Who Needs Algebra II?

In May this year, the National Center on Education and the Economy (NCEE) released its report, "What Does It Really Mean to Be College and Work Ready?" [1]. The report is in two parts: Mathematics and English Literacy. It is based on a national study of the proficiencies required and actually used for the most popular associate’s degree programs at two-year colleges. Just a few weeks earlier, Jordan Weissman published a piece in The Atlantic, "Here’s How Little Math Americans Actually Use at Work" [2]. That was based on a 2010 report written by Michael Handel at Northeastern University, "What Do People Do at Work? A Profile of U.S. Jobs from the Survey of Workplace Skills, Technology, and Management Practices" [3]. This large-scale survey includes an assessment of what mathematics is actually used in the workplace.

I hope that it will come as a surprise to no one that not everyone actually uses the contents of Algebra II in their work or that College Algebra taught in two-year colleges is essentially high school Algebra II. Weissman highlights Handel’s data that less than a quarter of all workers use any mathematics that is more advanced than fractions, ratios, and percentages. He raises the question whether requiring Algebra II for high school graduation is placing an unnecessary roadblock in the way of too many students. NCEE poses more nuanced questions. Why are so many students being hurried through the critical early mathematics that they will need to be work and college ready, especially fractions, ratios, and percentages, just so that they can get to Algebra II? Isn’t there a better way to prepare them for what they will need?

I will begin with the data. Handel divided the workforce into five categories:

• Upper White Collar (management, professional, technical occupations)
• Lower White Collar (clerical, sales)
•  Upper Blue Collar (craft and repair workers, construction trades, mechanics)
• Lower Blue Collar (factory workers, truck drivers)
• Service (food service, home health care, child care, janitors)

Generally, the best paying and most desirable jobs are Upper White Collar (UWC) and Upper Blue Collar (UBC). We should be equipping our students so that they can aspire to such jobs. It’s still not true that everyone needs Algebra II, but 35% of UWC workers reported using basic algebra, geometry, and/or statistics in their work. This level of mathematics is even more important for UBC workers, with 41% reporting using mathematics at the level of basic algebra, geometry, and/or statistics. This is still not Algebra II (which Handel lists as “complex Algebra” as opposed to “basic Algebra”), which was reported being used by 14% of UWC workers and 16% of UBC workers. Much less is it Calculus, which was reported by 8% of both UWC and UBC workers. But about 40% of those working in UWC or UBC jobs need a working knowledge of some high school mathematics, a higher bar than simply having passed the relevant courses. It is interesting to observe that UBC workers are more likely to use mathematics than UWC workers.

The NCEE report looked at the mathematics required for the nine most popular associate’s degree programs at two-year colleges: Accounting, Automotive Technology, Biotech/Electrical Technology, Business, Computer Programming, Criminal Justice, Early Childhood Education, Information Technology, and Nursing, as well as the General Track. This ties nicely to the Handel study because the nine are generally seen as preparation for UWC or UBC careers. NCEE selected seven two-year colleges in seven states and examined the texts, assignments, and exams in the introductory courses for these disciplines as well as for the mathematics courses required for these fields. There were three notable insights:

First, except for some work on geometric visualization, NCEE found no content in either College Algebra or Statistics, two college-credit bearing courses, that goes beyond the high school curriculum described in the Common Core State Standards in Mathematics (CCSS-M). They found that College Algebra had a large component of middle school topics, especially CCSS-M for grades 6–8 in Expressions and Equations, Functions, Number Systems, Geometry, and Ratios and Proportions. Statistics was a mix of CCSS-M middle and high school level statistics, with a significant component of grades 6–8 Ratios and Proportions and Expressions and Equations.

Second, the introductory textbooks in the disciplinary fields used nothing beyond Algebra I. Ratios and proportions are important as well as interpreting quantitative relationship expressed in tables, graphs, and formulae, but, as the report says,

When mathematics is present in the texts, equations are not solved, quadratics are absent, and functions are present but not named or analyzed, just treated as formulae. […] Students do not have to perform algebraic manipulations nor construct graphs or tables. […] The area of high school content with the highest representation in the texts, Number Systems, is found in six percent of the text chapters. [p. 16]

Third, the mathematical knowledge that was tested in these introductory courses in the disciplinary fields was far lower than what was in the textbooks. Not only was there nothing requiring Algebra II on the exams, the NCEE team could find nothing, or almost nothing, that reflected knowledge of Algebra I. Furthermore, the questions that were asked on examinations were of low difficulty. The NCEE team used the PISA (Program for International Student Assessment) Item-Difficulty Coding Framework with four levels. Examples of what is expected at each level include

• Level 0: perform simple calculations and make direct inferences;
• Level 1: use simple functional relationships and formal mathematical symbols, interpret models;
• Level 2: use multiple relationships, manipulate mathematical symbols, modify existing models; and
•  Level 3: solve multi-step application of formal procedures, evaluate arguments, create models.

