Degree of Mathematics Fluency and Success in Second-Semester

Research: Science and Education edited by

Chemical Education Research

Diane M. Bunce The Catholic University of America Washington, DC 20064

Degree of Mathematics Fluency and Success in Second-Semester Introductory Chemistry

Vickie M. Williamson Texas A&M University College Station, TX 77823

Doreen Geller Leopold* and Barbara Edgar Department of Chemistry, University of Minnesota, Minneapolis, MN 55455; *[email protected]

For a student not fluent in English, a linguistic barrier to understanding material presented in an American introductory chemistry class would be easily identified. For a student unknowingly lacking an adequate fluency in mathematics, however, an analogous impediment to learning may instead be attributed to the intrinsic complexity of the subject matter. One of the first college science classes in which students may consistently encounter this difficulty is second-semester introductory chemistry. In this course, the qualitative as well as quantitative aspects of topics such as kinetics, chemical equilibrium, entropy and free energy, acid–base chemistry, and electrochemistry are expressed in language liberally seasoned with conversational mathematics. We describe here a mathematics assessment administered to students during the first week of the Spring 2006 semester of Chem 1022 (Chemical Principles II). The quiz was designed to assess students’ basic fluency in the specific areas of mathematics used in lectures, labs, discussions, readings, and homework. For this reason, no calculators were allowed. Students’ responses, and their correlations with success in this course, are described, and several misconceptions revealed by these results are discussed.

Administration Procedures and Student Demographics The 20-question, 30-minute, calculator-free University of Minnesota mathematics assessment for second-semester chemistry (UnMMASSC, “unmask”) and detailed answer key are in the online supplement. This week-1 mathematics assessment (MA) consists of four sections, each with 4–7 questions, on logarithms, scientific notation, graphs, and algebra. The questions are all multiple choice, with all but 3 having 10 possible answers. Each question is worth one point, with no penalty for incorrect answers. The MA was administered as a surprise quiz, for which students did not deliberately study. The detailed answer key was subsequently posted online, with links to some useful mathematics review Web sites. Table 1 provides descriptive information concerning the students tested. These were about half of those enrolled in Chem 1022, whose prerequisite is a passing grade in the first semester (Chem 1021) course (see the online supplement). These courses have no college mathematics prerequisites. In general, high school mathematics requirements for admission as a freshman are currently four years for IT (Institute of Technology) and CBS

Table 1. Comparative Student Demographic Data and Mathematics Assessment Scores Demographic Variable Descriptions Total Students’ Scores,a N (Mean, Women’s Scores,a N (Mean, Men’s Scores,a N (Mean, Median, Standard Deviation) Median, Standard Deviation) Median, Standard Deviation) Chem 1022 Students Tested All students 360 (13.8, 14.0, 3.4) 175 (13.2, 13.0, 3.1) 185 (14.4, 15.0, 3.5) All students, degree-granting programs 325 (13.9, 14.0, 3.3) 163 (13.3, 13.0, 3.0) 162 (14.5, 15.0, 3.5) University Status Freshmen 124 (14.3, 14.0, 3.1) 62 (13.9, 14.0, 2.8) 62 (14.8, 15.0, 3.3) Sophomores 132 (13.6, 14.0, 3.4) 74 (12.9, 13.0, 3.1) 58 (14.6, 15.0, 3.6) Juniors 49 (14.1, 14.0, 3.5) 19 (13.6, 13.0, 3.3) 30 (14.4, 15.0, 3.6) Seniors 19 (12.4, 11.0, 3.0) 8 (12.4, 11.5, 2.8) 11 (12.4, 11.0, 3.3) Nondegree: adult 27 (12.5, 12.0, 4.3) 10 (10.8, 11.0, 3.9) 17 (13.5, 14.0, 4.3) Nondegree: high schooler 8 (15.0, 16.0, 2.7) —b —b Other 1b —b —b Student Distribution by College Programc IT (Institute of Technology) 95 (15.5, 15.0, 2.7) 29 (15.0, 15.0, 2.3) 66 (15.7, 17.0, 2.8) CBS (College of Biological Sciences) 82 (14.6, 15.0, 3.3) 50 (14.2, 15.0, 3.1) 32 (15.3, 16.0, 3.7) CLA (College of Liberal Arts) 104 (12.5, 12.0, 3.0) 64 (12.1, 12.0, 2.8) 40 (13.0, 13.0, 3.1) CNR, COAFESc 34 (12.4, 12.0, 2.8) 19 (12.1, 12.0, 2.5) 15 (12.7, 12.0, 3.2) CCE (Coll. of Continuing Education) 35 (nondegree) —b —b Other 10 (13.2, 11.5, 5.2) —b —b Home State Minnesota 281 (13.7, 14.0, 3.4) 126 (13.1, 13.0, 3.2) 155 (14.2, 14.0, 3.6) Wisconsin 56 (14.0, 14.0, 2.9) 33 (13.3, 13.0, 2.5) 23 (15.0, 15.0, 3.2) North Dakota and South Dakota 12 (14.4, 14.5, 4.1) —b —b Other 11 (15.2, 15.0, 3.8) —b —b could earn a total of 20 points possible on the MA. bTo protect student privacy, data are not provided for categories with small numbers of students. includes chemistry, chemical engineering, other science and engineering departments, and mathematics. University of Minnesota freshmen student admissions requirements currently include four years of high school mathematics for IT and CBS, and three years for CLA, CNR (College of Natural Resources), and COAFES (College of Agricultural, Food, and Environmental Sciences). cIT

aStudents

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Journal of Chemical Education • Vol. 85 No. 5 May 2008 • www.JCE.DivCHED.org • © Division of Chemical Education

Research: Science and Education Table 2. Distribution of Students’ Responses to a Multiple-Choice Mathematics Assessment Questiona Question Text # Log-Related Questions 1 What is the log of 100?

