Problem Solving: Tips For TeachersAuthor(s): Phares G. O'Daffer, Gene Maier and Ted NelsonSource: The Arithmetic Teacher, Vol. 34, No. 3 (November 1986), pp. 27-29Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193008 .

Accessed: 12/06/2014 13:28

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.

http://www.jstor.org

This content downloaded from 195.34.79.20 on Thu, 12 Jun 2014 13:28:21 PMAll use subject to JSTOR Terms and Conditions

Problem >ol'>ing T¡p> for Teacher) □ H 1QQ9

«88

November 1986 27

[^Strategy Spotlight^ "

I If I I ll '| Recalling an Image I ÎeMHHMHHHNÏMHMe HeHeeÏÏ [ÜHHHeeeeeee ¡III I

One reason for using a variety of manipulatives and visual I models while learning mathematics is to build a collection I of mental images that can be called forth, as needed, in I mathematical thinking. Recalling nonsymbolic images is a If she moved the shaded section of the original draw- I strategy that offers insight into many problems. ing on the left to the position shown in the sketch on the I ^ L1 right, the result was a 2-by-3 array of large squares of 100 I Problem: ^ L1 What patterns do you see in these equations that unjts each wjth a 4_by_6 array

array of 24 un¡ts d

squares ,¡ beneath I

would help you compute the products mentally? it Hence the product 24 array x 26 was equa| t0 2 x 3 or 6 I

hundreds plus 24, or 624. I 14x16= 224 5he rea|jzec| that similar sketches could be made for I 24 x 26 = 624 other two-digit products provided the tens digits were the I 34 x 36 = 1224 same and the units digits summed to ten. The following di- I 44 x 46 = 2024 agram for 13 x 17 was sufficient to convince her of the I

54 x 56 = 3024 generality of the procedure. I

One middle-grade student conjectured that the last two I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I digits of the product came from 4x6 and that the leading IIIIIIIIII IIIIIII IIIIIIIIIIIIIIIIIII "" I digits came from multiplying the leading digit by one more ~~i::

"" I

than itself (e.g., 1 x2 = 2, 2x3 = 6, 3x4 = 12). She IIIIIIIIIIIIIIIi: ~~i::

IIIIIIIIIIIIIIIIi:: I correctly conjectured that 94 x 96 = 9024, since 4x6= IIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIII I 24and9 x 10 = 90. IZZZZZZZZZZZZZZZZ I M I I I I I I ffl I

She asked her teacher why this method works. (If you IIIIIIIIIIIIIIIi: IIIIIII I were the teacher, what would you do?) The teacher 1 I I I I I I I i II I I I I I I I I I I I I I IJ I thought of the array model that she had used to introduce I multiplication. Sketching one specific example, 24 x 26, on The teacher used sketches to demonstrate the process I base-ten grid paper, she noticed several things. t0 the student. The student was also able to see that the I

process would not work if the tens digits were different or I the units digits did not add to ten. I Edited by Phares Phares G. G. O'Daffer O'Daffer By drawing a rough sketch or imagining the diagram, I Phares Phares G. G. O'Daffer O'Daffer

Nor°ma'lL6l76¡erSUy try to determine if the sam® property holds for three-digit I d 'j u ^ ™ • ^ ^ KT . products like 121 x 129. I Prepared d by u Gene Gene ^ Maier Maier ™ • and and ^ Ted Ted ^ Nelson KT Nelson . I ^ Gene Gene Maier Maier ™ • and and ^ Ted Ted ^ KT Nelson Nelson . Portland State University I Portland, OR 97207 ^ ■ , I (Continued ^ ■ , on next page) I

Phares Phares G. G. O'Daffer O'Daffer

^ Gene Gene Maier Maier ™ • and and ^ Ted Ted ^ KT Nelson Nelson .

This content downloaded from 195.34.79.20 on Thu, 12 Jun 2014 13:28:21 PMAll use subject to JSTOR Terms and Conditions

28 Arithmetic Teacher

G Tip Board

■ n ■ - - ■ -

n L ■ Problem c°™r - - .

U around »deas are xa M ,hat a// together h™ *he nu"*er ,¿11 1 ?' and 3 * 4. I

"--" | 1 I 1 1 I] ^| (suo!SU9LU!p )U9J9j I 1 I 1 I ''11111 1 1 ■

-¿!P sii 9JB Jaqtunu b io sjopbj em [suoisueiuip juejeyip omj gabij i i i i i i i i i i i i i titilli itili H

JOU S90p PUB eJBnbS B asís S| 9|6uBP9J SjlJl '9|ßUBpej 9U0 Á|PB g H -X9 SBM i! LjßnoL|;|B :9iuud b p9J9p¡suoo }ou S| i. jgqwnu 9L|i -9|6ub ^«^^

