Helping Students Make Sense of

Helping Students Make Sense of

Mathematics Wendy B. Sanchez

W PHOTOGRAPH BY WENDY B. SANCHEZ: ALL RIGHTS RESERVED

hy is x=−

b 2a

the axis of symmetry of a parabola? Why does b + know b2 − 4that ac the volsin2 x + cos2 x = 1? How do−you x =3 ume of a sphere is (4/3)pr ? 2a While these are exactly the kinds of questions that we should be encouraging students to ask, − b − b2 − 4 ac x = and Standards for . School according to the Principles 2a Mathematics (NCTM 2000), we do not always address them in our lessons. As teachers, we need 1 these − b + kinds b2 − of 4 aquestions. c − b − b2 − 4 ac  to be prepared to answer +  . Beyond preparing for them 2  in case 2athey are asked, 2a  we can plan for them to be asked; and if they are not asked, we can craft our lessons so that students are 1  −2 b  −b able to answer them. Mathematics, perhaps as much  2a  , or 2a . 2 or more than any other subject, makes sense. What a shame that so much of the population does not know how to make sense of mathematics. Sfard (2003, −b x = is .mounting to support p. 358) writes that evidence 2a the claim that “instruction that focuses on meaning can be expected to be more effective than instruction 4 3 then, can teachers that tries to circumventVit.” = How, πr 3 help their students make mathematics meaningful? In many mathematics classes in the United States, students take notes, copy examples, and leg  2  adj  opposite 2 2 sin x +oncos x =own + subsequently work problems their  hy yduring potenuse   hyp

opp adj opp + Vol. 100, No. 5 • December 2006/January 2007 | Mathematics = Teacher + 379= 2

2

2

2

hyp

hyp

Copyright © 2007 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. 2

hyp

2

hyp

guided practice and homework exercises. While practice in mathematics is imperative, what is lacking from this scenario is an opportunity for students to investigate mathematics and to make conjectures and draw conclusions on their own. Often the objective of mathematics lessons is to provide students with mathematical formulae, theorems, and processes and then have pthem practice using this new material2by1working exercises. A more power3 ful objective is 4to have students develop important mathematical formulae, and processes for q 6 5 theorems, 7 themselves and then8 practice using them. Two concerns immediately arise when this latter objective is embraced: How do I find the time? and How do I design lessons that will enable students to develop important mathematical ideas on their own?

In the sketch below, line l is parallel to line m. Trace the picture on patty paper, and lay the patty paper over the original sketch to determine which angles are congruent.

2

4

1

l

3 6

m

5 8 7

List all pairs of congruent angles in the space below:

TIME

Angles 1 & 5, 2 & 6, 4 & 8, and 3 & 7 are called ______________________________ angles. Angles 3 & 6 and 4 & 5 are called ______________________________ angles. Angles 1 & 8 and 2 & 7 are called ______________________________ angles.

In the sketch below, line p is not parallel to line q. Again, trace the picture onto patty paper and determine which pairs of angles are congruent.

2

p

1

D

4 3 q

5

6 8

7

4

Teachers are so pressed for time that allowing students to explore and reflect is often an impossible luxury. Certainly, policy needs to be revisited to allow for this essential feature of learning. But in lieu of major policy changes, what can teachers do in their classroom to afford students time to think about and understand mathematics? One strategy is to reduce the amount of time that students spend copying notes during class. Notes with blanks can be copied and handed to students to fill in during the lesson. The notes can be designed so that students construct meaning for mathematical facts and Aprocedures before they write them down inBtheir notes (see 2 fig. 1). This strategy requires students to 1 write less so that note taking takes less time. In the activity in figure 1, students use patty paper to determine that alternate interior angles, alternate exterior angles, and corresponding angles 3 are congruent when two lines 4 cut by a transversal are parallel and, just as important, C that they are not congruent when the lines are not parallel. The design of the handout allows students to spend valuable class time analyzing data and making conjectures rather than copying notes.

