Mathematical Readiness of Entering College Freshmen: An Exploration of Perceptions of Mathematics Faculty

Scaled items. Analysis of the scaled items was performed using SPSS software and standard statistical techniques (Green, Salkind, & Akey, 2000), including finding means, standard deviations, skewness, and tests for normality. Given the relatively small sample comprised in this study, it was decided to focus primarily on means and standard deviations in the analysis. Although the sample size was not sufficient for performing a factor analysis (Grimm & Yarnold, 1998), the four constructs identified during the content-validity process were revisited during the analysis process. Means and standard deviations were found not only for each item, but also for the four constructs (i.e., subject knowledge, number sense, measurement and data, and reasoning and generalization).

Free response item analysis. The free responses were coded selectively (Strauss & Corbin, 1990) according to the original constructs. Additionally, open and axial coding were used to identify themes not represented by the original constructs. After the items were classified, peer debriefing (Lincoln & Guba, 1985) was employed to validate the classification and coding of the qualitative data. If there were disagreements in coding, they were discussed until consensus was achieved. Along with calculating frequencies, the coded responses were inspected for patterns and themes that might further elucidate the perceptions of college mathematics instructors related to mathematical readiness.

Overall results of scaled items. When asked to rate the mathematical ability of an average freshman, the mean rating was 2.17 (SD 1.0316) across all items within the subject knowledge construct. On the 0-5 scale, this falls between the poor and average rating. When asked to rate the importance of topics within the subject knowledge construct, the mean rating was 1.516 (SD 0.666). On the 0-2 scale, this falls between somewhat important and very important. This demonstrates a potential disconnect between perceived mathematical ability (i.e., poor to average) and importance of mathematical topics (i.e., important).

Overall results of free response items. Participants mentioned topics related to subject knowledge in all three of the free response items. The first question asked, “Are freshman prepared for the mathematics requirements at your school?” Of the 22 respondents, only 2 (9.1%) said that freshman were prepared, and even these qualified their responses with statements such as, “generally”; 13 (59.1%) said that freshman were not prepared; another 4 (18.1%) said that some were prepared and some were not; 2 (9.1%) of the respondents did not answer this question. Within their responses to this question, participants mentioned a topic related to subject knowledge in 9 of the 22 responses (40.9%). Within that number, 8 made additional comments mentioning subject knowledge skills that were lacking. The second question asked, “What are some mathematical skills and topics that students are lacking when entering college?” Participants mentioned subject knowledge topics in 13 of 22 responses (59.1%). Of the 13 responses, 10 mentioned algebraic knowledge, with the remaining responses falling within geometry, statistics, and overall basic skills. The last free response item asked, “What are some of the strong mathematical skills that entering college freshmen have?” Again, 13 of 22 responses (59.1%) mentioned a topic in the subject knowledge construct. Across the three questions, the clear majority of responses related to subject knowledge.

Given results of both the scaled and free response items, it seemed clear that subject knowledge was perceived as important. To further uncover the perceptions of these faculty members, results from the analysis of subconstructs will be reported. As noted previously, results will focus primarily on those related to algebra — since it is the subject topic that stood out in the responses. Following this, results related to geometry, and then those related to calculus, trigonometry, and probability will be discussed. Next, results associated with reasoning and generalization and number sense will be described. Finally, the constructs of measurement and data, as well as the additional findings obtained through open and axial coding, will be reported.

Scaled response results. Within the 30 scaled response items, 6 related to algebra. Four of the items obtained a mean score within the poor range; 2 items fell within the very poor range (see Table 1 for means and standard deviations). Of these data, the highest mean of 2.9 (SD 0.852) in the entire subject knowledge construct occurred with the first item related to solving algebraic equations in one variable; this is between the poor and adequate range. The next highest rating within the algebra category related to the ability to graph functions (Mean: 2.86, SD 0.854) and ability to combine algebraic expressions (Mean: 2.76, SD 0.889). The faculty rated the importance of these two topics with means of 1.77 (SD 0.429) and 1.91 (SD 0.294), respectively. This shows that while these topics were rated within the very important range, related student ability was perceived within the poor to adequate range. Of the scaled ability level responses within algebra, the two lowest were finding the inverse for invertible expressions and solving algebraic expressions in two variables. Both fell between the very poor and poor ability levels, with the perceived importance rated as somewhat important to very important.