The team found that over 60% of the mathematical questions on the examinations given in introductory courses were at Level 0. Few rose to Level 2, much less Level 3. (This was not the case in College Algebra and Statistics where most of the examination items were at Level 1 or 2 and some attained Level 3. This suggests that even though the material of College Algebra and Statistics does not go beyond topics covered in CCSS-M, the level of expected proficiency may be higher than what is typically encountered in high school.)

NCEE did find three mathematical topics required for the introductory courses that are not covered in CCSS-M nor in the College Algebra or Statistics classes: complex applications of measurements, schematic diagrams (2-D schematics of 3-D objects and flow charts), and geometric visualization. They also found a much greater demand for knowledge of statistics, probability, and modeling (“how to frame a real-world problem in mathematical terms”) than is commonly taught in most mainstream high school mathematics programs today.

What makes the NCEE report even more depressing is that it restricted its attention to college-credit bearing courses. Most of the mathematics taught at two-year colleges is below the level of College Algebra (see Figure 1). The mathematical requirements for UWC and UBC jobs may not be high, but we do not seem to be doing a very good job of preparing students even for what they will need.

Figure 1. Fall term Mathematics course enrollments (thousands). “Introductory” includes College Algebra, Trigonometry, and Precalculus.

Source: CBMS.

All of this raises serious questions about whether Algebra II should be expected of all graduating high school students. This parallels the situation that has been my primary concern: Should Calculus be expected of all graduating high school students who are going directly into a four-year undergraduate program, especially those who may need to take Calculus in college? I would far prefer a student who can operate at PISA Level 3 in Algebra I over a student who cannot handle problems above Level 1 in Algebra II. I would prefer Level 3 in Precalculus over Level 1 in Calculus. When students are short-changed in their mathematical preparation simply so that Algebra II or Calculus appears on the high school transcript, with little regard to what that actually means, then neither they nor society as a whole are well served.

It also raises questions about what mathematics should be required for an associate’s degree. College Algebra constitutes a significant hurdle for most two-year college students. Should there be alternatives? In this case, I believe that most two-year college students would be better served with a program that combines demanding use of the topics of Algebra I with a college-level introduction to Statistics.

We are not at the point where we can demand Algebra II for high school graduation. To do so would either create unacceptable rates of high school failure or force us to change what we mean by “understanding Algebra II.” But I worry that if we simply lower our sights and decide that, since few of our students actually will use anything from Algebra II once they have graduated, it should not be expected for graduation, then that will actually weaken the preparation that occurs in the earlier grades. Elementary and middle school mathematics should be laying the foundation for a student to succeed in Algebra II. If we want our students to have a strong working knowledge of the high school mathematics that is needed for 40% of the UWC and UBC jobs, then we want them to have the mathematical preparation that would enable them to succeed in Algebra II.

References:

[1] National Center on Education and the Economy. 2013. What does it really mean to be college and work ready? The mathematics requirements of first year community college students. Washington, DC. Available at http://www.ncee.org/college-and-work-ready/

[2] Jordan Weissman. April 24, 2013. Here’s how little Math Americans actually use at work. The Atlantic. Available at http://www.theatlantic.com/business/archive/2013/04/heres-how-little-math-americans-actually-use-at-work/275260/

[3] Michael Handel. 2010. What do people do at work? A Profile of U.S. jobs from the Survey of Workplace Skills, Technology, and Management Practices. OECD (forthcoming). Available at http://www.northeastern.edu/socant/?page_id=366

MAA Calculus Study: Graphing Calculators and CAS

This column continues my report on results of the MAA National Study of Calculus I, Characteristics of Successful Programs in College Calculus. This month I am sharing what we learned about the use of graphing calculators (with or without computer algebra systems) and computer software such as Maple or Mathematica. Our results draw on three of the surveys:

• Student survey at start of term: We asked students how calculators and/or computer algebra systems (CAS) were used in their last high school mathematics class and how comfortable they are in using these technologies.
• Student survey at end of term: We asked students how calculators or CAS had been used both in class and for out of class assignments.
• Instructor survey at start of term: We asked instructors what technologies would be allowed on examinations and which would be required on examinations.

Our first question asked students how calculators were used on exams in their last high school mathematics class (see Figure 1). As in previous columns, “research” refers to the responses of students taking Calculus I at research universities (highest degree in mathematics is doctorate), “undergrad” refers to undergraduate colleges (highest degree is bachelor’s), “masters” to masters universities (highest degree is masters), and “two-year” to two-year colleges (highest degree is associate’s).

