Mathematical Ratio & Proportion: An Introductory Physics Teaching
Experiment Exposes a Serious Barrier to Learning Physics
and Validates an Effective Remedy.

Abstract. An experimental attempt to increase student understanding of topics in physics that involve mathematical proportion encountered an unexpected result: Students showed higher than normal difficulty learning the higher order generalization that the physics equations involving proportion:

1) Are elaborations of one general equation. (Example y = mx, where y & x are the proportional variables    )
2) Follow a consistent pattern wherein the constant m is nearly always an “invented” new concept defined by the ratio m =y/x.
3) Reveal the newly defined concept to be a physical property which we experience.
4) And thus are quantities we want to measure, talk about, and list as
numerical values in charts and handbooks.

Although many students (~ 10 %) were receptive to the above ideas, the majority of students (~90 %) had a greater than normal “barrier” to seeing/adopting the “obvious” generalizations presented. This article discusses the nature of this barrier and how to achieve “break-through” using a teaching method called the “Table of Proportional Quantities”. This method significantly reduced the learning barrier and thus paves the way for easier learning and improved student understanding of proportion, a mathematical system that is central to all of physics. The ability of students to understand and appreciate the above-mentioned generalizations apparently depends on “formal thinking” as opposed to “concrete thinking”.

Introduction and Overview.
In my efforts to improve student understanding in my Introductory Physics Classes, I noticed that: 1) ratio concepts, such as are mentioned above, occurred over 60 times, 2) And despite the fact that all 60 of them are similar mathematically, 3) Nevertheless physics text book authors introduce each new ratio concept with a wide variety of presentations.
Now.....AC Power, Nuts & bolts, and Highway Signs are standardized for very important reasons. They are easier to learn, plus convenience and effectiveness of use. For these same reasons, I decided to standardize how I introduce my students to ratio concepts.
My standardization efforts, combined with the ideas of other authors, developed gradually to into a new method, which guides students to an important goal: expand and generalize their understanding of proportion.4 However as the students are asked to “expand and generalize” their understanding of proportion, they clearly encounter a (hard to describe) “invisible barrier”. The only method that seems to “break-through” student blinders is a work sheet called the “Table of Proportional Quantities” (TPQ). If and when the students do break through the “barrier”, as shown by their writing, they are often positive and occasionally enthusiastic about the ideas gained. This “break-through” and positive response of these students over the years has confirmed the validity of (and the need for) what I am promoting.
The observations/conclusions stated here-in concerning student’s state of learning and attitude are the cumulative result of over 12 years of proportion teaching experiments combined with a very close study of student performance on lab reports, quizzes, special projects, and exams. Each successive year I would think carefully about the most recent student learning progress, bring in ideas gained from my current reading of Physics Education Research and alter (hopefully improve) my class teaching strategies for the next cycle of experimentation. I Although the results reported here-in involve the vague and problematic “leading edge” of student perception, I hope to provide enough “clues”, so that the physics teacher can confirm that these results are reliably present in their own classes year after year.
 

Fig. 1. Fall Semester Introductory Physics Labs Which Are Taught Using Roger’s Model.

“Summary Code” Is Shown for Rogers Example and Each Respective Lab.

        Name                           Proportion                  Definition New Concept           Equation

*Note: In this table and other places in this article, Firefox Internet Browser will display the definition symbol (three horiz bars) as a tiny zero.

**Note: All mathematical proportions have a meaningful constant ratio. However in the labs marked with a double star the ratio is not usually considered (and defined) to be a new concept.

Fig. 2. Rogers Model Adapted from his Textbook        

    PHYSICS FOR THE INQUIRING MIND

                      by ERIC M. ROGERS

Proportionality, the Key to Many Laws

In codifying our knowledge of nature in simple laws, we look first for constancy: the mass of a body remains constant; total electric charge remains constant; momentum is conserved; all electrons are the same. Almost as simple and equally fruitful is direct proportionality, when two measured quantities increase together in the same proportion: stretch of a spring with its load; force and acceleration; gas pressure and gas density.

We say that for a good spring (within the Hooke's Law range):

        STRETCH  is proportional to LOAD     or we write
              STRETCH        varies as        LOAD     or  we write
                        STRETCH      LOAD

Like percentages, proportion and variation are often given special treatment in elementary teaching that makes them seem mysterious and hard. Without this conditioning people would find them obvious pieces of common sense. So we shall discuss some simple examples.

