There is still a difference between something and nothing, but it is purely geometrical and there is nothing behind the geometry.
- Martin Gardner
I may be teaching intermediate microeconomics again in the not too distant future, this time to adult learners in some professional program offered here, quite possibly in a totally online or blended format. So I’ve been thinking a bit about how I might do that effectively. Intermediate micro requires math. In my view the appropriate content soft pedals calculus ideas in favor of making the basic economics points in a discrete choice setting. But graphs, the kind one did in analytic geometry in high school, are appropriate, even essential. There is much economic intuition in the construction and proper interpretation of these graphs. Indeed, some instructors who teach this stuff think of this course as essentially applied math. My view is otherwise, there is also much economic intuition in considering real world issues that demand an economic analysis and in applying an appropriate economic framework to that real context. So I favor a balanced approach where one does both, some math but also some interpretation of real situations.
When I was teaching this course to 18-22 year old undergraduates, the course was taught by a wide variety of instructors. It had a poor reputation among many of the students, the majority of whom were majors in the College of Business. Their perception was both that it was difficult and that it wasn’t particularly relevant. The former, I believe, was because of the math, although all the students had taken Calculus (or placed out of it) ahead of time. The latter because they didn’t perceive the economic ideas to be foundational for what they might learn in their other Business courses. This may be an example of the broader problem that students fail to see the forest for the trees with respect to their studies and hence typically adopt an extreme instrumental approach to what they learn. Ironically, the economics metaphor can be extremely helpful for students in enhancing their soft skills, since as I’ve argued elsewhere it is much easier to communicate ideas if those are cast within a simple framework and the economics metaphor is precisely that. The math emphasis, with its seeming focus on technical detail, (the instructor intent is to supply needed rigor to the analysis) obscures the simplicity and elegance of the metaphor. So, unfortunately, mostly the students don’t get the economics to which they are exposed.
Whether I could now do a better job with this undergraduate audience will remain an open question for me. Given what I’ve gained from the learning technology job and some successful experiments I tried when teaching Principles to Campus Honors students, I have more experience with which to address the issue how to tap into the basic motivation of students taking the Intermediate course. For now my focus will be on adult learners. While there still are questions about providing motivation with that audience, I believe those students will not be quite as impatient nor will they be prone to cheat or to take their preparation lightly. And these students will almost certainly have a richer set of real word experiences that we can tap into to add flavor to what we will study. So I hope to avoid some of the pitfall I experienced teaching Intermediate in the past. Sometimes I felt there was a kind of Gresham’s Law at work. So, for example, I got into designing assessments with random numbers in large part to discourage cheating – each student would get an individualized version of the questions. But those were harder to write and so effort put forth in that dimension meant less effort to design assessments that would be broadly educative.
As I am writing this post, there was a very favorable piece in this afternoon’s Chronicle Update about the UTeach Program at the University of Texas-Austin College of Education, which shows that others have been thinking hard about related questions – are the skills for doing effective teaching of this sort transferable and what mechanisms promote that transfer? Their Publications Page appears to have many interesting resources to aid in thinking about these questions. I will work through some of those as I go through designing the intermediate microeconomics site after the New Year. But I’ve got a more immediate reason for writing now.
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It’s now Thursday and my kids are on Holiday till after the New Years. Last week my younger son, an eighth grader and now somewhat of a history buff, bombed on an algebra exam. Many of his classmates did likewise, so the instructor sent home the graded exam as a practice test for a makeup that was given yesterday. I spent several evenings trying to coach my son into better understanding the math. It was an anxious time for both of us. I’m going to try to take some lessons from that experience and then circle back on the subject of this post.
Both my kids, like their entire generation, have been heavily into fantasy stories - Star Wars, Lord of the Rings, Harry Potter, Halo, etc. They have access to these stories in multiple media – DVD, Xbox, computer games, etc. The joy from these narratives led to other computer games that are historic fiction, not pure fantasy – Age of Empires, Civilization. The younger one, especially, learned a huge amount of historic detail as he encountered the narrative and challenges of playing these games. This is the benefit of immersive gaming environments that so many others have talked about. Interestingly, at least to me, the games have created a taste for historic narrative itself, whether that comes through a game or otherwise. The younger one is now an avid fan of the History Channel. It feeds on the learning need that the computer games created. And there are beginning to be signs that books will do likewise. Indeed, while the books are obviously slower to get through, they have that much more detail and a more intricate story. In other words, what was set in place as playful diversion with computer games has been transformed into learning via intrinsic motivation through a habit developed outside of school. After inwardly moaning that the games were a complete waste of time for several years, I’m very pleased with this more recent development.
