Concept #1. Goals for Presentation. I m going to be a mathematics teacher: Where did this stuff come from? Why didn t I know this before?


 James Gilmore
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1 I m goig to be a mathematics teacher: Why did t I kow this before? Steve Williams Associate Professor of Mathematics/ Coordiator of Secodary Mathematics Educatio Lock Have Uiversity of PA 00 Steve Williams Goals for Presetatio To provide participats with some of the iformatio that preservice secodary math teachers state was lackig i the eplaatios they received cocerig certai mathematical cocepts. To motivate participats to cosider a more coceptual way of commuicatig mathematics (it s ot always the studets fault). To motivate participats to cosider other mathematical cocepts where we have shortchaged our studets. To provide participats with some alterate ways of viewig certai math topics that are more coceptual tha traditioal ways of viewig them. To help participats begi to be able to see mathematical cocepts despite beig blided by the midless procedures. Where did this stuff come from? Seve years of teachig all levels of secodary mathematics Te additioal years of workig with preservice teachers ad talkig to them about potetial coceptual deficiecies or misuderstadigs Sevetee years of tryig to closely eamie as may secodary (ad elemetary) cocepts i as much detail as possible A sicere iterest to have my studets develop that Profoud Uderstadig of Fudametal Mathematics by lookig at cocepts i differet or otraditioal ways Noe of this stuff is ew, or eve difficult...it s just that most secodary math majors have ever bee asked to eamie may thigs coceptually, ad certaily ot i differet ways. Whe show these simple coectios or cocepts, most preservice teachers get upset with themselves ad state that they feel like idiots for ot seeig it before. I just remid them that it is t ecessarily their fault. Much of the fault rests o us as their teachers (ad o the curriculum we choose for traiig them). Cocept #1 Why the umbers we call the perfect squares are called such. is called a perfect square because its square root is a whole umber,. It is oly a procedural uderstadig. It leads to circular reasoig because they aswer why do we call raisig a umber to the secod power squarig the umber? with, because you are tryig to fid the umber that you could take the square root of to get back to the umber that you are squarig. 1
2 So what do I do? Costruct all of the rectagular arragemets that ca be made with objects 1 But what happes whe we try this with? These are all ice rectagles; however...the last rectagle is a special type of rectagle. 1 8 These are all rectagles... othig special A perfect square! Questio: Could we cosider all other umbers perfect rectagles? Cocept # It is perfectly permissible to simply divide the umerators ad divide the deomiators whe dividig fractios. You have to ivert ad multiply whe dividig fractios. It is oly a procedural uderstadig. It does t help studets uderstad why ivert ad multiply actually works. You really do t have to Cosider Most studets isist that you must ivert ad multiply i order to perform this operatio. However, this is simply ot true. It is perfectly permissible to divide the umerators ad divide the deomiators, just like multiplicatio of fractios. 1 = = = = Of course, they always wat to verify it = 7 Certaily, this is t always the most efficiet way to perform this operatio, but it certaily ca always be doe. 8 Most of the time, we eed to get a commo deomiator. 8 1 = = Oce studets get used to this method, they fid that it is t ay more time cosumig tha ivert ad multiply ad woder why o oe ever told them this before. Ad, of course, this is also the method that allows us to justify ivertig ad multiplyig = = = 1 1 1
3 Cocept # Why the umber 1 ot a prime umber. 1) Because the defiitio of a prime umber is a umber greater tha or equal to. ) I thought it was. ) I thik it should be, sice the defiitio of a prime umber is ay umber whose oly divisors are itself ad 1. What s wrog with these? The first oe provides o uderstadig of why prime umbers are defied that way. The secod oe is just icorrect. The third oe lacks a uderstadig of a importat coectio betwee mathematical cocepts. But it does get them thikig. What do I do? I ask my studets what the Fudametal Theorem of Arithmetic says. Every composite umber ca be writte as the product of prime umbers i eactly oe way (ecludig arragemet). Prime factor : This is the oly way. Prime factor if 1 is a prime umber : Cocept # The differece betwee square uits ad uits squared. From here, they get it. They have just ever bee asked to actually thik about it. square iches ad iches squared mea the same thig. It really is t correct. They thik that readig math from left to right is always correct. They forget that they ofte have to iterpret what they are readig. It does t distiguish betwee adjectives ad verbs. What do Geometry tetbooks say? Area is measured i square uits. From there o out, the tets oly use the symbol i, leavig the studets to read it as either iches squared or square iches ad thik the phrases are iterchageable, ad eve thik that iches squared is more correct sice they read from left to right. So, what s the differece?
