The first official sign of Fall!
Today appears to be the first Student Moving Weekend.
The first hint was the sudden appearance of U-Haul trucks over the last few days, some of which might have been people clearing out at the end of July, but students were clearly moving into rental houses around the area today.
Traffic accidents and under age parties won't be far behind.
This is the second of three articles concerning calculators and mathematics triggered by a blog article by Dean Dad, a community college dean who appears to be writing from another part of the country yet has the same problems we have at our CC. I have already commented on the blogspot version of this blog (more than once), which has collected a huge number of comments, but there are also a large number of comments on the IHE version of the same posting. I'll write as if you have at least read Dean Dad's article. The discussion has been quite wide ranging, often not bothering to make a distinction between the various levels of "calculator" available to students or the many levels of math classes they might be used in. (Definitions are given here to provide a common reference.)
In this article, I will take up the specific issue raised in Dean Dad's blog and the one I know the least about as an instructor - calculators and developmental mathematics - but also look at college algebra. I'm mainly interested in putting some of my thoughts on paper and seeing feedback I get from others about their opinions of the problem.
I'll start with a particularly telling comment from Dean Dad:...part of me wonders if we’re sacrificing too much on the altar of pencil and paper. It’s great to be able to do addition in your head and long division on paper -- yes, I know, I’m old -- but is it worth flunking out huge cohorts of students because their high schools let them use calculators and we don’t?
This isn't about LETTING them use calculators. Part of it is about spending time teaching them how to use a specific brand of calculator rather than how to do algebra. However, as I wrote in my first comment on DD's blog (9:11 AM time stamp about halfway into the 50 or so comments that are there now), I think this is mostly correlation without causation. The real problem lies elsewhere.
I think the problem starts in K-5 and gets compounded by pushing kids along into the next class and lying about the content level of that class. That is, I don't believe for a minute that a student I advised had passed a REAL pre-calc class in HS just a week before I talked to her. DD writes about similar cases:...students who have passed algebra and even pre-calc in high school frequently crash and burn when they hit our developmental math, because the high schools let them use calculators and we don’t.
I don't buy it, and here is why: Our placement test will put them into Intermediate even if they can't do arithmetic, provided their algebra score is high enough. And you can't work with logarithms and exponentials or trig identities (a given if it is really pre-calc) if you can't solve a simple linear equation written symbolically like I asked the student to do.
You might forget 6 months of math in a week, the newest stuff, but not 3 years of it. And if you do forget that much that fast, you should have failed that pre-calc class. You can't get to much new material if you spend most of the year re-teaching three years of previously-taught material de novo.
K-5 Curriculum:
I don't want to belabor this, but one reason they can't do arithmetic might be that they never learned it. I'm convinced this is the biggest problem we face because it also lies behind the existence of pre-calc classes that are really teaching basic algebra. I'm sure part of it is that teachers who never understood math and hate it with a passion are teaching it by-the-book following a curriculum none of you could possibly imagine anyone would use.
My analogy is to the "look say" approach to reading, where guessing replaced phonetic decoding of words and Johnny (not to mention a cousin with a high IQ who is now an senior engineer at the VP level) couldn't read. Using guessing to construct your own mathematics might work with someone like me (I feel eternal guilt for, AFAICT, being an unwitting subject in a math ed research project that was run before the days of IRB and informed consent where they deduced that this curriculum worked really well), but it is unlikely to work with someone who was not going to get a PhD in physics. Really good algorithms were developed 12 centuries ago and survived for a reason. As is illustrated here, for the Everyday Math curriculum, the most efficient methods are not taught first or (in some cases) are not taught at all in some schools. The starting point for Dean Dad might be to get out into the feeder systems for his CC and find what they are doing in fourth and fifth grade.
If I know anything about learning, it is that students always favor the first method they get taught. (That is one reason you have to really emphasize when conservation of energy or momentum should be used instead of Newton's Laws: they learned F=ma first so it is the first thing they want to try. I'm the same way.) That means it is a really bad idea to start with an inefficient method, but which some people find useful when doing 'mental math', and teach the more efficient one last.
Now think about how to teach synthetic division or multiplication of polynomials to someone who only knows partial quotients division or the lattice method for multiplication. Not pretty. Then consider that they might never have even heard about "invert and multiply". Not pretty at all.
Outcomes:
The answer to Dean Dad's fundamental question, whether students should be allowed to use calculators in an Arithmetic class, starts at the beginning - the first step of course design. What are the desired outcomes for this math course? If the outcome is to be able to do a certain amount of arithmetic with pencil and paper (not in their heads), then the only use of a calculator is to check your own work as you make up your own problem and solve it. Ditto if the purpose is to simplify fractions involving simple whole numbers as preparation for a similar skill with symbols. You need to change the outcomes before changing what you do in the class.
Personally, I'd be happy if they taught them how to do arithmetic on their Basic calculator. Seriously! My biggest complaint when teaching physics isn't that they can't do arithmetic (they can't), it is that they can't calculate worth a damn. Digital natives my ass. I was 22 when I got my first calculator and I am faster than most of them are, and I'm slower than I used to be. (Sure, I've been using one longer than they have been alive, but that only proves they haven't used the thing enough to be competent with it.) I mean, I've watched a student work out a product by multiplying two numbers, writing it down times all of the others, entering it again !!! and multiplying it by the next, etc etc.