Correct Answer to the Question 2

All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR

50 62 35 41 52 28 65 74 55 64 75 51 78 84 71 63 72 51 68 70 65

All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR

96 98 95 99 99 99 99 99 99 46 51 36 86 90 80

All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR

86 89 82 87 89 83 45 55 31 33 38 26

b quadruples (increases by a factor of 4) q=2 r=1

All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR

72 77 66 71 75 67

c = (ab/d )½

All (325) IT/CBS CLA/CNR All (325) IT/CBS CLA/CNR

89 92 85 56 67 41

2

What is the log of 0.0001?

3

What is the log of 104?

4

What is: log (ab)?

log a + log b

5

What is: log (ab)?

b log a

6c

Given: b = 10a/2, solve for a.

7

Which is larger: log (10) or ln (10)?

−4 4

Scientific Notation-Related Questions 8 Write the number 620 in scientific notation, with 2 significant figures.

a = 2 log b

ln (10)

6.2 × 102

9

Which is larger: 1.0 × 104 or 8.0 × 103?

1.0 × 104

10

Which is larger: 1.0 × 10–7 or 2.0 × 10–8?

1.0 × 10–7

11

What is: 5.00 × 104 – 3.00 × 10–2?

5.00 × 104

12

What is: (2 × 104) (3 × 102)?

Graphing-Related Questions 13 The equation of a straight line is y = mx + b. What are m and b in the graph below? (See Figure 2.) 14 The equation of a straight line is y = mx + b. What are m and b in the graph below? (See Figure 3.) 15 What function is graphed below? (See Figure 4.) 16

What function is graphed below? (See Figure 5.)

Algebra-Related Questions 17 Given: b = c2d If c is doubled and d is unchanged, how does the value of b change? 18 Given: b = c q d r When c is doubled and d is unchanged, then b quadruples. When c is unchanged and d is doubled, then b doubles. What are q and r ? 19 Given: Solve for c. ab d= c2 20

Given:

d=

a–b c –b

Simplify, if possible.

Students Correct by Program, %

6 × 106

m = –1 b = 10 m = 0.8 b=4 y = 10x y = log x

none of the above

Most Common Incorrect Answer,b % 10 0.01 100 (log a) (log b) (log a)b log (b/2)

log (10)

6.2 × 101

Students with This Answer, %

r Values

All (325) 44 IT/CBS 34 CLA/CNR 56 All (325) 27 IT/CBS 19 CLA/CNR 36 All (325) 15 IT/CBS 11 CLA/CNR 20 All (325) 27 IT/CBS 16 CLA/CNR 40 All (325) 9 IT/CBS 3 CLA/CNR 14 All (325) 8 IT/CBS 3 CLA/CNR 14 All (325) 32 IT/CBS 30 CLA/CNR 35

0.22

All (325) 3 IT/CBS 2 CLA/CNR 4

0.01

0.20 0.21 0.22 0.12 0.21

0.09

0.04 −0.07 4.97 × 104

All (325) 21 IT/CBS 22 CLA/CNR 20 All (325) 11 IT/CBS 7 CLA/CNR 15

0.19

All (325) 11 IT/CBS 10 CLA/CNR 13 All (325) 5 IT/CBS 6 CLA/CNR 4 All (325) 23 IT/CBS 25 CLA/CNR 20 All (325) 32 IT/CBS 37 CLA/CNR 25

0.18

b doubles (increases by a factor of 2) q=4 r=2


0.16

c =(ab/d )2


0.19

6 × 108

m=1 b = 10 m=4 b = 0.8 y = ex y = ln x

d = a/c

0.04

0.07 0.20 0.20

0.08

0.22

aThe 12 questions displayed in bold type in column 1 showed the highest correlations of the students’ correct response with their course grades, with correlation coefficients r = 0.16–0.22 as given in the last column (each with p < 0.005). Each question had ten answers listed, except for questions 7, 9, and 10, which had two answers. The percentage of students choosing the correct answer (column 4) or the most common incorrect answer (column 6) is shown for each question in row 1 for all 325 students in degree-granting programs, in row 2 for the 177 students in IT or CBS (which require four years of high school mathematics for incoming first-year students), and in row 3 for the 138 students in CLA, CNR, or COAFES (which currently require three years of high school mathematics for incoming first-year students). See Table 1 for college acronyms. bOther incorrect answers (%) chosen by at least 10% of one or more of the three student groups (all 325; IT/CBS; CLA/CNR/COAFES): 2, none of the above (19; 20; 19); 3, 10,000 (13; 9; 17); 11, none (of the answers listed) are correct to three significant figures (16; 16; 16) and 2.00 × 102 (10; 7; 15); 15, y = log x (13; 6; 23); y = ln x (8; 6; 12); 16, y = ex (9; 7; 12); y = −log x (7; 3; 12); 20, d = (a/c) – 1 (11; 9; 14); d = (a/c) – b (9; 8; 10) cQuestion 6 erroneously had a second correct answer, a = log (b2), chosen by 6% of the students, who are included above in the percentages answering correctly.

© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 5 May 2008 • Journal of Chemical Education

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Research: Science and Education

Students’ Responses to Mathematics Questions Table 1 reports that the average score for the 325 students in degree-granting programs was 13.9 of 20 possible points, and 13.8 for all 360, including the 35 CCE nondegree students. Average scores were higher for IT (15.5) and CBS (14.6) students than for CLA (12.5) and CNR or COAFES (12.4) students. Scores were about the same for first–third-year students. In subsequent analyses, we omit the 35 CCE (College of Continuing Education) students to focus on the less heterogeneous group of 325 students in degree-granting programs. Fig-

10

Students (%)

(College of Biological Sciences), and three years for other colleges, including CLA (College of Liberal Arts), CNR (College of Natural Resources), and COAFES (College of Agricultural, Food, and Environmental Sciences). To help assess the transferability of these results to other student populations, we note that the average composite ACT scores for freshmen admitted in Fall 2005 were 28.3 (IT), 27.8 (CBS), 25.8 (CLA) and 24.3 (CNR, COAFES) (see the online supplement). The 2005 Minnesota state average was 22.3 out of a possible 36 points (1). Students enrolled in the IT Honors course Chem 1032H are not represented in this study. These included 55 IT students and 16 students from other colleges, together 10% of the ~715 students who completed Chem 1022 that semester.

5

0

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20

Mathematics Assessment Score Figure 1. Graph of students’ MA scores (N = 325) out of 20 points possible (mean 13.9, median 14.0, standard deviation 3.3).

ure 1 shows the broadly distributed mathematics scores for this group. At the high end, 34% of students scored 16–20 points, missing ≤4 questions. Conversely, 36% of the students answered ≤12 of the 20 questions correctly. Table 2 summarizes MA responses of the 325 students. The percentage of students choosing the correct answer and the most common incorrect one are given for all students, for the IT or CBS students, and for the CLA, CNR, or COAFES students. Table 2 note b lists other answers from at least 10% of students

Table 3. Distribution of Students’ Responses to Related Mathematics Questions Program, Gendera

Criteria for Analysis

Students Correct, %b

Log-Related Questions Questions 1–7 and 16: Students with at least seven of these eight questions correct

Questions 1–3: Students with all three of these questions correct (common logs of 100, 0.0001, and 104)

Questions 1–3: Students with at least two “square root” incorrect answers (log 100 = 10, log 0.0001 = 0.01, log 104 = 100) Questions 1 and 5: Of students with question 5 correct (log question 1 correctly (log 100 = 2)

ab

= b log a), % also answering

Questions 1 and 16: Of students with question 1 correct (log 100 = 2), % also answering question 16 correctly (graph of y = log x)

All (325) IT/CBS CLA/CNR F, M All (325) IT/CBS CLA/CNR F, M All (325) IT/CBS CLA/CNR F, M All (325) IT/CBS CLA/CNR F, M All (325) IT/CBS CLA/CNR F, M

21 31 9 15, 31 42 17 25, 27 18 37 27, 55 65 38 50, 43 46 33 32,

All (325) IT/CBS CLA/CNR F, M All (325) IT/CBS CLA/CNR F, M All (325) IT/CBS CLA/CNR F, M All (325) IT/CBS CLA/CNR F, M

30 39 17 21, 19 26 9 11, 60 67 51 55, 36 44 25 30,

28

38

27

59

54

Other Questions Questions not log-related (8–15 and 17–20): Students with at least 11 of these 12 questions correct

Graphing questions (13–16): Students with all four of these questions correct

Ratios involving integral exponents (17–18): Students with both questions correct

Algebra questions (17–20): Students with all four of these questions correct

38

27

65

41

aFemale

denoted by “F”, male denoted by “M”. See Table 1 for college acronyms. bPercentages of student responses meeting the stated criteria among the 325 students in degree-granting programs and among the following subsets of this group: 177 students in IT or CBS (79 F, 98 M); 138 students in CLA, CNR, or COAFES (83 F, 55 M); all women (“F”, 163) and all men (“M”, 162). For example (last item), 36% of the 325 students correctly answered all four algebra questions (questions 17–20), including 44% of the students in IT or CBS, and 25% of the students in CLA, CNR, or COAFES.

