy enough where studentscan start experiments during the firstfew days of class. By taking the timeto show how density = mass/volumeis of the form a = b/c during the firstweek of the year, and by making sureevery student can manipulate it tosolve for any of the three variables, itis possible to move directly into atopic such as linear motion andanother familiar concept, speed. Eventhe weakest students will already beable to tell you what your averagespeed is if you travel 100 miles in twohours, or that if you are driving witha speed of 30 miles per hour for threehours you will travel a distance of 90miles. They already know thisthrough years of experience. Whatthey may not realize is, on paper, theyare manipulating an equation of theform a = b/c. We just happen tochange the letters and call it v = d/t.

Teaching Physics with Math to Weak Math StudentsMark Vondracek, Evanston Township High School, 1600 Dodge Avenue, Evanston, IL 60204; mfv@fnald.fnal.gov

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Teaching Physics with Math to Weak Math Students Vol. 37, Jan. 1999 THE PHYSICS TEACHER 33

Upon learning this and seeing theconnection to those drills they wereforced to do on a = b/c problems, alarge majority of students can imme-diately begin solving speed problems.And then they can begin solving sim-ple acceleration problems, and thenproblems with Newton’s second law,and so on throughout many of thetopics covered during the entireschool year. Students who do notimmediately catch on usually willover time, especially through groupwork with other students who have“mastered’’ a = b/c.

Beyond Plug and ChugThere is another benefit to having

students who understand the solu-tions to a = b/c, and that is the differ-ence between direct and indirect rela-tionships. Throughout the year, it ispossible to do experiments where noequations or relationships are provid-ed to the students before entering thelab. For example, students may cometo class one day and find some resis-tors, wires, and power supplies ontheir lab tables. After demonstratinghow to safely and properly set upsimple circuits, students begin tomeasure voltages and currents whencertain resistors are connected to thepower supply. Students will be able toderive Ohm’s law simply by observ-ing how the current depends on volt-age and resistance: currents increasewith larger voltages, and they tend todecrease with larger resistance. Theywill be able to write amps =volts/ohms without any problembased on their data. It is really quitethrilling to watch, knowing that at thestart of the year many of the studentsgave themselves zero chance ofunderstanding anything in a “hardclass” like physics.

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