{"id":491,"date":"2014-11-20T00:01:01","date_gmt":"2014-11-20T05:01:01","guid":{"rendered":"http:\/\/blogs.ams.org\/matheducation\/?p=491"},"modified":"2014-11-14T13:38:43","modified_gmt":"2014-11-14T18:38:43","slug":"proportionality-confusion","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/matheducation\/2014\/11\/20\/proportionality-confusion\/","title":{"rendered":"Proportionality Confusion"},"content":{"rendered":"<p><em>By Dick Stanley, Professional Development Program, University of California at Berkeley<\/em><\/p>\n<p>The notion of one quantity being proportional to another is certainly a very basic part of an understanding of mathematics and of its applications, from middle school through calculus and beyond. Unfortunately, the picture of proportionality that tends to emerge in school mathematics in this country is narrow and confused. Everyone learns the procedure of setting up and solving a proportion, but the connection of this to the idea of one quantity being proportional to another is tenuous.<\/p>\n<p>In support of this statement, I summarize below the results of participant responses given in a workshop attended by teachers, mathematics educators, and mathematicians. The surprisingly shallow responses show a striking lack of a common, mathematically coherent understanding in this audience of the subject of proportionality.<!--more--><\/p>\n<p><strong>A. \u00a0A simple problem<\/strong><\/p>\n<p>In the workshop, participants first worked to solve this problem:<\/p>\n<hr \/>\n<p><strong>Paper Stacks Problem:<\/strong><\/p>\n<p>Suppose you want to know how many sheets are in a particular stack of paper, but don&#8217;t want to count the pages directly. You have the following information:<\/p>\n<ul>\n<li>The given stack has height 4.50 cm.<\/li>\n<li>A ream of 500 sheets has height 6.25 cm.<\/li>\n<\/ul>\n<p>How many sheets of paper do you think are in the given stack?<\/p>\n<hr \/>\n<p>All 18 participants found the expected result (360 sheets) by setting up and solving a proportion.<\/p>\n<p><strong>B. What is proportional to what?<\/strong><\/p>\n<p>Next, participants were asked this question:<\/p>\n<hr \/>\n<p>Write down a sentence or two in response to this question:<\/p>\n<p><em>\u00a0 \u00a0 \u00a0 \u00a0 &#8220;In this paper stacking situation, is anything proportional to anything else?&#8221;<\/em><\/p>\n<hr \/>\n<p>The most natural response: &#8220;the number of sheets in a stack is proportional to the height of the stack&#8221; did in fact appear, but only in about a fifth of the responses. This response is in accord with a modern understanding of proportionality: a variable quantity <em>A<\/em> is proportional to a variable quantity <em>B<\/em> when there is an invariant <em>k<\/em> such that <em>A<\/em>\u00a0=\u00a0<em>kB<\/em>. In this situation the invariant is the number of sheets per centimeter.<\/p>\n<p>Other responses suggested that &#8220;the height of the small stack is proportional to the height of the large stack.&#8221; But the ratio of these heights (about 0.72) is particular to these two stacks, and is not an invariant of the paper stacking situation. These two heights are not proportional in a modern sense of the term. What is getting in the way in these other responses, we feel, is a view commonly put forth in school materials: a ratio can be formed only between quantities of the <em>same kind<\/em>. The relationship between the number of sheets and the height of the stack cannot then be proportional, since the required &#8220;ratio&#8221; is between quantities of different kinds.<\/p>\n<p>However, most disturbing is the number of responses that merely put together some scraps of remembered procedures, such as response number 4: &#8220;A proportion is the relationship of two ratios. The height of the two stacks is proportional since you are comparing one ratio to another; i.e. \\(\\frac{360}{4.5}=\\frac{500}{6.25}\\)&#8221;<\/p>\n<p><strong>C. What does &#8220;proportional to&#8221; mean in general? <\/strong><\/p>\n<p>Finally, participants were asked this question:<\/p>\n<hr \/>\n<p>Write down a brief answer to this question:<\/p>\n<p><em>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 &#8220;What does it mean in general to say that one quantity is proportional to another quantity? Be as precise as you can.&#8221;<\/em><\/p>\n<hr \/>\n<p>The 18 responses are interesting enough that they are included in full:<\/p>\n<ol>\n<li>proportional relationship means that when one quantity in a relationship changes another will change according to some specific pattern (which won&#8217;t change in time \/ vary)<\/li>\n<li>&#8220;a&#8221; is prop. to &#8220;b: means that if b is altered by a factor (e.g., multiplied by t), then a is altered the same way.