# Category Archives: Liberal arts

## The inverted classroom and student self-image

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This week I’ve been immersed in the inverted classroom idea. First, I gave this talk about an inverted linear algebra classroom at the Joint Meetings in New Orleans and had a number of really good conversations afterwards about it. Then, this really nice writeup of an interview I gave for MIT News came out, highlighting the relationship between my MATLAB course and the MIT OpenCourseware Project. And this week, I’ve been planning out the second iteration of that MATLAB course that’s starting in a few weeks, hopefully with the benefit of a year’s worth of experience and reflection on using the inverted classroom to teach technical computing to novices.

One thing that I didn’t talk much about at the Joint Meetings or in the MIT interview was perhaps the most prominent thing about using the inverted classroom model on a day-to-day basis: how students react to it and change as a result of it. I was actually quite surprised that nobody at my Joint Meetings talk asked me a question about this, because honestly, the inverted classroom sounds great on paper, but when you start to imagine the average college student walking in on the first day of class and having this method of instruction described to him, it becomes clear that a significant amount of work is going to have to be done in order to get students — who are already resistant to any change from their accustomed modes of instruction — on board with the plan.

Students do tend to resist the inverted classroom at first. Some forms of resistance are more benign than others. On the benign end of the spectrum there are students with little experience with the course material or its prerequisites who get bogged down on the basic podcast viewing (which takes the place of in-class lectures in this model) or the accompanying guided practice, and instead of actively seeking a resolution to their question will wait for the instructor to clear it up — in class. On the other end is the student who simply doesn’t believe I’m serious when I say there won’t be any lecturing, who then doesn’t do the work, assuming I’ll bail him out somehow — in class. But in the inverted model, students are held responsible for acquiring basic competencies before class so that the hard stuff — what we refer to as assimilation — is the primary focus of the class time.

I break this distinction down for students, but not everybody buys into it. Those who don’t will have to undergo a learning process that usually looks like shock — shock that I won’t reteach them the material they were supposed to have viewed and worked on, while the lab assignment based on that material is going on. This can get very ugly in ways I probably don’t need to describe. Let’s just say that you had better not use the inverted classroom model if you aren’t prepared to put out a constant P.R. effort to convince students of the positive benefits of the model and constantly to assuage student concerns.

I’ve often wondered why students sometimes react so negatively to the inverted classroom model. I’ve come to believe it’s the result of a invasive, false belief that can arise in students about their ability to learn things independently of others — namely, that they simply cannot do so. I have had students tell me this to my face — “I can’t learn [insert topic] unless you lecture to me about it in class first.” Clearly this is not true. Toddlers learn their native language without formal instruction, just by assimilating (there’s that word again) the language going on naturally in their background. We all learn things every day without sitting in a classroom; we may seek out training data first through printed instructions, worked-out examples, YouTube videos, etc., but it’s almost never in a classroom setting. Learning new things on our own initiative and without formal instruction in a classroom setting is as natural to humans as breathing. Indeed you could say that it’s the capacity to learn in this way that makes us human. But somehow many students think otherwise.

Where does this belief come from? I think that it comes from its own instance of assimilation, namely the assimilation of a culture of programmed classroom instruction that takes place from roughly the first grade through the twelfth grade in this country. Students have so few experiences where they pursue and construct their own knowledge that they simply come to believe that they are incapable of doing so. And this belief is propagated most rapidly in mathematics. I’ve been reading in Seymour Papert‘s book Mindstorms: Children, Computers, and Powerful Ideas, and this quote hits this issue right on the head:

Difficulty with school math is often the first step of an invasive intellectual process that leads us all to define ourselves as bundles of aptitudes and ineptitudes, as being “mathematical” or “not mathematical”, “artistic” or “not artistic”, “musical” or “not musical”, “profound” or “superficial”, “intelligent” or “dumb”. Thus deficiency becomes identity and learning is transformed from the early child’s free exploration of the world to a chore beset by insecurities and self-imposed restrictions.

That last sentence (emphasis added) sums it up, doesn’t it? Deficiency becomes identity. Eventually, if a student is robbed of experiences of self-motivated learning, the student eventually adopts a self-image in which she is incapable of self-motivated learning. It is a false self-image that is ultimately dehumanizing.