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point at x = 10, y = 1) and knowledge of common log values. This question was answered correctly by 33% of the students. Even among the students who correctly answered question 1 (log(100) = 2), only 43% also answered question 16 correctly. The most common incorrect answer, y = ln x, again had the correct functional form, yet was not consistent with the numerical values of the coordinates provided. These results indicate that many students have difficulty extracting quantitative information from graphs of exponential and logarithmic functions. Overall, 19% of the students answered all four graphing questions correctly. Part IV comprises four algebra questions. Questions 17 and 18 (b = c 2d and b = c q d r, respectively) are modeled after 12

y-Axis

10 8 6 4 2 0

0

2

4

6

8

10

12

x-Axis Figure 2. Graph for MA question 13 (see Table 2).

12

y-Axis

10 8 6 4 2 0

0

2

4

6

8

10

12


y-Axis

1000 800 600 400 200 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0


2

y-Axis

in these three groups. The 2006 results (online supplement) displayed a reasonable degree of reproducibility the following year. Table 3 describes, for sets of related questions, the responses of the students, grouped by college program and by gender. See the online supplement for further discussion of gender and these results. Part I of the MA consists of seven questions concerning logarithms. Questions 1–3 ask for the common, base 10 logarithms of 100, 0.0001 and 104. A note (conspicuously printed in a larger, bold font, just above question 1) points out that in these questions, “log means log10” and “ln means natural log (base e = 2.71828...)”. The answers listed are integers, and do not require identification of the correct number of significant figures. As noted in Table 3 (row 2), all three questions were answered correctly by 31% of the students, including 42% of the students in IT or CBS, and 17% of the students in CLA, CNR or COAFES. Thus, only a minority of the students demonstrated a solid understanding of the meaning of the common log (if 10a = b, then a is the log of b). In contrast, 78% knew that log (ab) = b log a. Of the students who correctly answered question 5, only 55% also knew that log (100) = 2 (#1), although one assumes that all knew that 102 = 100. As noted in Table 3 (row 3), the most common incorrect answers to questions 1–3 were the square roots (log 100 = 10, log 0.0001 = 0.01, log 104 = 100), and 27% of the students chose at least two of these three answers. These results reveal that many college students are familiar with the formal rules for manipulating logs, but have forgotten the actual meaning of the log. The responses suggest that many may think of logs as being equal or similar to square roots, or as being fundamentally defined in terms of the formal rules for their manipulation. Thus, even common logs may be viewed as inherently arcane entities not expected to have obvious numerical values. These misconceptions might hinder students’ understanding of acid–base equilibria, electrochemistry, and other introductory chemistry topics involving common logs. Part II of the MA has five questions on the use of scientific notation. Questions 8–10 are intentionally easy, and correct answers were chosen by ≥96% of the students. These three questions, together with questions 12–14 and 19, provided seven easy questions to reduce the possibility that the MA might backfire, and excessively discourage some students already anxious about the adequacy of their mathematics skills. Although the students did well on questions 8–10 and 12 on scientific notation, a perusal of their exam papers revealed that many chose to write out the answers in more familiar notation (e.g., 20,000 × 300 = 6,000,000) and then convert back to scientific notation to select the correct answer. These observations indicate a lack of comfort with the direct interpretation of numbers in scientific notation and of the simple arithmetic rules for manipulating them (see the online supplement). When students do these types of problems using calculators, a procedure that may involve human error, many may have limited ability to use estimation to assess the reasonableness of the results. Part III includes four questions requiring the interpretation of graphs, which are shown in Figures 2–5. To facilitate the identification of question 15 (Figure 4), the graph includes points at x = 3, y = 1000 and x = 2, y = 100. This function was correctly identified as y = 10x by 45% of the students, with many (23%) choosing instead the related function y = ex. In question 16 (Figure 5), the identification of the function y = log x requires both the interpretation of the graph (which includes a revealing

1 0 ź1 ź2 ź3

0

5

10

15



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initial rates problems in chemical kinetics. Both questions were answered correctly by 60% of the students. Thus, 40% would benefit from additional instruction and review of problems involving ratios of numbers with integer exponents, before adding the additional complexities of chemical applications. For question 20 (56% correct), the students’ diverse responses also emphasize the need to review the meaning of a common factor and to practice simplifying rational expressions. All four algebra problems were answered correctly by 36% of the students.

all students women men

Students (%)

15

10

5

0

0

1

2

3

4

5

6

7

8

9

10

11

12

Mathematics Assessment Score Figure 6. Scores on the mathematics assessment’s subset of 12 questions showing the highest correlations with course grades (bold type in Table 2) for 325 students in degree-granting programs (dark gray bars, mean 7.1, median 7.0, standard deviation 2.7), including 163 women (light gray bars, mean 6.6, median 6.0, SD 2.5) and 162 men (white bars, mean 7.6, median 8.0, SD 2.9).

Average Course Grade

4.0

3.0

2.0

1.0

0

2

4

6

8

10

12

Mathematics Assessment Score Figure 7. Average final course grade in Chem 1022 vs MA score (questions correct out of the 12 in bold in Table 2) for 325 students in degree-granting programs. Course grade scale: A = 4.00, A− = 3.67, B+ = 3.33, B = 3.00 ... F or withdrawal after 8th week = 0. Combined data are plotted for students with mathematics scores of 0–2 (11 students). The other 10 points each show the average course grade obtained by students having the same score (15–40 students per point). The line shows regression results (y-intercept 1.63±0.13, slope 0.14±0.02). See the online supplement for the tabulated data.