Suppose that in providing potatoes to feed a camp, we find the weekly needs are
         for a camp of 100 men,   400 pounds of potatoes
              "               200,          800         "         
               "              300,         1200        "
               "              500,         2000         "

The mass of potatoes increases proportionally with the size of the household to be fed. Here is the simple type of relationship that we meet so often  in physics. We shall give several wordings to describe it:

  (i) MASS OF POTATOES is proportional to NUMBER OF MEN
 (ii) MASS OF POTATOES varies (directly) as NUMBER     OF MEN
 (iii) MASS OF POTATOES     NUMBER OF MEN (This is mathematical shorthand for (i) or (ii)
 (iv) MASS OF POTATOES  = (constant) NUMBER OF MEN

Versions (i) and (ii) and their mathematical form, (iii) are simply attempts to say the simple obvious things, "The two go up in the same proportion. If we double one, the other doubles; if we triple one, the other triples; and so on." With that view in mind, you can easily solve problems, without any calculating or using the "constant" of version (iv). For example, given that 100 men need 400 lbs of potatoes, how much do 600 men need? Six times the number need six times as much food, 2400 lbs.

Tests of Proportionality

How can we recognize simple proportionality in analyzing measurements? In the potato example, the numbers given make the relationships obvious, but we need easy tests for more obscure cases. Here they are:

TEST A.
 Divide one measurement by the other and look for a constant result. In our example:

NUMBER                MASS OF                  MASS OF POTATOES PER
 OF MEN,               POTATOES              NUMBER OF MEN

  100                        400 lbs                        4 lbs/man

  200                        800 lbs                        4 lbs/man

  300                      1200 lbs                       4 lbs/man

and so on,……always giving the result .…  4 lbs/man.

The "Proportionality Constant"

Here is an unfailing test of direct variation or proportionality. It works either way of course: if we divide NUMBER OF MEN by MASS OF POTATOES, we get another constant answer, 1/4 man per pound.

 MASS OF POTATOES   =  (constant)  NUMBER OF MEN

is much the same as version (iii), but to the scientific eye it does not emphasize the idea of relationship so brightly; therefore, you should avoid it where you can use common sense, as in the two examples above.

      Clearly in each of the data given for potatoes

 MASS OF POTATOES  = (4) NUMBER OF MEN, N

so we can include all the examples in the statement P = 4N.

The essence of this last statement is the relationship: not so much the actual value 4 as the fact that the 4 stays the same, remains 4, remains constant. (Actually, it is the individual consumption, 4 lbs per man.) Since the 4 is constant, we can say

                    P = (constant) N

This general statement should also apply to camps of men from a different district where all are great potato‑eaters, consuming 10 lbs of potatoes a week. Then the statement would take the numerical form P = (10) N. (Of course, if we mix the two types of potato‑eater arbitrarily, the whole story fails‑and we must beware of the corresponding danger in scientific laws.)
END of  Rogers Model as adapted from his Textbook

My Alternate Approach: Expanded and Generalized Proportion Following the Rogers Model.
 The ideas and relationships that I call Expanded and Generalized Proportion were gradually evolved and became an integral part of my instruction in my Introductory Physics classes.5  I borrowed heavily from the presentation techniques of physics text Physics for the Inquiring Mind by Eric M. Rogers6. Preexisting laboratory sessions, listed in Fig. 1 above, were revised so the discussion, data, tables, graphs, ideas, and meaning closely followed the “Rogers Model” as spelled out in Fig 2. The overall learning experience, for each successive laboratory, demonstrated the complete Rogers’ message and meaning of proportion.
The preexisting lectures and homework assignments, provided additional opportunities to illustrate proportion following the Rogers Model, but were considerably abbreviated. As was the case for laboratory sessions, the lectures featured real physical demonstrations concurrent with the presentation of the symbols and meaning of each new proportion. Of course the names of the variables (and constant) were appropriately changed. The arrival of each new proportion thus followed the same pattern. The pattern was modified to handle topics that involved inverse proportion or proportions involving more than two variables.
I worked very hard to help my students understand how equations of proportion are really just variations (or elaborations7) of the equation y = mx where m is the proportionality constant defined by the ratio y/x. I have also tried to show the students that “there is a repetitive system being used here" and if they were to understand (and appreciate) this ‘system’, their understanding of physics8 would be much easier and more powerful. I told the students this understanding would help them when they encountered each (of the sixty) new proportions, especially the more esoteric ratio-concepts, such as Electric Field, Magnetic Field, & Entropy.9 I tried to get the students to pay attention to the definition process and see that each new proportionality constant is actually a new concept “creation” (i.e. invention) and the new concept is actually something we experience in our daily life.

