Problem Solving Resource Units, grades 1-8. grades 1-3, grades 4-6, grades 7-8Review by: Virginia HorakThe Arithmetic Teacher, Vol. 33, No. 3 (November 1985), pp. 56-57Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194095 .

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Etcetera Edited by Carol Novillis Larson University of Arizona Tucson, AZ 85721

Easy-to-Make Awards and Certifi- cates of Merit . . . Using Your Office Copier. ¡984, 78 pp. Caddy lak Publishing, 60 Shames Dr., Westbury, NY 11590.

This booklet contains seventy-eight examples of awards and certificates of merit that are appropriate for various occasions and for all levels of education. The format of the awards is an 8^" x И" sheet that can be photocopied and then distributed. Each of the sample awards identifies the specific area of merit. These areas include outstanding ability, certificate of achievement, extraordinary attitude, most en- ergetic, and most thorough.

Many other types of awards are also sup- plied. Each particular award includes a relevant graphic, as well as a place for the recipient's name, the date, and an appropriate signature. These awards are useful for individuals as well as for groups. Their format and quality also make them suitable for display.

This awards booklet is appropriate not only for education but also for personal use and special business occasions. The 8|" x 11" certif- icates are professionally prepared. They are designed for easy reproduction and distribu- tion. Many of the seventy-eight awards are more appropriate for noneducational settings. However, classroom teachers generally, and mathematics teachers specifically, may find many useful certificates. The one chief detrac- tor is the similarity of the awards. Although the certificates contain a variety of graphics, bor- ders, and outlines, not many differences are apparent among them. It would be questionable whether a school system could effectively con- tinue to use this system, since the similarity of the certificates would tend to lessen their over- all significance to concerned students.

This booklet is useful to teachers in various situations. The collection is very extensive, and

one can find a certificate or award form appro- priate for almost any situation. Overall, this book should be a welcome addition to many professional collections. - Willis J. Horak, Uni- versity of Arizona, Tucson, AZ 85721.

Problem Solving Resource Units, grades 1-8. 1983, grades IS, $8 ea.; grades 4-6, $10 ea.; grades 7-8, $12 ea. (plus $2 minimum shipping and handling charge). Stark County Department of Education, 7800 Columbus Rd., N.E., Louisville, OH 44641.

This collection is a comprehensive program for teaching problem solving in grades 1-8. It is based on a strategies approach, with three to five strategies identified and taught at each grade level. Some of the fifteen to twenty strategies included in the units are act it out, guess and check, find a pattern, make a model, check for hidden assumptions, and write an open sentence.

Each unit begins with an introduction to the teaching of problem solving, which contains general suggestions concerning methods of instruction, grouping, expectations, and evalua- tion. For each strategy identified at grades 3-8,

From the F"«

J Computation Ì

COMPLEMENTARY SUBTRACTION

Students enjoy "strange algorithms." Here is one for subtraction that is sometimes called the "method of complements."

Example 1: 6132 -1937

Step 1 : Copy the top number (the minuend). » 6132 Step 2: Subtract each digit in the bottom number (the subtrahend) from 9

-^ ̂ »- 8062

and write the digits under the minuend. 7 and 2 are complements, - -^ ̂

and so on. Step 3: Add the numbers.

6132 Step 4: Subtract 1 from the first digit on the left of the sum and add 1 to tn©^^^ 8062

remaining number to obtain the answer. ^^^^^^ ^4194 Other examples Ì-1

78326 78326 39261 39261 4195 - 9825 174 -1520 9479

й/8500 6 3 О48740 6 + 1 3 + 1 68501 38741

Challenge your students to figure out why the method works. The following is an explanation for example 1 .

6132 - 1937 = 6132 + (9999 - 1937) - 10000 + 1 = (6132 + 8062) - 10000 + 1 = (14194 - 10000) + 1 = 4194 + 1 = 4195

From the file of Gail H. Adele, University of Idaho, Moscow, ID 83843

- Readers are encouraged to send in two copies of their classroom-tested ideas for "From the File" to the Arithmetic Teacher for review-

56 Arithmetic Teacher

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one or two large-group lessons have been designed. These are followed by small-group activities, which include from ten to thirty problems whose solutions lend themselves to the strategy under study. The third section of the unit is an individual problem-solving set, which contains another ten to fifty problems of varying difficulty. For grades 4-8 a comprehen- sive end-of-year evaluation is provided, with suggested alternative methods of assigning credit for work completed. For grades 1 and 2, the units focus primarily on large-group instruc- tion in which the teacher is to lead a guided- discovery lesson about a given problem-solving strategy.

The units for grades 7 and 8 also contain a section called "Problem Solving with the Com- puter,1' which is also based on large-group introductory problems followed by approxi- mately thirty problems to be solved by students individually and in small groups. It is assumed that all students are comfortable with program- ming in BASIC and have a knowledge of the elementary functions and statements used in the language. This section supplies numerous computational problems whose their solutions rely mainly on the calculating and recursive abilities of computers. The section would be more useful if a larger variety of problem- solving approaches were required.

Each unit is a source of problems appropriate for the given grade level. Many of the problems are challenging and thus will provide the oppor- tunity for sustained small-group activity. Al- though many problems are well-known and found in other sources, some are new or at least have been given a novel twist. Since each unit is designed to use and reinforce previously learned strategies, the seventh- and eighth- grade units contain problems to which a variety of strategies can be applied. One strength of these units is the convenience of having a readily accessible source of usable problems. All problems are typed in a ready-to-copy for- mat, with solutions given on a separate sheet or on the back of the problem page.

