Gas station without pumps

2013 October 8

Duct tape sandals

Filed under: Uncategorized — gasstationwithoutpumps @ 17:26
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For relaxation, I often read “There I Fixed It“, a blog of funny bad repairs, often featuring duct tape.

My wife recently added our own contribution to the genre.  Her favorite sandals had a strap fail (disconnecting from the sole).  She wanted to replace the sandals, but the style she likes is no longer made.  Rather than switch to some less comfortable style, she took some red duct tape and reattached the strap:

duct-taped-sandalsAside: I’ve always preferred the term “duct tape” to “duck tape”, believing that the latter was a vulgar corruption of the former.  I heard from several sources that “duck tape” is actually the older term, and checked with Google ngrams:

Sure enough, around WWII "duck tape" was the more common usage, but the term "duct tape" became that standard term around 1970 and now clearly dominates.

Sure enough, around WWII when this style of adhesive taped first became available, “duck tape” was the more common usage, but the term “duct tape” became the standard term around 1973 and now clearly dominates.

2013 October 7

Thinking about decimals

Filed under: Uncategorized — gasstationwithoutpumps @ 10:14
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In Money and decimals [TDI 2] | Overthinking my teaching, Christopher Danielson brings up some questions about whether money is really a good example for teaching about decimals.  Unfortunately, he has closed comments on the blog post, preferring to have tiny snippet discussions on Twitter or having people join some course website on, neither of which appeals to me.  So I’ll discuss his points on my own blog instead.

The most interesting question he raises is the following:

 Is it possible, as Max Ray suggests below, that the conception people tend to carry in their minds is of dollars and cents as separate units, as they do feet and inches?

@Trianglemancsd though I am curious what students use to make sense of cents prior to fractions. A dollars unit and a pennies unit, right?

— Max Ray (@maxmathforum) September 27, 2013

I report my height as 6 feet 1 inch. I do not report it as 6\frac{1}{12} feet, although I know that I could. Likewise I don’t think of 1 hour and 5 minutes as 1\frac{5}{60} hours, although I know this to be correct.

Is it possible that many people think of $1.25 as 1 dollar and 25 cents, rather than as 1\frac{25}{100} dollars?

Maybe students are thinking of dollars and cents as different units that have a nice conversion rate, rather than of dollars as the natural unit and cents as a partitioning of that unit.

Follow-up questions: (a) Might Max’s insight help to explain the errors in the gallery of misplaced decimals? (b) What are the implications of this for using money to teach decimals?

I think it is certainly the case that most people think of money in terms of dollars and cents, not in terms of dollars and fractions of a dollar, and many of the notational mistakes one sees in ads supports this interpretation.

The Mad Hatter's hat was priced at "ten and six"—10 shillings and 6 pence, which was half a guinea.  Incidentally, I've never really understood why there was a unit for 21 shillings, when 20 shillings was a pound.

The Mad Hatter’s hat was priced at “ten and six”—10 shillings and 6 pence, which was half a guinea. Incidentally, I’ve never really understood why there was a unit for 21 shillings, when 20 shillings was a pound. (According to the Wikipedia article about guineas, it started out as a one-pound coin, but fluctuated in value according to the relative prices of gold and silver, before getting fixed at 21 shillings in 1717.)

It used to be that many monetary systems were not based on a decimal system—I’m old enough to remember the former British system of pounds, shillings, and pence (20 shillings to the pound, 12 pence to the shilling) with prices given with a slash separating shillings and pence.

The colloquial use of “and” to separate shillings and pence or dollars and cents is also indicative of the general thought pattern that these are separate units, neither composed nor divided in the way that Christopher likes to think of units. People don’t think in terms of dollars and fractional dollars, nor in terms of all prices in cents, but in terms of mixed unit of dollars and cents. (This may also help explain why 99¢ seems much cheaper to people than $1, while 99 and 100 are close—though the tendency for people to truncate rather than rounding is probably a bigger part of the story.)