  Figure 1. GC = graphing calculator. CAS = graphing calculator with computer algebra system capabilities (e.g. TI-89 or TI-92).

There are several interesting observations to be made from this graph. First, not surprisingly, almost all Calculus I students reported having used graphing calculators on their exams at least some of the time (“always” and “sometimes” were mutually exclusive options). Second, there is a difference by type of institution. Students at undergraduate colleges were most likely to have used graphing calculators on high school exams (94%), then those at research universities (91%), then masters universities (86%), and finally two-year colleges (77%). The differences are small but statistically significant. My best guess is that these are reflections of the economic background of these students. A second observation is that for most students, access to a graphing calculator was not always allowed. However, it is still common practice in high schools (roughly one-third of all students) to always allow students to use graphing calculators on mathematics exams.

Another striking observation from Figure 1 is that the percentage of students who were always allowed to use graphing calculators on exams is almost identical to the percentage of students who were always allowed to use graphing calculators with CAS capabilities on exams. For all categories of students, over half of them were allowed to use graphing calculators with CAS capabilities at least some of the time, which suggests that over half of the students in college Calculus I own or have had access to such calculators.

The next graph (Figure 2) shows how students at the start of the term reported their comfort level with using graphing calculators or computer algebra systems (Maple and Mathematica were provided as examples of what we meant). The most interesting feature of this graph is that students at two-year colleges are much more likely to be comfortable with Maple or Mathematica than those at four-year programs. I suspect that the reason behind this is that most Calculus I students at two-year colleges are sophomores who took pre-calculus at that college the year before. This gave them more opportunity to experience these computer algebra systems.

Figure 2. Student attitude toward use of graphing calculator or CAS on a computer such as Maple or Mathematica.

The graphs in Figures 3–5 show what students reported at the end of the term about use of technology. For the graph in Figure 3, students were asked how frequently each of these occurred in class. Percentage shows the fraction of students who responded “about half the class sessions,” “most class sessions,” or “every class session.” We note large differences in instructor use of technology generally (for this question, “technology” was not defined), and especially sharp differences for instructor use of graphing calculators or CAS (with Maple and Mathematica given as examples). It is interesting that students are most likely to encounter computer algebra systems in undergraduate and two-year colleges, much less likely in masters and research universities.

 Figure 3. End of term student reports on frequency of use of technology (at least once/month). For this question, CAS refers to a computer algebra system on a computer, such as Maple or Mathematica.

The first two sets of bars in Figure 4 show student responses to “Does your calculator find the symbolic derivative of a function?” The first set gives the percentage responding “N/A, I do not use a calculator.” The second set displays the percentage responding “yes.” Looking at the complement of these two responses, we see that across all types of institutions, roughly 50% of students taking Calculus I own a graphing calculator without CAS capabilities. The third set records the percentage responding “yes” to the question, “Were you allowed to use a graphing calculator during your exams?” Note that there are some discrepancies between what students and instructors report about allowing graphing calculators on exams (Figures 4 and 6), but the basic pattern that graphing calculators are allowed far less frequently at research universities than at other types of institutions is consistently demonstrated.

 Figure 4. End of term student reports on calculator use. No calculator = do not use a calculator. Calculator with CAS = use a calculator with CAS capabilities. Calc allowed on exams = graphing calculators were allowed on exams.

We also asked how often “The assignments completed outside of class time required that I use technology to understand ideas.” Again, we see much less use of technology at research universities, the greatest use at undergraduate and two-year colleges.

 Figure 5. Frequency with which technology (either graphing calculators or computers) was used for out of class assignments. Almost never = less than once per month (includes never). Sometimes = at least once per month but less than once per week. Often = at least once per week.

The last two graphs (Figures 6 and 7) are taken from the instructor responses at the start of the term: what technology they would allow on their exams and what technology they would require on their exams. Again, we see a clear indication that technology, especially the use of graphing calculators without CAS capabilities, is much less common at research universities than other types of institutions.

It is interesting to observe that there are large numbers of instructors who allow but do not require technology on the exams. At research universities, 26% require the use of some kind of technology, and a further 25% allow but do not require the use of some sort of technology. For undergraduate colleges, 38% of instructors require technology, an additional 42% allow it. At masters universities, 42% require, and a further 33% allow. At two-year colleges, 52% require, and an additional 36% allow.

 Figure 6. Start of term report by instructor of intended use of technology on exams. GC = graphing calculator. Most of those who checked “other” reported that they allowed graphing calculators on some but not all parts of the exam. Some reported allowing only scientific calculators.

Figure 7. Start of term report by instructor of intended use of technology on exams. GC = graphing calculator. Most of those who checked “other” reported that they required graphing calculators on some but not all parts of the exam. Some reported requiring only scientific calculators.