Once one has what seems to be a good algorithm for learning one wants to try to learn everything that same way. Further, if the learning seems effortless and happens en passant, the learner develops an expectation that can continue to happen regardless of context. It makes it hard to admit that a subject is difficult and even harder when there aren’t coping skills in place to manage the learning more effectively. Math, particularly the algebra my son is being taught now, is quite different from history. There is no narrative into which to weave the various facts and procedures. There is only abstraction and the seeming arbitrariness of the various rules. These eight grade kids have experienced what appear to be arbitrary rules before (“i” before “e” except after “c” or when sounding like “a” as in weigh or neighbor). So there is a tendency to want to learn the math the same way they learned to spell – through rote. But that is wrongheaded. The problem is there is nothing else to ready them for a better approach. (My kids did some computer programs in the very early grades to learn spelling and arithmetic (perhaps those were in the Reader Rabbit series), but those apps were not nearly as compelling as the games they played later.)
The fundamental value of the math – something that will serve the kids big time later in life if they get it during Middle School and High School – is that there are multiple ways to represent the same idea. A big part of what we call critical thinking is to find a convenient representation, reason through it, and then translate the results back to some other representation that motivated the inquiry at first. But to an eighth grader who has yet to see the value of multiple abstract representations for his own thinking, the math just seems like an obstacle, one that might limit his GPA and thereby thwart some ambition for yet unknown down-the-road achievements. That induces stress qua performance anxiety that exacerbates the problem.
There is a further complication, one that I’ve witnessed repeatedly in my intermediate microeconomics teaching, that the math currently being taught rests on a foundation, one where the student’s understanding is shaky and vague. For my son, the critical conceptual base is arithmetic with negative numbers. If done slowly and patiently, he can reason through it well enough. But he has nowhere near the fluidity with such arithmetic as compared to his mastery of the historical facts about battles in the Civil War. This lack of comfort with the basics contributes to missing the forest for the trees and feeds into the desire to want to learn the math by rote.
Also, it appears nobody has coached him previously on the recording and record keeping of his thinking through the algebra. He writes an equation down and then immediately manipulates that. It doesn’t occur to him to write the equation again and manipulate the reproduction, keeping the original pristine. This matters not if the thinking is correct all the way through. But it matters a great deal if errors are made and one has to back track to find the error. If the steps are rendered distinctly, then each can be checked for correctness. I believe that learning how to spot errors and correct them is an additional crucial skill, one that doing math well encourages. Even very bright folks who can keep a lot of information in their heads can benefit on occasion from clear record keeping of the intermediate steps. Doing that lessens the cognitive load and allows for quicker recognition of the source of the error. It is a must for a student who is struggling with the concepts.
Now I want to switch gears slightly and recall a presentation that Julie Evans made at the annual ELI Conference last January, where she reported on the technology habits and desires of then current High School students. My recollection is that she found: (a) the students were alienated from the curriculum in general, (b) the students would like to see more technology used in their classes because it would better match their informal learning outside of school, and (c) Math, in particular, was the subject matter they thought would benefit the most from being taught with computers. I had the thought of Julie’s presentation in the back of my mind as I was trying to help my son.
So, as is my passion even though I knew timing-wise this effort would not help my son in this particular instance, I made an interactive spreadsheet that is a tutorial on determining the equation of a line given the coordinates of two points on the line. It is by no means perfect; for example, it can’t deal with vertical lines, some of the formatting is only so-so (particularly the mixed fraction for the Y intercept), and in the method it uses where items are hidden till they are needed, in a few cases that is done imperfectly.
But there are certain parts of it that appeal to me nonetheless and I want to emphasize those here. For each coordinate of the points, students can choose integer values from -20 to +20. This gives them an ample set to practice with so they can try this until they feel comfortable that they understand. There is an attempt to make the various stages at the right coincide with the plotting of some component in the graph on the left. This is meant to capture the dual representations of the same idea. To compute the run and the rise, there is enough information so students can actually count the answer. I believe that counting is more basic than arithmetic and allowing the students to count should help them feel comfortable about the calculations they make. Further, the data entry is by pushing a button rather than by typing in a number. They can hold down the button and watch the graph instead of the button, to see the consequence of their action. This gives them a qualitative sense of what is going on. Indeed the calculation of the Y intercept is deliberately done to encourage them to eyeball the graph. If they want to for the sake of their own understanding, they can in addition compute the Y intercept on a piece of paper to verify that they get the same answer either way.
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Of course, since I’m writing this post quite soon after I completed the spreadsheet I’ve not tested this content on students and this being the holiday I doubt very much I’ll get my son to try it. So what I say next is pure conjecture. It doesn’t have use data to back it up. Nonetheless, I believe this is the type of content that the students Julie Evans surveyed would regard as useful and engaging for learning math. And what I want to argue here is that we should be authoring more of this type of content (and then get the students themselves to author even more of it both to aid their own learning and to help others). Unfortunately, we don’t see much content of this sort. What we do see is flatter, less interactive content, a lot of multiple choice questions, stuff that may be ok to assess the student’s understanding, but is much less useful in helping the students gain that understanding. This puts the technology in the role of the stick. We need more carrots, but we’re not getting them. Here’s my argument for what this the case.