Because people think of dollars and cents as separate units, using money as a metaphor for explaining decimal notation is not as useful as it might be. With separate units, the conceptual process is one of unit conversion and scaling, not of place value. Unfortunately, more useful models (like using cm, mm, and meters; grams and kg; mL and liters) are not open to US educators, because of the archaic units (inches, feet, and miles; tons, pounds, and ounces; pints, cups, tablespoons, and teaspoons) still used here.

A lot of teachers claim that decimals are hard—harder than fractions—but I’ve never understood that claim. Place value always seemed easy and natural to me, so having more places to the right seemed like a trivial extension.  None of the arithmetic algorithms changed (other than having to keep track of where the decimal point ended up), so decimals always seemed like a simpler extension to whole numbers than fractions were.

Of course, fractions are a more powerful and more fundamental concept than decimals—not all rational numbers can be expressed as simple decimals, and the notational conventions for repeating decimals are not obvious nor easily explained. So the rather complicated relationship between fractions and decimal notation is understandably difficult to teach.  But I would have thought that the problem comes more from the fractions than from the decimals.

2013 October 6

MathTwitterBlogosphere, mission 1

Sam Shah and other math bloggers have started a challenge to encourage more math-teacher blogging Mission #1: The Power of The Blog | Exploring the MathTwitterBlogosphere:

You are going to write a blog post on one of the following two prompts:

  • What is one of your favorite open-ended/rich problems? How do you use it in your classroom? (If you have a problem you have been wanting to try, but haven’t had the courage or opportunity to try it out yet, write about how you would or will use the problem in your classroom.)
  • What is one thing that happens in your classroom that makes it distinctly yours? It can be something you do that is unique in your school… It can be something more amorphous… However you want to interpret the question! Whatever!

I’m not a math teacher blogger—looking back over my posts for the past couple of years, I only see a few that are really about math education:

I use math all the time in my classes (complex numbers, trigonometry, and calculus in the Applied Circuits class; probability and Bayesian statistics in the bioinformatics classes), and I do reteach the math the students need, as I find that few students have retained working knowledge of the math that they need.  But it has been quite a while since I taught a class in which math education was the primary goal (Applied Discrete Math, in winter 1998).

So I fell a little like an imposter participating in this blogging exercise with the math teacher bloggers.

I don’t have any “favorite” open-ended or rich problems.  Most of the problems that I given in my classes have a heavy engineering design component, in either the circuits course or the bioinformatics courses.  Any good engineering design problem is an open-ended, rich problem.  If I had to pick a favorite right now, it would be from my circuits class: either the EKG lab (look for many posts about the design of that lab in the Circuits Course Table of Contents) or the class-D power amplifier (see Class-D power amp lab went smoothly and other posts).  But these are not the sort of “open-ended” problems that the MathTwitterBlogosphere seem to be interested in—the engineering design constraints that make the problems interesting are too restrictive for them, and a lot of them prefer videos to text (for reasons that seem to me to be based mainly on assumptions of the functional illiteracy of their students, though a few times a sounder justification is given). In any event, I doubt that any of the problems that I give to students would be appealing to math teachers, so they are not really germane to the MathTwitterBlogosphere challenge that Sam Shah put out.

It is hard to say what I do as a teacher that is “unique”. It is not a goal for me to be a unique teacher—I’d like to see more teachers doing some of the things I do, like reading student work closely and providing detailed feedback, or designing engineering courses around doing engineering design.

I may be unique in the School of Engineering in how much emphasis I put on students writing well, and how much effort I put into trying to get them to do so.  I created a tech writing course for the computer engineers and scientists back in 1987 and taught it until 2000.  More recently, I have provided many bioengineering students feedback on their senior theses, reading and giving detailed feedback on five drafts from each student in 10 weeks.   In my bioinformatics classes, I read the students’ programs very closely, commenting on programming style and the details of the in-program documentation—these things matter, but students get very little feedback on them in other classes. In the circuits course, I require detailed design reports for each of the 10 weekly assignments (though I encourage students to work in pairs for the labs and reports).  I evaluate the students almost as much on their writing as on their designs—engineers who can’t write up their design decisions clearly is pretty useless in the real world.