We see a pattern of very heavy use of graphing calculators in high schools, driven, no doubt, by the fact that students are expected to use them for certain sections of the Advanced Placement Calculus exams. They are still the dominant technology at colleges and universities, but there the use is as likely to be voluntary as required. This implies that in many colleges and universities questions and assignments are posed in such a way that graphing calculators confer little or no advantage. The use of graphing calculators at the post-secondary level varies tremendously by type of institution. Yet even at the research universities, over half the instructors allow the use of graphing calculators for at least some portions of their exams. 

The MAA national study of calculus, Characteristics of Successful Programs in College Calculus, is funded by NSF grant no. 0910240. The opinions expressed in this column do not necessarily reflect those of the National Science Foundation.

MAA Calculus Study: Progressive Teaching

Last month (MAA Calculus Study: Good Teaching) I discussed the student-described attributes of instructors that were highly correlated with improvements in student confidence, enjoyment of mathematics, and desire to continue to study mathematics. This month I will discuss a second set of instructor attributes that we are labeling "Progressive Teaching" because they are generally associated with approaches to teaching and learning that focus on active engagement of the students.

Here the evidence for improved results is less clear. In particular, Sadler and Sonnert discovered a strong interaction with the attributes we are calling "Good Teaching": teachers who rated high on Good Teaching improved student outcomes if they also rated high on Progressive Teaching. But if they rated low on Good Teaching, then a high rating on Progressive Teaching had a strongly negative effect on student confidence. This might have been expected. Good Teaching describes student-teacher interactions, including the degree to which students feel encouraged to participate in class and supported by the instructor. It is not surprising that students who are encountering unfamiliar approaches to classroom learning react negatively if they believe that that the instructor is not encouraging or supportive.

We also have evidence of some consistently positive effects from Progressive Teaching. Even with a low score on Good Teaching, Progressive Teaching was seen to be helpful in convincing students to continue the study of mathematics. Our conclusions are that:

a.       Good Teaching and Progressive Teaching are independent clusters of student perceptions of instructor behaviors,

b.      Good Teaching is more important to student persistence than Progressive Teaching,

c.       both can serve to improve student outcomes, and

d.      teaching is most effective when instructors rate high on both measures.

There were 12 student responses that clustered into what we are calling Progressive Teaching:

My calculus instructor frequently
1.      Assigned sections of the textbook to read before coming to class.

2.      Had students work with one another.
3.      Had students give presentations.
4.      Asked students to explain their thinking in class.
5.      Required students to explain their thinking on homework assignments.
6.      Required students to explain their thinking on exams.
7.      Held whole class discussions.

My calculus instructor did not frequently
8.      Lecture.

Assignments completed outside of class
9.      Required that I solve word problems.
10.  Were problems unlike those done in class or in the book.
11.  Were often submitted as a group project.
12.  Were returned with helpful feedback and comments.

With one exception, the following graphs show the percentage of students who reported that their instructors employed each of these practices often or very often (a 5 or 6 on a Likert scale from 1 = not at all to 6 = very often). The exception is practice #8. Here we record the percentage of students who responded 1, 2, or 3 on the same scale to the question, "During class time, how frequently did your instructor lecture?".

We see that for most of the instructor behaviors (practices 1 through 8), the undergraduate colleges and two-year colleges are where these are most likely to be employed. The relatively large percentage of instructors at masters universities who had students give presentations in class (13% as opposed to 6% at all other types of institutions) is still small and may be an artifact of the relatively small number of responses from students at masters universities (305 students at 18 institutions).  The research universities are where we find the most challenging problems being posed on assignments, either word problems or those unlike those done in class or in the book. Instructors at two-year colleges provide the most helpful feedback on assignments, instructors at research universities the least helpful feedback.

Figure 1: Instructor practices 1 through 3 and 8

Figure 2: Instructor practices 4 through 7

Figure 3: Instructor practices 9 through 12

The MAA national study of calculus, Characteristics of Successful Programs in College Calculus, is funded by NSF grant no. 0910240. The opinions expressed in this column do not necessarily reflect those of the National Science Foundation.

MAA Calculus Study: Good Teaching

One of the primary goals of the MAA Calculus Study, Characteristics of Successful Programs in College Calculus (NSF #0910240), has been to identify the factors that are highly correlated with an improvement in student attitudes from the start to the end of the calculus course: confidence in mathematical ability, enjoyment of mathematics, and desire to continue the study of mathematics. To this end, Phil Sadler and Gerhard Sonnert of the Science Education Department within the Harvard-Smithsonian Center for Astrophysics constructed a hierarchical linear model from our survey responses to identify these factors. The factors reside at three levels: institutional, classroom, and individual student. Not surprisingly, most of the variation in student attitudes can be explained by student background, but there are influences at the institutional and classroom level. We have been particularly interested in what happens at the classroom level where there is the greatest opportunity for improvement.

