The question whether algebra deserves its prominent role in the high school curriculum was raised once again on July 28 by Andrew Hacker in a New York Times opinion column, “Is Algebra Necessary?’’ [1]. His piece echoes the argument made a year ago by Sol Garfunkel and David Mumford in the opinion pages of the New York Times, “How to Fix our Math Education” [2]. It also resonates with Mike Shaughnessy’s comments in his President’s Column for the NCTM newsletter, “Endless Algebra—the Deadly Pathway from High School Mathematics to College Mathematics’’ [3].
There have been many thoughtful responses to the Hacker article. For a clear explanation why algebra is important, the best is still Zal Usiskin’s piece from 1995, “Why is Algebra Important to Learn?” [4]. Daniel Willingham posted a good reply to Hacker on his blog, “Yes, algebra is necessary” [5]. In a private communication [6], Dan Kennedy of the Baylor School in Chattanooga describes the beauty of algebra and laments the fact that we are doing such a very poor job of communicating that beauty.
I want to focus this column on a theme raised in Lynn Steen’s response, “Reflections on Mathematics and Democracy” [7]. His article was conceived as a reaction to the Garfunkel and Mumford editorial, but he also discusses Hacker. As Steen points out, Hacker’s argument is not that algebra is not important; it is that algebra is not working in the curriculum. The problem—as Steen distills it—is the fact that many if not most students never learn how to transfer the knowledge and skills that are taught in algebra. They are trapped within a perception of algebra as a system of arcane manipulations with no relevance to anything outside the mathematics classroom.
Steen’s solution focuses on the curriculum: embedding applications into mathematics courses, team-teaching cross-disciplinary courses, and employing project- or career-focused curricula. I fully agree that the curriculum can be and has been an obstacle to learning how to transfer one’s knowledge and skills. I also recognize that the problem is greater than just the curriculum.
Teaching for transference is one of the most difficult tasks we face as educators. I would like to share two personal stories that serve as touchstones for me.
In my last year at Penn State, 1993–94, I taught a yearlong honors course in calculus using the Project CALC materials developed by David Smith and Lang Moore. CALC stands for Calculus As a Laboratory Course. The course met five days a week, two of those days in computer labs where the students explored and applied the ideas of calculus. The classroom was used both to prepare students for the laboratories and to reflect on and distill what had happened there. Some students put up initial resistance to such an unconventional approach, and a few switched to a regular section at winter break, but most came to enjoy learning this way. They believed they were getting much more than they would have from traditional instruction.
Midway through spring semester, one of my students came breathlessly to my office. She had just completed an engineering exam where, as she told me, she had forgotten the formula needed to solve one of the problems. But then she remembered what she had learned in my course, and she figured out how to solve the problem with what she knew from calculus.
I do not know what the problem was or what tools she used, but she made it clear that she was relying on the ideas, on the conceptual knowledge she had acquired. I knew then that my job with her was done. Whatever specific information she might still learn from me, nothing would equal the power that came from recognizing that she did not have to rely on memorized procedures, that she was capable of applying the principles of calculus to derive solutions to problems that mattered to her. Not that she always would do this, but now she knew that she could.
The second story is about myself. During the spring of 7th grade, I met periodically after school with my math teacher, Mr. Checkley. He presented and challenged me with bits of mathematics. One of these consisted of the rules for determining divisibility by 3, 9, or 11 by considering the digits of the integer. He asked me to find an explanation why these rules work. I struggled, convincing myself that they are always valid but unable to frame a proof.
The following year I took Algebra I. I had some difficulty at first with this strange kind of mathematics, as everyone did, until I realized that what I was learning in this class was exactly the language that I needed to provide Mr. Checkley with his proof. Once I knew that algebra is simply a language for exploring and explaining mathematical patterns, a language that I could use to answer mathematical questions of interest to me, it became easy.
There are several lessons that I take from these experiences. First, transference need not be to a real-world application. Nor is it about the need to use one’s mathematical knowledge in a career. It is important because of the power that comes from discovering that one can rely on one’s own reasoning to recover a forgotten formula or uncover the logic behind an unexpected pattern.
Second, I have learned that what works in a particular setting with one instructor and a particular group of students will not necessarily work when these parameters are changed. My Penn State class consisted of University Scholars, selected from the top 3% of the student body. I was highly motivated to make it work. Project Calc was used at Duke with mixed results. No curriculum by itself will ever be sufficient.
Third and finally, learning for transference is a process that takes time. A student begins with new knowledge, either acquired through personal discovery or introduced and explained by a teacher. This is followed by an opportunity to apply this knowledge in a fresh context. The next steps are critical and far too often neglected. The student must now reflect on what worked and what difficulties were encountered, then distill the significant features that made it work. Even now the process is not done. There must be a fresh attempt at application, followed once again by reflection and distillation. I have found that most students need to progress through this cycle several times before they have real control of a new piece of knowledge. I still struggle with the difficulties of engaging my students in this process while balancing the demands of the course. A continuing challenge for me is to decide which concepts deserve this much attention.
I work with privileged and highly motivated students. The difficulties inherent in accomplishing learning for transference in our public schools are far greater, but the goal should be the same. Lynn Steen has asked us to “organiz[e] the curriculum to pay greater attention to the goal of transferable knowledge and skills.” I would go beyond this. We need to organize the very way we teach so that we keep this end in mind.
[1] Hacker, A. 2012. Is algebra necessary? New York Times. July 28. http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?_r=1&smid=fb-share
[2] Garfunkel, S. and Mumford, D. 2011. How to fix our math education. New York Times. August 24. http://www.nytimes.com/2011/08/25/opinion/how-to-fix-our-math-education.html?scp=1&sq=fix%20our%20math%20education&st=cse
[3] Shaughnessy, J.M. 2011. Endless Algebra—the Deadly Pathway from High School Mathematics to College Mathematics. Summing Up. February 2. National Council of Teachers of Mathematics. Reston, VA. www.nctm.org/about/content.aspx?id=182
[4] Usiskin, Z. 1995. Why is algebra important to learn? American Educator. Vol. 19. pp 30–37.
