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456 Science Hands-on Teaching-Learning Activities of Elementary School Teachers Harold Harty School a/Education Fort Valley State College Fort Valley, Georgia 31030 Peter Kloosterman School of Education Indiana University Bloomington, Indiana 47405 Jack Matkin Science Department Northeast Texas Community College Mount Pleasant, Texas 75455 Background The use of hands-on materials for the teaching of elementary school science has been promoted for a number of years. Hands-on materials help to immerse students in science, making them feel science is something one does rather than something that one just learns about. As Kotar (1988) noted, "the paramount element of an excellent science program is the child’s involvement" (p. 40). Science instruction with curricula which use hands-on materials has been shown to increase achievement in most students (Bredderman, 1982; Shymansky, Kyle, & Alport, 1982). Elementary school students in science programs using hands-on materials have much more positive attitudes about the nature of science and their ability to learn science than do students in traditional textbook-oriented science programs (Bredderman, 1982; Kyle, Bonnstetter, & Gadsden, 1988; Shymansky et al., 1982). In addition, Rutherford (1987) has gone so far as to say that science textbooks should be removed from the elementary school as they actually get in the way of good science instruction. The issue of how to assess learning of children who use hands-on science programs has been given considerable attention in recent years (Hein, 1987). In brief, the issue of whether to teach elementary school science using hands-on materials is timely and the case for using hands-on materials is strong. School Science and Mathematics Volume 89 (6) October 1989

Science Hands-on Teaching-Learning Activities of Elementary School Teachers

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Science Hands-on Teaching-LearningActivities of Elementary SchoolTeachersHarold HartySchool a/EducationFort Valley State CollegeFort Valley, Georgia 31030

Peter KloostermanSchool of EducationIndiana UniversityBloomington, Indiana 47405

Jack MatkinScience DepartmentNortheast Texas Community CollegeMount Pleasant, Texas 75455

Background

The use of hands-on materials for the teaching of elementary school sciencehas been promoted for a number of years. Hands-on materials help toimmerse students in science, making them feel science is something one doesrather than something that one just learns about. As Kotar (1988) noted, "theparamount element of an excellent science program is the child’s involvement"(p. 40). Science instruction with curricula which use hands-on materials hasbeen shown to increase achievement in most students (Bredderman, 1982;Shymansky, Kyle, & Alport, 1982). Elementary school students in scienceprograms using hands-on materials have much more positive attitudes aboutthe nature of science and their ability to learn science than do students intraditional textbook-oriented science programs (Bredderman, 1982; Kyle,Bonnstetter, & Gadsden, 1988; Shymansky et al., 1982). In addition,Rutherford (1987) has gone so far as to say that science textbooks should beremoved from the elementary school as they actually get in the way of goodscience instruction. The issue of how to assess learning of children who usehands-on science programs has been given considerable attention in recentyears (Hein, 1987). In brief, the issue of whether to teach elementary schoolscience using hands-on materials is timely and the case for using hands-onmaterials is strong.

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Unfortunately, in some elementary school classrooms, the question ofwhether to use hands-on materials is not an issue because teachers in theseclassrooms do not teach science at all. Manning, Esler, and Baird (1982)surveyed 191 elementary school teachers in Florida, and found that 25%reported they taught no science at all. In a study of 121 elementary schoolstudent teachers. Gates, Krockover, and Wiedermann (1987) found that 92%claimed they taught science although the amount of time devoted to scienceinstruction had decreased in comparison to previous years. Of those teachingscience, 80% taught it after 1:30 p.m. when students were tired, and thus notable to concentrate as well as they might have earlier in the day (Gates et al.,1987). These data are somewhat disturbing given the importance of science inour society. The data are too limited, however, to be generalizable acrossentire states or regions of the country.

Final issues of importance relating to the use of hands-on materials in theteaching of elementary school science are how and when hands-on materialsare used. Materials that are used strictly for "fun" and are not well related toscience concepts are probably not very useful in helping students understandscience. An intriguing question is whether intermediate grade teachers utilizemore hands-on science instruction than primary grade teachers. Despite theimportance of hands-on materials in the teaching of science, data on thesequestions are incomplete and out-of-date.