I’ve not done much about math writing, though a good class on mathematical writing (using Halmos’s How to Write Mathematics) would be a great thing for the university to teach. I have blogged before about writing in math classes, in my post Out In Left Field: Two ways to ensure learning, which is a response to a post by Katherine Beals: Two ways to ensure learning.  In my post, I distinguished between writing mathematics and the sort of mushy writing about mathematics that many high school teachers favor these days.

Centering engineering courses on doing engineering design is a very important thing, but it is not a unique contribution—I’m not the only professor in the School of Engineering who puts the lab experience at the center of a course design. Gabriel Elkaim’s Mechatronics course is a good example, as are most (all?) of the lab courses that Steve Petersen teaches.  In think that, in general, the Computer Engineering department does a good job of highlighting design in their courses, as does the Game Design major.  I just wish that more of the engineering classes did—especially those where it is much easier just to teach the underlying science and hope that students pick up the engineering later.

At the end of this post, I’m feeling the lack of a good conclusion—I don’t have any open-ended problems to share with math teachers, and I don’t have anything really unique about my teaching that will make math teachers want to emulate me.  I just hope that even a weak contribution to “Mission 1” is useful, if only to make other participants feel better about their contributions.


2013 October 5

Balancing fun and fundamentals

In Computing is a Liberal Art » Automatons and Entertainers, Keith O’Hara writes

One oft-cited axiom in the MOOC debate is that Math and CS are easier to MOOC-itize than other fields. This is one fact that crosses the intellectual aisle. MOOC-leaning scientists and anti-MOOC humanists both take for granted that the teaching of math and programming should be the first to be automated. As you might have guessed, I whole-heartedly disagree. If you can’t automate the teaching of writing you can’t automate the teaching of math. You can automate multiplication drills, but you can also automate spelling drills. Humanists don’t consider spelling writing, Mathematicians don’t consider multiplying math. And for the record, computer scientists don’t consider programming language syntax computer science. Spelling is necessary to write. Multiplication is necessary to do math. Writing grammatically correct programs is necessary to study algorithms. But those aren’t the interesting parts of those disciplines, for exactly that reason, they can be automated. Academics are concerned with new knowledge, and that necessarily lives on the boundary of what is known and what is unknown. And if we can automate something, we know it very well.

I agree with him whole-heartedly.  The Java syntax courses that pass for first programming classes are a lot like spelling, grammar, or arithmetic drills.  They are essential skills, but boring as hell. Details matter, but details are not the whole picture.

One approach that gets used a lot in K–12 education is to drop the drills and concentrate on the “fun” parts.  This has dominated English teaching at the elementary and secondary school levels for a while now, so that many students entering college cannot spell and have only the vaguest notions of what a grammatical sentence is.  They’ve also only written self-reflections and literary analysis—styles of writing that have little existence outside English classrooms. Some math curricula have gone the same way, and students are entering college unable to multiply or add fractions and with only vague ideas about algebra, trigonometry, or complex numbers.  I can’t support a system in which fundamental concepts are ignored in this way.

Another approach is not to allow kids to do the fun stuff until they have mastered the fundamentals (the finish-your-spinach-or-no-dessert approach).  Unfortunately, the result is that many students never get to the fun stuff, and end up believing that huge swaths of knowledge are inaccessible and uninteresting.  Note: this approach dominates many engineering schools, which do a bottom-up approach teaching years of math, physics, and “fundamentals” before getting to engineering design, which is the heart of the field—the result is often a very high attrition rate and “engineers” produced who can’t actually do any engineering.

I think that a balanced approach, that mixes fun stuff in from the beginning but continues to teach the boring details, is essential to effective teaching in any field.