[5] Willingham, D. 2012. Yes, algebra is necessary. 7/30/2012. http://www.danielwillingham.com/1/post/2012/07/yes-algebra-is-necessary.html
[6] Kennedy, D. 2012. Response to Hacker’s editorial. (private communication posted with permission of the author) http://www.macalester.edu/~bressoud/misc/Hacker%Reply.pdf
[7] Steen, L.A. 2012. Reflections on mathematics and democracy. To appear in MAA Focus. http://maa.org/pubs/FOCUSoct-nov12_Steen.html
Saturday, September 1, 2012
Wednesday, August 1, 2012
Barriers to Change
As promised in my last column (Learning from the Physicists, July 2012), this month I will look at work from the Physics Education Research (PER) community on the barriers to adoption of empirically validated teaching practices. I will conclude with three steps that the mathematics community needs to undertake if we are to improve the teaching and learning of undergraduate mathematics.
The primary source for this article is the report by Henderson et al. [1] of a survey completed by 722 physics faculty from the United States, but I also will draw on some of the other literature and I recommend the slides from two talks, one given by Melissa Dancy [2] at the conference on Transforming Research in Undergraduate STEM (Science, Technology, Engineering, and Mathematics) Education and the other given by Charles Henderson [3] at the American Society for Engineering Education, both given in June of this year.
For their study, Henderson et al. questioned faculty about their awareness and use of twenty-four different research-based instructional strategies (RBIS) for which there are published studies establishing their effectiveness. These include a variety of materials produced in the 1990s and 2000s as well as Mazur’s Peer Instruction, developed at Harvard. The 722 respondents represent just over 50% of those who were contacted. While there may be some selection bias, only 12% had no knowledge of any of these strategies, and an impressive 72% had used at least one of them. But 32% of those who had used at least one strategy no longer used any of them. At least in physics, the issue is not getting people to try proven but innovative approaches to teaching and learning; it is enabling them to stick with it.
In trying to understand what influences faculty decisions about the use of innovative approaches, Henderson et al. looked at twenty possible explanatory variables that range from class size, type of institution, gender, rank, and number of years in the profession to research productivity (measured separately by presentations, articles, and research grants within the past two years), to departmental encouragement and discussions with peers about teaching, to goals for teaching (the importance of developing conceptual understanding and problem-solving ability) and interest in research in instructional strategies.
At the first level, simple familiarity with some of the instructional strategies, two of the strong predictors are attendance at talks or workshops on teaching and regular reading of journal articles about teaching. The strongest predictor is attendance at one of the New Faculty Workshops (NFW), 3½-day workshops held by the American Association of Physics Teachers (AAPT) for new physics faculty. The intent of these workshops is to introduce new faculty to the results of Physics Education Research, so it is not surprising that those who have participated in NFW are over ten times as likely to be aware of these strategies as the average faculty member. Two other variables were also significant: level of satisfaction that one’s goals for teaching were being met and whether one’s position was full- or part-time. At the next level, actually trying one of these strategies, a new but not surprising variable becomes important: interest in using RBIS.
I am intrigued by the list of variables that did not come into play. These include class size, type of institution, faculty rank or years in position, or research productivity. None of these significantly impacted knowledge of or willingness to try innovative strategies.
The critical stage however, is not the decision to try RBIS but the willingness to stick with it. The only survey variables that were significant at this stage were a desire to try more RBIS and gender: Women were significantly more likely to continue the use of RBIS than were men. Henderson et al. cite four other studies that also showed that women are far more likely to have student-centered physics classrooms than are men. The authors speculate that gender may be serving as a proxy for a collection of beliefs about and attitudes toward teaching.
Many researchers have studied the phenomenon of trying and then abandoning innovative approaches to teaching. The Henderson et al. article includes an extensive bibliography. The hurdles that faculty face include student complaints, inability to cover the same quantity of material, and weaker than promised student outcomes. One of the issues is that implementation frequently is not faithful. As I have seen first-hand both at Penn State and at Macalester, if the course that students experience is not cohesive, if bits and pieces are poorly articulated or motivated, students will complain and resist.
That does not mean that faculty dare not try to modify a received instructional strategy. It does mean that whatever changes they incorporate into their classes must be reflected upon and monitored for effectiveness. In the Henderson, Beach and Finkelstein [4] review of the literature on facilitating change in undergraduate STEM education, one commonly observed phenomenon is the importance of evaluation and feedback, an aspect of change implementation that often is omitted.
Among the interesting findings of the survey by Henderson et al. were the variables that demarcated the distinction between high and low users of RBIS. Low users were defined as using one or two of these strategies, high as three or more. This is the first time that some of the traditionally assumed barriers come into play: research productivity, type of institution, and class size. Highly productive faculty at research universities teaching large classes of introductory physics are significantly less likely to be high users of RBIS. What is surprising is that this is the first time that these variables arise. These highly productive research faculty were no less likely to know about or try some of these innovative strategies, just less likely to be high users.
Also intriguing is the fact that age, as implied by rank and number of years in service, had nothing to do with knowledge of or willingness to try new approaches to teaching. Young faculty who have attended a New Faculty Workshop are more likely to be familiar with the education research and to have tried some of these approaches, but they are no more likely than older faculty to stay with them.
What are the implications for Undergraduate Mathematics Education?
First, we need to conduct a comparable study of mathematics faculty. I suspect that knowledge of what is being done and the evidence of its effectiveness is weaker among mathematicians than it is in the physics community. The combination of the near-universal acceptance of the Force Concept Inventory as a measure of student conceptual knowledge and the high profile work of Eric Mazur on the use of clickers for Peer Instruction make it far more difficult for a physicist to be ignorant of everything that is happening in Physics Education Research. Part of the study of awareness of and responses to Research in Undergraduate Mathematics Education will need to be a thorough assessment of the effect of Project NExT. We have mountains of anecdotal evidence of the importance and effectiveness of this program, but now we need to learn what impact it really has had. The New Faculty Workshops in physics do raise awareness of what can be done, but appear to have little long-term impact on adoption of these strategies. Project NExT is different in many respects, most notably providing a much broader introduction to what it means to be a professional mathematician in an academic position. It also builds communities and provides mentorships. Project NExT is almost twenty years old. We should be able to get good data on its long-term influence.