Sadler and Sonnert ran a factor analysis of the classroom-level variables, clumping those responses that were highly correlated. They discovered that the responses broke into three distinct clusters, which we are labeling “technology,” “progressive teaching,” and “good teaching” because these seem to describe the characteristics of the instruction. By far, the most important of these in terms of high correlation with improved attitudes is “good teaching.” Listed below are the 21 student-reported characteristics of instruction that are highly correlated with each other and highly correlated with improvements in student attitudes, characteristics that collectively we are calling “good teaching”:

My calculus instructor:

1. Asked questions to determine if I understood what was being discussed.
2. Listened carefully to my questions and comments.
3. Discussed applications of calculus.
4. Allowed time for me to understand difficult ideas.
5. Helped me become a better problem solver.
6. Encouraged students to enroll in Calculus II.
7. Acted as if I was capable of understanding the key ideas of calculus.
8. Made me feel comfortable asking questions during class.
9. Encouraged students to seek help during office hours.
10. Presented more than one method for solving problems.
11. Made class interesting.
12. Provided explanations that were understandable.
13. Was available to make appointments outside of office hours, if needed.

My calculus instructor did not:

14. Discourage me from wanting to continue taking calculus.
15. Make students feel nervous during class.

My instructor often or very often:

16. Showed how to work specific problems.
17. Asked questions.
18. Prepared extra material to help students understand calculus concepts or procedures.

In addition:

19. My calculus exams were a good assessment of what I learned.
20. My exams were fairly graded.
21. My homework was fairly graded.

The good news is that most calculus instructors rated highly on most of these characteristics. This good news needs to be tempered by two facts: Instructors could and in many cases did elect not to participate even though other instructors at their institution were involved in the study, and these responses were all collected at the end of the term. They reflect the opinions of the students who had successfully navigated this course, predominantly students who were earning an A or a B in the course (roughly 40% A, 40% B, 20% C).

It is interesting and informative to see how students at different types of institutions rated their instructors on these criteria. We followed CBMS in categorizing post-secondary institutions by the highest mathematics degree offered at that institution. I am using “research” to designate universities that offer a PhD in Mathematics (predominantly large state flagship universities), “masters” if the highest degree is a master’s (predominantly public comprehensive universities), “undergrad” if it is a bachelor’s degree (predominantly private liberal arts colleges), and “two-year” if it is an associate’s degree (predominantly community and technical colleges). As shown in the graphs at the end of this article, instructors at research universities got the lowest ratings on every characteristic except “showed how to work specific problems.” For most of these characteristics, instructors at undergraduate colleges were the next lowest, then masters universities, and most of the time instructors at two-year colleges received the highest ratings. 

There were a few notable exceptions. Instructors at undergraduate colleges received the highest ratings in some of the areas where one would expect them to be strong:

• Acted as if I was capable of understanding the key ideas of calculus.
• Encouraged students to seek help during office hours.
• Was available to make appointments outside of office hours, if needed.
• Did not make students feel nervous during class.

Masters universities scored highest in often or very often showing how to work specific problems, and just barely edged out two-year colleges in “listened carefully” and “my exams were fairly graded.”

There are a number of possible explanations for the weaknesses of research universities and the strengths of two-year colleges. One is class size. The largest classes are found at the research universities where average class size is 53, the smallest at two-year colleges where the average is 21. However, average class size at masters universities is larger than at undergraduate colleges, so class size cannot be the only explanatory variable. Some of the discrepancies between institution types may be explained by student expectations. This is because SAT scores and high school mathematics GPA are highest for research universities, then undergraduate colleges, then masters universities, and lowest for two-year colleges. Better students may have higher expectations of their instructors, or they may be more discouraged by encountering difficulties in this course. The differences may also have something to do with age and thus maturity of the students. The youngest students are at research universities, the oldest at two-year colleges. They also may be related to the relatively large number of instructors at research universities who teach calculus but have little or no interest in teaching this course, as opposed to two-year colleges where the interest is very high (see my November column, MAA Calculus Study: The Instructors). Nevertheless, it is discouraging that students at research universities seem to be getting calculus instruction that has a worse effect on student attitudes than instruction at other types of institutions.

Figure 1: Instructor Characteristics 1–5.

Figure 2: Instructor Characteristics 6–10.

Figure 3: Instructor Characteristics 11–15.

Figure 4: Instructor Characteristics 16–18.

Figure 5: Instructor Characteristics 19–21.