Which is why I put such stock in the inverted classroom model. I think this method of teaching, along with other learner-centered modes of instruction like problem-based learning, is on the front lines in reversing students’ negative ways of thinking about how they learn. Students may (will?) chafe at the inversion at first. But in the MATLAB course at least, something really cool happened at the end of the semester. I made up a slideshow for students called “Five myths about how you think you learn that CMP 150 has busted”. Among the myths were “I can’t learn unless a professor lectures to me” and “I can’t learn on my own initiative”, and I gave concrete examples of work that the students had done in the class that contradicted these messages. In the end I showed them that through this inverted classroom process they had taken majors strides toward being confident, independent, skill learners and problem-solvers rather than just people who can play the classroom game well. And even the most skeptical students were nodding in agreement. And I think that makes it all worthwhile for everyone.

## Misunderstanding mathematics

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Robert Lewis, a professor at Fordham University, has published this essay entitled “Mathematics: The Most Misunderstood Subject”. The source of the general public’s misunderstandings of math, he writes, is:

…the notion that mathematics is about formulas and cranking out computations. It is the unconsciously held delusion that mathematics is a set of rules and formulas that have been worked out by God knows who for God knows why, and the student’s duty is to memorize all this stuff. Such students seem to feel that sometime in the future their boss will walk into the office and demand “Quick, what’s the quadratic formula?” Or, “Hurry, I need to know the derivative of 3x^2 – 6x +1.” There are no such employers.

Prof. Lewis goes on to describe some ways in which this central misconception is worked out in our schools and in everyday thinking. The analogy between mathematics instruction and building construction, in which he compares current high school mathematics instruction to a building project where the scaffolding is constructed and then abandoned because we think the job is done, is pretty compelling. The whole essay is well worth reading.

I do think that it’s a bit too easy to lay the blame for the current state of mathematics instruction at the feet of American high schools, as Lewis does multiple times. Even if high schools do have flawed models of math instruction, certainly they are not alone in this. How many universities, even elite institutions like Fordham, have math classes or even entire curricula predicated on teaching math as rote mechanics? And what about the elementary math curricula? Pointing the finger at high schools is the natural thing to do for college professors, because we are getting students fresh from that venue and can see the flaws in their understanding, but let us not develop tunnel vision and think that fixing the high schools fixes everything. Laying blame on the right party is not what solves the problem.

Lewis brings up the point that we should be aiming for “genuine understanding of authentic mathematics” to students and not something superficial, and on that I think most people can agree. But what is this “authentic mathematics”, and how are we supposed to know if somebody “genuinely understands” it? What does it look like? Can it be systematized into a curriculum? Or does genuine understanding of mathematics — of anything — resist classification and institutionalization? Without a further discussion on the basic terms, I’m afraid arguments like Lewis’, no matter how important and well-constructed, are stuck in neutral.

Again coming back to higher education’s role in all this, we profs have work to do as well. If you asked most college professors questions like What is authentic mathematics?, the responses would probably come out as a laundry list of courses that students should pass. Authentic mathematics consists of three semesters of calculus, linear algebra, geometry, etc. And the proposed solution for getting students to genuinely understand mathematics would be to prescribe a series of courses to pass. There is a fundamentally mechanical way of conceiving of university-level mathematics education in which a lot of us in higher ed are stuck. Until we open ourselves up to serious thinking about how students learn (not just how we should teach) and ideas for creative change in curricula and instruction that conform to how students learn, the prospects for students don’t look much different than they looked 15 years ago.

## In defense of big universities

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I’d like to take back something that I said in my post last week on the UCF cheating scandal (my emphasis):

[T]he more this situation unfolds, the more unhealthy it makes the whole educational environment surrounding it seem. Class sizes in the multiple hundreds: Check. Courses taught mainly through lecture: Check. Professor at a remove from the students: Check. Exams taken off the rack rather than tuned to the specific student population: Check. And on it goes. I know this is how it works at many large universities and there’s little that one can do to change things; but with all due respect to my colleagues at such places, I just can’t see what students find appealing about these places, and I wonder if students at UCF are thinking the same thing nowadays.