 I naturally expected students to meaningfully integrate the specifics of the Rogers Model in to their knowledge and I gave quizzes to check their progress as well as provide pressure to become proficient. Some students readily progress toward this goal and are positive to the ideas gained thus showing the value and validity of this approach to proportion. However I found out that the average student had more than moderate difficulty (resistance10) in memorizing and reporting even a minimal “mathematical code” summarizing the definition process as is illustrated in any one horizontal line of Fig.1. Moreover these average students actually resented being asked to memorize and write a brief summary such as is given in items 1) through 4) in the Abstract above. One year, finding very low quiz performance, I repeated the proportion lecture several times with even more emphasis as to the importance of my presentation. I gave quizzes more frequently than normal and required students to write even longer paragraphs about the important features of proportion. No improvement resulted. The “barrier” remained strong as ever!

 Because the resentment was obvious and the resistance so much more than other physics topics, prudence dictated a less confrontational approach. Laboratory sessions continued as stated above. Lecture discussions concerning proportion stated all the ideas, but in very abbreviated form. Quizzes requiring students to write about the important features of proportion were discontinued. Quizzes requiring students to report the “mathematical code” summarizing the most recent lab continued, but student apathy to doing so was evident.
In retrospect, I deduce that the students, on average, were “blind” to what I was trying to tell them, and of course this would leave the student helpless on the quiz. Students never mentioned their reservations, but their behavior speaks volumes: “ Give me a break. All the professor’s noise on this topic sounds like a broken record. Just give me the equation, that's all I need (or want to know).”11
The semester long effort to move the students toward a generalized and expanded understanding of proportion, exposes/ reveals those students who for a variety of reasons can not or will not move in that direction. There appears to be a “Barrier”.
There is, however, a successful way to “penetrate” the barrier and move students toward “deeper” understanding of proportion, a topic to which we now turn.

Increasing Student Understanding of Proportion and Proportionality Constants Through the Table of Proportional Quantities.
 The Table of Proportional Quantities (TPQ), Fig 3, is a two page exercise sheet upon which the student writes, in an organized table, the proportion equations of the Fall (or Spring) Semester Physics
The student is asked to follow the instructions and the examples supplied (at the top left & top center) of the first page and then fill in the empty spaces.
For over 12 years my students have completed the TPQ as an extra credit review sheet at various (experimental) times of the semester. By long experience I discovered that:
1) The TPQ actually is a good way for the student to review for the final exam.
2) The week prior to final exams was found to be the best time to have the students work on the TPQ.
3) The pressure of the final exam forces the student to desire an overview of the whole semester and limited time propels the student to   
      seek ways to organize and compactify information.
4) The students, when preparing for the final exam, will pay attention to what the TPQ offers, whereas offered earlier in the semester the
      TPQ is trivialized or ignored.
5) Moreover the effort of filling out the TPQ12 actually helps students break the above-mentioned barrier.

An Organized Study of the Learning Benefits of the Table of Proportional Quantities.
 The positive effects of the TPQ were first discovered through informal study of student feed back. For the purposes of this paper, a more organized study was conducted using the APHY 201 Final Exam Fall 1997. Students were supplied with copies of the TPQ similar to that shown in the appendix with the following instructions:
           This exercise sheet will help you prepare for the Final Exam.
     Effective and important learning will happen, if and only if, you complete this
     exercise sheet with your own efforts. You will be allowed to have these sheets
     during the final exam, if they are completed satisfactorily.  You may write anything
     else you want to on these “review sheets” (The last two unusual provisions were to
     insure that all students would complete the exercise. This was done only my last )
     year of teaching in preparation for this research article.

The final exam added a 12% credit proportion question. The students were given the following directions:
“You are allowed to have with you during this exam the two page review sheet, provided you filled it out yourself primarily using your own knowledge and skill. Describe how you went about filling this review sheet. Considering the part of the table that deals with proportion, please describe how the “filling out process” actually progressed and whether this effort resulted in any new or different or increased understanding of proportion. You may have other important facts or ideas or ways of understanding proportion which you may wish to include in your discussion to convince me you “really understand” this important topic.”