The MAA national study of calculus, Characteristics of Successful Programs in College Calculus, is funded by NSF grant no. 0910240. The opinions expressed in this column do not necessarily reflect those of the National Science Foundation.

Mathematics in 2025

The National Research Council of the National Academies has just released the preliminary version of its report, The Mathematical Sciences in 2025 [1]. This was produced in response to a request from the National Science Foundation. It comes as the latest in a series of glimpses into the future of mathematics that go back to the “David reports” of 1984 and 1990 [2,3] and the “Odom study” of 1998 [4]. This report is important because it will influence the direction NSF takes as it plans for the future.

The emphasis of the report is on the central role that the mathematical sciences are taking within research in areas as diverse as biology, finance, and climate science. Traditional disciplinary boundaries are blurring. There is an increasing need for scientists who are well grounded in mathematical sciences, especially the statistical and computational sciences, as well as other disciplines. This goes two ways. It means opening courses and programs in the mathematical sciences, especially at the graduate level, to those in other fields of study, and it means ensuring that students graduating in the mathematical sciences are prepared to work in this interdisciplinary world.

This has implications right down the line of mathematics education. The authors of the report question whether, in a scientific world that is dominated by big data and the challenges of large-scale computation, the traditional calculus-focused curriculum is the most appropriate for all students. As they say, “Different pathways are needed for students who may go on to work in bioinformatics, ecology, medicine, computing, and so on. It is not enough to rearrange existing courses to create alternative curricula; a redesigned offering of courses and majors is needed [my emphasis].” (NRC 2013, p. S-9)

The report also stresses the importance of attracting more women and students from traditionally underrepresented minorities to the mathematical societies. This is the one place where I disagree with the report, for it asserts that, “While there has been progress in the last 10–20 years, the fraction of women and minorities in the mathematical sciences drops with each step up the career ladder.” (NRC 2013, p. S-10). I don’t question the drop. I question whether there has been progress over the last 10–20 years.

If we look at mathematics majors (bachelor’s degrees) by gender, we see that over the period 1990 to 2011 the number of men majoring in mathematics grew by 25% while the number of women grew by only 10% (Figure 1). As a result, the percentage of bachelor’s degrees in the mathematical sciences going to women has dropped to 43.1%, the first time it has been this low since 1981. This is having knock-on effects for graduate programs. The percentage of bachelor’s degrees in mathematics that went to women peaked in 1999 at 47.8%. The percentage of master’s degrees in mathematics that went to women peaked in 2004 at 45.1% and has since dropped back to 40.9%. The percentage of doctoral degrees in mathematics that went to women peaked in 2008 and ’09 at 31.0%. It has since dropped back to 28.6%. The good news is that the past decade has seen strong growth in the number of mathematics majors, but two-thirds of the growth since 2001 has been in the number of men.

We see an even more discouraging pattern among Black students (Figure 2). The number of Black mathematics majors is essentially back to where it was twenty years ago despite the number of bachelor’s degrees earned by Black students almost tripling over this period. The number of Black mathematics majors peaked in 1997 at 1,089. It was back down to only 840 in 2011. The number of ethnically Asian mathematics majors has been growing strongly over the past decade. Even so, the number earning undergraduate degrees in the mathematical sciences has only doubled since 1990, while the number earning bachelor’s degrees has tripled. The growth in the number of Hispanic mathematics majors looks good, having slightly more than tripled in twenty years, until you realize that the number of Hispanic students graduating from college is almost five times what it was in 1990 (154,000 versus 33,000). Where we do see strong growth, especially since 2007, is in the number of non-resident aliens majoring in mathematics, which now stands at 7% of all US mathematics majors.

I must emphasize that the NRC report does highlight the importance of increasing the participation of women and members of underrepresented groups. It includes the following specific recommendation:

Recommendation 5-4: Every academic department in the mathematical sciences should explicitly incorporate recruitment and retention of women and underrepresented groups into the responsibilities of the faculty members in charge of the undergraduate program, graduate program, and faculty hiring and promotion.  Resources need to be provided to enable departments to adopt, monitor and adapt successful recruiting and mentoring programs that have been pioneered at other schools and to find and correct any disincentives that may exist in the department. (NRC 2013, p. 5-18)

I have only touched on a few of the topics covered in the NRC report. It also discusses the increasingly important role of the mathematical sciences institutes, the issue of maintaining online repositories of mathematical research such as arXive, and the threats to mathematics departments as more instruction—especially for the service courses that often provide the justification for a large mathematics faculty—is moved online. This is a report well worth reading and pondering. 

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[1] National Research Council. 2013. The MathematicalSciences in 2025. Washington, DC. The National Academies Press.

[2] NRC. 1984, Renewing US Mathematics: Critical Resource for the Future. Washington, DC. The National Academies Press.