Methodology

The present study was designed to assess the extent of manipulative use overa large sample of schools. General questions addressed by the study includedthe extent to which teachers had hands-on materials available to them, howoften teachers used hands-on materials, whether manipulatives were used morefrequently in the lower or the upper grades, and whether hands-on materialswere used more for building science process skills or for promotingunderstanding of broad science concepts.To examine the influence of the use of hands-on materials on elementary

school science instruction, an instrument was designed, validated and thenmailed to elementary school principals throughout a midwestern state.Elementary school principals were selected to respond to the instrument; as itwas expected, they would be the individuals with the best overall picture ofinstructional practices within a given elementary school. The instrument andpostage paid return envelopes were mailed to 414 elementary school principals.Instructions for completing the instrument were included in a cover letter andon the instrument itself. In addition, the cover letter suggested that theprincipal poll his or her teachers or ask teachers for help in completing theinstrument to provide an accurate picture of practice in the school. Seventy-three percent (73%) of the instruments were completed and returned (301 outof 414).

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Instrument

Questions concerning use of hands-on materials for teaching elementaryschool science were written with a multiple response format. For eachquestion, the respondent was to provide separate answers for the subcategoriesof: (a) science in grades K through 2, and (b) science in grades 3 through 5.As only one instrument was sent to each principal in the sample, responseswere to be generalizations or averages for all classes and grade levels in aschool. A space for comments was added to the end of the instrument.The instrument was given to a six member validation panel for reaction

concerning the content validity of the questions and the ability of elementaryschool principals to adequately respond to the questions. Members of thepanel who reacted to the questions included college professors and graduatestudents in science and elementary education, elementary school principals,and inservice elementary school teachers. In general, most of these individualswere very positive about the items as written although minor modifications tothe questions were made based on suggestions from this group. In addition tosuggestions for revision, the content validation panel was asked to rate, on afive choice Likert-type scale, the items on the instrument. Validation scalerating categories or dimensions were: (1) representativeness of the items fromthe total pool or universe of items dealing with use of manipulatives, (2)degree of congruence between the substance of the items and the underlyingconstruct, (3) degree of clarity of the items for elementary school principals,(4) potential for the findings to influence future teacher training andcurriculum development, and (5) degree of overall usefulness of knowledgeproduction from the study. A Scott’s coefficient of interrater agreement(Scott, 1955) was calculated across the five validation dimensions and the sixcontent validators. The coefficient computed was 0.86 reflecting respectablecontent validation for the instrument.

Test-retest reliability (stability reliability) is the degree to which individualsgive consistent responses to an instrument over time. High test-retest reliabilityon an instrument is an indication that the instrument is measuring wellformulated knowledge or opinions rather than momentary thoughts. Test-retest reliability was computed for the instrument by having seven doctorallevel graduate students with teaching or administrative experience respond tothe instrument with respect to their most recent elementary school experiences.These respondents were asked to complete the instrument again five weekslater. A test-retest reliability coefficient was calculated at 0.93 (p < 0.05)using the Spearman-Brown correlation technique. This coefficient was highenough to insure that the responses to the questions were stable over time.

Findings

Questions on the instrument and rationale for asking those questions wereas follows:

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1. What percentage of your teachers have commercially made "hands-on"materials/physical models available for use in your school?

Examples given were thermometer, balance, candles, live specimens, etc. Thefive possible responses for each subcategory (K - 2 and 3-5) and the percentresponse and frequency count (in parentheses) for each subcategory were:

Less than 10%10% to 39%40% to 60%61% to 89%90% to 100%No Response (Missing Data)

K-23-5

17%(52)10%(30)25%(74)16%(47)16%(49)21%(63)16%(49)24%(72)24%(72)28%(83)2%(05)2%(06)

2. What percentage of your teachers have teacher-made or teacher-collected"hands-on" materials/physical models available for use in your school?

Examples given were leaf collection, tin cans, insect collection, jars, rockcollection, rope, paper bags, etc. Response options for Question 2 were thesame as those for Question 1.

Less than 10%10% to 39%40% to 60%61% to 89%90% to 100%No Response (Missing Data)

K-23 -5

10%(29)5% (16)18%(54)17% (51)21%(64)21% (63)25%(76)30% (89)24%(71)25% (74)2%(07)3% (08)

The rationale for including Questions 1 and 2 in the survey was that nolarge scale data sets are available which document the extent to whichhands-on materials can be found in elementary school classrooms. It isentirely possible that a lack of hands-on materials use is the result, in part, ofa lack of materials. Hands-on and teacher-made materials were distinguishedin Questions 1 and 2 on the assumption that teacher-made manipulativeswould be less costly, and therefore more readily available in most schools.