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Of the 12 questions not related to logarithms (8–15 and 17–20), 30% of the students answered at least 11 correctly, including 39% of those in IT or CBS. Correlations of Mathematics Assessment Scores with Chem 1022 Final Course Grades Students were permitted to use scientific (including graphing) calculators for all chemistry exams and labs. Course grades were based 80% on multiple-choice exam grades and 20% on lab grades; they are reported here using a 0–4 scale, with the highest grade an A (4.00). To investigate possible correlations between the students’ responses to the MA questions and their final course grades, we calculated, for each of the 20 questions, the corresponding Pearson product–moment correlation coefficient, r (see the online supplement). This correlation coefficient is a measure of the degree of linear relationship between the two variables (2). As shown in Table 2, questions 5, 7, 8–10, 12, 14, and 18 had the lowest r values (r ≤ 0.12). These included 8 of the 11 easier questions, which were each answered correctly by about 70% or more of the students. The remaining 12 questions, indicated by bold type in Table 2, had higher r values of 0.16–0.22. Results for these questions are examined further below. For this 12-question subset of the MA, Figure 6 displays a bar graph showing the distribution of scores for the group of 325 students. The average score was 7.1 out of 12 points, with a broad distribution having a standard deviation of 2.7 points. Gender differences are observed, as in Tables 1 and 3 (see the online supplement). Again, the average scores were higher for the 177 students in IT (8.4) or CBS (7.6), colleges that require four years of high school mathematics for admitted first-year students, than for the 138 students in CLA (5.8), or CNR and COAFES (5.9 combined), which currently require three years. The relationship between the 325 students’ course grades and their total scores on the 12-question subset of the MA shows a correlation coefficient of r = 0.41. For our sample size, this r value indicates a significant correlation between these two variables at the p < 0.001 level. That is, if the true correlation were zero, then the probability (p) of observing r ≥ 0.41 (or r ≤ ‒0.41) would be below 0.1% (2). The corresponding square of the correlation (coefficient of determination), r2 = 0.17, indicates that about 17% of the variation in course grades is predictable from the mathematics scores (2). This modest value reflects the large range in course grades earned by students with the same MA score, as shown by the additional data in Table 1 and Figure 1 of the online supplement. Nevertheless, these r and r2 values are comparable to those reported for some tests specifically designed for chemistry course placement and diagnostics (3–5). The importance of mathematics skills as predictors of success in college chemistry has been stressed in these and other studies (3–11). Figure 7 displays a clear correlation (i.e., a nonzero slope) for the average course grades earned by the 325 students as a function of their scores on the 12-question subset of the MA. Combined data are plotted for the 11 students with mathematics scores of 0–2. Each of the other 10 points shows the average course grade earned by students with the same score (15–40 students per point). The slope of the regression line is 0.14±0.02, where the error bar is ±1 standard error. Consistent results (0.16±0.02) were obtained for students in Chem 1022 the following year (see the online supplement).



The relationship between average chemistry course grades and students’ scores on the 12-question subset of the MA is, of course, particularly pronounced upon comparing the high and low extremes. For example, the 15 students who answered all 12 mathematics questions correctly earned an average chemistry course grade of A−, and the 94 students with high MA scores of 9–11 points earned an average grade of B. Conversely, the 73 students with low MA scores of 4–5 out of 12 averaged a C+, and the 30 students with MA scores of 0–3 averaged a C. Examining these relationships in reverse, one finds that the 66 students earning an A or A− in the course had an average MA score of 8.7 out of 12 points, whereas the 44 students who received C− or below, or who withdrew during the second half of the semester, had an average MA score of 4.9. These trends can be sufficiently convincing to motivate some students to strengthen their mathematics skills, while the wide variations in course grades earned by students with the same MA score can offer reassurance that, with sufficiently diligent studying, one can excel in this course despite initial deficiencies. Discussion To fluently translate the many quantitative examples and problems used throughout the semester in lectures, labs, and discussions and extract their conceptual meanings, we believe that students need a mathematics background adequate to enable correct responses to all or nearly all of these very basic questions. The results described above revealed that the majority of the students in our second-semester chemistry course lacked this prerequisite degree of mathematics fluency. Misconceptions have been shown to play a powerful, adversarial role in science pedagogy (12, 13). Once flawed concepts become integrated into one’s cognitive structure, they can be resistant to change and can interfere with subsequent learning (12). As discussed above, the students’ responses revealed mathematical misconceptions, such as the interpretation of a logarithm as a square root. The results of this study also highlight additional misconceptions that can interfere with learning. These are pedagogical misconceptions, and are held by many teachers as well as by their students. Pedagogical Misconceptions One such misconception is: “The students already know this simple material.” To the surprise and dismay of many of our faculty and teaching assistants, only about one-third of the students tested did as well on the MA as we had anticipated. Thus, we found these results quite useful to familiarize us with our students’ backgrounds. For many students as well, their responses exposed specific deficiencies in areas they thought they understood. Such misconceptions can persist, in part, because the incessant use of calculators can “mask” deficiencies in mental mathematics skills and conceptual understanding, both from teachers and from the students themselves. We need to remember that the demonstration of sophisticated, calculator-based skills does not necessarily imply the mastery, or diminish the importance, of the more basic mathematics skills. A second type of pedagogical misconception is, “If they didn’t get it by now, they won’t get it”, or, as some students bemoan, “I’m bad at math”. We found that many bright students did not do well on the MA. Yet, these questions are simple enough to be solved by students with the requisite knowledge,