The quality of the units is not consistent. Some units provide a thorough, well- written introductory explanation, whereas others leave it to the individual teacher to supply this aspect of the lessons. In some units the problems follow the strategy being studied very closely, but in others a different (or even better) strategy could have been used. The format of all the units is basically the same, but the presentation on the written page is easier to understand and use in some units than in others.

The Problem Solving Resource Unit repre- sents a worthwhile effort to produce materials whose primary purpose is to teach problem solving. It does not encompass the entire top- ic-for example, no section in the units deals with having too little or too much information in a problem or requires students to gather their own data (information) before a problem can be solved. The units as a whole, however, give students the opportunity to learn many useful problem-solving strategies while applying them to numerous novel and challenging problems. -

Virginia Horak, University of Arizona, Tucson, AZ 85721.

A Touching Way to Teach Math, '/>" videotape ($425), training manual ($2), and classroom starter set ($38 ea., with special discount for quantity orders). Jan Bullock, Sandy Pierce, and Libby McClellan. Touch Math, P.O. Box 7402, Colorado Springs, CO 80933.

This program is designed to teach children in grades K-4 a procedure to be used in perform- ing basic computations with whole numbers. The starter set consists of a teacher's manual, a student workbook with reproducible practice pages, posters for display, desk-top cards for students, and an hour-long training presentation on videotape for teachers.

This computational procedure is based on identifying "touch points" on each of the nu- merals from 1 to 9. Children are then taught to use counting as they touch the points to deter- mine the answers to computational problems. Counting forward, backward, and "sequence counting, two through nines" are used. The procedure can be applied to the basic facts of addition, subtraction, multiplication, and divi- sion, as well as to the standard algorithms for each of these operations.

The video presentation is well done and pro- vides a multimodal training experience for those desiring to learn about the programs. The presentation is designed to give viewers the opportunity to practice as they watch.

The content of the "Touch Math" program stresses a mechanical procedure for finding answers to computational problems. Little if any emphasis is placed on helping children understand how the structure of the number system is applied in the various algorithms. No activities are suggested to help learners develop an understanding of the operations through manipulation of concrete models.

In summary, the program helps children learn a rote procedure for finding answers to addi- tion, subtraction, multiplication, and division examples. No work is devoted to helping chil- dren develop an understanding of the relation- ship between the algorithms and physical mod- els. - Edward D. Brown, University of Arizona, Tucson, AZ 85721.

Use Ur Head Math Games: Unit 1, Addition and Subtraction; Unit 2, Multiplication and Division. Peg onion Fisher. Game boards, cards, grids, dice, mark- ers, base-ten blocks, and play money. $22 ea., plus $2 shipping for each unit. Use Ur Head Math Games, 212 Estill St., Berea, KY 40403.

This set of materials consists of two units that reinforce the learning of basic facts through games. Unit 1 contains games dealing with addition and subtraction facts. Multiplication and division games compose unit 2. Three game boards are provided for playing the games in both units. In "Ur Home Safe," the players move around a baseball diamond. A player advances by rolling a die and correctly answer- ing a basic-fact question. For example, in the game Add To and Take Away, subtraction-fact cards are placed in one pile and addition-fact cards in another. The board is constructed so

that a player must draw from one of the piles and solve a problem on each turn. In calculating the answer, the player can use manipulatives or do mental computation. Manipulatives must be provided by the teacher or parent, as they are not included in the set.

A second game board is "On Ur Way." During the game, the players move from a house to a bank, landing on a cent, dime, or dollar space in the process. If a player lands on a cent space, she or he must solve an addition problem and, if correct, is entitled to take the sum in pennies. Landing on a dime space and correctly solving a problem entitles the player to the sum in dimes. A dollar space would result in earning dollars. Play money is supplied with the set of games.

A spinner wheel with numbers from 1 to 12 is used as the third game board. The player spins the spinner and uses the number on which it stops as part of a required computation. Each particular game determines the problem to be solved. For example, the object of Whirlaway to Ten is to determine which number must be added to the number on the spinner to equal 10. A number grid is provided if so needed by the player.

Each of the units has a booklet that explains how the games are played and what materials are needed. Teachers or parents might need to clarify some of the procedures.

Unit 1 contains fifteen games with multiple variations dealing with addition and subtraction facts. The playing cards included in this unit are divided into seven subsets that allow a player to work at his or her level. The games in unit 1 appear to be suited for use mostly in grades 1-5. The variety of games should appeal to most students.

Multiplication and division games make up unit 2. Thirteen basic games with several varia- tions are included. The activities reinforce the learning of multiplication and division facts from 2 x 1 through 12 x 12 and 2/2 through 144/12. Sets of playing cards are grouped into five stages of difficulty. Thus, the learner can progress through each level or practice at his or her own pace. The materials in unit 2 can be used for grades 3-6.

The strength of Use Ur Head Math Games is that needed practice is presented in a fun game format. - Richard L. López, Jr., University of Arizona, Tucson, AZ 85721. W

^-PROFESSIONAL DATES -x NCTM 64th Annual Meeting 2-5 April 1986, Washington, D.C.

NCTM 65th Annual Meeting 8-11 April 1987, Anaheim, Calif.

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

For a listing of local and regional meet- ings, contact NCTM, Dept. PD, 1906 Association Dr., Reston, VA 22091, Telephone: 703-620-9840; CompuServe: 75445,1 161; The Source: STJ228.

November 1985 57

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