“I Don’t Have To Memorize Proportion Equations Anymore, I can figure them Out!”.
Student responses concerning their work in filling out the Table of Proportional Quantities. Thirty-two students completed the exam. Transcribed below are selected portions13 of the students writing which show that the TPQ actually helps students increase (and generalize) their understanding proportion:

Jennifer....The more rows I completed, the easier it became. I could now see and understand the pattern mentioned
                                 previously.....After the process of filling out the sheet, I began to feel as if a weight had been lifted
                                 from my  shoulders ...... It became easier to see how terms related to each other.....
Brandi.....The way the sheet was set up made is fairly easy to see the relationships in the proportions and before long I had a system
                                set up.........It definitely helped me......
Kathy.......AHA ! After I discovered this [pattern], it made it much easier to whip out equations.......
Chuck.......I came to realize that they [proportions] really are all set up in the same fashion.
Darryl.......The process became routine, and it was much easier to “see” how they fit together [by]
                                filling this [work sheet] out than “seeing” it during the course........
Andrew.....In tracing it [one of the examples], I saw the process & therefore acquired the pattern......
Jeff............It seems that all proportions use the same idea.........
Rachel......The process expanded my ability to recognize proportion   ....  Now I realize that.... [Here Rachel gives complete discussion
                                  of proportion as it applied in an other class.]
Paul...........I found that even though I couldn’t remember all the equations necessary, some understanding
                                  of the proportions could help in getting it.........

Transcribed below are more extensive portions from student’s written responses.
The student responses given above, were of course, the very points I had been emphasizing all semester. But clearly they did not "see", until they did the needed work using the TPQ. Duplicated below are the more extensive of the student’s written TPQ responses. These responses are from the top students in the class, and are selected because they best show how far the student’s understanding of proportion is able to progress and indicate the role of the TPQ in their efforts.
 

Sarah...........I found that as I went down the page the proportions got easier and easier to do. The “proportion”, “definition”, & “practical equation” method was very helpful. I began to recognize proportions just by the equation. Like density, at the beginning of the semester, I knew what proportion meant, but didn’t know how to use it or recognize it. Now when I see m=dv or a similar equation I know right away that m     v just by looking at it. The worksheet made it very simple to derive equations when given proportion. Follow these three easy steps, you can derive an equation from any proportion.”

Joslyn........I really started to figure what you wanted us to get from this [work sheet]. You wanted us to know how to get the equation and not just to memorize them. If you know that two or more things are proportional then you can figure out the definition and the useful equation. .... Its pretty simple until you get to more than two variables. But it's not that bad if you take it Step by Step........

Brandon...  The most identifiable theme on the sheets is the proportions. I noticed several things while filling them out. First, it is amazing to see the amount of real world principles of physics that are simply proportions. All of the ideas, from velocity to the Universal law of gravity are found in the same manner. Another obvious fact is that new ideas [concepts] are found by using two or more things that are proportional........

Harry..........While completing the proportion sheet I learned how scientists of yesterday and today figure out relationships between different phenomena. I saw in Column 3, over and over, how a person sees and finds that there is a direct relationship or “proportion” between two different things -- for example; mass     volume. Then I saw how this proportion can lead to the “definition”of a whole new property of matter like density. Since everything has a mass and volume [and] mass     volume [then] density is defined by mass/volume and as [a] result you have just created a totally “new” property of matter! ............... I saw the above pattern in all the equations on the [proportion] worksheet. I really felt this worksheet made one look at equations in a totally different and interesting way. Thanks Dr. Gurr.......

Dan............I did fill out an amazing amount with my own knowledge. Contact, .........It was amazingly helpful in increasing my understanding of proportionality. It went so fast, when I saw how they fit together! It was like poetic rhythm. Another “contact” as this hit me - this makes the understanding of the definitions + equations SO MUCH EASIER to comprehend. This review sheet was infinitely helpful to my comprehension. - THANK YOU!! ........

Latasha...........I followed the directions. But what I realized is that there is a pattern to the proportion, constant, and equation. The density equation for example, started off with mass     volume, then a constant d = m/v, and ending with m=dv. It helped me when I finally figured out this [pattern]........... This same idea applied [to the definitions of] power, the spring constant, frequency, pressure, surface tension etc. By the time I noticed this [pattern] I flew through filling out the sheet. It was easier once I realized this........Overall, I would say that I have better understanding of proportion, constants, and the ideas they relate...........

Heath..........Whenever I got stuck, I used the “set up” to figure out what [was] supposed to be in [the] definition & [ the] equation..........As I did this several times (after checking to make sure I was getting right equations). Filling the tables out became AUTOMATIC. [It was] good to go over all important concepts & calculations -- outlining physically [on the worksheet] was helpful in preparing for exam.........

Alicia..........Almost everything in the world involves some sort of proportion and / or relationship with something else. Throughout the semester we were taught to see things in physics this way. Filling out the sheet was easy. With a little thought, I moved from the proportion box, to the constant box and to the equation box. This was far easier than memorizing formulas & helped cement the ideas in my mind.........