without requiring novel or complex problem-solving skills that would favor the more mathematically gifted students. Therefore, the present results probably reflect, at least in part, current methods of teaching mathematics in American high schools (14–17), which may not emphasize calculator-free quantitative problem solving as an important priority. In addition to enabling the evaluation of functions whose meaning one does not understand, excessive calculator dependency can decrease students’ practice using basic algebraic methods (e.g., simplifying expressions as in question 20). Therefore, for most of our students, a poor MA score is not a symptom of being intrinsically math-challenged. The simple mathematics skills essential to this course can be mastered by virtually every student with sufficiently focused practice. Negative mathematics self-efficacy beliefs form a type of misconception that can seriously impede students’ science education and may be more difficult to eradicate. The influence on mathematics performance of the beliefs that people hold about their own competence has been reported to be as strong as that of general mental ability (18). We have observed that some students are significantly hindered by a sort of mathematical nocebo response (19). This course can help to erode this psychological obstacle by deliberately developing internalized mathematics skills needed to approach chemistry problems with confidence. A third misconception might be expressed by some teachers as: “It’s not my job to teach high school math” or, from the student’s viewpoint, “I’m taking this class to learn chemistry, not math”. Naturally, it would be preferable for our students to embark upon college chemistry with a firmer grasp of these basic topics. However, our challenge as teachers lies in genuinely educating our present students, with full cognizance of the skill sets they now have. Just as one might fix a kink in a hose that impedes the flow of water, we need to help students identify and strengthen the specific mathematics skills whose deficiencies we realize will seriously hamper their efforts to learn chemistry. References to questions like those on the MA when covering the relevant chemistry topics in class can be helpful in leading our students to share this realization and motivating them to correct these deficiencies. A fourth pedagogical misconception might be summarized, “Mathematics tests prohibiting calculators are irrelevant to predicting success in this course, since the chemistry exams allow calculators”. As is illustrated in Figure 7, students with more fluent mathematics skills were found to be more likely, on average, to do well in Chem 1022, although calculators were allowed during exams. Moreover, with regard to the relationship between students’ mathematics fluency and their abilities to achieve the firm conceptual understanding that constitutes true success in this course, we believe that the correlation may be stronger than is suggested by the data in Figure 7, since many chemistry exam questions can successfully be answered through the mechanical application of formulaic methods that are only poorly understood. The MA results are meaningful not only in the direct sense that these specific mathematics skills are used in solving chemistry problems, but also—we believe primarily—because they reflect underlying thinking and learning patterns that are correlated with success in this course. The suitability of these simple questions to reveal students’ level of basic cognitive skills is particularly clear for questions 1–3 and 11. Although


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these four questions could easily have been answered with calculators, which were allowed on all of our chemistry exams, they were among those showing the highest correlations with course grades. The correct answers to these questions, in an unannounced and calculator-free quiz, may reflect an inclination to think of mathematical entities in terms of their underlying meanings (e.g., the log is an exponent) rather than in a more superficial operational manner (e.g., the log is what you get when you press the “log” key). The former approach, which should be assimilated in pre-college mathematics classes, is also likely to lead to greater success in a vertically structured, mathematicsbased knowledge domain such as second-semester chemistry. In contrast, if we encourage our students to dive into quantitative chemistry problems without a firm mastery of basic mathematics skills, we risk inadvertently reinforcing plug-andchug problem-solving strategies and students’ confusion of superficial with genuine understanding. Indeed, such pedagogical gaffes are common. For example, we might expect students who cannot evaluate the pH of a 0.0001 M HCl solution without a calculator (question 2) or quantitatively interpret logarithmic graphs (question 16) to calculate pH values at various points along a titration or a buffer curve. Or, we might require students who cannot solve problems involving ratios of numbers with integer exponents by inspection (questions 17 and 18) to solve for non-integer exponents using log–log plots in initial rates laboratory kinetics experiments. It is hardly surprising that many students feel that they have little choice except to approach these as mechanically formulaic exercises, and glean little conceptual insight from their efforts. For these students, introductory chemistry classes may succeed in transmitting mainly superficial knowledge that displays little transferability to other contexts and fails to provide a robust foundation for future courses (20–23). Strategies To Improve Students’ Mathematics Fluency To help foster the fluency required to really understand chemistry in the language of mathematics, a useful strategy includes the increased use of numerical problems whose exact solutions require only mental or written calculations. The advantages of calculator-free quantitative problem solving have been emphasized by instructors of introductory chemistry (24, 25), physics (26), and astronomy (27) courses. For grade-school children, many pedagogical experts concur that building a repertoire of automated skills is essential for the optimum development of their higher-order problem-solving abilities and conceptual understanding of mathematics (14, 28–31). This mental mathematics fluency subconsciously guides one’s intuitions and frees more of one’s attention to focus on the steps involved in complex problem solving. Similarly, for college chemistry students, improved mathematics fluency expedites the development of a matrix of mental associations of the sort that can generate the inspired guesswork that characterizes experts (25). We need to encourage our students to put aside their calculators and associated cyborgian thinking patterns, so they can surpass their calculators’ capabilities and learn to think conceptually and creatively about quantitative chemistry. Improving basic mental mathematics skills would also enable students’ reflections on their own learning and thinking processes to be more mathematically articulate. Metacognitive skills have been identified as key to “deeper, more durable, and more transferable learning” and are vital for mathematical prob730