I am trying to create such courses for the bioengineering majors at UCSC: the applied circuits course, for example, and a new freshman design seminar. The bioengineering major at UCSC probably has the least engineering design of any of the majors in the School of Engineering (except for Technology and Information Management, which I don’t think belongs in the School of Engineering).  I want to fix that flaw in the curriculum, but it is hard to overcome the “you have to know all this before you can do anything” attitude of both students and faculty.  Fitting in all the prerequisite chains for math, physics, chemistry, biology, programming, and statistics makes it very difficult to schedule courses for freshman—students need to get prereqs done early enough that they can finish in 4 years.

I think that the computer engineering department at UCSC does a good job of mixing in engineering design down to at least the 2nd year courses (I’ve not looked at their freshman courses lately—they may be doing well there also).  I’m less impressed with what computer science has done, though the large sizes of their courses make good teaching and design content harder to incorporate.  Their game-design major does get into design much sooner than the standard CS course sequence, I believe. I’m decidedly unimpressed with EE, where there is no design content at all until the 3rd or 4th year, even where it could have been easily incorporated.

Note that engineering design is damn hard to MOOCify.  The essence of design is that the answers are not known in advance—there are many ways to achieve desired design goals.  There are also many different tradeoffs to make in setting the design goals.  Students not only have to come up with designs, but have to build and test them—the real world is very important in engineering, and simulation is rarely an adequate substitute. (Note, I’m not saying that engineering students should not use simulations. Learning how to use simulators properly and what their strengths and limitations are is an important part of engineering education—but simulation-only education is not sufficient.)

One problem I am facing in trying to improve the bioengineering curriculum is that most of our bioengineering students are in the biomolecular engineering track.  Molecular engineering is decidedly slower and needs more science background than most other fields of engineering (which is why it is mainly a graduate field elsewhere). It is particularly hard to provide freshman with design experience in molecular engineering.

UCSC has one honors course that attempts to provide this experience to freshmen, but the capacity in the course is only about 20 (shortage of wet-lab space and teaching resources), and probably only 3 or 4 bioengineering students qualify for the honors course—what do we do with the other 50–100 bioengineering freshmen?

The freshman design course I’ll be creating this winter will be able to handle maybe half of them, but it will be focusing on designing low-cost do-it-yourself instrumentation, not molecular engineering.  (I’m hoping that we can entice more students into the bioelectronics and rehabilitation tracks, and reduce the load on the biomolecular track.) The course is just 2 units, not 5, so that students can add it to a nominally full schedule, without delaying any of their required courses. That was all I thought I could get students to take with the current curriculum.

I’d like to have a required 5-unit course with substantial engineering design in the freshman year, and not just a 2-unit optional course, but I don’t currently see how to fit that in even with a revised curriculum—it would require reducing the chemistry requirements for the degree substantially, and the Chem department is unlikely to create a faster route to biochem.


2013 October 3

Future with less driving?

Filed under: Uncategorized — gasstationwithoutpumps @ 19:21
Tags: ,

It seems that we have hit the peak of the mania for driving in the US, and have started a gradual return to sanity.  I found a graph of annual per capita miles driven for the past 40 years at SSTI to Transport Officials: Start Planning for a Future With Less Driving:

Annual, per-capita vehicle miles traveled by Americans have been declining for eight years. Image: State Smart Transportation Campaign

Annual, per-capita vehicle miles traveled by Americans have been declining for eight years.
Image: State Smart Transportation Campaign
Caption: Angie Schmitt

Note: this doesn’t mean that total driving has started heading down yet, since population is still rising. Nor can I  defend 9,500 miles per capita, since that is still a huge amount of driving, but it seems that there is a definite sustained downward trend.  It may take another 40 years to get back to more sane levels of driving (which I put at around 2,000 miles per capita per year), but I see some hope for the future.

Thanks to Richard Masoner, whose post  on pointed me to the post.

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