A comparison of sentential logic skills: Are teachers sufficiently prepared to teach logic?
Easterday, Kenneth E
The Curriculum and Evaluation Standards for School Mathematics (Commission on Standards for School Mathematics, 1989) provide a guideline for improving mathematics instruction and serve as an excellent resource for mathematics educators for making curricular decisions and evaluating current needs. One of these standards recommends for “grades 9 – 12, the mathematics curriculum should include numerous and varied experiences that reinforce and extend logical reasoning skills so that all students can . . . follow logical arguments; judge the validity of arguments; construct simple valid arguments; .. .” (p. 143).
Mathematics educators now have the responsibility of seeking ways to implement the recommendations set forth in the Standards. However, before they can be implemented, there needs to be an understanding of the current level of content and pedagogical knowledge of mathematics teachers. Such understanding would assist educators in planning efficient and beneficial inservice and teacher education programs.
How important is the teacher’s level of knowledge of mathematics? Porter and Brophy (1988) noted that good teachers are knowledgeable about their content area and appropriate strategies for teaching. Stein, Baxter and Leinhardt (1990) found that teachers with “more explicit and better organized knowledge” (p. 641) about their content area were more likely to present lessons leading to conceptual understandings rather than simply a collection of facts. These researchers also suggested that teachers with a weak content knowledge base were more likely to overemphasize procedural rules. After viewing the literature on effective teaching, Anne Reynolds (1992) concluded that among other skills, beginning teachers should enter the first year of teaching with a strong knowledge base of the subject matter they will teach. As Brophy and Good (1986) noted, this knowledge affects how the teacher presents the material, paces the curriculum and turns unexpected questions or events into meaning instruction. So if teachers are expected to emulate and teach logical reasoning skills, they need to possess at least a basic understanding of mathematical logic and how it can be applied.
Easterday and Henry (1978) examined the relationship between maturation or education and understanding of sentential logic by comparing scores on a test of sentential logic (Eisenberg & McGinty, 1974) of junior high and high school students and secondary preservice teachers given in 1976. Results indicated that maturation did appear to be a factor in the development of certain sentential logic skills.
In 1986 and 1992, the same test of sentential logic was given to comparable groups in the same schools used in the 1976 study. All administrations of the test were given during the spring of the year indicated. Results of all three studies will be compared. Questions motivating the researchers of this study include:
1. Are beginning teachers equipped with the necessary training to implement the math standards relating to logic and formal reasoning?
2. Is maturation still a key factor in the understanding of sentential logic?
3. Has a lack of emphasis on formal logic instruction during the last 15 years had an effect on the understanding of sentential logic?
Although these questions will not be answered by this study, an examination of the trend of the scores on the sentential logic test should help in the development of future studies.
Method
The test instrument used in this study was reported in a study by Eisenberg and McGinty (1974). The 30 item test consisted of questions which were equally divided into five categories to represent different logical forms (see Table 1). In each category, the question types are divided into groups of two based on the wording of the two part hypothesis. Therefore each category had two questions with both parts of the hypothesis worded affirmatively, two questions with one part affirmative and one part negative, and two questions in which both parts of the hypothesis were worded negatively. Such test question structure allowed the researchers to examine the effect of the wording of the problem on logical reasoning.
Reliability coefficients of .69 and .75 were reported by Eisenberg and McGinty (1974) for the samples of elementary school children and college students, respectively. Reliabilities for the 1986 and 1992 data collections ranged from .62 to .68 for groups with N>40 (reliabilities were not available for the 1976 data collection).
The test was administered to 7th, 8th and 12th grade students in a local school system. For each period of data collection, the students taking the advanced coursework in each grade level were used. In 1976, the most advanced 7th grade (Pre-algebra), 8th grade (Algebra) and 12th grade (Calculus and Pre-calculus) classes were used. In 1986, the two most advanced 7th grade, 8th grade and 12th grade classes were involved. In 1992, the local school system offered Geometry to the most accelerated 7th and 8th grade students. There were two geometry classes used and these classes contained a mixture of 7th and 8th graders. For the 12th grade, the two calculus classes were used. Presumably the students questioned in 1992 had a much stronger mathematics background than did students questioned in 1976.
The test was also administered to three groups of secondary mathematics education majors at a southern university with a very diverse faculty and student population drawn from every state and many foreign countries. One group (CTD 401) consisted of students studying middle school teaching methods while the second group (CTS 403) consisted of students in the second-quarter of a high school methods sequence. Students in the third group were at the end of their internship in secondary mathematics (CTS 425). The CTD 401 and CTS 403 groups correspond with SED 405 and SED 410 of the Easterday and Henry (1978) study, respectively. Changes at the university since 1976 would imply a “better” student for each data collection. Since 1976, there has been an increase in the average ACT scores of students entering the university and there are more selective grade requirements for admission to the teacher education program.