lem solving in chemistry (32–34). By practicing calculator-free quantitative problems, students will be better able to tell when their chemistry calculations “make sense” or “feel right”, and will acquire a greater sense of “personal ownership” (24) toward them. When doing calculator-based problems, they will be better able to estimate the answers and to assess the reasonableness of the results. A variety of pedagogical strategies might be employed to promote students’ mathematics fluency during, and prior to, their participation in second-semester college chemistry, and we discuss some of these in the online supplement. For authors of chemistry textbooks, these include a greater use of quantitative problems amenable to mental or pencil-and-paper solutions (perhaps identified by a “no calculator” symbol), the inclusion of more problems of this type in multiple-choice test banks, and the addition of practice questions (with answers) to appendices on mathematical methods. In many cases, existing chemistry questions can be converted into calculator-free versions by artful choices of numerical values to simplify the mathematics. The pedagogical benefits can outweigh the minor drawback of using less-accurate values (26). For chemistry teachers, these textbook improvements would encourage the use of calculator-free exams, which would help to further motivate students to develop these skills. For high school mathematics teachers as well, the use of cumulative calculator-free numerical exercises can help students maintain the repertoire of basic skills needed to achieve the requisite mathematics fluency. We hope that the results reported here will help motivate additional efforts among educators to develop effective methods to solve this problem. Conclusion We have reported the contents and results of a mathematics assessment administered at the start of the second-semester introductory chemistry course required for science and engineering majors at the University of Minnesota. This calculator-free, 20-question, 30-minute, multiple-choice quiz (see the online supplement) includes questions selected specifically for their relevance to this course, concerning logarithms, scientific notation, graphs, and algebra. Significant correlations were observed between these MA scores and success in this course, as measured by performance on exams (for which scientific calculators were permitted) and final course grades. Thus, responses to the simple questions on this unannounced quiz appear to have some predictive utility as signatures of underlying thinking and learning patterns associated with success in this course. In addition, we have argued that these results indicate an inadequate degree of mathematics fluency for the majority of the students tested, which can seriously impede their abilities to develop a firm conceptual understanding of quantitative introductory chemistry. Several mathematical and pedagogical misconceptions highlighted by these results have been discussed, and possible methods suggested (see the online supplement) to enhance students’ mathematics fluency during, and prior to, their participation in this course. American students’ mathematics skills and scientific literacy are currently issues of national concern (35, 36). The knowledge gained in this and other studies can help empower chemistry teachers to motivate students to improve their mathematics fluency as applied to the quantitative description of chemical phenomena. Thus, in teaching the specific topics covered in