3

DESIGNING LESSONS

2 1

List all pairs of congruent angles in the space below: 3.1415

–6.28319 –3.1415

Conclusions:

6.28319

–1 –2

When two parallel lines are cut by a transversal, –3 –4

Fig. 1 Time-saving strategy for taking notes

380 Mathematics Teacher | Vol. 100, No. 5 • December 2006/January 2007 A B 2 1

Teaching that focuses on why mathematical procedures work rather than merely on how to do procedures can help students make sense of mathematics. Mathematics is not magic, and the way we introduce students to mathematics can convince them that the processes and procedures really do work. To this end, three strategies are helpful when designing lessons: discover, prove, and verify. If students discovered, proved, and verified everything they learned in mathematics, imagine how powerful their mathematical knowledge would be. Although it is not reasonable to think this approach is possible, it is reasonable to attempt to use at least one of the three strategies for all new material.

y = −3x 2 1 y=− x x+2 5 3 5then meaGraph the following 3 sets of ylines on the grids below and = −3x y = x+3 sure the angles between the two lines using a protractor. 2 y=

Discover When students discover mathematical facts, they collect and/or analyze data and then make and test conjectures. The activity worksheet in figure 2 allows students to use their knowledge of graphing lines to discover that perpendicular lines have slopes that are negative reciprocals of each other. The activity also allows students the opportunity to write a mathematical conjecture and then generate examples in order to test their conjecture. While this activity can be used without technology, geometry exploration software such as The Geometer’s Sketchpad will enable students to graph the functions and use the tools of the software to measure the angles between the two lines. Teachers might even want to modify this activity so that instead of equations, students are given sets of ordered pairs and then are asked to develop the equations themselves. This activity is aligned with both the Reasoning and Proof Standard and the Communication Standard, since students are required to “make and investigate mathematical conjectures” (NCTM 2000, p. 342) and to “use the language of mathematics to express mathematical ideas precisely” (NCTM 2000, p. 348). The activity described in figure 3 provides an opportunity for students to develop the formula for the axis of symmetry of a parabola. After developing and using the quadratic formula, students know b xx == −− b 2 that the quadratic function 22aa y = ax + bx + c intersects the x-axis at

1 x+2 3 y = −3x y=

2 y=− x 5 5 y = x+3 2

2 y=− x 5 5 y = x+3 2 y = 4x − 2 1 y = − x +1 4

y = 4x − 2 1 y = − x +1 4 If

a c b d = , then = . b d a c

Let

a c = . b d

a c b d If = , then = . b d a c 1 1 1 y = − x +1 • ad = • bc , 4 a What conclusion can you draw about the two lines? a a c Let = . Describeathe crelationship b dbetweenb thedslopes of the two lines in each bc If = , then = . pair. d= , b d a c a 1 1 • ad = you • bc , Make a conjecture about the pattern observed. a a a c 1 1 bc Let = . •d = • Generate three more pairs of lines that fit the pattern you observed. . b d c c ofa lines List your three pairs of lines on your paper. Graph your pairs bc d = , holds. to determine whether your conjecture a 1 1 d b • ad = • bc , = , a a c a reciprocals Fig. 2 Discovering that the slopes of perpendicular lines are negative 1 1 bc •d = • . c c a b c which isdpart AB ≅ DC and AD ≅DBC. = of , the Algebra Standard, and it helps a students “understand how mathematical ideas d b to produce interconnect and build on one another 2 = , − b + b 2 − 4 ac AB || DC and AD || BC. c p.a354), which is xx == − b + b − 4 ac 1 1 bc a coherent whole” (NCTM 2000, •d = • . 22aa part of the Standard. The Reasoning c Connections c a 4 b and and Proof and CommunicationAB Standards areAD also≅ BC.AC ≅ AC ≅ DC and x=− 2 2a −−bb − bbb2 −− 44ac addressed ac . d inbthis activity. 3 xx ==x = − b . = , Many mathematical facts, formulae, and procedures x =22a − 2aa AB || DC and AD || BC. c a are well suited for discovery activities.2 When students − b + b2 − 4 ac 2a x = They can observe discover mathematical facts for themselves, their 2 bb+22 − b42of − parabolas 4 ac 2−ab +−graphs 2 − 4 ac  11 from − b − bthat AB ≅ DC and AD ≅ BCAC . 1 − b + b− b−+4aacbc 2between 4 ac . x = ≅ AC ++−−4bac− bthe− two the axis of symmetry  lies halfway  .  22  22aa 2a 22aa x =   x-intercepts. Using the midpoint2formula, they can a What are the zeros of the quadratic function y = − b − b2 − 4 ac determine AB+ ||c?DC and AD || BC. x =that the x-coordinate . 2 of the midpoint ax2 + bx 3.1415 6.28319 –6.28319 –3.1415 2a − b − bis − 4 ac between the two1x-intercepts  − 2 b − b 2 x = . 1  −2 b  , or −ab . b − 4 ac −b 2 –1 . 22  22aax=, or 22aa .2a WhereAC is the axis of symmetry of a parabola ≅ AC 2 2   1 − b + b − 4 ac − b − b − 4 ac with respect to the zeros of the function that + . –2  aca − b − b2 − 4 ac  defines the parabola? 2 2a1  − b + b2 − 42 2 2 + .  −b  xx ==2−b1.. − b 2+a b − 4 ac + − b 2−a b − 4ac . –3   2 a This, of course, simplifies 2a 2a2  to 2a  Determine the axis of symmetry of the parabola 1  −2 b  −b , or . defined by the quadratic function–4y = ax2 + bx + c.   −b 2  2a  142− a2 b  4 3 , or . V V ==23 ππ12rra3−2 b  2a − b , or . Create a quadratic function and use the formula 3   2  2a  2a you discovered above to determine the axis of −b x =the equation . Therefore, of the axis of symmetry 2 is 2 symmetry 2 − b 2a     2 of the parabola defined by the function leg 2x = opposite leg leg  +  adjacent adjacent leg .−22bx =  opposite 2 x + cos sin you created. Graph your parabola and verify that sin x +  +  yypotenuse x2cos =a x. =  hy hy potenuse  hypotenuse hypotenuse  of symmetry you determined is correct. the axis 2 a 4 2 2 2 2 opp V = π r3 opp2 + adj adj2 = opp opp2 ++ adj adj2 = 4 3 = + = 2 2 2 This activity supports development of anhyp 2 Fig. 3 Discovering the formula for the axis of symmetry of hyp V =students’ π 4r 3 hyp hyp2 properties, hyp2 hyp a parabola understanding of functions their 2 V3= and π r 3 hyp 2 2 3 = hyp =2 1  opposite = hy 2 = 1  adjacent leg  leg 2 y p 2 sin 2 x + cos2 x =  + Vol. 100, 2No. 5 • December 2006/January 2007 | Mathematics Teacher 381 hy yp    opposite leg  2 adjacent leg  ypotenuse   hypotenuse 2  hy 2 sin x + cos x =   opposite leg+   adjacent leg  2 2 2 2 2 2 2 hy y potenuse hypotenuse  adjx = opp + adj   +  x + cos π sinopp  y = 4x − 2