[3] NRC. 1990. Renewing US Mathematics: A Plan for the 1990s. Washington, DC. The National Academies Press.

[4] NSF. 1998. Report of the Senior Assessment Panel for the International Assessment of the US Mathematical Sciences. Arlington, VA. National Science Foundation.

The Red Herring of Grade Inflation

Two things happened in the week before Christmas that got me thinking about grade inflation. The first was that I graded the final exams for my multivariable calculus class. I have never before seen my students do so well. Out of 33 students in the class, 22 received an A. For my class, an A requires earning more than 92% of the total possible grade. The last time I graded on a curve was over 20 years ago.

This past semester I had worked these students hard. They were responsible for and graded on:

•        Reading Reflections (three times per week, reading the section and answering questions about the material before we discussed it in class). 

•       Two sets of homework each week (about 12 fairly straightforward questions on WeBWorK due on Thursdays and three challenging multi-part problems due on Mondays). 

•       Seven short projects developed by Tevian Dray and Corinne Manogue as part of their Bridge Project (see http://www.math.oregonstate.edu/bridge/). These were started in groups of three or four, but each student was responsible for writing his or her own three to five page report of the solution. For the first report, I required a first draft that was critiqued and returned for revision and resubmission.

•       A major project based on the Hydro-Turbine Optimization chapter in Applications of Calculus [1]. The project was started in groups. Each student was responsible for an 8–12 page paper explaining the solution. The papers were turned in, critiqued, and returned for revision and resubmission. LaTeX and pdf files of my version of this project are available here.

•       Two examinations during the semester and a final exam. After each exam during the semester, students were required to write about the problems they had missed points on, explain what they did wrong, and explain how to do it correctly. They could earn back half the points they had lost. For the final exam, they had to explain what they were doing to solve the problems, not just give an answer.

I was available to my students every afternoon, and I also had a great undergraduate preceptor (teaching assistant) who held help sessions Sunday and Thursday evenings, before the homework assignments were due. By the end of the semester, over half the class was coming to each of these, and so she organized them into groups working with each other on the homework while she circulated to help the groups that were stuck.

Not surprisingly, in the end of semester course evaluations my students wrote about how much work they had done for this course. And yet, when asked specifically whether or not they agreed with the statement, “The general workload was appropriate for this level course,” only five of my 33 students disagreed. One student comment that summarized the tenor of the end of course evaluations stated, “I would say that the course is difficult and a lot of work, but very rewarding, because if you put in a lot of time and effort then you can see yourself understand the material and do well. Although the course can be really hard at times, there is always somewhere to go for help.”

The second thing that happened this past week was my discovery of How Learning Works: 7 Research-Based Principles for Smart Teaching [2]. This collaborative effort, published in 2010, translates what has been learned by those engaged in research in undergraduate education into practical guidance for those of us in the classroom. What the authors call principles, I see more as facets of teaching to which I need to pay attention. This is my own paraphrasing of these principles or facets:

1.      The need to understand the variety of prior knowledge that my students bring to my class and how it helps or hinders them.  
2.      The importance of how students organize the knowledge they are acquiring and the need for me to understand common misalignments and to help them make the necessary connections.
3.      The critical role of student motivation and my responsibility to strengthen it.
4.       The need to develop automaticity in basic skills and the fact that learning how to integrate and apply these skills requires guidance and directed practice from me.
5.       How important it is that I provide useful feedback that is targeted at improving performance.
6.       The role of the social, emotional, and intellectual climate in my classroom.
7.        The need for me to guide students in practicing metacognition, monitoring what they are doing and why.

The book discusses the relevant research, but is also full of examples of traps we can fall into and strategies for dealing with these principles or facets in order to improve our teaching.

One trap discussed under #3 describes the teacher who, with the intent of spurring his students to work hard, warned them at the start of the course that they could expect that a third of them would not pass. This had exactly the opposite effect. With the expectation that they would not do well regardless of how much effort they put into the course, a large proportion of the students directed their time and energy to other courses.

The issue here is motivation, getting the students to put in the effort needed to learn the material. I believe that I did succeed particularly well this past semester in motivating most of my multivariable calculus students.  How Learning Works identifies three levers that motivate students to work hard. The first is value. They have to believe that what I want them to learn will be of value to them. Personal enthusiasm on my part goes a long way toward building this sense of value. The second is a supportive environment. They have to believe that the course is structured in such a way as to help them be successful, rather than throwing up obstacles to their success. Starting the projects and encouraging them to share their understanding of homework problems within groups, providing feedback and multiple opportunities to demonstrate understanding (as with WeBWorK and the chance to earn back points lost on exams), and the availability of myself and my preceptor build the sense of support. The third is self-efficacy, belief that one is capable of achieving success.