Second, MAA and the mathematics community need to work on expanding knowledge of what works, including the development of web resources comparable to AAPTs PERusersguide.org. As I explained last month, this will be a much more challenging undertaking than it has been in physics. However, the next CUPM Curriculum Guide—now in its early stages of development and slated for publication in 2015—will address some of this lack.
Third, as the literature amply demonstrates, simply developing effective strategies and putting them out there for people to find is not sufficient. As I travel around the country and talk with faculty at a broad range of colleges and universities, I encounter a lot of dissatisfaction with the way undergraduate mathematics instruction is now conducted and frustration with the difficulty of maintaining quality in the face of budget cuts. I also see a lot of uncertainty about how to proceed and a fear of undertaking radical changes that might prove disastrous. MAA and the mathematics community need to develop mentorship programs that promote small, coherent steps and provide instruments for collecting feedback and monitoring the effectiveness of these efforts. This is part of the critical task of equipping faculty for continual development and refinement of their teaching. We also need to offer the kind of flexible guidance that enables each institution, ideally each instructor, to take ownership of these changes, allowing for modifications that meet local needs and preferences while signposting the dangerous mutations that would destroy the integrity and coherence of the strategy.
The third step will be complex and difficult, yet absolutely critical. I am optimistic that we can accomplish this. There is broad recognition that the teaching and learning of undergraduate mathematics needs to improve. Corporate, foundational, and governmental resources are lining up to bring about improvements. We also know much more than we ever have before about the barriers to change. In cooperation with the Physics Education Research community and all those working on STEM education, we can do this.
[1] Henderson, C., M. Dancy, and M. Niewiadomska-Bugaj. 2012. The Use of Research-Based Instructional Strategies in Introductory Physics: Where do Faculty Leave the Innovation-Decision Process? Accepted for publication in Physical Review Special Topics - Physics Education Research.
[2] Dancy, M. 2012. Educational Transformation in STEM: Why has it been limited and how can it be accelerated? Available at http://www.chem.purdue.edu/Towns/TRUSE/TRUSE docs/TRUSE Talks 2012/Dancy TRUSE Talk .pdf
[3] Henderson, C. 2012. The Challenges of Spreading and Sustaining Research-Based Instruction in Undergraduate STEM. Available at http://homepages.wmich.edu/~chenders/Publications/2012HendersonASEETalk.pdf
[4] Henderson, C., A Beach, and N. Finkelstein. 2011. Facilitating Change in Undergraduate STEM Instructional Practices: An Analytic Review of the Literature. Journal of Research in Science Teaching. Vol. 48, no. 8, pp. 952–984.
The primary source for this article is the report by Henderson et al. [1] of a survey completed by 722 physics faculty from the United States, but I also will draw on some of the other literature and I recommend the slides from two talks, one given by Melissa Dancy [2] at the conference on Transforming Research in Undergraduate STEM (Science, Technology, Engineering, and Mathematics) Education and the other given by Charles Henderson [3] at the American Society for Engineering Education, both given in June of this year.
For their study, Henderson et al. questioned faculty about their awareness and use of twenty-four different research-based instructional strategies (RBIS) for which there are published studies establishing their effectiveness. These include a variety of materials produced in the 1990s and 2000s as well as Mazur’s Peer Instruction, developed at Harvard. The 722 respondents represent just over 50% of those who were contacted. While there may be some selection bias, only 12% had no knowledge of any of these strategies, and an impressive 72% had used at least one of them. But 32% of those who had used at least one strategy no longer used any of them. At least in physics, the issue is not getting people to try proven but innovative approaches to teaching and learning; it is enabling them to stick with it.
In trying to understand what influences faculty decisions about the use of innovative approaches, Henderson et al. looked at twenty possible explanatory variables that range from class size, type of institution, gender, rank, and number of years in the profession to research productivity (measured separately by presentations, articles, and research grants within the past two years), to departmental encouragement and discussions with peers about teaching, to goals for teaching (the importance of developing conceptual understanding and problem-solving ability) and interest in research in instructional strategies.
At the first level, simple familiarity with some of the instructional strategies, two of the strong predictors are attendance at talks or workshops on teaching and regular reading of journal articles about teaching. The strongest predictor is attendance at one of the New Faculty Workshops (NFW), 3½-day workshops held by the American Association of Physics Teachers (AAPT) for new physics faculty. The intent of these workshops is to introduce new faculty to the results of Physics Education Research, so it is not surprising that those who have participated in NFW are over ten times as likely to be aware of these strategies as the average faculty member. Two other variables were also significant: level of satisfaction that one’s goals for teaching were being met and whether one’s position was full- or part-time. At the next level, actually trying one of these strategies, a new but not surprising variable becomes important: interest in using RBIS.
I am intrigued by the list of variables that did not come into play. These include class size, type of institution, faculty rank or years in position, or research productivity. None of these significantly impacted knowledge of or willingness to try innovative strategies.
The critical stage however, is not the decision to try RBIS but the willingness to stick with it. The only survey variables that were significant at this stage were a desire to try more RBIS and gender: Women were significantly more likely to continue the use of RBIS than were men. Henderson et al. cite four other studies that also showed that women are far more likely to have student-centered physics classrooms than are men. The authors speculate that gender may be serving as a proxy for a collection of beliefs about and attitudes toward teaching.
Many researchers have studied the phenomenon of trying and then abandoning innovative approaches to teaching. The Henderson et al. article includes an extensive bibliography. The hurdles that faculty face include student complaints, inability to cover the same quantity of material, and weaker than promised student outcomes. One of the issues is that implementation frequently is not faithful. As I have seen first-hand both at Penn State and at Macalester, if the course that students experience is not cohesive, if bits and pieces are poorly articulated or motivated, students will complain and resist.
That does not mean that faculty dare not try to modify a received instructional strategy. It does mean that whatever changes they incorporate into their classes must be reflected upon and monitored for effectiveness. In the Henderson, Beach and Finkelstein [4] review of the literature on facilitating change in undergraduate STEM education, one commonly observed phenomenon is the importance of evaluation and feedback, an aspect of change implementation that often is omitted.
