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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 September 9

MOOC failures in remedial math

Filed under: Uncategorized — gasstationwithoutpumps @ 07:10
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By now, probably everyone interested in the MOOC debate has read one or more articles about San Jose State’s experiment with Udacity last Spring.  The best article I’ve found about why San Jose State terminated the pilot is from the San Jose Mercury News: MOOC mashup: San Jose State University — Udacity experiment with online-only courses fizzles. That article explains that the experiment was stopped for the same reason that clinical trials are often stopped—there was clear evidence that the new treatment was worse than the standard one:

In the Udacity Remedial/Developmental Math course there was a disappointing 29 percent pass rate compared to an 80 percent pass rate in the regular face-to-face SJSU course. Only 12 percent of non-SJSU students in the Udacity version of the course passed, including students from Oakland Military Institute, the college-prep charter school.

Likewise, in the online College Algebra course, only 44 percent of San Jose students achieved the required C pass rate compared to a 74 percent C pass rate in the face-to-face version. Here again, only 12 percent of non-SJSU students in the online version achieved a C.

Finally, in the statistics class, which Udacity expected to produce far superior results, only 51 percent of students achieved a C pass rate, in stark contrast to the 74 percent C pass rate students accomplished in the face-to-face version of the same course.

I’ve been planning to post this for almost 2 months now, but I didn’t have much to add to the Mercury News article.  Today I was reading another blog post on Daniel Collins’s Angry Math blog, Reasons Remedial is Rough, which explains some of the reasons why the failure rate in remedial math classes is so high. Unless MOOCs can address some of these problems, they will not be able to reduce the failure rate.

  1. Lack of math skills from high school. Many students simply don’t have the requisite skills from high school, or really junior high school (algebra), or in many cases even elementary school (times tables, long division, estimations, converting decimals to percent, etc.). It’s hard to make up many years of deficit in a single semester.
  2. Lack of language skills from high school. What’s dawned on me in the last year or so, in the context of applied word problems, is that many students may actually be worse at English than they are at the basic math. Grammar is not taught any more, so students can’t parse a sentence in detail, can’t identify the noun or verb in a sentence, and so forth. This cripples learning the structure of any new language, algebra included.
  3. Lack of logic skills from high school. Basically, no one is taught basic logic anymore, so students can’t parse If/Then, And, Or, Not statements, which form critical parts of our mathematical presentations and procedures.
  4. Lack of study skills or discipline. Almost none of my students do any of the expected homework from our textbook. (On the one hand, I don’t collect or award points for homework, so you might say this is unsurprising; but my judgement is that the amount of practice students need to do greatly exceeds the amount of time I could possibly have to mark or assess it.)
  5. Lack of time to study. Certainly most of our community college students are holding full-time jobs, or caring for children, or supporting parents or other family members. The financial aid system actually requires a full-time course load for benefits; combine that with a full-time job—really, the equivalent of two 40-hour jobs at once—and you get a very, very challenging situation. (Side note: In our lowest-level arithmetic classes, I find that work hours are positively correlated with success, but not so in algebra or other classes.)
  6. Learning disabilities like dyslexia and dyscalculia. All I can do is speculate as to what proportion of remedial students would exhibit such problems if we could institute comprehensive screening. But I suspect it’s quite high. When students are routinely mixing or dropping written symbols, then disaster results. Unlike other languages, concise math syntax has no redundancies to enable the “you know what I meant” safety net.
  7. Emotional problems or contempt for the class. I put this last, because it’s probably the least common item in my list—but common enough that it usually shows up in one or two students in any remedial classroom; and a single such student can irrevocably damage the learning environment for the whole class. Some students who actually know some algebra start the course thinking that it’s beneath them, and become regularly combative over anything I ask them to do, sabotaging their own learning and that of others. It’s pretty self-destructive, and the pass rate for “know-it-all” students like these seems to be about 50/50.

Dan does give the caveat that these are personal observations from teaching remedial math, not large studies by a sociologist.

Of these problems, MOOCs only address the last one (disruptive students), since there is no classroom for them to disrupt—though MOOCs that rely on forums can be easily have the environment destroyed by a couple of trolls, so they don’t even solve this problem.

I think that the key observation is that “It’s hard to make up many years of deficit in a single semester. “ The whole premise of remedial math education is flawed—we are spending enormous amounts on trying to rescue students damaged by poor prior education, with a very low success rate.  It would be far better (and probably cheaper) not to damage the students in the first place.

Of course, it is easy to say “first do no harm”, but that is hard to achieve in practice.  It is very rare for a teacher or school to deliberately harm a student, but many are coming out of the system harmed anyway.

The purist ideological positions that people staked out during the math wars are probably contributing to the harm—students do not benefit from pure drill-and-kill nor from discover-all-of-math-by-yourself (to use the pejorative descriptions of the two extreme positions).

Math teachers need to use enough drill in the early years that students can have “automaticity”—a terrible eduspeak word that means that they are fluent with basic arithmetic and don’t have to think about it.  This is point 1 above—students need to develop the prerequisite skills to the point where they don’t need to think about them any more, and can work on the higher level thinking skills.