Summary. Although proportions are the simplest of all equations and follow a simple and quite repetitive pattern, there are evident barriers that prevented many of my introductory physics students from seeing this pattern. I make a hypothesis: The students, who readily see this pattern, may be identified as “formal thinkers”. They will also see and appreciate the deeper meanings in the ratio concept definition process in all proportions. Students who resist, and / or fail to see this overall pattern, are probably concrete thinkers. The Table of Proportional Quantities is a good end of semester review exercise, and reliably helps many students to identify “the pattern” that pervades all proportions. A semester long emphasis on the importance of proportion helps the student to “think proportion” concerning real world quantities/ variables and to become much more aware how in proportion, there is truly a constant.
 
As is true for most university courses, physics presents of a series of ideas (trees). As the course develops it is hoped and assumed that relations between ideas (forest) become evident. These more global views, despite their importance, are hard to express in lecture, difficult to work into an already overloaded course, and even harder to test for. The results reported, show that helpful and needed generalizations, in case of proportion, do not “spontaneously happen” but require considerable forceful exposure. The fact that this generalization encounters so much resistance in students may account for the non-existence of such efforts in present day physics textbooks, and possibly even mathematics books.


Footnotes:
1 Purists may object on grounds that many topics, e.g. velocity, acceleration, are variables hence do not involve proportion. However close examination of introductory texts will show most such “variables” are actually first introduced (or shown as an example) as constants and only later are treated as variables by hand waving or an abbreviated calculus.
2 How we reason with proportion is discussed by David Bohm on page 59 of Shirley Sugarman (ed), Evolution of Consciousness (Wesleyan University Press, Middletown, CN, 1976) 1st ed. pp 51-67.
3 The author wishes to thank Harold Kelley for this historical note and many important suggestions concerning this manuscript in development.
4 As I worked to standardize proportion and ratio, my own appreciation increased. I became interested in the somewhat subtle techniques we use in science to bootstrap (from nothing!!!) concepts, such as Electric Field, Magnetic Field, and Entropy.
5 My Introductory Physics Course, lectures and labs, followed the content and ordering of the excellent text College Physics by Franklin Miller now unfortunately available only by special reproduction.
6 Eric M. Rogers, Physics for the Inquiring Mind, (Princeton University Press, Princeton, NJ, 1960) 1st ed. See pp 198.
7 All sixty defined ratio concept proportions of Introductory Physics, whether direct, squared, inverse, inverse squared, etc build by successive elaboration from a good understanding of the elementary case of direct proportion y=mx. There are some ten additional equations involving proportion where, for a variety of reasons, a new ratio concept is not defined.
8 The study of other mathematical sciences such as chemistry, geology, economics, psychology would likewise benefit.
9 I had additional motives: 1) Students need to understand WHY we define new concepts as ratios. 2) The typical “we shall define”...of many Physics Texts...entirely trivializes very important and crucial intellectual steps which are fundamental in creating basic concepts for our science. 3) As illustrated by Rogers in Fig.2, students must be able to “think proportion” concerning real world quantity/variables: “As this quantity goes, so does this other quantity in proportion, be it direct, inverse, inverse squared etc. 4) Students invariably think all symbols in any equation are variables, and thus need to become much more aware how in proportion, there is truly a constant, which absolutely can not be a variable, without the application of calculus.
10 Professor Arnold Arons has extensive and valuable discussion of how to teach ratio and proportion as well as discussion of student resistance to learning how to think with proportion. Pages 4-98 are highly recommended, most especially p 4-16, 23-28, & 97, in Arnold Arons, Teaching Introductory Physics, (Wiley, New York, 1997).
11 Various forms of “giving the equation” or “we shall define” as is done in many physics texts, is a sure way to defeat the meaningful learning of physics. In my experience students do not have the preparation/ experience to use such statements in constructing a meaningful understanding of physics at the level needed.
12 This is a clear case where the active effort of the student, is more productive than passively hearing (these) ideas from the professor!
13 In the interest of brevity, much of each student’s writing has been omitted. The omitted student text, in practically all cases supports, rather than detracts, from the statements given herein. The transcription preserves the student's original punctuation, capital letters, and other means of emphasis. A series of dots ......... indicates omission of portions of student writing. [Brackets] indicate H. Gurr editing.
This exam question must have been fairly interesting (engaging) to the students because discourse on this question averaged around 170 words. The full exam response of all students, as well as student responses from previous years, will be supplied upon request.
14 Herbert Ginsburg, Piaget's Theory of Intellectual Development; an Introduction.