3. About how many days per school year do pupils use "hands-on"materials/physical models (commercial or teacher-made)?

Response categories and percents/frequencies for Question 3 were as follows.As this question was quite straightforward, no illustrative examples weregiven.

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K-2 3-5

Less than 10 Days10 to 21 Days22 to 41 Days42 to 89 Days90 or More DaysNo Response (Missing Data)

9%(28)6% (18)27%(81)21% (62)31%(92)34% (102)22%(66)28% (85)8%(25)8% (24)3%(09)3% (10)

4. When "hands-on" materials/physical models (commercial or teacher-made) are used in (he classroom, to what extent are they used to help pupils"learn (he rules’* for computation, measuring, estimating, etc. rather thanunderstand how or why these rules work?

Response categories and percents/frequencies were:

K-2 3-5

Unable to AnswerNot Used at AllRarely, Once in a Great WhileSometimes but Not OftenOften but Not AlwaysMost if Not All of the TimeNo Response (Missing Data)

6%(19)5%(15)5%(16)4%(11)25%(76)16%(47)31%(94)42%(125)26%(78)29%(86)4%(12)3%(10)2%(06)2%(07)

Examples given for Question 4 were that hands-on materials could be used tolearn the rules for graphing, operationally defining, variable identification,classification, etc.

5. When "hands-on" materials/physical models (commercial or teacher-made) are used in the classroom, to w^hat extent are (hey used to get pupils tounderstand broad concepts or to solve problems which require substantialcreative thinking?

Response options for Question 5 were the same as those for Question 4.Examples provided were that manipulatives are used for explaining inductiveand deductive approaches, setting up controlled experiments, doing sciencefair projects, etc.

K-2 3-5

Unable to AnswerNot Used at AllRarely, Once in a Great WhileSometimes but Not OftenOften but Not AlwaysMost if Not All of the TimeNo Response (Missing Data)

3%(09)1% (04)6%(17)1% (03)29%(86)18% (55)35%(106)40% (119)20%(61)31% (93)5%(16)7% (20)2%(06)2% (07)

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The rationale for including Questions 4 and 5 was that hands-on materialsmay not aid learning in classrooms where they are used in ways other thanthose specified by materials developers. Science materials are usually intendedto help pupils understand broad science concepts.

6. How much did your school look for texts that use "hands-on"materials/physical models in your most recent textbook selection?

Response options and percents/frequencies for Question 6 were:

K-2 3-5

Unable to AnswerWas Not a FactorConsidered OccasionallyConsidered OftenConsidered ExtensivelyWas the Main FactorNo Response (Missing Data)

8%(24)7% (20)5%(15)3% (09)17%(52)16% (48)29%(86)30% (91)31%(92)32% (96)9%(26)10% (30)2%(06)2% (07)

Question 6 was included as science textbooks w^ere in the adoption processstatewide while this survey was being undertaken. Text decisions in theelementary school are very important because of the expense involved as wellas the fact the text selection does not take place again for a number of years.Thus, school personnel that felt hands-on materials were important factors intext selection probably had more of a commitment to the use of suchmaterials than did those who did not feel manipulative materials were animportant factor in text selection.

7. On the average, how many minutes per week are devoted to the"hands-on" teaching of science in each classroom?

Response options and percents/frequencies for Question 7 were:

K- 2 3-5

None1 to 59 Minutes60 to 119 Minutes120 to 240 MinutesMore than 240 MinutesNo Response (Missing Data)

4%(12)3% (08)66%(200)57% (171)19%(58)25% (75)6%(18)10% (30)1%(03)2% (05)3%(10)4% (12)

8. On the average, how many minutes per week are devoted to (he

"non-hands-on" teaching of science in each classroom?

Response options for Question 8 were the same as those used for Question 7.Both questions were included in the instrument to obtain a global view of the

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amount of class time devoted to manipulative as opposed to non-manipulativeactivities in elementary school science classes.