But math teachers need to teach more than just skill drills.  Students who can do routine algorithms when told to can still struggle mightily when trying to figure out which algorithm to apply. And it isn’t just the math teachers responsible for teaching this. As described in both points 2 and 3 above: students need to be able to read and comprehend fairly sophisticated English in order to do algebra.  The discarding of grammar from most English instruction in the US means that many students have only vague ideas about how sentences are constructed and interpreted, and so can’t translate what they read into the precise concepts needed for doing math.

Points 4 and 5, the lack of study skills and the lack of time, go hand in hand.  Students with little time and no idea how to use that little time efficiently are going to struggle to learn anything unfamiliar.  Having a regularly scheduled study group of students of roughly the same ability can help with time management—it is easier to block out time for a regularly scheduled meeting than to handle multiple priorities and make sure that studying comes to the top of the list sufficiently often.  The social pressure from a group of fellow students who are seriously trying to learn can also make a big difference in how much time is spent studying and how effective that studying time is.  This may be one of the biggest advantages that in-person classes have over MOOCs.

Learning disabilities and attitude problems are always going to be difficult to handle, and may require one-on-one attention, not just remedial classes.

In summary, I think that elementary schools, middle schools, and high schools need to look carefully at their programs—especially if a large fraction of their graduates are requiring remedial math in college.  Do the elementary students and middle school students have sufficient facility with arithmetic and fractions to be able to use those skills freely when taking secondary school math?  Do they have enough grammar to be able to pick apart a sentence and translate it precisely into mathematical terms?  Have they practiced doing so?  Secondary schools need to be sure that they are remediating the flaws of their feeder schools, and not just kicking the can down the road to the colleges.

The sooner problems are fixed, the smaller the fixes needed.  MOOCs do not seem to be a rational attempt to fix the real problems.

2012 July 1

Out In Left Field: Two ways to ensure learning

Filed under: Uncategorized — gasstationwithoutpumps @ 22:48
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Katherine Beals, in her Out in Left Field blog, had a post about evaluating whether a student has actually mastered the skill they are supposed to be learning: Two ways to ensure learning.

She starts with the observation that with multiple-choice questions (the favorite format for online and clicker questions) it is easy to be misled by students choosing the right answer with only very limited understanding. I believe that most teachers are aware of this problem, though few know how difficult it is to make really good multiple-choice question that has distractors for all the more common misunderstandings.

She goes on, though to point out that the current math teacher “obsession with having them explain their answers to math problems” is not the right solution to the problem.  To see whether students really understand the math, one should “construct math problems whose solutions are unlikely to be found by any means other than by using the skill in question.”  Not only are such problems easier to check for correctness, they  also do not discriminate against kids who can do the math but struggle with English writing.

She also points out that showing your work requires “problems complicated enough that there’s actual work to show; work that, if written out systematically, helps the students at least as much as the teacher.”

I’ve noticed that my son, who used to strongly resist writing out his work in math classes, now does quite competent, clear write-up for his Art of Problem Solving classes (both the precalculus and the calculus classes).  I think that there were at least three factors leading to this change:

  • The AoPS problems were difficult enough that you couldn’t just write down the answer from solving the problem in your head on on a calculator.  Multi-step reasoning is needed for most AoPS problems.
  • The teacher for his AoPS classes gave detailed feedback on the problem set—not just right or wrong, or a simple “show your work”, but feedback on the writing style and on the math shown (like what assumptions were made but not written down and why the lack of the assumption would make the math wrong).
  • The AoPS homework forum also gave students a place to present their work to people (not just the teacher) who were interested in understanding the solutions.  Writing clearly was rewarded by fellow students expressing their appreciation of a clear solution. Students also saw examples of both good and bad style and learned to recognize the difference in their own writing.

Note that the AoPS math writing is very different from the multi-page papers that I saw from the Math Academy at my son’s former school (he was not in the Math Academy).  Those papers were rather light on math and heavy on descriptive writing—good writing exercises, but not much math content.  The AoPS homework write-ups were very intense mathematics, and the writing was judged by how clearly it expressed the math to a mathematician.

2012 May 26

Free calculus book

Filed under: home school — gasstationwithoutpumps @ 22:02
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There is a free calculus book (by Gilbert Strang) available on line though the MIT OpenCourseWare.  The book covers all of single-variable calculus and most of multi-variable calculus, with over three times as many pages for the single-variable stuff as for the multi-variable stuff.

My son just finished single-variable calculus a month ago through the Art of Problem-Solving online course and took the AP Calculus BC class, and so I’m thinking about what math he should do next year.  Doing the MV calculus from a book like this would be one reasonable choice.

Although multi-variable calculus is the most common course to follow single-variable calculus, I don’t think I’ll push him into it next year (though he could take it if he chooses to).  I think that it would be more valuable for him to take Applied Discrete Math (combinatorics, proof by induction, Boolean algebra), probability, and statistical inference courses before going further in calculus.

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