This last is the main reason I will never again grade on a curve. The message sent by grading on a curve is that the proportion of failures has been determined in advance, regardless of how much work students are prepared to invest in the course. It is also why I am disturbed that in our national survey of calculus, faculty at the start of the term were able to predict, almost perfectly, what their grade distributions would be at the end of the term (see the last bullet under Instructor Attitudes in The Calculus I Instructor, Launchings, June 2011). Going into this course, I would never have predicted 67% A’s. I am delighted that what I did worked so well with so many of my students. [3]

Which brings me back to the issue of grade inflation. Grade inflation is a red herring because it misdirects our attention from what should be our true concerns: What do our grades mean in terms of expectation of student achievement and understanding? And how can we support as many students as possible to meet our highest expectations?

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[1] Straffin, P. D., Jr. 1996. Hydro-Turbine Optimization. Pages 240–250 in Applications of Calculus. P.D. Straffin, Jr., editor. Classroom Resource Materials. MAA. 
 [2] Ambrose, S. A., M. W. Bridges, M. DiPietro, M. C. Lovett, M. K. Norman. 2010. How Learning Works: 7 Research-Based Principles for Smart Teaching. Jossey-Bass.
[3] Not all my students did well. The class GPA was 3.5. What was important was that I had explicit expectations of what would constitute A work, that I clearly communicated what was required to meet those expectations, that students saw them as challenging but achievable, and that my students really were graded according to these expectations.

Mathematics and the NRC Discipline-Based Education Research Report

This past spring, the National Research Council of the National Academies released its report, Discipline-Based Education Research: Understanding and Improving Learning in Undergraduate Science and Engineering [1]. The charge to the committee writing this report was to synthesize existing research on teaching and learning in the sciences, to report on the effect of this research, and to identify future directions for this research. The project has its roots in two 2008 workshops on promising practices in undergraduate science, technology, engineering, and mathematics education.

Unfortunately, between 2008 and 2012 undergraduate mathematics education dropped out of the picture. The resulting report discusses undergraduate education research only for physics, chemistry, engineering, biology, the geosciences, and astronomy. Nevertheless, it is an interesting report with useful information—especially the instructional strategies that have been shown to be effective—that is relevant for those of us who teach undergraduate mathematics.

The studies that are described are founded on the assumption that students must build their own understanding of the discipline by applying its methods and principles, and this is best accomplished within a student-centered approach that puts less emphasis on simple transmission of factual information and more on student engagement with conceptual understanding, including active learning in the classroom.

The great strength of this report is the wealth of resources that it references and the common themes that emerge across all of the scientific disciplines.  A lot of attention is paid to the power of interactive lectures. Given that most science and mathematics instruction is still given in traditional lecture settings, finding ways of engaging students and getting them to think about the mathematics while they are in class is essential for increasing student understanding.

The recommendations of effective practice range from simple techniques, such as starting each class with a challenging question for students to keep in mind, to transformative practices such as collaborative learning. A common intermediate practice involves student engagement by posing a challenging question, having students interact with their peers to think through the answer, and then testing the answer. In some respects, this is more easily done in the sciences where student predictions can be verified or falsified experimentally. Yet it is also a very effective tool in mathematics education where a well-chosen example can falsify an invalid expectation and careful analysis can support correct understanding. But  most important is that it forces to try to use what they have been learning.

In large classes, this type of peer instruction can be facilitated by the use of clickers. The report does include the caveat, with supporting research, that merely using clickers without attention to how they are used is of no measurable benefit.

The greatest learning gains that have been documented occur when collaborative research is incorporated into the classroom. The NRC report includes many descriptions of how this can be accomplished in a variety of scientific disciplines. It also references the research that has established its effectiveness. Again, attention to how it is done is an important component of effective practice.

Two of the areas that are identified as needing more research are issues of transference (see my September column on Teaching and Learning for Transference) and metacognition. Usefully, the authors point out that there are two sides to transference: the ability to draw on prior knowledge and the ability to carry what is currently being learned to future situations. Metacognition is an important issue in research in undergraduate mathematics education, especially for those studying the difference between experts and novices engaged in activities such as constructing proofs. Experts monitor their assumptions and progress and are prepared to change track when a particular approach is not fruitful. Novices are more likely to choose what to them seems the likeliest approach and then ignore alternatives.

In sum, this is a useful and thought-provoking report. I wish that it had included undergraduate mathematics education research, but perhaps that omission can be corrected as we move forward.

[1] National Research Council. 2012. Discipline-Based Education Research: Understanding and ImprovingLearning in Undergraduate Science and Engineering. S.R. Singer, N.R. Nielsen, and H.A. Schweingruber, eds. Washington, DC. The National Academies Press.