I’m coming at that statement as somebody who’s spent the last 14 years in small liberal arts colleges. The idea of 600-student lecture classes, using prefabricated tests from a test bank, and so on is completely alien to how I conceive of teaching and learning in higher education. The larger the university, the easier it is to adopt such depersonalized (even dehumanizing) “teaching” techniques. But I think I painted with too broad of a brush here. Because the fact is, there are going to be faculty who employ depersonalized approaches to education no matter how big or small the institution is. There are small colleges who willfully, even readily, employ such approaches to teaching on an institutional scale even though they are small enough to do better. And on the other side, there are large universities that, despite their largeness, still manage to treat undergraduate education with the care and skill it deserves.

I’d like to point out a couple of such large research universities with which I’ve had direct experience who, to me, really get undergraduate education right, or are at least trying to do so.

First is Vanderbilt University, where I did my graduate studies and got my first taste of teaching. Vanderbilt has a real culture of teaching and learning that pervades the entire academic structure of the university. It has a fabulous Center for Teaching where I was privileged to spend a year as a Master Teaching Fellow during my last year of grad school, working with other graduate teaching scholars and university faculty to help them get better at their craft. And what always impressed me at Vandy was that a lot of professors were interested in getting better. It’s a great research university, but the profs there — at least the ones I knew, with the exception of a few entrenched math people — all took teaching seriously and really wanted to work at getting better. And it shows in the quality of undergraduates Vanderbilt produces. I can definitely see why a high school kid would want to go there.

The other example is The College of Engineering at the University of Wisconsin – Madison. I spoke there recently to a group of faculty and support staff who are involved with a program called Engineering Beyond Boundaries, an ambitious program to transform the teaching, learning, and practice of engineering in response to key shifts in the discipline and the culture around it. The people involved with that program are embarking on an all-out effort to push the culture in the engineering school toward one that adopts a more modern approach to teaching and learning, including the renovation of learning spaces, work with innovative instructional techniques, and creating opportunities for cross-disciplinary work. They’re just getting started with this program, but I think some interesting things are ahead for them as they proceed in terms of teaching and learning.

Do you have other examples of big universities that are doing a good job with undergraduate education? Brag on them in the comments.

## Want a job? Major in what you enjoy.

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Excellent blog post in the NY Times website this morning telling us that the choice of college major is not as important as we think. The author shares this research finding:

A University of Texas at Austin professor, Daniel Hamermesh, researched career earnings data sorted by choice of major and concluded that:

“Perceptions of the variations in economic success among graduates in different majors are exaggerated. Our results imply that given a student’s ability, achievement and effort, his or her earnings do not vary all that greatly with the choice of undergraduate major.”

A study conducted by PayScale Inc. found that history majors who pursued careers in business ended up earning, on average, just as much as business majors.

The author goes on to cite four reasons why a liberal arts major would be a fine choice for career-minded college students, including the development of transferable skills and the value — both personal and professional — of majoring in something you truly enjoy rather than something you don’t enjoy but think might be useful someday.

I’m reminded of this great post over at Cal Newport’s blog from last year in which he advises prospective business majors not to major in business but rather to choose a classical liberal arts major and then take 4-6 math courses on the side. That amounts to majoring in something like economics or history and then getting a math minor. The liberal arts major will show employers that you are broadly educated and have those transferable skills, such as the ability to do research and communicate clearly in oral and written forms. Then the math minor adds a significant amount of training to show that you can handle quantitative information — a skill sorely lacking among a huge portion of new job marketeers today — and that you’re not in the liberal arts major to avoid hard work, which is unfortunately a common public perception of liberal arts programs. (That perception is something that we who work in the liberal arts colleges are partially responsible for perpetuating by not communicating the value of the liberal arts clearly enough.)

This combination doesn’t always work — engineering, for instance, really does require a degree in engineering at some point — and the student who goes this route takes on a double responsibility for making sure his or her liberal arts degree is really academically rigorous and for being ready to explain to hard-headed employers that they have the skills that will make them viable in the long term as employees. But I think it’s right to tell students to first study what they love, and then worry about the career part a little later. I’m certainly advising my own students to that effect. And given that most jobs are going to require new employees to learn on the fly the things they need to know anyway, it makes sense to develop students’ passions for learning and abilities to learn on their own, which is IMO one of the major things a liberal arts education is good for.