Among the interesting findings of the survey by Henderson et al. were the variables that demarcated the distinction between high and low users of RBIS. Low users were defined as using one or two of these strategies, high as three or more. This is the first time that some of the traditionally assumed barriers come into play: research productivity, type of institution, and class size. Highly productive faculty at research universities teaching large classes of introductory physics are significantly less likely to be high users of RBIS. What is surprising is that this is the first time that these variables arise. These highly productive research faculty were no less likely to know about or try some of these innovative strategies, just less likely to be high users.
Also intriguing is the fact that age, as implied by rank and number of years in service, had nothing to do with knowledge of or willingness to try new approaches to teaching. Young faculty who have attended a New Faculty Workshop are more likely to be familiar with the education research and to have tried some of these approaches, but they are no more likely than older faculty to stay with them.
What are the implications for Undergraduate Mathematics Education?
First, we need to conduct a comparable study of mathematics faculty. I suspect that knowledge of what is being done and the evidence of its effectiveness is weaker among mathematicians than it is in the physics community. The combination of the near-universal acceptance of the Force Concept Inventory as a measure of student conceptual knowledge and the high profile work of Eric Mazur on the use of clickers for Peer Instruction make it far more difficult for a physicist to be ignorant of everything that is happening in Physics Education Research. Part of the study of awareness of and responses to Research in Undergraduate Mathematics Education will need to be a thorough assessment of the effect of Project NExT. We have mountains of anecdotal evidence of the importance and effectiveness of this program, but now we need to learn what impact it really has had. The New Faculty Workshops in physics do raise awareness of what can be done, but appear to have little long-term impact on adoption of these strategies. Project NExT is different in many respects, most notably providing a much broader introduction to what it means to be a professional mathematician in an academic position. It also builds communities and provides mentorships. Project NExT is almost twenty years old. We should be able to get good data on its long-term influence.
Second, MAA and the mathematics community need to work on expanding knowledge of what works, including the development of web resources comparable to AAPTs PERusersguide.org. As I explained last month, this will be a much more challenging undertaking than it has been in physics. However, the next CUPM Curriculum Guide—now in its early stages of development and slated for publication in 2015—will address some of this lack.
Third, as the literature amply demonstrates, simply developing effective strategies and putting them out there for people to find is not sufficient. As I travel around the country and talk with faculty at a broad range of colleges and universities, I encounter a lot of dissatisfaction with the way undergraduate mathematics instruction is now conducted and frustration with the difficulty of maintaining quality in the face of budget cuts. I also see a lot of uncertainty about how to proceed and a fear of undertaking radical changes that might prove disastrous. MAA and the mathematics community need to develop mentorship programs that promote small, coherent steps and provide instruments for collecting feedback and monitoring the effectiveness of these efforts. This is part of the critical task of equipping faculty for continual development and refinement of their teaching. We also need to offer the kind of flexible guidance that enables each institution, ideally each instructor, to take ownership of these changes, allowing for modifications that meet local needs and preferences while signposting the dangerous mutations that would destroy the integrity and coherence of the strategy.
The third step will be complex and difficult, yet absolutely critical. I am optimistic that we can accomplish this. There is broad recognition that the teaching and learning of undergraduate mathematics needs to improve. Corporate, foundational, and governmental resources are lining up to bring about improvements. We also know much more than we ever have before about the barriers to change. In cooperation with the Physics Education Research community and all those working on STEM education, we can do this.
[1] Henderson, C., M. Dancy, and M. Niewiadomska-Bugaj. 2012. The Use of Research-Based Instructional Strategies in Introductory Physics: Where do Faculty Leave the Innovation-Decision Process? Accepted for publication in Physical Review Special Topics - Physics Education Research.
[2] Dancy, M. 2012. Educational Transformation in STEM: Why has it been limited and how can it be accelerated? Available at http://www.chem.purdue.edu/Towns/TRUSE/TRUSE docs/TRUSE Talks 2012/Dancy TRUSE Talk .pdf
[3] Henderson, C. 2012. The Challenges of Spreading and Sustaining Research-Based Instruction in Undergraduate STEM. Available at http://homepages.wmich.edu/~chenders/Publications/2012HendersonASEETalk.pdf
[4] Henderson, C., A Beach, and N. Finkelstein. 2011. Facilitating Change in Undergraduate STEM Instructional Practices: An Analytic Review of the Literature. Journal of Research in Science Teaching. Vol. 48, no. 8, pp. 952–984.
Sunday, July 1, 2012
Learning from the Physicists
This is a continuation of my personal responses to and musings upon the report Engage to Excel from the President’s Council of Advisors on Science and Technology (PCAST), this month focusing on the first recommendation:
- Catalyze widespread adoption of empirically validated teaching practices.
As the PCAST report points out, there are many techniques for improving classroom interaction that are known to improve student performance. Table 2 on page 17 of the report illustrates several of these and references the research literature. Their list includes small group discussion, one-minute papers, clickers, and problem-based learning.
The Physics Education Research (PER) community, through the American Association of Physics Teachers, has done a nice job of organizing a website of 51 Evidence-based teaching methods that have been demonstrated to be effective: PERusersguide.org. The site is organized to make it useful for the instructor: a brief description and each method and six searchable cross-listings that describe
- Level: the courses for which it is appropriate, usually introductory physics,
- Setting: whether designed for large lecture, small classes, labs, or recitation sections,
- Coverage: whether it requires studying fewer topics at greater depth,
- Effort: low, medium, or high,
- Resources: what is needed, from computer access to printed materials that must be purchased to classrooms with tables,
- Skills: what students are expected to acquire, usually including conceptual understanding, but also possibly problem-solving skills and laboratory skills.
In addition, each of the methods includes a list of the types of validation that have been conducted: what aspects of student learning were studied, what skills the method has been demonstrated to improve, and the nature of the research methods.
I wish for a comparable site for mathematics. Inevitably, a site developed by MAA and the Research in Undergraduate Mathematics Education (RUME) community would need to be both more comprehensive and more complex. Introductory physics is a relatively straightforward course with clear goals, a restricted clientele, and only two flavors: calculus or non-calculus based. In addition, most of the content is new to most of its students.