K -2 3-5

None1 to 59 Minutes60 to 119 Minutes120 to 240 MinutesMore than 240 MinutesNo Response (Missing Data)

3%(08)2%(06)26%(79)13%(39)42%(125)40%(119)23%(69)38%(115)3%(08)4%(11)3%(08)4%(11)

The results of the statewide survey are also broken down and analyzedaccording to the: (1) quantity of manipulatives, (2) use of manipulatives, (3)implications of manipulatives, and (4) manipulatives and textbooks. Thefindings are discussed by way of mean responses, standard deviations, andpercentage of response per rating category. Also, tests of significance betweengrades K - 2 (lower level) and grades 3-5 (upper level) responses arediscussed.

Quantity of Manipulatives

When considering the amount of commercial hands-on materials available(Table 1), it was found that upper elementary school classrooms (3 - 5) havesignificantly more (M = 3.44, SD = 1.32) manipulatives than lowerelementary classrooms (M = 3.06, SD = 1.45). A t-value of 7.09 was foundto be significant at the 0.001 level. In grades K through 2, the distribution ofresponses was fairly uniform as 42% of the respondents indicated less than40% of their teachers had commercially-made manipulatives available and41% indicated more than 60% of their teachers had commercially-mademanipulatives available. In grades 3 through 5, responses were skewed moretoward teachers having commercially-made manipulatives as 55% of therespondents indicated 60% or more of their teachers had commercially-mademanipulatives available for use.Teacher-made manipulatives for science teaching were available somewhat

more often than commercially-produced materials (Table 1). Upper elementaryschool teachers (M = 3.52, SD = 1.20) have significantly more (t == 3.34, p< 0.001) teacher-made manipulatives than lower elementary school teachers(M = 3.35, SD = 1.30). Only one-fourth of the respondents, however,reported that all of the teachers in their schools possessed teacher-mademanipulatives for the teaching of science.

Use of Manipulatives

The use of manipulatives during science teaching aspect of the instrumentincluded questions on: (1) days per year pupils use hands-on materials, (2)time per week spent on hands-on teaching, and (3) time per week spent on

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Table 1

Science Hands-on Teaching-Learning Utilization

463

^^’�-^^^ Perceptions

Dimensions ^^^"^^^^Percent of Teachers HavingCommercially-Available "Hands-on" Manipulatives in the Classroom

Percent of Teachers HavingTeacher-Assembled "Hands-on"Manipulatives Accessible

Days/Year Pupils Use "Hands-on"Manipulatives

Application of "Hands-on"Experience toward "Learning theRules" Rather than Understandingthe Concepts

Application of "Hands-on"Experience toward UnderstandingConcepts or Solving ProblemsCreatively

Extent to Which "Hands-on"Models Were Considered in MostRecent Textbook Selection

Time/Week Spent on "Hands-on"Teaching

Time/Week Spent on Non-"Hands-on" Teaching

Grades K-2 Grades 3-5t- p df

M SD M SD value

3.06 1.45 3.44 1.32 7.09 0.001 290

3.35 1.30 3.52 1.20 3.34 0.001 288

2.93 1.11 3.12 1.04 3.51 0.001 286

2.98 0.98 3.14 0.88 3.34 0.001 270

2.90 0.99 3.25 0.88 8.18 0.001 280

3.22 1.04 3.34 1.00 3.66 0.001 266

2.31 0.70 2.49 0.80 4.50 0.001 285

2.96 0.87 3.29 0.82 7.88 0.001 285

non-hands-on teaching (Table 1). The number of days per school year thatpupils use hands-on materials, manipulatives, or physical models (eithercommercial or teacher-made), in upper elementary classrooms (M = 3.12, SD= 1.04) was significantly greater (t = 3.51, p < 0.001) than the number ofdays that lower grade level classes used such materials (M = 2.93, SD =

1.11). Approximately 30% of all elementary teachers employed hands-onmaterials for 22 to 41 days per year; only 8% of all teacher utilizedmanipulatives 90 or more days. Ten percent of the teachers at the lower gradelevels and 6% at the upper level employed hands-on materials less than 10days during a given school year.