Filed under Education, Higher ed, Liberal arts, Student culture, Vocation

## What (some) engineers think about liberal education

I’m currently at the American Society for Engineering Education conference and symposium in Louisville. There is a lot to process as I attend sessions on student learning, technological literacy, liberal education, and so on, all from the perspective of engineers and engineering educators. There is an entire division (a sort of special interest group) within the ASEE for Liberal Education, and I attended one of their paper sessions this afternoon.

Engineers have a quite different perspective on liberal education than those in “liberal arts” disciplines (by which we usually mean social sciences, arts, humanities) and those of us math/science people working in liberal arts colleges, but surprisingly — at least for the engineers I hung out with in the session — the two conceptions largely agree. We all conceive of liberal education as education that integrates multiple perspectives into understanding what we study and do. We believe in the importance and relevance of disciplines other than our own and seek to learn about other disciplines, connect with practitioners and colleagues in other disciplines, and incorporate other disciplines in meaningful ways into our courses. We believe in teaching students metacognitive skills and preparing them to be human beings, not just workers.

Of course there are engineers who don’t think this way and in fact look down on other disciplines in direct proportion to their methodological distance from engineering (the less data and design involved, the greater the disdain). But consider too that there are also poets, philosophers, historians, mathematicians, sociologists, and so on who feel the same way about their own disciplines. The departmental silos exist all over campus.

Particularly enlightening was a parallel given in a talk by Cherrice Traver and Doug Klein of Union College (a liberal arts college known for its strong and historically-rooted engineering programs) between the criteria for ABET accreditation of engineering programs on the one hand, and the learning outcomes of Liberal Education and America’s Promise (or LEAP; a prospectus from the American Association of Colleges and Universities) on the other. Here are ABET’s Program Outcomes and Assessment criteria:

Engineering programs must demonstrate that their students attain the following outcomes:
(a) an ability to apply knowledge of mathematics, science, and engineering
(b) an ability to design and conduct experiments, as well as to analyze and interpret data
(c) an ability to design a system, component, or process to meet desired needs within realistic
constraints such as economic, environmental, social, political, ethical, health and safety,
manufacturability, and sustainability
(d) an ability to function on multidisciplinary teams
(e) an ability to identify, formulate, and solve engineering problems
(f) an understanding of professional and ethical responsibility
(g) an ability to communicate effectively
(h) the broad education necessary to understand the impact of engineering solutions in a global,
economic, environmental, and societal context
(i) a recognition of the need for, and an ability to engage in life-long learning
(j) a knowledge of contemporary issues
(k) an ability to use the techniques, skills, and modern engineering tools necessary for
engineering practice.

The entire accreditation document is here (PDF).

Compare those with the LEAP outcomes:

Beginning in school, and continuing at successively higher levels across their college studies, students should prepare for twenty-first-century challenges by gaining:

Knowledge of Human Cultures and the Physical and Natural World

Through study in the sciences and mathematics, social sciences, humanities, histories, languages, and the arts

Focused by engagement with big questions, both contemporary and enduring

Intellectual and Practical Skills, Including

Inquiry and analysis
Critical and creative thinking
Written and oral communication
Quantitative literacy
Information literacy
Teamwork and problem solving
Practiced extensively, across the curriculum, in the context of progressively more challenging problems, projects, and standards for performance

Personal and Social Responsibility, Including

Civic knowledge and engagement—local and global
Intercultural knowledge and competence
Ethical reasoning and action
Foundations and skills for lifelong learning
Anchored through active involvement with diverse communities and real-world challenges

Integrative and Applied Learning, Including

Synthesis and advanced accomplishment across general and specialized studies
Demonstrated through the application of knowledge, skills, and responsibilities to new settings and complex problems

As the presenters mentioned, you can make an exercise of lining these two lists of learning outcomes side by side (in fact, they gave us a handout where this was done) and draw lines connecting learning outcomes in LEAP with corresponding, or even identical, criteria from ABET’s list.

What this means, I think, is that there is a strong base of support for liberal education among engineers. One might even say that those in charge of accrediting engineering programs want engineers to be liberally educated. Some engineers, like the ones in the session I attended, will even say that themselves.

What nobody seems able to explain just yet is the implicit and sometimes explicit resistance to liberal education among many engineers and engineering programs. For example, why do most engineering programs require monumental depth in a single engineering discipline — as undergraduates — with only token amounts of university-required coursework outside of engineering? The electrical engineering degree at one university, for example, requires 68 credit hours just in freshman and electrical engineering courses. Then 33 hours in math and science, and a 3-hour mechanical engineering course. Eighteen hours total are left over for electives outside math, science, or engineering — and six of those are prescribed courses (composition and communication), leaving just 12 hours to be chosen from General Education elective blocks.