IMI H

-J09J 9U0 Á|PBX9 9ABL| SJ9qOinU 9LUUCl Sngi] -SU0ISU9LUIP UJ9J9JJIP | ^«^^ | | | | |

--- H

OMi ÁIPBX9 SBM ÏBMI J9qujnu b S| jgqoinu 9wud v "J9qujnu b |O joj M l I I 1 I I I I I i i l I l l l I I I H -ob^ puB jgqwnu 9iuud ^o sidgouoo 9i|i ^o 96blu! ub S9p|A0jd suois H

-U9LUIP JI9MJ puB s9|6ubp9j Lj^M sj9quunu Qu've'ooss'/ :a¡o¡'¡) , , ?. . 9 ■

, 1 , 1 1 I I . *- / Mil ^H >2 S| SU0ISU9LU lllll *- MM H

-!p jq6¡9 Á|pBX9 SBLj }Bmj jgqiunu is9||Blus 9iji "suoisugiuip }U9J8^ i i i i i i iTI i i i i i i n i i i i i i 1 H

-^!P 99JM1 Á|PBX9 9ABM 6 PUB f SJ9qilinU 9L|} :SU0!SU9LU!p JU9J9JJIP H

OMJ Á|PBX9 9ABq |. '. pUB 7 '9 'g '£ SJ9qilinU 9Lj} :SU0!SU9LU|p OM} S fr € 3 I- H

UBU,; J9M9^ SBLJ }BL|J J9qwnU Á|UO 9LJJ S| L ¡61 PUB '¿l l£l l9|dUJB _^_r-p_^_, R~] I -X9 JO^ '9|ßUBP9J 9U0 Á|PBX9 9ABL| 21. UBLj} J9JB9JÔ SJ9qiUnU ÁUBLU MIMI

_^_r-p_^_, (-t-i-t-, lllll

CDD DI] D H ■ Ajd^ju^U! :9|6ubp9J 9U0 ÁIPBX9 9ABL| u puB l¿ 'g 'e '2

' l sj9qtunu lllll (-t-i-t-, ■

9l|l 'Sdì!) J9^9i JO 9A|9MJ ßuilMBJUOO S9|ßUBP9J ¡O U0!P9||00 9L|) U| 'ZV~V SJ9qLUnU 9L|) JO^ S9|ßUBP9J 9Lj) 9JB 9J9H H

siuauiLuoo pue suoßnios H

This content downloaded from 195.34.79.20 on Thu, 12 Jun 2014 13:28:21 PMAll use subject to JSTOR Terms and Conditions

November 1986 29

rÍT^^ • actw»t.es can help 1 H

I | Background Ideas L I ' •

oi prob'ertvsoW*n9 ^e oncept erf surface H Mathematics has the reputation of being an ■ ■ 1 This sequence oi

P ̂ ^age °* ̂e H H abstract subject where symbols reign and vi- H I 1 students deve P ̂ H sualization plays little part- except in geom- I I 1 area. cube, scissors, as

a ^ce ^1 H etry However, one interesting survey of the H I

1 a Give each studen ^a^e size as the g ^a_

H H way scjent¡sts ancj mathematicians do their H I

1 a i havinq each sq „.it 'ackets trom the g ^a_ H creative thinking concluded that "practically H ■

PMheecube i havinq

&^^^**°** each

w« „.it <*

'ackets

some «*<* 1 h

creative all of them...

thinking use concluded

vague images that "practically

that are" I I ■

t*' when iolöed, w« <* aWe jackets lor

some «*<* h _ most frequently visual but they may also be ■ ■

I Per

h Ask the students The

to nu ^/^

aWe sQUares

Discuss »n

ttie^ _ H

Qf ̂̂ ^ kjnetjc„ (Hadamard 1 g54) ■ ■

1 1 h h'ocK D Ask

shapes- The nu oi s^ape. Discuss H

H Here are SQme vjews on the importance of ■ I 1 1 these h'ocK D

's surface area oi

^^ H

havjng ¡mages> that one can caN forth as ■ ■ I jacket 's ai»« t eaCh ligure. M¿7' H needed, to learn or think mathematically. ■ ■ 1 surface ai»« rni-« /~7Tl 3 H ■ H I Z^7' I- 4^-tI fi J ^ ^H Anything is easy if you can assimilate it to your ■ ■ 1 ~-r-r-~7~~7' (¿?7y I ' 1/ '

^B collection of models. If you can't, anything can be H H

1 f~Ç''V U- ^ he ■ painfully difficult. (Papert 1980) ■ ■ ' ' - J - 1 -

twelve tw cubes. Ask t ^B h

A mathematical statement leaves the hearer cold ■ I ' i r-lWiS each « student tw

au .uerent rectangu» h h when ¡t evokes no ¡mages or associations - ■ ■ i c. r-lWiS Give « HoW many au ^Wch of h

Emptying ideas of their sensuality does not pro- ■ ■ ' ioWowir^Q quesi1 . w^ tweWe cuu

'' ̂ e ^eW. _ duce meanjngfu| learning or discovery. . . . What ■ ■ 1 boxes can be torn»

w sUrtace area v • '' test H must be criticized is not abstraction itself ... but ■ ■