y = 4x − 2 y = 4 x1 − 2 y = − x +1 1 y = − 4 x +1 4

yy ==5−x +x 3 2 y = 45x − 25 y = x 5+ 3 y21 = x+3 y = y−= 4x2x+−12 4 a c b yd=y 4=x−−12x + 1 If = , then = . y = 44x − 2 a c b d 1c b d If b = d , then a =Ifyca=. = +1 = . b d a c b y−=4d−x, 1then x + 1a c a c b d a c If = 4 , then = . Let = . b d a c a c aa cc b d Let b = d . If = , then = Let = . a c b .d Thenb add= bc, since cross bIfb dproducts ain ac=propor= , then . d a c a c tion Then Letb =d . 1 are equal. 1 • ad = • bc , b d 1 a • ad = 1 a • bc, a 1c 1 • ad ==a •.bc Let c, a a b ad= . a Let 1 1 b d• bc, bc • ad = d =the ,multiplication aproperty by a of equality. Simbac 1 bc 1 d = , yields plifying bc, 1 ,= 4 • 1 a y = 1 x + 2 da=• ad a • ada= • bc, 3 b c 1 1 bc da= ,3 a • d = y •= −3 .x a 1c 1c bc a. 1 bc1 bc •d = • •= d , d = . b•c 2 the c multiplying c a and both sides question by 1/c a=c of c d ,a 1 2 1 1 bc a d b y=− x yields •d = • . = , 5 c c a dc ab 3.1415 6.28319 1 1 bc –6.28319 –3.1415 d b = , • 1 .bc , –1 c a y = 5 x + 3 cc =• d1a= • dc= a • . 2 AD ≅ BC dc . b c a AB ≅ DC and = –2 , Hence, c . a AB ≅ DC and ADAB ≅d BC ≅bDC and AD ≅ BC. y = 4x − 2 =d , b–3 AB || DC and AD ||cBCa. = , 1 AB DC and AD ≅ BC. c . ≅a–4 AB || DyC=and || BC − AD x + 1AB || D C and AD || BC. 4 AB ≅ DC and AD ≅ BC. and the reflexive property of equality, AC ≅byAC AB||≅DDC andAD AD||≅BC BC AB C and .. AC ≅ AC AC ≅b ACd a c If = , then AB || =DC .and AD || BC. b d a ≅||cAC AB DC and AD || BC. AC Fig. 4 Proof that AC ≅ ACof equal fractions are equal a the c reciprocals

Let

=

.