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An additional question on the survey addressed the issue of the averageminutes per week (Table 1) devoted to the hands-on teaching of science.Students at the upper elementary school level (M = 2.49, SD = 0.80) usedmanipulatives significantly more (t = 4.50, p < 0.001) minutes per week thanlower elementary school pupils (M = 2.31, SD = 0.70). Respondentsindicated that approximately 10% of the lower grade level and 60% of theupper grade level students used hands-on materials but used them less than 60minutes per week. Less than 2% of all elementary teachers employ an inquiryapproach for more than 240 minutes per week.As shown in Table 1, the number of minutes per week that students

experienced non-manipulative science teaching was somewhat greater than thenumber of minutes of hands-on instruction. In most cases non-manipulativeinstruction consisted of reading about science in a textbook series. Students atthe upper elementary school level (M = 3.29, SD = 0.82) experiencedsignificantly more (t = 7.88, p < 0.001) non-hands-on science time thanlower level classrooms (M = 2.96, SD = 0.87). Approximately 40% of allelementary students’ classrooms engaged in non-manipulative science activitiesfor 60 to 120 minutes per week. Respondents also indicated that 27% of thelower and 44% of the upper elementary school students experienced morethan 120 minutes per week of non-inquiry-oriented science. Sadly enough, 3%of the lower and 2% of the upper level students had no science at all. Thesedata reflect additional sad tones when considering less than an hour of scienceper week was taught in 30% of the lower and 15% of the upper elementaryschool classrooms reported.

Applications of Hands-on Instruction

Two basic areas (Table 1) where there are direct implications from the useof manipulatives are: (1) to facilitate the understanding of concepts, and (2) toenhance the problem solving process. With respect to the first area,information was sought concerning the extent to which manipulatives are usedto help pupils "learn the rules" for measuring, estimating, etc., rather thanunderstand how or why these rules work. Specifically, during science activitieshands-on materials might be used to learn the rules for graphing,operationally defining, variable identification, classification, etc. Teachers atthe upper elementary school level (M = 3.14, SD = 0.88) employedmanipulatives significantly more (t = 3.34, p < 0.001) to learn rules thanteachers at the lower elementary level (M = 2.98, SD = 0.98). Six percent(6%) of the respondents did not feel they could respond to this dimension and4% said their teachers never employed hands-on materials for the purpose oflearning rules. Surprisingly, 4% of the schools utilized manipulatives only forlearning rules.The use of manipulatives to enhance the problem solving process (Table 1)

might specifically include situations where materials are used for explaininginductive and deductive approaches, setting up controlled experiments, doing

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science fair projects, etc. Teachers at the upper elementary school level (M =3.25, SD = 0.88) had experienced significantly more (t = 8.18, p < 0.001)activities where manipulatives were used to promote problem solving thanstudents at the lower grade levels (M = 2.90, SD = 0.99). Roughly 4% of allelementary teachers were not in a position to respond to the concern, and anadditional 3% had never used manipulatives for this purpose. On the positiveside approximately 6% of all elementary teachers employed hands-onmaterials to enhance problem solving ability most, if not all, of the time.

Manipulatives and Textbooks

An additional dimension of the study (Table 1) focused on the amount ofinfluence hands-on materials, manipulatives, or physical models had ontextbook selection. Classrooms at the upper elementary school level (M =3.34, SD = 1.00) possessed textbooks which were significantly influencedmore (t = 3.66, p < 0.001) by the incorporation of activities involvinghands-on experience than classrooms at the lower grade levels (M =3.22, SD= 1.04). Some 7% of the respondents were unable to determine whether thiswas a consideration in their teAt selection. Also, on a negative note, anadditional 4% of the schools reported that manipulative usage was not afactor considered during the selection process. On the brighter side,approximately 9% of the respondents noted that the incorporation andpromotion of manipulatives was the main factor in selecting a textbook.

Conclusions

When taking into account all of the concerns about manipulatives orhands-on materials considered in this study, the percent of teachers havingteacher-assembled manipulatives accessible was rated the highest for both thelower grades (K - 2) and the upper grades (3 - 5). About 50% to 60% ofelementary school teachers have these materials to use. Also, contrary topopular opinion, most elementary schools report teaching science in somecapacity. Grade level differences in use of manipulatives were apparent on allstudy dimensions. Upper grade level teachers used materials and taught sciencesignificantly more than lower grade level teachers.Comments provided on some of the returned instruments reflected more

concern about hands-on materials availability than was apparent fromresponses to specific questions. For example, one respondent noted thatcommercial hands-on materials are beyond the budget of elementary schools,and another reported that several teachers have indicated that they would liketo use more manipulatives and models if the funding were available for thepurchase of them or the materials to make them.