Introductory college-level mathematics is far more diverse and serves a broad set of disciplines that place often quite specific and disparate demands on these courses. On top of this, we in the mathematics community are plagued by the fact that almost nothing commonly taught in the first year, even Calculus I and II, is completely fresh to these incoming students. At the same time, too many of these students enter without the conceptual knowledge of mathematics and skill in using it that are needed to thrive in this first college course. For those of us who teach college-level mathematics, pressures for coverage are greater and gaps in student preparation are more acute and problematic than they are for introductory physics.
Compounding the difficulties of conducting research in methods of undergraduate mathematics is the fact that the RUME community is only one small part of Mathematics Education Research, which studies the learning of all mathematical knowledge beginning with early childhood recognition of small counting numbers as cardinalities. RUME fights for dollars and publication space against well-established research programs with methods of validation that have been honed over decades but are often inappropriate for understanding the complexities inherent in the learning of higher mathematics.
Nevertheless, the mathematical community does have research evidence for instructional strategies that work. There is a long history of studies of Emerging Scholars Programs, active learning strategies, and computer-aided instruction (see my column Lessons for Effective Teaching, November 2008). Sandra Laursen and her group at UC-Boulder have studied Inquiry Based Learning and documented its benefits (see my column The Best Way to Learn, August 2011).
Unfortunately, the experience of the physicists demonstrates that the existence of research based instructional strategies together with documentation of their effectiveness is not sufficient to guarantee their widespread adoption. Why not?
Again, the PER community is ahead of the RUME community in this regard. At the recent interdisciplinary conference, Transforming Research in Undergraduate Science Education (TRUSE), held at the University of Saint Thomas in Saint Paul, MN, June 3–7, Melissa Dancy of UC-Boulder spoke on Educational Transformation in STEM: Why has it been limited and how can it be accelerated?
Much of the work described in Dancy’s talk can be found in the preprint by Henderson, Dancy, and Niewiadomska-Bugaj [1]. This paper will be the topic of my August column. The work that they have done via surveys of physics faculty demonstrates that the greatest problem is not in making faculty aware of what has been done, or even in getting faculty to try different approaches to teaching. The greatest problem is in getting faculty to stick with these strategies.
What Henderson et al. have to say resonates with my own experience. As I reported in Reform Fatigue, June 2007, the use of calculators, computers, writing assignments and group projects in Calculus rose during the 1990s, but dropped off sharply between 2000 and 2005. The 2010 survey conducted under MAA’s study of Characteristics of Successful Program in College Calculus showed a continuing but modest decline in the use of graphing calculators and computers. Yet there was a bright spot. The use of group projects has rebounded (see Graph 1). Much depends on the quality of the group projects and how they are used, but the data suggest that the mathematics community is not totally immune to empirically validated teaching practices.
 |
| Graph 1. Percentage of sections of Calculus I that use group projects for part of the course grade. Data for 1990 through 2005 from CBMS surveys. For 2010, data from MAA survey of Characteristics of Successful Programs in College Calculus. |
Friday, June 1, 2012
Response to PCAST
MAA has just made public its official response to the report
to President Obama from the President’s
Council of Advisors on Science and Technology (PCAST), Report
to the President, Engage to Excel: producing one million additional college
graduates with degrees in Science, Technology, Engineering, and Mathematics. This response to John Holdren and Eric Lander, the
co-chairs of PCAST, is on MAA’s Science
Policy page and can be
downloaded here. One aspect of the response that I particularly like is
the appendix, which lists many MAA activities that align with the PCAST
report’s recommendations. Specifically, my synthesis of these recommendations is
1. To draw on the available research and empirical evidence to improve undergraduate education in mathematics, science, and engineering;
2. To improve attraction and retention of students by engaging them in activities where they get to discover the science or mathematics; and
3. To encourage greater collaboration between mathematics and the science and engineering programs.
For this column, I
will focus on the subheading to the title of the PCAST report: producing one
million additional STEM graduates. As I explained in my March column, On Engaging to Excel, this impressive
number is taken over a decade and includes associate’s degrees. Nevertheless,
it translates into the still impressive-sounding goal of an additional 75 to 80,000
bachelor’s degrees in STEM fields each year. How ambitious is that goal?
Graph 1 shows the
number of full-time freshmen arriving each fall with the intention of majoring
in engineering, as well as the number who graduated with a bachelor’s degree in
engineering the previous spring. The most recent number of intended majors is
for fall 2011. For actual bachelor’s degrees awarded, the most recent number is
for spring 2010.
The most striking
feature of this graph is the remarkable consistency in the number of intended
Engineering majors from 1980 through 2007 and the dramatic increase since then,
from 102,000 in fall 2007 to 184,000 in fall 2011. There are our extra 80,000
STEM majors, if we can keep them. [3]
It is too soon to be
able to tell how well we are doing at retention. The first big increase, with
the incoming class of fall 2008, has only just seen the graduation of those who
completed their degrees in four years. It will be two more years before the US
Department of Education releases the spring 2012 graduation numbers. But the
situation in the physical and biological sciences can shed some light on what
we might expect.
In both cases, the
number of intended majors took off following the year 2000, more than doubling
over the following decade. In the biological sciences, the annualized rate of
growth in the number of bachelor’s degrees from 2005 to 2010 has almost exactly
matched the rate of growth in the number of incoming students five years
earlier, at about 6% per year. In the physical sciences, the percentage rate of
growth in the number of degrees from 2005 to 2010, at 4% per year, is half of
the 8% per year growth rate in the number of incoming physical science students
five years earlier. It is reasonable to assume that engineering may be more
similar to the physical sciences than the biological sciences.
In short, the
problem is not with attracting students to STEM fields. The issue will be to
retain them.
[1] Higher Education
Research Institute. The
American Freshman. UCLA.
[2] National Center
for Education Statistics. Digest
of Education Statistics. US Department of Education.
[3] The increased
interest in engineering programs is almost certainly a result of the recession
that began in 2008 and the high unemployment rate since then. I discussed this
connection in an earlier column, A Benefit of High Unemployment, November 2010.
Note that while enrollments in the biological and physical sciences began to increase
following the class of 2000, the rates of increase accelerated after 2007.