That’s just four courses the student gets to pick out of sheer curiosity and personal interest for his or her entire college education! Can that possibly be in line with what ABET — or for that matter, the engineering community and its clients — really want?

## Active learning is essential, not optional, for STEM students

This article (1.2 MB, PDF)  by three computer science professors at Miami University (Ohio) is an excellent overview of the concept of the inverted classroom and why it could be the future of all classrooms given the techno-centric nature of Millenials. (I will not say “digital natives”.) The article focuses on using inverted classroom models in software engineering courses. This quote seemed particularly important:

Software engineering is, at its essence, an applied discipline that involves interaction with customers, collaboration with globally distributed developers, and hands-on production of software artifacts. The education of future software engineers is, by necessity, an endeavor that requires students to be active learners. That is, students must gain experience, not in isolation, but in the presence of other learners and under the mentorship of instructors and practitioners.  [my emphasis]

That is, in the case of training future software engineers, active learning is not an option or a fad; it is essential, and failure to train software engineers in an active learning setting is withholding from them the essential mindset they will need for survival in their careers. If a software engineer isn’t an active learner, they won’t make it — the field is too fast-moving, too global, too collaborative in its nature to support those who can only learn passively. Lectures and other passive teaching techniques may not be obsolete, but to center students’ education around this kind of teaching sets the students up for failure later on.

One could argue the same thing for any kind of engineer, or any of the STEM disciplines in general, since careers in those disciplines tend to adhere to the same description as software engineering — a tendency toward applications (many of which don’t even exist yet), centered on interaction and collaboration with people, and focused on the production of usable products.

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## Turning questions into learning

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The hardest thing about teaching the MATLAB course — or any course — is responding to student questions. Notice I do not say “answering” student questions. Answers are not the issue; I’m no MATLAB genius, but I can answer 95% of student questions on the spot. The real issue is whether I should. If my primary task is to teach students habits of mind that translate into lifelong learning — and I earnestly believe that it is — then answers are not always the best thing for students.

I’ve noticed four types of questions that students tend to ask in the MATLAB course, and these carry over pretty seamlessly to my other courses:

1. Informational questions that have nothing to do with the problem they’re working on or the material. Example: When are your office hours? When is this lab due? When is the final exam?
2. Clarifying questions that seek to make sense of the specifications of a problem. Should we use a script M-file or a function M-file here? A FOR loop or a WHILE loop? Do I have to make this plot from the command line or can I use the Plot Tools window?
3. Functional questions that are generally of the form, How do I [insert task here]?. How do I plot a function in MATLAB? How do I get the plot to be red instead of blue? How do I get this FOR loop to work?
4. Interpretive questions that seek the meaning of syntax, a command, or an error. What does MATLAB mean when it says I should ‘pre-allocate’ this variable? Why are there all these different ways to call the MAX command? Why do I have to use num2str in some situations but not in others?

I’ve tried to list these question types in increasing order of cognitive complexity, although that ordering doesn’t always hold. (Some clarifying questions can get quite complicated, for instance.) How these types of questions are ranked in this way points me in the direction of how to respond to them.

Formal informational questions are easy to answer. I always give the same, direct answer to these: It’s (on the syllabus || in the calendar || printed on your assignment). Students learn pretty quickly that this is the same answer to this kind of question all the time, so they tend to stop asking and just look it up instead, which is precisely what they should be doing.

I’m happy to give straight answers to clarifying questions, although sometimes it’s better not to. For example, if a student team is working on a program that needs a loop, and they want to know if they should use a FOR or a WHILE loop, then the best way to respond is not to tell students what to do but rather to lay out the pros and cons of each approach and let them choose.

It gets very tricky when we get to the last two types.