<\/li>\n<li>One quantity is proportional to another means the comparison is relating equal ratios.<\/li>\n<li>\\[\\frac{a}{b} = \\frac{c}{d}\u00a0\\hspace{3em}\u00a0ad = bc\\]<\/li>\n<li>To be proportional means to have the same ratio in simplest form. The relationship between the two things is the same (in the real world like sugar:flour)<\/li>\n<li>As the numbers in the proportion change \u2026 there is a constant pattern of increase or decrease \\[\\frac{1\\times 4}{2\\times 4} \\hspace{4em} \\frac{4}{8} \u00a0\\hspace{4em} \\frac{10\\div 5}{5 \\div 5} \u00a0\\hspace{4em} \\frac{8 \\div 2}{4\\div 2} \u00a0\\hspace{4em} \\frac{1}{2}\\]<\/li>\n<\/ol>\n<ol start=\"7\">\n<li>It means that a fraction is equal to a fraction or that the two ratios are equal.<\/li>\n<li>As one part of the proportion changes the other part changes in the same relational way.<\/li>\n<li>If one quantity increases, the other quantity also increases. Or If one quantity decreases, the other quantity decreases<\/li>\n<li>It means that quantity &#8220;A&#8221; changes in a fixed or quantifiable manner as quantity &#8220;B&#8221; changes.<\/li>\n<li>The two ratios are equal. of, cross products are =<\/li>\n<li>As one quantity increases or decreases by a specified amount, the similar quantity also increases or decreases by the same amount.<\/li>\n<li>The rate of change between the two quantities is constant.<\/li>\n<li>quotients of 2 quantities are equal \/ constant if proportional<\/li>\n<li>The ratio of parts of each term is the same \\(\\frac{1\\ \\text{sheet}}{\\text{ height}}\\) is same for both<br \/>\n(each piece of the proportion is made up of like parts)<\/li>\n<li>amount of an item will have a relation to another item<\/li>\n<li>As one quantity grows the other quantity also grows; it is a multiplicative relationship; ratio is constant; what about inversely proportional?<\/li>\n<li>When one thing is proportional to another, we can set up two fractions that are equivalent.<\/li>\n<\/ol>\n<hr \/>\n<p><strong>D. What has gone wrong? <\/strong><\/p>\n<p>The confusing jumble of responses here is disturbing. At the very least it points to a lack of a common understanding within the school mathematics community of this very basic and important subject. It would certainly be wrong to blame teachers. Rather, I believe the culprit is a general lack of mathematically sound grade-level appropriate presentations of proportionality that have been available to teachers. In addressing this lack, mathematicians must certainly play a major role.<\/p>\n<p><strong>E. Comments<\/strong><\/p>\n<p>The subject of proportionality in school has a long, complex, and fascinating history. Here, I will simply suggest the range of relevant issues.<\/p>\n<ol>\n<li><strong> Euclid<\/strong><\/li>\n<\/ol>\n<p>All school approaches to proportionality have their origins in Euclid&#8217;s treatment in Book V of <em>Elements<\/em>. This is where the brilliant treatment of ratio by Eudoxus appears. However, Euclid&#8217;s treatment of proportionality is essentially that of <em>discrete<\/em> quantities: four magnitudes that have the same ratio are called proportional. (See <a href=\"http:\/\/aleph0.clarku.edu\/~djoyce\/java\/elements\/bookV\/defV5.html\">Definition 6<\/a>.) Today, proportional relationships are understood as being between two <em>variable<\/em> quantities. In my view, the inadequate understanding of proportionality shown by many responses in the workshop is due to the failure of school mathematics materials to sufficiently stress the role of variable quantities in a modern understanding of proportionality. We elaborate on this idea in the next section.<\/p>\n<ol start=\"2\">\n<li><strong> Proportions and missing the crucial invariant<\/strong><\/li>\n<\/ol>\n<p>Finding the numerical solution to a problem such as the paper stacks problem by setting up and solving a proportion is fully reasonable, and we all do it. However, the mathematically interesting point in a situation such as this is that there is an <strong><em>invariant,<\/em><\/strong> namely the number of sheets per centimeter (80 sheets per cm).<\/p>\n<p>In an approach that focuses only on setting up and solving a proportion, this invariant never needs to be found. All that is found is an unknown (360 sheets) in one particular situation. This means that the crucial relationship between the variable quantities <em>n<\/em>\u00a0=\u00a0number of sheets and <em>h<\/em>\u00a0=\u00a0height of a stack is never seen: \u00a0\\[n\u00a0=\u00a080h.