In addition, the data reported here indicate that manipulative activities arebeing used to help pupils build conceptual models of science ideas. A responseto such concerns cannot be determined from a large-scale survey of this type.It seems somewhat apparent from this study that non-manipulative scienceinstruction is still the norm in the area where data were collected. The extent

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to which teachers and principals see a need to improve instruction through anappropriate introduction of manipulatives is unclear, although comments

provided on some of the instruments provide cause for optimism. Onerespondent noted that teachers are ... "book" oriented and need to usehands-on materials much more than they do.

Based on the above findings and suggested inferences, the followingconclusions have been drawn:

1. On the average, about half of the teachers have manipulatives availablein their classrooms. The percentage of classrooms with commercially availablescience manipulatives is a little higher than the percentage of classrooms withteacher-assembled science manipulatives. In addition, teachers in the upperelementary grades have more science manipulatives than lower grade levelteachers. There is also considerable variation from school to school in termsof the findings.

2. On the average, teachers use hands-on science materials about 30 days ayear; however, this figure varies considerably from school to school.

3. Hands-on science materials are used to help students both "learn therules" for a procedure and to understand broad concepts. The materials areused for these purposes a little more in the upper grades than in the lowergrades.

4. On the average, lower grade level teachers spend about 70 minutes perweek teaching science with manipulatives and 90 minutes per week teachingscience without manipulatives. For teachers at the upper grade levels, theseamounts are slightly higher. Anecdotal data reflect that these figures may betoo high.

References

Bredderman, T. (1982). Activity science: The evidence shows it matters.Science and Children, 20(1), 39-41.

Gates, R. W., Krockover, G. H., & Wiedermann, R. (1987). Elementarystudent teachers’ perceptions of science in their classrooms: 1985-86. SchoolScience and Mathematics, <^?7(8), 633-643.

Hein, G. E. (1987). The right test for hands-on learning. Science andChildren, 25(2), 8-12.

Kotar, M. (1988). Firsthand experience = firsthand knowledge. Science andChildren, 25(8), 40.

Kyle, W. C., Bonnstetter. R. J., & Gadsden, T. (1988). An implementationstudy: An analysis of elementary students’ and teachers’ attitudes towardscience in process-approach vs. traditional science classes. Journal ofResearch in Science Teaching, 25(2), 103-120.

Kyle, W. C., Bonnstetter, R. J., McCloskey, J., & Fults, B. A. (1985).Science through discovery: Students love it. Science and Children, 25(2),39-41.

Manning, P. C., Esler, W. K., & Baird, J. R. (1982). How much elementaryscience is really being taught? Science and Children, 7P(8), 40-41.

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Rutherford, F. J. (1987). The character of elementary school science. Scienceand Children, 24(4), 8-11.

Scott, W. A. (1955). Reliability of content analysis: The case of nominal scalecoding. Public Opinion Quarterly, J9(3), 321-325.

Shymansky, J. A., Kyle, W. C., & Alport, J. M. (1982). How effective werethe hands-on science programs of yesterday? Science and Children, 20(3),14-15.

Tobin, K. (1986). Student task involvement and achievement in process-oriented science activities. Science Education, 70(1), 61-72.

Yager, R. E., & Bonnstetter, R. J. (1984). Student perceptions of scienceteachers, classes, and course content. School Science and Mathematics,84(5), 406-414.

Authors Sought for NCTM Yearbook onCalculators in Grades K-14

Potential authors are invited to submit manuscripts for the 1992Yearbook, Calculators in Mathematics Education: Impact and Potential.The Yearbook will address the influence of present and emergingcalculator technology on mathematics content, pedagogy, and assessmentmethods in grades K-14.

This Yearbook will provide a forum for discussion of how the universalavailability of calculators in schools brings a new order of priority andemphasis among traditional topics, leads to the inclusion of newmathematical content and methods, and also alters conventionalapproaches to teaching and assessment of student learning.The Yearbook Advisory Panel is interested in reviewing brief descriptive

accounts of successful calculator-based teaching and testing activities inaddition to longer position papers and reports. The Yearbook editor isJames T. Fey, University of Maryland, College Park, Maryland.

Guidelines for authors, which include a more complete description of thetopics to be addressed and instructions for preparing manuscripts, may be,

obtained from:

Christian R. HirschGeneral Yearbook EditorDepartment of Mathematics and StatisticsWestern Michigan UniversityKalamazoo, Michigan 49008

The deadline for submission of manuscripts is March 1, 1990.

School Science and MathematicsVolume 89 (6) October 1989