AC ≅ AC

b d knowledge of that content is more robust, and they will likely be able to retain their knowledge longer. 1 1 • ad = • bc , a Prove a

Many times, and not only in geometry, deductive reasoning canbcbe used to convince students that mathd= , ematical procedures are valid. For example, consider a the example shown in figure 4, which demonstrates that if two 1 ratios1arebcequal, then their reciprocals are •d = • . equal. Depending c c ona the level of the course in which this property is addressed, teachers could either demonstrate the proof or have students develop the proof on d b their own. Either = , way, students are exposed to the fact c a makes sense and is not mere magic. that mathematics Many fundamental properties in mathematics AB ≅ DC and AD ≅For BCexample, . can be proven deductively. consider the example shown in figure 5, which shows a proof that opposite sides of .a parallelogram are ABthe || D C and AD || BC congruent. Students can come to recognize “reasoning and proof as fundamental aspects of mathematACare ≅ AC ics” if they used throughout the mathematics curriculum (NCTM 2000, p. 342).

Given the parallelogram ABCD, prove that AB ≅ DC and AD ≅ BC. A

B 1

2

4

D

382 Mathematics Teacher | Vol. 100, No. 5 • December 2006/January 2007

C

Since ABCD is a parallelogram, AB || DC and AD || BC. ∠1 ≅ ∠3 because parallel lines AD and BC cut by the transversal AC yield congruent alternate interior angles. ∠2 ≅ ∠4 because parallel lines AB and DC cut by the transversal AC yield congruent alternate interior angles. AC ≅ AC, since congruence is reflexive. Therefore, nADC ≅ nCBA by ASA. Hence, AB ≅ DC and AD ≅ BC, since corresponding parts of congruent triangles are congruent. Fig. 5 Proof that the opposite sides of a parallelogram are congruent

Radius

Volume

2 cm

33.510

3 cm

113.097

4 cm

268.082

5 cm

523.598

6 cm

904.778 b x=− Determine a pattern in the2chart above and give a a formula for the volume of sphere given the radius r. − b +for bthe−volume 4 ac of a sphere Fig. 6 Determining the formula 2

x=

2a sphere. Imagine trying to discover this formula from the information given in figure would − b − 6. b2 It −4 ac be the x notice, = . you take rare genius who would “Oh, look—if 2a 4/3 times p times the cube of the radius, you would get the volume!” The proof of this formula is beyond 1  −geometry b + b2 − class. 4 ac So − bhow − b2 − 4 ac  the scope of a high school +  . are we to convince students that 2 2athe formula really 2a  works? This is where the verify strategy is useful. Teachers can use a transparent plastic sphere  −2 b  − b commerthat can be filled with1water, ,available or .   2  2solids. a 2a sets usually cially in sets of geometric These contain both a sphere and a hemisphere that have the same radius. Teachers− bcan have the students = hemisphere . measure the radius ofxthe in centime2a ters, and then use the formula

Verify There are cases in which discovering a formula is unreasonable. For example, consider the volume of a