Tuesday, May 1, 2012
Are Textbooks Better Online?
The March 30 issue of Science included a letter from psychologists David Daniel and Daniel Willingham [1] that provides an excellent overview of what is known about how students use online textbooks, including an account of what we know about the strengths and the weaknesses of switching from print to electronic delivery. I find that what they have to say is in line with my own experiences. For the past year, I have taught our Single Variable Calculus class using MAA’s online textbook Calculus: Modeling and Application, 2nd edition, by David Smith and Lang Moore. This is a direct descendant of their Project CALC materials that I have used and loved. The current edition is only available as an online textbook.
The overwhelming advantage of online publishing is the cost savings. The cost to my students is $25 apiece. This is charged as a lab fee. In exchange, my students get to download the entire textbook onto their own computers, a benefit that is relatively uncommon among online mathematics texts but which my students appreciate because they are not restricted to using their textbook only when they have internet access.
Smith and Moore’s book is written in html, using MathML for the mathematics (which effectively restricts the web browser in which it is read to FireFox). The authors have done a thoughtful job of separating the chapters into sections that each fit fairly comfortably on a single web page. Several of my students have commented on how much that helps with the readability of the text. But my students have found that reading an online textbook does require a period of adjustment.
I have yet to encounter a student who prefers reading web pages instead of printed pages. Daniel and Willingham cite three different studies that confirm that most students prefer traditional print books. Intriguingly, online texts work very well for young readers. In fact, those learning how to read often do better with online books. The difficulties seem to arise when students need to study and learn from the text, a characteristic that is especially true of mathematics books. For reasons that we do not fully understand but which are well documented, careful reading of an electronic text takes longer and is more fatiguing than trying to learn the same material from a printed text. The research also has found that this effect of greater difficulty with electronic texts is independent of the level of familiarity and experience with e-books.
One of the greatest benefits of online textbooks is the ability to embed links to definitions, animations, and software programs. Smith and Moore’s textbook is liberally sprinkled with links to explorations that are available in Maple, Mathcad, and Mathematica. There also are links to WeBWorK, where a library of problems linked to the sections of their book is available.
My own experience is that students frequently ignore the links to explorations unless I specifically assign them. This is in line with the findings reported by Daniel and Willingham: Students often find such ancillary material distracting at best, confusing at worst. Following these links often leads to loosing the thread of the conceptual development in the text. Another characteristic of e-textbooks that can cut both ways is the ability to link to networking sites where they can exchange thoughts about the mathematics and insights into each other’s difficulties. While that can be very beneficial, there is also the danger that these students will be tempted by the distraction of social media that is equally close at hand.
As Daniel and Willingham point out, online textbooks are easily corrected and updated. For the authors of such a text, that means that the job of working on the book is an ongoing task that is never completed, working against the ability to keep the cost down.
The bottom line is that we do not yet know how best to take advantage of online textbooks. Doing it right is clearly not as simple as putting the text on line and inserting links.
[1] David B. Daniel and Daniel T. Willingham. 2012. Electronic Textbooks: Why the Rush? Science. 335. 30 March, 2012. 1570–71.
http://www.sciencemag.org/content/335/6076/1569.full?sid=ba9ce743-5704-4c50-aaef-3f933a3b26ba
The overwhelming advantage of online publishing is the cost savings. The cost to my students is $25 apiece. This is charged as a lab fee. In exchange, my students get to download the entire textbook onto their own computers, a benefit that is relatively uncommon among online mathematics texts but which my students appreciate because they are not restricted to using their textbook only when they have internet access.
Smith and Moore’s book is written in html, using MathML for the mathematics (which effectively restricts the web browser in which it is read to FireFox). The authors have done a thoughtful job of separating the chapters into sections that each fit fairly comfortably on a single web page. Several of my students have commented on how much that helps with the readability of the text. But my students have found that reading an online textbook does require a period of adjustment.
I have yet to encounter a student who prefers reading web pages instead of printed pages. Daniel and Willingham cite three different studies that confirm that most students prefer traditional print books. Intriguingly, online texts work very well for young readers. In fact, those learning how to read often do better with online books. The difficulties seem to arise when students need to study and learn from the text, a characteristic that is especially true of mathematics books. For reasons that we do not fully understand but which are well documented, careful reading of an electronic text takes longer and is more fatiguing than trying to learn the same material from a printed text. The research also has found that this effect of greater difficulty with electronic texts is independent of the level of familiarity and experience with e-books.
One of the greatest benefits of online textbooks is the ability to embed links to definitions, animations, and software programs. Smith and Moore’s textbook is liberally sprinkled with links to explorations that are available in Maple, Mathcad, and Mathematica. There also are links to WeBWorK, where a library of problems linked to the sections of their book is available.
My own experience is that students frequently ignore the links to explorations unless I specifically assign them. This is in line with the findings reported by Daniel and Willingham: Students often find such ancillary material distracting at best, confusing at worst. Following these links often leads to loosing the thread of the conceptual development in the text. Another characteristic of e-textbooks that can cut both ways is the ability to link to networking sites where they can exchange thoughts about the mathematics and insights into each other’s difficulties. While that can be very beneficial, there is also the danger that these students will be tempted by the distraction of social media that is equally close at hand.
As Daniel and Willingham point out, online textbooks are easily corrected and updated. For the authors of such a text, that means that the job of working on the book is an ongoing task that is never completed, working against the ability to keep the cost down.
The bottom line is that we do not yet know how best to take advantage of online textbooks. Doing it right is clearly not as simple as putting the text on line and inserting links.