\\] \u00a0And in fact, seeing proportionality as involving a relationship between <em>variable<\/em> quantities was the key point missing from most responses in the workshop. Repeating this point from Part B above:<\/p>\n<blockquote><p>A variable quantity <em>A<\/em> is proportional to a variable quantity <em>B<\/em> when there exists an invariant <em>k<\/em> such that <em>A<\/em>\u00a0=\u00a0<em>kB<\/em>.<\/p><\/blockquote>\n<p>This statement includes two hard but very important mathematical ideas, the idea of an invariant, and the idea of a variable quantity. It is my feeling that work toward bringing out these ideas should begin as soon as the language of proportionality is introduced in middle school.<\/p>\n<ol start=\"3\">\n<li><strong> Analogy: the Law of Sines<\/strong><\/li>\n<\/ol>\n<p>To make an analogy, consider the Law of Sines for triangles: \\(a\/\\sin\\alpha = b\/\\sin\\beta = c\/\\sin\\gamma\\). These three ratios are not only equal, but their common value is an important invariant of a triangle: the diameter of the circumcircle. Bringing out the invariant and its meaning is an essential part of a fully mathematical treatment of the Law of Sines. A focus on the invariant as the common value of a set of ratios should be an essential part of a mathematical treatment of proportionality as well.<\/p>\n<ol start=\"4\">\n<li><strong> The Common Core State Standards in Mathematics<\/strong><\/li>\n<\/ol>\n<p>The approach to proportionality suggested in the <a href=\"http:\/\/www.corestandards.org\/Math\/Practice\/\">Common Core State Standards in Mathematics<\/a>\u00a0promises to be of real help, since the emphasis is directly on proportional relationships and the constant of proportionality. In fact, the approach is remarkable in that the term &#8220;ratio and proportion&#8221; does not appear at all, nor does the idea of &#8220;setting up and solving a proportion.&#8221; Instead, the central concept is proportional relationships themselves.<\/p>\n<p>However, my observation is that old habits are hard to break. It will not be easy to overcome tradition in developing and implementing this far more reasonable approach.<\/p>\n<p><strong>F. Conclusion<\/strong><\/p>\n<p>I think we would all agree that a reasonable treatment of proportionality should lead to students being able to understand a statement such as this:<\/p>\n<blockquote><p>The gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance separating them.<\/p><\/blockquote>\n<p>This requires a rather flexible understanding of the idea of \u201cproportional to.\u201d We have argued that a traditional approach to proportionality that focuses on setting up and solving a proportion is not adequate. Instead, what is needed is an approach that emphasizes the role of variable quantities and their invariant ratio. The responses in the workshop seen in Section C above would have been rather different if these ideas had been more prominent in school materials.<\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" ><\/div>","protected":false},"excerpt":{"rendered":"<p>By Dick Stanley, Professional Development Program, University of California at Berkeley The notion of one quantity being proportional to another is certainly a very basic part of an understanding of mathematics and of its applications, from middle school through calculus &hellip; <a href=\"https:\/\/blogs.ams.org\/matheducation\/2014\/11\/20\/proportionality-confusion\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" data-url=https:\/\/blogs.ams.org\/matheducation\/2014\/11\/20\/proportionality-confusion\/><\/div>\n","protected":false},"author":76,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[27],"tags":[102,101,103],"class_list":["post-491","post","type-post","status-publish","format-standard","hentry","category-classroom-practices","tag-proportion","tag-proportionality","tag-ratio"],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p6C2AC-7V","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/posts\/491","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/users\/76"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/comments?post=491"}],"version-history":[{"count":16,"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/posts\/491\/revisions"}],"predecessor-version":[{"id":511,"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/posts\/491\/revisions\/511"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/media?parent=491"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/categories?post=491"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/tags?post=491"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}