It should be noted that the school system and the university from which subjects were chosen are in the same city. Furthermore, the junior high and high schools used a the only public schools for this level in the city. Thus, the schools used in this study serve as the primary educational institutions for the children of the university faculty and to other residents more typical of the region. While the researchers acknowledge that this may result in an atypical school for this area, the influence of the university provides a more diverse student population.
Results and Discussion
A summarization of the raw score means for all three years of data collection is shown in Table 2. (Tables 2 and 3 omitted). It appears that maturation and/or education still has some effect on sentential logic skills; however, the effect appears to be less pronounced in the more recent data collections. The differential between the average raw scores of seventh graders and college students at the end of their teacher training has decreased with each collection of data. In 1992, the average raw score for student interns differed by less than one point from the average score of the seventh grade students. These preservice teachers did not even score as well, on average, as the high school seniors.
Raw scores among the 7th, 8th and 12th graders have generally improved since 1976. This is not surprising since at each data collection, the level of mathematics taught for these groups had increased. In contrast, the preservice secondary mathematics teachers’ scores have decreased since 1976. In 1992, as in 1976, the preservice teachers at the end of their mathematics education training had the lowest average raw score among the college students. This decrease leads to the question: Is there a point at which mathematics education has a negative effect on sentential logic skills? In an effort to encourage preservice teachers to become more aware of the impact that their use of logic has in the classroom, do mathematics educators interfere with student understanding of the use of formal logic statements? Another concern is that it appears that preservice teachers have minimal levels of knowledge about formal logic above that of the students they may soon be teaching. This should be considered if implementation of increased logical reasoning at all levels of the mathematics curriculum is expected.
Due to a small group of 7th graders in 1992, the scores for the 7th and 8th grade groups were combined and a weighted average obtained. The same process was also used to combine scores for all college students into one group. The mean percent correct for the three groups to be examined are provided in Table 3. In this table the results are broken down by question type.
For Type I questions, 7th and 8th graders improved dramatically from 1976 to 1992 while the scores of the 12th graders and the college students stayed virtually the same. Figure 1 provides an analysis between the groups based on the wording of the question. (All figures omitted).
In general, all groups did better on affirmatively worded questions and scored the lowest on negatively worded questions for this type. The 12th graders scored more consistently across the three wordings than did the other two groups. Except for the 1992 testing, the college students were better at dealing with questions in which one part was worded affirmatively and the other part was worded negatively. In 1992, the 7th and 8th grade students had higher scores on this form and on the Type I overall.
As in the 1976 study, maturation and/or education still had an effect on Type II questions. The 7th and 8th graders consistently performed below the chance level while the means for 12th graders increased over the three data collection periods. The college students outperformed the other groups substantially in 1976 and 1986, however, the performance of the college students decreased for each data collection By 1992, the preservice teachers barely outperformed the 12th grade students. It would seem that the semantic form of the question has little impact on the scores for the Type II problems (see Figure 2). This would indicate that the semantic form does not interfere with the understanding of this problem type.
Type III, the contrapositive, results are almost the reversal of the results for Type II questions (see Figure 3). The 7th and 8th grade students were the lead performers for all semantic forms. While the group relative performance does not alter between wordings of the question, the negatively worded items appear to have been a problem for 12th graders and college students. Since both of these groups have a substantial drop for this semantic form, it seems reasonable to conclude that there is a negative relationship between mathematics training and performance on this form. The importance of the contrapositive in reasoning and proof is well established which makes these results of more interest.
Type IV questions were apparently difficult for all groups. Scores for groups were below the chance level for the affirmatively and mixed wording problems. The overall performance of the 7th and 8th grade students, as well as that of the college students, decreased since the 1976 study while the 12th graders were relatively consistent for the three collection periods. Figure 4 indicates that although all groups performed better on the negatively worded items, none of the groups did very well on this question type.
Students were more successful with Type V questions than Types II – IV but not as successful as with Type I. For the Type V questions, the college students outperformed both of the other groups. Figure 5 shows that students scored well on the positively and negatively worded questions; while the mixed wording was the most difficult for the students. It would seem that students have a better understanding of the logical “or” when the given is false.