3

V=

4 3 πr 3 2

 opposite leg   adjac sin x + cos x =  + ypotenuse   hypo  hy 2

2

2a

b x=− b 2 x=− a − b − b2 − 4 ac2a D x= . 2a − b + b2 − 4 ac x = − b + b2 − 4 ac to compute the ofa the sphere in cubic ceny = sin2 x + cos2 x x =volume 2 2 2 2 a   timeters, Next, 1 − b + which b − 4 aisc equivalent − b − b to − 4milliliters. ac 4 +water,2 and then pour .  fill the water 2 the sphere 2a with 2 a − b − b − 4 ac  3 x =measure into a beaker to − b − 2b2aits − 4volume ac . in milliliters. x = surprised when . the measure2 Students are often 2a   1 − 2 b − b ment is very close to the calculated volume. 1 , or  . 2  1a − bthere + bis −more 4 ac than − b −oneb2way − 4 ac to 2 In2many a  cases, 2 2 12 − b +facts, 4 ac + − b −and2b2a − 4 ac  . 3.1415 6.28319 –6.28319 –3.1415 2b a −formulae, verify mathematical proce+  . –1 2 2 a 2 a dures. Consider,  for example,bthe trigonometric  −b = −There are several ways identity –2 x= .sin2 x + cos2 x =x 1.  −2 b  1 −identity. b2a 2 a to verify this trigonometric Three dif, or . –3 12 −22ab  −2ba ferent verifications shown algebraic, . below: 2  2a are  , or − b + b − 4 ac 2 2 a –4 4 numerical, x= V = π r 3 and graphical. 2a 3 −b Algebraic x = −2ba . Fig. 7 Graphical verification that sin2 x + cos2 x = 1 x= . 2 2 2 − b − b − 4 ac a x = leg 2opposite adjacent . leg  sin 2 x + cos2 x =  Any one of these three strategies is convinc +2a hypotenuse  ypotenuse  4hy ing, but the combination of the three is particuV = 4 π r3 3π2r 3 adj2 opp2 + adj2 V = opp larly powerful. Using all three approaches helps = 3 2 +1  − b2 += b2 − 4 a2 c − b − b2 − 4 ac  address hyp hyp hyp +   . 2 more student learning styles and requires 2 2 2a 2a students   to make more connections and use more 2  opposite leg 2  adjacent leg 2hyp 2 2 sin = x +2cos = 1 x = opposite leg  + adjacentrepresentations. leg  ypotenuse  +  hypotenuse p cos2 x =  hy sin 2hy xy+   hypotenuse  NCTM’s Learning Principle states that “stuypotenuse hy 2 −b 2  2 2 1  −2 bopp adj opp + adj , or . In this verification, students use the+ definition of   2 2 = 2 2 dents must learn mathematics with understanding, = a 2 opphyp 2  2aopp  2 2adj + adj π cosine and the2Pythagorean sine theorem actively building new knowledge from experience = hyp2 + hyp = to 2 2 x = and . 2 hyp hyp 6 hyp hyp demonstrate the identity. and prior knowledge” (NCTM 2000, p. 20). Using 2 =1 2 −=bhyp class time to engage students in activities that will hy y p x = = . 2 =1 Numerical help them develop a meaningful understanding of yp 1 3 2ahy sin x = values and for cos x x to = verify the identity: Choose mathematics is more than important—it is essential 2 2 π as we seek to embrace the vision outlined in Prinx = . 4 1 π6 2 V3= π r 3 sin 2 x = Letand x =cos . x= ciples and Standards. As teachers introduce mathe4 4 3 6 matical topics, using strategies of discovering, prov1 3 1 3 ing, and verifying can only help students develop a + =1 2 sin x = 1 and cos x = 3  opposite leg   adjacent leg  2 4 4 2 2 2 sin x = 2 and cos x = sin x + cos x =  +more robust understanding of mathematics. 21 2 3  hy ypotenuse   hypotenuse  2 2 sin x = 1 and cos x = 3 x+3 2 REFERENCES 4 opp2 adj2 opp + adj2 ? sin 2 x = 4 and cos2 x = = + = 3 National Council of Teachers of Mathematics 4 hyp2 hyp2 1 3 4 hyp2 (NCTM). Principles and Standards for School Math14 + 34 = 1 2 + =1 hyp ematics. Reston, VA: NCTM, 2000. = =1 4 4 2 hy ypmore The more values students chose for x, the Sfard, Anna. “Balancing the Unbalanceable: The x+3 convinced they would be that the identity is true. NCTM Standards in Light of Theories of Learning x +3 3 ? ? Mathematics.” In A Research Companion to Princiπ 3 x= . Graphical ples and Standards for School Mathematics, edited by 6 In the graphical verification (see fig. 7), students Jeremy Kilpatrick, W. Gary Martin, and Deborah graph the function y = sin2 x + cos2 x and observe that Schifter. Reston, VA: National Council of Teachers 1 3 the result is the same assin thexconstant of Mathematics, 2003. ∞ = andfunction cos x = y = 1. The verify strategy is useful2 in many cases2to 1 3 2 convince students that sin certain WENDY B. SANCHEZ, wsanchez@ x = procedures and cos2 and x= 4 4 processes work. It can also be useful to convince kennesaw.edu, is an associate profes1 3 students that certain procedures do not work. For sor of mathematics education at Ken+ =1 example, how many times your students tried 4 have 4 nesaw State University in Kennesaw, to “cancel the 3s” in an expression like GA 30144. She teaches undergraduate and x+3 ? 3

Substituting numbers for x can help students understand that this strategy does not work.

graduate-level mathematics and mathematics education courses. Her primary research interests are in assessment and professional development. Photograph by Greg Sanchez; all rights reserved

Vol. 100, No. 5 • December 2006/January 2007 | Mathematics Teacher 383