Free response results. The free response items helped to underscore the scaled response results. In responding to the question, “Are freshmen prepared for the mathematics requirements at your school?”, many participants observed insufficient algebra manipulation and insight. In addition, the participants remarked that the lack of readiness resulted in students taking remedial mathematics courses in college in order to learn material that they thought should have been covered in high school. For example, one participant observed that “51% [of entering freshmen] had to take remedial math because of low SAT scores and failing the placement test.” Another participant said, “Roughly 85% of incoming freshmen place into a remedial math class.” Yet another participant shared, “Too many need to retake second, and even first, year high school algebra. Those who don't need those remedial classes are typically just adequate in algebra skills except incoming math and science majors.” These responses indicate that these faculty members perceive that the average incoming freshmen do not enter with the proper algebraic skills to complete the required mathematics courses at the participating institutions.

Participants offered opinions about relationships between high school coursework and mathematical readiness. For example, one participant said, “Those who took 4 years of math in high school and who took rigorous courses are well prepared. Others who did not take a senior math or did not go beyond geometry are not well prepared.” Another participant conjectured that even with four years of high school math, many students may be unprepared for college math, suggesting that they may be rushed through algebra, trigonometry, and geometry in order to get to calculus. Overall, responses seemed to support the literature recommending that students in high school should take four years of mathematics (Schmidt et al., 2005), but with some caution that students not be rushed through key courses and concepts that could ground their mathematical understanding. Specific skills and topics that may be important for this grounding will be described next.

The second free response question, “What are some mathematical skills and topics that students are lacking when entering college?”, elicited an overwhelming majority of responses dealing with algebra. A total of 10 out of the 13 responses (77.0%) within the subject knowledge construct dealt with algebraic topics. Five participants mentioned general algebraic skills or algebraic manipulations as a major weakness of average incoming freshmen. Specific skills that were mentioned included manipulating inequalities, understanding where algebraic formulas come from, the quadratic equation and applications, the distributive power over addition, function notation, and exponents. Although many of these concepts are ones included in high school algebra courses, they are clearly perceived as weaknesses of average college freshmen.

The final free response question asked, “What are some of the strong mathematical skills that entering college freshmen have?”. The majority of responses, 10 out of 13 (76.9%), included algebraic skills. However, the skills seen as strengths were of much lower level than those noted as lacking. For example, strong skills included the following: a mild ability to manipulate algebraic expressions, basic algebra skills, memorizing formulas without understanding, the slope-intercept form of linear equations, memorizing the quadratic formula but being unable to use it, plotting points, the Pythagorean theorem; solving linear equations in one variable, and the ability to solve some quadratic equations but inability to solve in unfamiliar contexts. Two responses stated that students who placed out of remedial college courses were competent in algebraic manipulations. Looking across responses, a majority of the stronger skills identified (61.5%) related to very basic-level algebra. This shows that across the free response items, the faculty perceived that average students were prepared for some topics in basic algebra courses, but were weaker with topics in intermediate algebra. There appears to be a discrepancy between perceived algebra skills and expectations for readiness to engage in college mathematics.

Scaled response results. Three of the 30 scaled response items related to geometry, specifically, analyzing two-dimensional and three-dimensional objects and the topic of similarity (see Table 2). Mean skill levels were found to be poor to adequate for finding the area of two-dimensional objects and from very poor to poor for three-dimensional analysis. According to national and state standards, area of two-dimensional objects should be mastered by middle school, and three-dimensional properties should be covered in high school (Goranson, 1999; NCTM, 2000); yet, faculty are reporting poor performance for these at the beginning of college. Responses related to similarity revealed perceptions that students have poor to adequate abilities; yet these skills were ranked as important.

Free response results. The following weaknesses were noted by one or more of the respondents: general geometry, and, specifically, in finding perimeter, area, and volume of objects. Strengths noted included converting angles from radians to degrees, finding the area of triangles and circles, and basic geometry. These results are consistent with those of the scaled items in that greater strengths were noted in procedural skills and greater weaknesses were noted in more conceptual skills, especially those related to analyzing three-dimensional objects. There were some inconsistencies with responses for geometry, including finding areas of triangles and the general concepts within the discipline of geometry. This may be due to the fact that average freshmen do not take college level geometry classes. This should be explored in future studies.

Scaled response results. Overall, trigonometry, probability, and calculus were seen as less important than algebra and two-dimensional geometry (see Table 3). Understanding trigonometric relationships was the lowest-ranked skill within the subconstruct and the third lowest-ranked overall among all subject knowledge items. However, this skill was not seen as important, as other skills across subjects (mean importance 0.67). Similarly, both the skill and importance levels for finding the probability of dependent and independent events were among the lowest for subject knowledge. In contrast, while understanding the concept of a limit ranked in the bottom four for skill level, it was ranked as relatively important (mean importance 1.38).