Given the similarity of the results of the Type IV and Type V questions over the three data collections, it seems reasonable to ask a question raised by Easterday and Henry (1978) of whether students understand that both parts of the logical disjunction may be true simultaneously. In a study of college students and their understanding of logical conjunction and disjunction problems, Vest (1981) concluded that students use interpretations of these question types that differ from the definitions for logic. Also note that age and education do not appear to improve the understanding of the disjunction.
Conclusions
While small sample sizes limited analysis, a disturbing trend does seem to be indicated by this data analysis. Since the first comparisons in 1976 between junior high, high school and college students, changes have occurred in the local school system and the university so that students have a stronger mathematics background for the more recent data collections. The results reported here seem to contradict this and suggest that the effect of age and/or mathematical education has lessened. College students are barely performing better than children whom they may one day teach. Since 1976, the degree requirements for the college students included 10 additional quarter hours in mathematics. The combination of increased mathematics requirements and a very strong offering in discrete mathematics courses at this college increased the likelihood that the college students in the two more recent data collections had taken courses in discrete mathematics. Yet, as the results for the preservice teachers indicate, there appears to be a factor interfering with the understanding of sentential logic.
Sentential logic has traditionally been taught in geometry as a means of helping students learn to think about and understand proofs. However, to implement the Standards’ call for “numerous and varied experiences that reinforce and extend logical reasoning skills” (Curriculum and Evaluation Standards for School Mathematics, 1989, p. 143), instruction of logic and good modeling of logical reasoning by teachers are obvious beginnings. Retzer(1985) presented a rationale for shifting sentential logic instruction to algebra. Gregory and Osborne (1973) “identified the frequency of teacher use of logic as a significant variable in children’s acquisition of logic” (p. 35). Both studies provide the framework for training teachers to deal with usage of logic and incorporating logical reasoning in all mathematics classes, not just geometry.
While generalizing to a broader population is limited, a potential weakness for mathematics education programs does seem indicated. Some review and emphasis on logic and how it can be incorporated into the mathematics classroom needs to be addressed in teacher education programs.
Possibly this is only a small indication of a broader concern; that is, mathematics coursework is possibly not providing the same level of conceptual understanding for preservice teachers that it once did. Mathematics educators need to assess the mathematical understandings and/or misconceptions of their students. Teachers must have strong mathematical conceptual frameworks themselves before they can competently implement the Standards.
Many questions remain. It does appear, however, that further detailed study is justified. It is hoped that these results will stimulate mathematics educators to consider their own skills in logic and increase their effort to emulate a correct logical reasoning model in the classroom.
Table 1. Logical forms used in instrument
Type Form
I If p, then q; p; therefore q.
II If p then q; q; therefore not necessarily p.
III If p then q; not q; therefore not p.
IV p or q; p; therefore not necessarily q.
V p or q; not q; therefore p.
References
Brophy, J., & Good, T.L. (1986). Teacher behavior and student achievement. In M.C. Wittrock (Ed.), Handbook of Research on Teaching (3rd. edition). New York: Macmillan, pp. 328-375.
Commission on Standards for School Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Easterday, K.E., & Henry, L.L. (1978). The effect of maturation or education on sentential logic. Journal for Research in Mathematics Education, 9, 67-69.
Eisenberg, T.A., & McGinty, R.L. (1974). On comparing error patterns and the effect of maturation in a unit on sentential logic. Journal for Research in Mathematics Education, 5, 225-237.
Gregory, J.W., & Osborne, A.R. (1973). Logical reasoning ability and teacher verbal behavior within the mathematics classroom. Journal for Research in Mathematics Education, 5, 26-36.
Porter, A.C., & Brophy, J. (1988). Synthesis of research on good teaching: insights from the work of the institute for research on teaching. Educational Leadership, 45, 74-85.
Retzer, K.A. (1985). Logic for algebra: new logic for old geometry. Mathematics Teacher, 76, 457-464.
Reynolds, A. (1992). What is competent beginning teaching? A review of the literature. Review of Educational Research, 62, 1-35.
Stein, M.K., Baxter, J.A., & Leinhardt, G. (1990). Subject-matter knowledge and elementary instruction: A case from functions and graphing. American Educational Research Journal, 27, 639-663.
Vest, F. (1981). College students’ comprehension of conjunction and disjunction. Journal for Research in Mathematics Education, 12, 212-219.
Note: Kenneth Easterday’s address is Department of Curriculum and Teaching, 5040 Haley Center, Auburn University, Auburn, AL 36849-5212. Linda Galloway’s address is Department of Learning Support and Development Studies, Macon College, 100 College Station Drive, Macon, GA 31297.
Copyright School Science and Mathematics Association, Incorporated Dec 1995
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