1 *L^cp boxes has tne w »n^t 'acKet)? T he g' H

_ abstraction at the expense of the senses. (Som- ■ ■

est number boxes

oi squares »n^t _ meM978)

expense

| | 1 sU rface area? ^^^^^^^^B The words or the language, as they are written or H H

1 sU rface ^^^^^^^^^^^^^^^^^^^^ spoken, do not seem to play any role in my mech- H H

1 ^- ̂

anism of thought. . . . The physical entities which H H ^-

^ seem to serve as elements in thought are certain H H gm^^^^^^_l_l^_^^^_^_g___^____ll^___ signs and more or less clear images which can be H H ^

- i voluntarily reproduced and combined. (Albert Ein- H H I rillllll/fflllíTIIIIllllllllílllll stein, quoted in Hadamard [1954]) H H H [))))))) 'v ) ) ) L '* 's sa'cl tnat seventy-five percent of us have a H H H v // / V

) ) )) dominant visual memory, twenty-five percent an H H

H auditory one. As for me, mine is quite visual. H H ^1 B9JB 90Bpns jsb9| 9Lji J9Ô oj '||B |0 9dBL|S pBdiuoo jsouj 9U,} '9J9u,ds B 0}U| When I think about mathematical ideas, I see the ^B ^M ■ LU9MI 9dBMs pinoo 9M «Ab|0 ¡o 9pBLU 9J9M S9qno jno || B9JB 90B¿jns JSB9| abstract notions in symbolic pictures. (Ulam 1976)

J ■ H H dl'' 9ABU. HJM 9dBU,S PBdUJOO )SOLU 9L|) '9LUn|0A 9LUBS B'J'' |O Spj|OS JO| '|B _^^_^^____^^^_^___^____H H

H -J9U96 U| *9LUn|0A 9LUBS 9q) 9ABq S9X0q 9S9U.) ¡0 JnO} ||B )Bljl 9)0|S| S9JBnbS ^ ̂̂ ^^^^^^^^^^^^^^^^^^^^^^^^^^^M I

H OMJ-ÂJJiqi SBL| J9>jOB[ SJ! :B9JB 90B|jnS JSB9| 9U.J SBL| XOq PBdlilOO JSOIJU 9U.1 ^ ̂̂^^^^^^^^^^^^^^^^^^^^^^^^^^^M 1 H

H j9>10b[ s'' u| S9JBnbs Áyi| SBu, i! ÍB9JB 90Bpns jsoLu 9qj sbi| xoq u|u,i ßuo| yK/KA rn T] n H H 9qi S9qno 9A|9mj qiiM p9LUJ0| 9q ubo S9xoq jnoj ßu|M0||0| 9qi o Pnknkn I I

I I I

I I I I I I I I I I I

H LJ ' _ _ _ LJ I I I I I I I '

' '

I I I I II

I I I I j S9JBnbs puß xis |0 |bjoj b S9J¡nb9j ^9>job[ qoBg p9J9A00S!p H

H ' » I - I - I ' - ' - ' II I I I I j 9ABq sju9pnjs ye'j{' suj9UBd 9luos 9jb 9J9h Á|Pbx9 9qno H

■ Á|9A!P9dS9J '8 1 pue 'h '9 1. 9U0 JeA0° II!M S;9>l°Bi luaj9Ji!P ̂0 Al9|JBA 9P'M V "» I H lQl 9JB SB9JB 9qi 9Jnßü qOB9 JOJ }9>jOBf 9U0 |O 9|dUJBX9 UB SI 9J9H <7 SIU9LULU00 pUB SUO'in¡OS I

Part of the Tip Board is reserved for techniques that you've found useful in teaching problem solving in your class. Send your ideas to the editor of the section. W J

Authors Sought for Elementary- Grades Yearbook

The NCTM's Educational Materials Committee announces that the 1989 NCTM Yearbook will be entitled Elementary School Mathematics: Issues and Directions and will be edited by Paul Trafton of the National College of Education. The Yearbook Advisory Panel is now seeking manuscripts for the yearbook and is interested in both substantive papers addressing issues or directions in the teaching of elementary school mathematics and relatively short papers relating classroom practices to these issues or directions. Guidelines for the preparation of manuscripts are available from Albert P. Shulte, General Editor, Oakland Schools, 2100 Pontiac Lake Road, Pontiac, MI 48054.

^-PROFESSIONAL DATES-. NCTM 65th Annual Meeting

8-11 April 1987, Anaheim, Calif.

NCTM 66th Annual Meeting 6-9 April 1988, Chicago, III.

NCTM 67th Annual Meeting 12-15 April 1989, Orlando, Fla.

For a listing of local and regional meet- ings, contact NCTM, Dept. PD, 1906 Association Dr., Reston, VA 22091, Telephone: 703-620-9840; CompuServe: 75445,1161; The Source: STJ228. ^ 75445,1161; The Source: STJ228. J

This content downloaded from 195.34.79.20 on Thu, 12 Jun 2014 13:28:21 PMAll use subject to JSTOR Terms and Conditions