[1] David B. Daniel and Daniel T. Willingham. 2012. Electronic Textbooks: Why the Rush? Science. 335. 30 March, 2012. 1570–71.
http://www.sciencemag.org/content/335/6076/1569.full?sid=ba9ce743-5704-4c50-aaef-3f933a3b26ba
Monday, April 2, 2012
MAA/NCTM Joint Position on Calculus
I am very pleased to report that MAA and NCTM (National Council of Teachers of Mathematics) have just approved the joint position statement on calculus that comes at the end of this column. It was crafted in response to the recognition of three serious problems:
[1] US Department of Education. 2008. National Education Longitudinal Study of 1988 (NELS:88). nces.ed.gov/surveys/NEL
[2] National Science Board. Science and Engineering Indicators: 2010. National Science Foundation. Arlington, VA. Appendix Table 1-20. www.nsf.gov/statistics/seind10/appendix.htm
[3] The MAA study Characteristics of Successful Programs in College Calculus found that 61% of the students in college Calculus I had studied calculus in high school, and 58% of all the students starting Calculus I expected to earn an A in the course. Confidence was the greatest single casualty of the course, dropping by almost half a standard deviation from the start to the end of the course.
- The first problem is the general perception among high school students and their parents that preparation for college should, if at all possible, include studying calculus in high school. The result is that too many students short-change their preparation in algebra, geometry, trigonometry, and other mathematical topics in order to stay on a fast track to calculus. Too many otherwise talented students arrive at university without the mathematical foundation that is needed to succeed in the college-level mathematics required for their intended major.
This problem is documented particularly clearly in two national longitudinal studies conducted by the US Department of Education. From the high school class of 1992, 31% of those who completed a course of calculus in 12th grade or earlier then enrolled in precalculus when they got to college.[1] From the high school class of 2004, 17% (one in six) of those who had completed a calculus course in secondary school reported taking remedial mathematics when they got to college.[2]
The solution is to insist that the first priority should be to establish a solid foundation in mathematics, rather than to get into calculus while still in high school.
- The second problem is related to the first. It is that too often the calculus course that is taught in high school is a version that trains students in techniques of differentiation and integration and in procedures for solving certain standard problems without developing their understanding of calculus. While this builds familiarity with the language of calculus, it misrepresents the true nature of college-level mathematics and creates a false sense of confidence. One of the most dramatic findings of the MAA’s national study of Calculus I instruction in college and university is the high confidence level of entering students and how precipitously it drops as a result of encountering the reality of college-level expectations for calculus.[3]
The solution is to insist that when calculus is taught in high school, it is taught as a course whose content really is equivalent to a mainstream college course.
- The third problem is an outgrowth of the first two. Today’s reality for most students headed into STEM careers is a double dose of introductory calculus, once on each side of the high school to college divide. At the very least, this wastes student time, especially when the college course is taught as if this material were being experienced for the first time. For those students who, for whatever reason, have had a bad or simply uninspiring experience of calculus in secondary school, the prospect of repeating their experience can dissuade them from continuing their study of mathematics. Even for those who enjoyed their secondary school calculus, repeating this course can lead to boredom and poor study habits, often resulting in poor performance and a move away from mathematics intensive disciplines.
The solution is to insist that college faculty recognize their audience and deal with it, modifying how calculus is taught and offering alternatives to calculus for entering students.
Recommendations that address these problems lie at the heart of this position statement. Most satisfying from my point of view is the recognition that the issue of when and how calculus is taught is important and requires the efforts of both MAA and NCTM.
I want to thank all those who have worked to make this joint position statement possible. First is Mike Shaughnessy, President of NCTM, who agreed to make the adoption of such a statement one of his personal priorities. Next is Gail Burrill, former NCTM President and current Chair of the MAA-NCTM Joint Committee on Mutual Concerns who created the framework in which it was possible to craft this statement and who provided much useful feedback during its creation. And finally I want to thank the three people who worked with me to actually write the statement: Mike Boardman at Pacific University, Tom Kilkelly at Wayzata High School in Minnesota, and Dane Camp at New Trier Township High School in Illinois.
Question: How should secondary schools and colleges envision calculus as the course that sits astride the transition from secondary to postsecondary mathematics for most students heading into mathematically intensive careers?MAA/NCTM Position Although calculus can play an important role in secondary school, the ultimate goal of the K–12 mathematics curriculum should not be to get students into and through a course in calculus by twelfth grade but to have established the mathematical foundation that will enable students to pursue whatever course of study interests them when they get to college. The college curriculum should offer students an experience that is new and engaging, broadening their understanding of the world of mathematics while strengthening their mastery of tools that they will need if they choose to pursue a mathematically intensive discipline.
In particular, the fact that calculus is a college mathematics course that increasingly is taught in high school requires that—
- Students who enroll in a calculus course in secondary school should have demonstrated mastery of algebra, geometry, trigonometry, and coordinate geometry;
- The calculus course offered in secondary school should have the substance of a mainstream college-level course;
- The college curriculum should acknowledge the ubiquity of calculus in secondary school, shape the college calculus curriculum so that it is appropriate for those who have experienced introductory calculus in high school, and offer alternatives to calculus.
Faculty in our colleges and secondary schools should work together to re-envision the role of calculus in secondary and postsecondary mathematics education. Faculty on both sides of the transition from secondary to college mathematics should work together to strengthen the mathematics curriculum so that students who intend to pursue a mathematically intensive career can acquire the mathematical knowledge and capabilities needed for such a career. College faculty and secondary teachers should define what it means for a student to be ready for college-level mathematics. After a student has matriculated in college, they should assess the effect of college-level mathematics offered in secondary school. They also should clarify and broaden what is meant by college-level mathematics for secondary school. They should also work to achieve a better understanding of the mathematical strengths and weaknesses of matriculating students, assess the effectiveness of placement programs for collegiate mathematics, and clarify and broaden what the first year of college mathematics can and should entail.
MAA and NCTM are committed to taking appropriate action within the structure of their organizations to assist in guiding the implementation of these recommendations.
[1] US Department of Education. 2008. National Education Longitudinal Study of 1988 (NELS:88). nces.ed.gov/surveys/NEL
[2] National Science Board. Science and Engineering Indicators: 2010. National Science Foundation. Arlington, VA. Appendix Table 1-20. www.nsf.gov/statistics/seind10/appendix.htm
[3] The MAA study Characteristics of Successful Programs in College Calculus found that 61% of the students in college Calculus I had studied calculus in high school, and 58% of all the students starting Calculus I expected to earn an A in the course. Confidence was the greatest single casualty of the course, dropping by almost half a standard deviation from the start to the end of the course.