4 Cosider i as " five iches squared." This implies that we should take five iches ad square it Cosider i as " five square iches." This implies that we show five square iches. i 1 i 1 i i Area = i i i But the tetbook said that area was measured i square uits. This is what i really meas! i should be read as five square iches. Cocept # The defiitio of arithmetic average (or mea) is NOT the umber you get whe you add up all of the umbers ad divide by the umber of terms preset. That this procedure is actually the defiitio. However, moder tetbooks do use this phrase as the defiitio of the mea. It s ot a defiitio. The defiitio of defiitio is a statemet of the meaig of a word or a statemet epressig the essetial ature of somethig. At best, it s a procedural defiitio which tells us othig about the cocept of mea. Studets do t kow what it meas whe their average i my course is 8%. The questio remais: What does it mea for a umber to be the average of a set of umbers? Secodary math majors are stued whe they do t kow the aswer to this questio. They claim they have ever bee asked. 190 tetbook: The average of a set of umbers is a umber which ca be put i place of them without chagig the sum. Thik about what this meas for the average of,, ad 1. The average of,, ad 1 is a umber which ca be put i place of,, ad 1 without chagig the sum. Put oe umber i place of each of them so the sum does t chage (i.e. rearrage the squares you have so each colum has the same umber i them but keep the same umber of colums). There are 7 i each colum, so the average is 7. Secodary math majors have ever looked at the average like this before.
5 My compromise defiitio of average The umber that represets a equal distributio of the total amog all umbers preset. So whe a studet gets a 8% i my course, it meas that 8% is the grade that they would have received o each assigmet if they had doe the eact same o all of them. Cocept # You CAN multiply the bases whe you have a m a. a m a = a m+ a a = What s wrog with this? Of course, there is o real error i thikig here. It just does t take ito cosideratio oe of the most commo studet errors: = 9 8 = a 8 = ( ) 8 Of course, studets rarely say that a a a. 8 What do I have my studets do to eamie this cocept? I wat my studets to closely eamie the rules ad procedures they have always used; sometimes havig them develop ucommo rules ad procedures of their ow. I ask my studets if it is okay to multiply the bases together whe preseted with a m a. I remid them that I did ot ask m+ m them if a m a = (a a) m+. I a a = ( a a) asked them if they could 1 multiply the bases together. (I 7 1 = 9 = 9 = the have to ecourage them to develop their ow rule.) That s somethig I ever thought about! They always say o! Cocept #7 A liear fuctio has a costat first differece, a quadratic fuctio a costat secod differece, ad a cubic fuctio a costat third differece, with respect to their tables of values, whe the idepedet variable icreases by 1. The have o idea what I m talkig about because they ve ever actually aalyzed the table of values of a fuctio. Studets oly recogize fuctios by either their graph or from their equatio. It leaves out a etirely differet aalytical techique that brigs out some good mathematics.