Free response results. The free response data included only a few responses for trigonometry, probability, and statistics. When asked if students were ready for college mathematics, one participant conjectured that trigonometry was rushed in order for students to get to calculus. One participant observed descriptive statistics as a weak skill for students; another participant observed trigonometry as a strong skill for the average incoming freshman.

Scaled response results. Within this construct, the mean response for student ability was a 1.717 (SD 0.944), with an importance rating of 1.743 (SD 0.5136) (see Table 4). These scores represented the lowest mean ability paired with the highest mean importance. This indicates that while these college mathematics faculty members view reasoning and generalization as the most important skill for incoming college freshmen, average students' abilities with this skill were perceived among the lowest. In fact, this construct produced the only unanimous response among the faculty members, with faculty giving a mean rating of 2.00 (SD 0.00) when asked the importance for students to be able to problem solve. Students were rated as having a mean score of 2.19 (SD 0.873) for this skill, ranking between the very poor and poor ratings. Even more telling was that seven out of the nine skills for reasoning and generalization scored within the very poor to poor range for student ability — the ability to find connections between ideas, reflect on their own reasoning, prove a given conjecture, apply a given method for solving problems in multiple contexts, develop their own conjectures, use various forms of reasoning to problem solve, and develop some form of proof given a theorem.

Free response results. The free response items concur with the scaled response data. Thirteen out of 22 responses (59.1%) named reasoning and generalization as a weakness; three participants (13.6%) named student reasoning and ability to solve word problems as major weaknesses. The same number of participants reported that students could not think about mathematics beyond memorization of facts and procedures. For example, a participant noted that students do not have an “[a]wareness that one can understand math at a level beyond rote formulas.” This suggests that the participants perceive that average incoming freshmen at their institutions are not living up to the expectations set by NCTM and the international community (Martin & Mullis, 2005; NCTM, 2000).

Three participants mentioned the inability of students to make connections between mathematical ideas and apply concepts to a variety of situations. For example, one participant noted the “[i]nability to make connections between different mathematical representations (for example, solving f(x) = 0 but not realizing that this gives the x-intercepts of the graph of f(x)).” Inability to make these connections is of concern, since connecting and applying mathematical ideas have been identified as necessary for building higher level mathematical thinking (Gray et al., 1999). Further weaknesses named by participants included the inability to understand why mathematical procedures and theorems were accurate. For example, responses indicated that students did not understand the concept of a proof or how to use mathematical guesses to reason whether or not an answer makes sense.

Scaled response results. The scaled response data contained five questions related to number sense (see Table 5). The most important result from this construct is that it included the item with the highest ability rating for all 30 items. The mean rating of 3.20 relates to knowing the identity relationship for multiplication and addition. According to the literature, this skill should be covered prior to high school mathematics and does not require higher level mathematical thinking (Goranson, 1999; Gray et al., 1999; NCTM, 2000). The other skills in this construct all fell within the very poor to adequate range, with the pull toward the poor rating. These results indicate that the participants perceive that average incoming students do not have a sense of different number systems or relationships within number systems.

Free response results. The free response items showed additional insights into perceptions of freshmen's understanding of numbers. When asked if average freshmen were prepared for the requirements at the participant's institution, two responses touched upon number sense — specifically the inability for students to perform arithmetic, knowing multiplication tables, and performing calculations with rational numbers. Two participants mentioned the inability to perform basic computation as a weakness. In addition, the inability to use positive and negative numbers, logarithms, and radical exponents were seen as weaknesses.

Consistent with concerns expressed by the recent NMAP report (NMAP, 2008), the inability to use and understand fractions was the most frequently coded topic within number sense. One participant went as far as to say that students were “afraid of fractions.” Another participant mentioned the students' inability to manipulate and simplify fractional expressions. Only one participant mentioned number sense as a strength — namely knowing odd and even numbers, positive and negative numbers, and understanding the comparison of sizes between numbers. These skills are ones that should be mastered prior to high school algebra or covered within the beginning of an elementary algebra course (NCTM, 2000).

Scaled response results. This construct focused on skills in measuring and correctly representing data, as opposed to statistical analysis (see Table 6). The item of interest was the ability to use rulers, protractors, or compasses. While the ability level fell within the adequate range, the importance level fell within the not important to somewhat important range. This was the second highest ability rating overall, but the second lowest importance rating of all responses.

Free response results. Responses coded for this construct were found related only to the skills that were lacking — for example, inability to extract relevant information from word problems or determine reasonableness of their answer, and inability to visualize and manipulate data to solve problems. The results indicated that physically measuring objects was not deemed important for college-level mathematics; however, representing data and using correct scales and units were seen as somewhat important to important by most faculty members.