Thursday, March 1, 2012
On Engaging to Excel
The President’s Council of Advisors on Science and Technology (PCAST) has just released its report to President Obama on undergraduate Science, Technology, Engineering, and Mathematics (STEM) education: Report to the President, Engage to Excel: producing one million additional college graduates with degrees in Science, Technology, Engineering, and Mathematics.
As the title suggests, the bottom line is to increase the number of undergraduate degrees (Bachelor’s or Associate’s) earned in STEM fields by one million. As with budget savings, the numbers are made more impressive by making this a ten-year goal. Even so, that amounts to an extra 100,000 STEM majors per year. From the most recent year for which we have data, 2009, there were 241,000 Bachelor’s degrees awarded in STEM fields and 70,000 Associate’s degrees. We are thus looking to increase the number of STEM degrees by about one-third.
This is a report that matters. Last fall PCAST did a thorough survey of all federal grant programs that support undergraduate work in STEM fields, focusing on the National Science Foundation (NSF) but including programs of the Departments of Education and Labor. This report carries specific recommendations on how this money should now be directed. The NSF programs Transforming Undergraduate Education in STEM (TUES), STEM Talent Expansion Program (STEP), and Widening Implementation and Demonstration of Evidence-Based Reforms (WIDER) have come in for particular attention.
There are four principal recommendations:
After acknowledging the central role of mathematics in preparing students for STEM careers, the report then asserts that “introductory mathematics courses often leave students with the impression that all STEM fields are dull and unimaginative” [p. vi] and later complains that “Discipline-based education on effective undergraduate mathematics teaching also appears less developed when compared with other STEM fields.” [p. 27] The report describes at some length the problem of mathematically under-prepared students and recommends a sustained effort to increase the production of high school mathematics teachers by 100,000 (presumably over ten years) and a joint effort by NSF and the Departments of Labor and Education to “support a national experiment in mathematics undergraduate education.”
This national experiment has four components:
In fact, there have been and continue to be many promising, sustained, and national efforts to reform undergraduate mathematics education. The mathematical community is learning from the successes that have occurred in physics, chemistry, and biology, but the lessons learned are not always easy to apply. Because of its central role across the disciplines, the sheer number of students involved, and the complexity of the issues that reach right down into pre-kindergarten, accomplishing measurable improvement in undergraduate mathematics education on a national scale is extremely difficult. We welcome the help of the other STEM disciplines, but ultimately this is the responsibility of the mathematics community.
I will return to this report in future columns. For now, I’d like to close with one observation and a question. In the fall of 2007, 276,000 of the entering freshmen in full-time four-year undergraduate programs declared that they intended to major in a STEM field. This number had remained essentially unchanged since 2001. Thanks to the economic downturn, by this past fall the number of entering freshmen intending to major in a STEM discipline had grown to 424,000 [data from The American Freshman]. There are our extra 100,000 STEM majors and then some. They are out there if the incentives are right. Now, can we keep them?
As the title suggests, the bottom line is to increase the number of undergraduate degrees (Bachelor’s or Associate’s) earned in STEM fields by one million. As with budget savings, the numbers are made more impressive by making this a ten-year goal. Even so, that amounts to an extra 100,000 STEM majors per year. From the most recent year for which we have data, 2009, there were 241,000 Bachelor’s degrees awarded in STEM fields and 70,000 Associate’s degrees. We are thus looking to increase the number of STEM degrees by about one-third.
This is a report that matters. Last fall PCAST did a thorough survey of all federal grant programs that support undergraduate work in STEM fields, focusing on the National Science Foundation (NSF) but including programs of the Departments of Education and Labor. This report carries specific recommendations on how this money should now be directed. The NSF programs Transforming Undergraduate Education in STEM (TUES), STEM Talent Expansion Program (STEP), and Widening Implementation and Demonstration of Evidence-Based Reforms (WIDER) have come in for particular attention.
There are four principal recommendations:
- Catalyze widespread adoption of empirically validated teaching practices.
- Advocate and provide support for replacing standard laboratory courses with discovery-based research courses.
- Launch a national experiment in postsecondary mathematics education to address the mathematics-preparation gap.
- Encourage partnerships about stakeholders to diversify pathways to STEM careers.
After acknowledging the central role of mathematics in preparing students for STEM careers, the report then asserts that “introductory mathematics courses often leave students with the impression that all STEM fields are dull and unimaginative” [p. vi] and later complains that “Discipline-based education on effective undergraduate mathematics teaching also appears less developed when compared with other STEM fields.” [p. 27] The report describes at some length the problem of mathematically under-prepared students and recommends a sustained effort to increase the production of high school mathematics teachers by 100,000 (presumably over ten years) and a joint effort by NSF and the Departments of Labor and Education to “support a national experiment in mathematics undergraduate education.”
This national experiment has four components:
- Summer and other bridge programs for high school students entering college;
- Remedial courses for students in college, including approaches that rely on computer technology;
- College mathematics teaching and curricula developed and taught by faculty from mathematics-intensive disciplines other than mathematics, including physics, engineering, and computer science; and
- A new pathway for producing K-12 mathematics teachers from undergraduate and graduate programs in mathematics-intensive fields other than mathematics.
In fact, there have been and continue to be many promising, sustained, and national efforts to reform undergraduate mathematics education. The mathematical community is learning from the successes that have occurred in physics, chemistry, and biology, but the lessons learned are not always easy to apply. Because of its central role across the disciplines, the sheer number of students involved, and the complexity of the issues that reach right down into pre-kindergarten, accomplishing measurable improvement in undergraduate mathematics education on a national scale is extremely difficult. We welcome the help of the other STEM disciplines, but ultimately this is the responsibility of the mathematics community.
I will return to this report in future columns. For now, I’d like to close with one observation and a question. In the fall of 2007, 276,000 of the entering freshmen in full-time four-year undergraduate programs declared that they intended to major in a STEM field. This number had remained essentially unchanged since 2001. Thanks to the economic downturn, by this past fall the number of entering freshmen intending to major in a STEM discipline had grown to 424,000 [data from The American Freshman]. There are our extra 100,000 STEM majors and then some. They are out there if the incentives are right. Now, can we keep them?
Subscribe to:
Posts (Atom)