I let the following basic philosophy guide me: I don’t answer functional questions on labs during lab sessions or on homework while the homework is still not yet turned in. Once the lab or homework is over, then I’ll usually answer directly. But otherwise, my goal is to guide students into turning their functional question into an interpretive question. I do this through a series of Polya-like questions to the students that flows a little something like this (click to enlarge):

This is less complicated than it looks. Basically, if a student asks a functional question, I first see if the student’s done what they’re asking before. If so, they go refresh their memories. If not, they look it up in an appropriate help file until they find something that looks like what they want. Then they play with it for a minute or two to get the basic idea. Then, by that point, they either know what they’re supposed to do, or else they have a deeper, more cognitively complex question to ask, not What do I type in to do this? but Why does this work the way it does? In a freshman-level class like this one, any time I can get students to elevate themselves from functional questions to interpretive/clarifying questions, I consider it a win.

What students get out of this process is the ability to move beyond needing the professor to tell them what to do. They become self-feeders. This is important because the professor is not going to be there when they really need this stuff, two or three courses down the line or when they’re out on the job (for which they were hired because they, and nobody else around them, has these kinds of computer skills). They are getting the ability to learn on their own, which is what I consider really to be the single, primary life skill.

Unfortunately students tend to resist this process. It is not what they are used to. They are used to teachers telling them the answers to their questions, regardless of what kind of question it was, and to them a failure of a teacher to give a straight answer to their questions is tantamount to either incompetence or indifference. So this process requires a constant P.R. effort and constant clarifying about why we do things this way. And that P.R. effort doesn’t always work. I still have students who complain that I don’t answer their questions; who feel belittled when they identify that they’ve seen a command before and are asked to go back and review it; who feel questions are pointless because I’m just going to ask them more questions in return.

These are freshmen, used to a transactional model of education predominant in American high schools. The fact that this model — the teacher tells the students what to do; students follow teacher’s directions; students get good grades — is the predominant one is a serious problem in our schools, but that’s another issue. Whatever the case may be, I am getting these folks in the final four years of their formal schooling (for the most part) and if I don’t get them thinking on their own, they will crash and burn in the real world.

And I think that even if these students go on never to use MATLAB again after graduation but have a well-practiced and fluid ability to learn new and complicated things on their own, I consider that the biggest win of all. And it’s a good reason to take the MATLAB course in the first place.

## MATLAB and critical thinking

My apologies for being a little behind the curve on the MATLAB-course-blogging. It’s been a very interesting last couple of weeks in the class, and there’s a lot to catch up on. The issues being brought up in this course that have to do with general thinking and learning are fascinating, deep, and complicated. It’s almost as if the course is becoming only secondarily a course on MATLAB and primarily a course on critical thinking and lifelong learning in a technological context.

This past week’s lab really brought that to the forefront. The lab was all about working with external data sets, and it involved students going to this web site and looking at this data set (XLS, 33 Kb) about electoral vote counts of the various states in the US (and the District of Columbia). One of the tasks asked students to make a scatterplot of the land area of the states versus their electoral vote counts. Once you make that scatterplot, it looks like this:

The reaction of most students to this plot was really surprising. Almost unanimously and without consulting each other, the reaction was: “That can’t be right.” When I’d ask them why not, they would say something like: It looks strange; or, it’s not like scatter plots I’ve done before; or, it just doesn’t look right.

The first instinct of those who felt like they had made a critical error in their plot was to ask me to verify whether or not they had gotten it right. That’s understandable, but it doesn’t go very far because I have a rule that I don’t answer “Is this right?” questions in the lab. (See the instructions in the lab assignment.) Student teams are responsible in the labs for determining by themselves the rightness or wrongness of their work. So it’s time for critical thinking to take center stage — which in this context would refer to using your brain and all available tools and information to self-verify your work. (I wrote about the idea of self-verification here using Wolfram|Alpha.)

Some of the suggestions I gave these teams were:

• Have you checked your plot against the actual data? For example, look at the outliers. Can you find them in the data set itself? And look at the main cluster of data; given a cursory glance through the data set, does it look like most states have a land area less than $10^6$ square miles and an electoral vote count of between 5 and 15?
• Have you tried to create the same scatterplot using different tools? For example, everybody in the class knows Excel (because we teach it in Calculus I); the data are in Excel already, so it would be virtually no work to make a scatterplot in Excel. Have you tried that? If so, does it look like what MATLAB is creating?
• Have you taken a moment just to think about the possible relationship between the variables, and does the shape of the data match your expectations? Probably we don’t really expect much of a relationship at all between the land area of a state and its electoral vote count, even with the outliers trimmed out, so a diffuse cloud of data markers is exactly what we want. If we got some sort of perfectly lined-up string of data points, we should be suspicious this time.