6 Eamie a table of values for liear fuctio y = + the Eamie a table of values for the quadratic fuctio y = y diff 0 1 y st diff d diff 0 1 Eamie a table of values for the cubic fuctio y = y st diff d diff rd diff This is usually a completely ew way of lookig at these fuctios, eve though we always have studets use a table to graph. Cocept #8 The Law of Cosies is a geeralizatio of the Pythagorea Theorem...or...the Pythagorea Theorem is a special case of the Law of Cosies. The Pythagorea Theorem is used for right triagles ad the Law of Cosies is used for oblique triagles. This is, of course, true...just ot complete, sice the Law of Cosies ca techically be used for both. The coectio is simple ad straightforward... Law of Cosies : a = b + c bc cosα o Now let α = 90 o a = b + c bc cos90 ( 0) a = b + c bc a = b + c c c α α a b a b
7 Cocept #9 The power rule for epoets is NOT a epoet raised to a epoet. 1 ( ) = is the rule for "raisig a epoet to a epoet." Although may tetbooks phrase it this way, this is NOT raisig a epoet to aother epoet. It causes real problems whe tryig to relate the rules of epoets to the rules of logarithms to a iquisitive studet. Have studets read the phrase raisig a epoet to a epoet carefully ad write dow what it is really sayig. is "raisig a epoet to a epoet." 1 = This may seem iocet eough, util you try coectig these rules to the rules for logarithms to the iquisitive studet. ( ) m m We usually coect log m = log m to = ad say somethig like "sice a logarithm is a epoet" that this rule of logarithms is similar to the rule of epoets i which you "multiply epoets" whe "raisig a epoet (log) to aother epoet." But aiquisitive studet commeted that the ( ) correlatig logarithm should look like this: log m... ad they were right. But ( log m) log m The Solutio... Stop phrasig this rule of epoets as, raisig a epoet to aother epoet. The rule should be phrased as, whe you raise a base that has bee raised to a epoet to aother epoet, you multiply the epoets. log m ca the be iterpreted as "fidig the logarithm of somethig that has already bee raised to a epoet." log( ) A iterestig ca of worms... TI 8 family TI 8 log() log TI 8 family () = ( log) ot log( ) = log log TI 8 () = log( ) = log correct, right? 7
8 Cocept #10 The meaig of the slope of a lie as a umber. Slope is rise over ru or the chage i yvalues divided by the chage i values. They are both very procedural defiitios. This is all studets are left with from their tetbooks...ay may teachers. It leaves out ay real coceptual uderstadig of oe of the most importat Algebra I cocepts. It abuses a very importat uderstadig of fractios: as a umber. What is slope (as a umber)? Slope: The amout that the depedet variable chages whe the idepedet variable icreases by oe uit. Cosider y = 8 How much does the yvalue chage whe the value chages by 1? What is the value of this fuctio whe = 7? 1 What is the value of this fuctio whe = 8? 1 What is the value of this fuctio whe = 9? 19 The slope is. Cosider y = + How much does the yvalue chage whe the value chages by 1? What is the value of this fuctio whe =? 1. 7 What is the value of this fuctio whe =?. What is the value of this fuctio whe = 7?. The slope is. Note that if this were cotiued for a chage of o the value, the yvalue would chage by  (over four ad dow rather tha dow ad over ). The abuse of fractios depedet variable y=(/) Uderstadig slope as oly "the chage i divided by the chage i studets to oly look at the fractio from a "umeral" perspective ad ot as a "umber" by itself. y values values" actually forces the   idepedet variable 8
9 Other topics Why the method of addig two equatios together works whe solvig a system of equatios. How a radia is defied. The differece betwee a umber ad a umeral. How the area formulas are derived. How the Quadratic Formula is derived. What is Ad so may more Coclusios Secodary mathematics majors (ad may iservice teachers) usually have ot bee provided the opportuities to closely eamie may of the basic cocepts that they take for grated ad will oe day have to teach. The traditioal curriculum to prepare studets to teach secodary mathematics feeds ito this gap i teachers kowledge. There is usually o course that studets take i which to discuss these potetial deficiecies. We, as their teachers, should admit some of the resposibility i helpig to cultivate these deficiecies. Cotact Iformatio Steve Williams Associate Professor of Mathematics/ Coordiator of Secodary Mathematics Educatio Lock Have Uiversity of PA 9
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