I put in somewhere between 12 and 20 hours to author this tutorial. And for a similar tutorial on Long Division I made a couple of years ago, I spent the better part of a weekend in the construction. This is not because I was ignorant of the math or of the design issues. It is simply because there is a need to conceptualize in the construction – the layout and sequencing of the presentation matter as does the math and one has to see those pieces interplay well. I did this free form starting from a blank Workbook in Excel. I don’t believe there are too many people with the wherewithal to make content of this sort in this manner. They might be many others with an interest to author stuff of this sort, but they would need to rely on templates so they can focus on the content. Unfortunately, to my knowledge there are few if any templates of this sort for content creation. (The software our Physics department uses for student homework has some of this type of capability but it requires authors to submit Perl scripts. That works in Physics but is too much of an entry barrier to be more broadly applicable.)
We are now more than 12 years after I got started with learning technology and back in 1995 there was the promise to deliver on what I’m talking about right here on my campus with CyberProf and Mallard; the CAPA system from Michigan State was another good possibility. The more recent commercial course management systems and even most of the newer open source alternatives have delivered less in this dimension than might have been hoped. Part of that is because they’ve not yet integrated current technologies (Ajax) into the assessment tools, so you can’t yet do what I’ve got in my Excel tutorials – students do data entry that shows up in the graphs, which are rendered dynamically.
But another part of this is conceptually and I believe even the early assessment engines made this mistake, a mistake which parallels a conceptual error we’ve seen in the Banner student system, where the basic element is the section. That approach goes against the culture here where the basic unit is really the course – we have many multi-section courses as well as courses taught in a single section but that are cross listed.. We spent a good part of the first year after Banner was implemented here reconciling our ordinary business processes concerning instruction with the peccadilloes of how Banner manages these things. (For example, the professor may have had access only to the lecture section, not to the discussion section, but it was the latter where the grades were to be entered.) The same sort of issues occurs with how the LMS deals with assessments.
There the basic element is the question, with question one essentially unrelated to question two except in the order in which they are presented to the students. In my tutorial each stage of the process coincides approximately to a question. That all the stages can be rendered in one unified view (I designed the tutorial with that goal in mind) is a huge benefit compared to where one must scroll to see subsequent or prior questions. Most of the learning management systems don’t even allow a numerical parameter to be passed between questions. Instead, with each subsequent question the parameters are set anew. And, most definitely, students can’t set the parameters themselves so they can practice. The notion of practice is in the number of attempts that are allowed, but not otherwise found in the quiz tool. Further most of these assessment engines don’t allow the provision of instant feedback in quiz mode (meaning the assessment is for credit) but rather only allow it for self-assessment. Instructors are apt, however, to design quizzes because they know students will do those to receive course credit. (Although it is also true that many students “do” the quizzes in a manner aimed at circumventing the intent. For example, with multiple choice that can be submitted many times, they simply try the various letter alternatives till the get the thing correct.)
All of this is unfortunate. The question is what to do about it. One possibility is to extend the argument I made about 18 months ago and add this to the wish list for what we need in an LMS. But now I’m less inclined to think that is the best answer. My sense is that we need to bring the discussion of what is educative content outside of any container that might deliver it. And I’d like to engage all my friends and colleagues who embrace a Constructivist approach toward learning to reflect and comment on this question, particularly as it pertains to the teaching of math content, especially when that math is used in a College level setting, such as the Intermediate Micro course that is my focus. To aid in that, I will close with a critique of my own approach.
In my view, the difference between what is construction “from scratch” and the reproduction of “spoon fed knowledge” has to be considered relative to the current understanding of the learner. In my tutorial the sequencing of the steps in built in, predetermined for the student’s point of view. In some cases there may be many different possible sequences that will produce the same result and in that case there is the possibility that such a tutorial creates the impression of a single correct approach. That would be unfortunate. In other cases the sequencing might matter more but it might be an important part of the construction for the learner to come up with the sequencing on his own. Then having it provided, as I’ve done in the tutorial, amounts to spoon feeding.
My retort is that tutorials of this sort represent early steps toward allowing the student to deal with complexity in the math. Students need to get past the approach in the tutorial onto harder stuff, but some students won’t be able to get there in one fell swoop. In any event, I hope we can separate out the question of what works for the students from the issue of whether we can generate this content given the limited free time of those who are likely to make it. For my part, I’ll try to make the more of this sort of stuff, targeted at microeconomics instead of eighth grade algebra and see how it works with my own students.