Once you phrase it like this, students pretty quickly gain confidence in their results. But, importantly, most of them have never been put into situations — at least in the classroom — where this sort of thing has been necessary. If critical thinking means anything, it means training yourself to ask questions like this and pursue their answers in an attempt to be your own judge of your work.

I was particularly surprised by the rejection of any scatter plot that doesn’t look like points on the graph of a function. “Authentic instruction” is a term without an operational definition, a lot like the term “critical thinking”, but here I think we may have a clue to what that term means. Students said their scatterplots didn’t “look right”, meaning they didn’t look like what their textbook examples had looked like, i.e. the points didn’t have an overwhelmingly strong correlation despite the existence of a few token outliers. In other words, students are trained by the use of made-up data that “right” means “strong correlation”. So when they encounter data that are very much not correlated, the scatter plot “looks wrong” rather than “looks like there’s not much correlation”. Students are somehow trained to place value judgements on scatter plots, with strong correlation = good and weak correlation = bad. I’m not sure where that perception comes from, but I bet if we gave students real data to work with, it would never take root.

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## The suckage of being an engineering student

A blog post at Wired claims to give the Top 5 Reasons It Sucks to be an Engineering Student. Discussion is in the comments there and at this lively thread at Slashdot. The reasons given at the Wired blog are (in reverse order):

1. Awful textbooks
2. Professors are rarely encouraging
3. Dearth of quality counseling
4. Other disciplines have inflated grades
5. Every assignment feels the same

It sounds to me like the blogger at Wired is stereotyping, based on what goes on at large research universities. A student could avoid #2, #3, and maybe #5 just by doing a 3+2 program where the first three years are done at a liberal arts college (…shameless plug alert…).

As for the grade inflation, I admit there’s no solution to this short of doing the right thing and forcing real academic standards on some of the touchiest-feeliest portions of the liberal arts world. But I think that would lead to mass chaos, as the stability of many liberal arts college depends on having some department on campus to be the “good cop” which offers refuge to students who just aren’t that interested in getting good at something difficult. All I can offer is some sympathy, that math and science professors are often eviscerated on course evaluations by those very students, who are shocked — SHOCKED — that deadlines would be enforced, hard material would be on tests, and so forth.

So to all engineering students out there, keep on keepin’ on. It might suck a little in the short term, but when it’s over you get to run our entire society!

## What is a classical education approach to mathematics?

Following up on his three posts on classical education yesterday, Gene Veith weighs in on mathematics instruction:

I admit that classical education may be lagging in the math department. The new classical schools are doing little with the Quadrivium, the other four liberal arts (arithmetic, geometry, astronomy, and music). The Trivium, which is being implemented to great effect (grammar, logic, and rhetoric), has to do with mastering language and what you can do with it. The Quadrivium has to do with mathematics (yes, even in the way music was taught).

This, I think, is the new frontier for classical educators. Yes, there is Saxon math, but it seems traditional (which is better than the contemporary), rather than classical, as such.

Prof. Veith ends with a call for ideas about how mathematics instruction would look like in a classical education setting. I left this comment:

I think a “classical” approach to teaching math would, going along with the spirit of the other classical education posts yesterday, teach the hypostatic union of content and process — the facts and the methods, yes (and without cutesy gimmicks), but also the processes of logical deduction, analytic problem-solving heuristics, and argumentation. Geometry is a very good place to start and an essential to include in any such approach. But I’d also throw in more esoteric topics as number theory and discrete math (counting and graph theory) — in whatever dosage and level is age-appropriate.

At the university level, and maybe at the high school level for kids with a good basic arithmetic background, I’d love to be able to use the book “Essential College Mathematics” by Zwier and Nyhoff as a standard one-year course in mathematics (and in place of the usual year of calculus most such students take). It’s out of print, but the chapters are on sets; cardinal numbers; the integers; logic; axiomatic systems and the mathematical method; groups; rational numbers, real numbers, and fields; analytic geometry of the line and plane; and finally functions, derivatives, and applications. You have to see how the text is written to see why it does a good job with both content and process.

(I took out the mini-rant against the gosh-awful Saxon method.)

Any thoughts from the audience here?