second-semester college chemistry, we are also contributing to our students’ education in the broader sense of offering a sort of language immersion course that can provide an essential key to their future scientific literacy. As Wittgenstein has said, “the limits of my language mean the limits of my world” (37). Acknowledgements We thank Barbara Barany, Ken Leopold, Sandy Weisberg, John Edmundson, Lee Penn, Michelle Driessen, Stephen Miller, Beau Barker, Alex Wagner, and Jaye Warner for valuable advice and discussions; Karen and Allison Leopold for help with the data analysis; the reviewers for suggestions; and Bernadette Caldwell for editorial assistance. We are also grateful to our Chem 1022 students for their willing participation in this study. Literature Cited 1. Minnesota Office of Higher Education, Insight Newsletter, August 2005. http://www.ohe.state.mn.us/pdf/enrollment/Insight/ InsightAug05.htm (accessed Feb 2008). 2. Moore, D. S.; McCabe, G. P. Introduction to the Practice of Statistics, 3rd ed.; W. H. Freeman: New York, 1999; pp 73, 127, 144, 458. 3. Bunce, D. M.; Hutchinson, K. D. J. Chem. Educ. 1993, 70, 183–187. 4. Russell, A. A. J. Chem. Educ. 1994, 71, 314–317. 5. Flapan, E.; Gonzalez, B. L. Enhancing the Mathematical Understanding of Students in Chemistry: 2004 Evaluation Report. http:// www.math.pomona.edu/flapan_nsf/AnnlEval06.04.pdf (accessed Feb 2008). 6. Tai, R. H.; Ward, R. B.; Sadler, P. M. J. Chem. Educ. 2006, 83, 1703–1711. 7. Pienta, N. J. J. Chem. Educ. 2003, 80, 1244–1246. 8. McFate, C.; Olmsted, John III. J. Chem. Educ. 1999, 76, 562– 565. 9. Spencer, H. E. J. Chem. Educ. 1996, 73, 1150–1153. 10. Ozsogomonyan, A.; Loftus, D. J. Chem. Educ. 1979, 56, 173– 175. 11. Pickering, M. J. Chem. Educ. 1975, 52, 512–514. 12. Nakhleh, M. B. J. Chem. Educ. 1992, 3, 191–196. 13. Misconceptions as Barriers to Understanding Science. In Science Teaching Reconsidered: A Handbook; Moore, C. B., Chair; National Academy Press: Washington, DC, 1997; Ch. 4, pp 27–32. http://www.nap.edu/readingroom/books/str/4.html (accessed Feb 2008). 14. Ball, D. L.; Ferrini-Mundy, J.; Kilpatrick, J.; Milgram, R. J.; Schmid, W.; Schaar, R. Reaching for Common Ground in K–12 Mathematics Education, 2005. http://www.maa.org/commonground/ (accessed Feb 2008). 15. Mathematically Correct. http://www.mathematicallycorrect.com/ (accessed Feb 2008). 16. Illinois Loop. Understanding the Battles over Math. http://www. illinoisloop.org/math.html (accessed Feb 2008). 17. New York City HOLD. http://www.nychold.com/ (accessed Feb 2008). 18. Pajares, F. Gender Differences in Mathematics Self-Efficacy Beliefs. In Gender Differences in Mathematics; Gallagher, A. M., Kaufman, J. C., Eds.; Cambridge University Press: Cambridge, UK, 2005; Chapter 14, pp 294–315.

19. Harvard Mental Health Letter, Harvard Medical School, March 2005. http://www.health.harvard.edu/newsweek/The_nocebo_response.htm (accessed Feb 2008). 20. Pickering, M. J. Chem. Educ. 1990, 67, 254–255. 21. Lythcott, J. J. Chem. Educ. 1990, 67, 248–252. 22. Nakhleh, M. B.; Mitchell, R. C. J. Chem. Educ. 1993, 70, 190–192. 23. McDermott, L. C. Am. J. Phys. 1993, 61, 295–298. 24. Hoffman, M. Z. Teaching General Chemistry Concepts to One (Or Maybe Two) Sig Figs. In Book of Abstracts, 219th ACS National Meeting, San Francisco, CA, March 26–30, 2000, CHED193; American Chemical Society: Washington, DC, 2000. 25. Barouch, D. H. Voyages in Conceptual Chemistry; Jones and Bartlett: Sudbury, MA, 1997. 26. Sherfinski, J. Physics Teacher 2001, 39, 184–185. 27. Ruiz, M. J. Physics Teacher 2004, 42, 530–533. 28. Wu, H. Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education. American Educator/American Federation of Teachers, Fall 1999, pp 1–7. http://www.aft.org/ pubs-reports/american_educator/fall99/wu.pdf (accessed Feb 2008). 29. Seeley, C. National Council of Teachers of Mathematics, News and Media, Past President. Do the Math in Your Head! http:// www.nctm.org/about/content.aspx?id=928 (accessed Feb 2008). 30. Mighton, J. The Myth of Ability: Nurturing Mathematical Talent in Every Child; Walker: New York, 2003. 31. Royer, J. M.; Tronsky, L. N.; Chan, Y.; Jackson, S. J.; Marchant, H., III. Contemp. Educ. Psych. 1999, 24, 181–266. 32. Rickey, D.; Stacy, A. M. J. Chem. Educ. 2000, 77, 915–920. 33. Hollingworth, R. W.; McLoughlin, C. Australian J. Educ. Technol. 2001, 17, 50–63. http://www.ascilite.org.au/ajet/ajet17/hollingworth.html (accessed Feb 2008). 34. Hong, E.; O’Neil, H. F.; Feldon, D. Gender Effects on Mathematics Achievement. In Gender Differences in Mathematics, Gallagher, A. M., Kaufman, J. C., Eds.; Cambridge University Press: Cambridge, UK, 2005; Chapter 13, pp 264–293. 35. National Academies Press. Rising Above the Gathering Storm. http://books.nap.edu/catalog.php?record_id=11463#toc (accessed Feb 2008). 36. U.S. Department of Education, National Mathematics Advisory Panel. http://www.ed.gov/about/bdscomm/list/mathpanel/index. html (accessed Feb 2008). 37. Wittgenstein, L. Tractatus Logico-Philosophicus (1922); Routledge: London, 1962.

Supporting JCE Online Material http://www.jce.divched.org/Journal/Issues/2008/May/abs724.html Abstract and keywords Full text (PDF) Links to cited URLs and JCE articles Supplement Suggestions for promoting mathematics fluency

Administration procedures; population and sampling methods

Statistical analysis of the study’s results

Results from Spring 2007 class

Mathematics assessment and answer key

Bibliography of online mathematics review resources


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Degree of Mathematics Fluency and Success in Second-Semester

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