I know they use their calculators a lot in our Algebra (meaning College Algebra) classes, but it must all be with simple whole numbers like we used back when there were no calculators. That is the only possible explanation for their struggles with 3 and 4 digit decimal or scientific notation numbers or their mysterious belief in rounding intermediate answers.
Algebra:
There really shouldn't be many numbers in an algebra class, IMHO. Somehow the appearance of Graphing calculators changed the curriculum to emphasize numbers and, curiously, de-emphasize graphing. Since you can't actually read a graph on a TI display screen, let alone interpolate on it using a ruler, they don't appear to know how to make or read an actual graph rather than a cartoon of a graph. This is a nightmare in the physics lab, but also in the classroom when data are supplied in a graphical representation.
I don't believe anyone has tried teaching algebra with a Scientific calculator and graph paper in decades. I doubt if anyone other than the textbook and calculator companies have studied it, and studies like that are notorious for the difficulty in controlling the student mix or the instructor effect. However, stories about students who finally got algebra in a class where only symbols were used - no calculators needed - are common enough to make one wonder how it would work. The studies (see some comments toward the end) seem to lack a smoking gun in favor of the primitive Graphing calculators used today.
There is another side effect. Since they don't know how to use their calculators, particularly concerning order of operations, they use parentheses like they were the only operator known to man. ((3)(2))/((5)). One result is they don't see the key role of the parenthesis to denote "function of". I've seen calculus students who think x(t) means x*t, although this could be partly due to the fact that x is never a function in calculator-based Algebra. I have to wonder out loud if this would improve if they all used HP calculators instead of TI calculators. Also see my next comment below.
Commenting on the comments:
Some Anonymous, writing at 5:56 PM, about 3/4 of the way into the comments, writes:A kid that is getting good marks in algebra screws up their physics equations. i check with their math teacher, and they don't make those mistakes in math class. So I test them myself, and they can manage algebra just fine when x, y, and z are variables and a, b, c, and d are constants. Anything else and they're lost.
This could be a result of using graphing calculators. The TI-83 will only plot Y(X) unless it is in one of the other modes (where it is similarly limited). Parametric mode, the only place where you can do X(T), does not appear to be used at all until they get to Calc III. This really bugs a chemistry colleague, because they are always plotting the log of this versus the sqrt of that, neither of which is X or Y. Similarly, we start out in physics by plotting x on the Y axis and t on the X axis and it blows their minds. I'm tempted to start by doing only y(t) problems at the start of fall, then moving to x(t).
Mthgeek, aka timfc, writing at 7:45AM of the second day of comments, listed several references. The first of these wasThe Arithmetic Gap
I'd like to know what grade level this was, but it sounds like K-8 from the title. As for the second one that was listed, I don't ever pay attention to something like a meta analysis of 42 other papers that span middle school through calculus. Apples, oranges, confounding variables, design differences, and systematic errors make a tasty goulash but don't help with teaching Basic algebra. One other reference, discussing "computer assisted instruction" would appear to be irrelevant to this discussion. You can use computers as an instructional aid (instant HW feedback, for example) without using a graphing calculator - or any calculator at all.
Educational Leadership, v61 n5 p55 Feb 2004
Summary: The students using calculators in school classrooms result in lower math scores than students who never use them.
However, the last reference from Mthgeek is rather interesting. It is a link to an article (Refocusing Introductory College Mathematics Courses) that has a link to a new textbook (Contemporary College Algebra: Data, Functions, Modeling) that implements some of the ideas from the study. That is, the study and the textbook are closely coupled, but I recognize some things in that report that are reflected in what we do in our Intermediate class with a different book. (I'll have to ask around, but we might have made this choice because Intermediate is a pseudo-terminal course for many majors in our curriculum. The situation discussed in the article does not apply as much to our college Algebra course, because it normally leads to business calculus or trig. The statement in that report that biological sciences don't go beyond Algebra is patently false in our curriculum. They have to take Calculus even if they don't ever use it.) That said, I strongly criticize the textbook author for conflating a graph on a Graphing calculator with a graph produced on a computer. There is no comparison in detail or quality.
Some Anonymous, writing at 11:21AM on the first day, said (in part):1) middle school math is more focused on algebra as early as 7th grade. So students don't have enough mastery of fractions, percents etc
...
3) More students attending college- so the lack of good high school prep is more evident.
4) Content of dev math courses in college are aimed at preparing students for a precalc/calculus track. But those going into sociology or psychology ...
Developmental math at my CC is about preparing students for 9th grade math, not pre-calc. The Intermediate course barely prepares them for real Algebra, and certainly not for the calculus track. (Our failure rate is spectacular at every one of those steps.) Besides. students going into Psychology need a real, college-level statistics class that has college Algebra as a pre-req. Criminal Justice, on the other hand, has no real math requirements and our statistics show that the combination of our Developmental and Intermediate classes does a GREAT job of preparing them to pass the basic financial math class that constitutes their "college level" math requirement while teaching them about compound interest.
The fact that they have not learned arithmetic or fractions by the time they get to 7th grade (which is when we started Basic algebra when I was growing up) is the real problem. Three years should be enough if the curriculum and teachers were any good, but if they aren't or the kids don't learn it in 3 years, our schools track those kids away from Basic algebra for another year, or more. But this does help strengthen my point that the problem is really in the K-5 classroom.
I don't buy the "more students" argument because the fraction going to college has not changed that much in the last few decades.
Finally ...
If you have read this far, thank you. I want to close by saying that the problem really is deeply rooted in our educational system and very frustrating for all involved. The high failure rate in Developmental classes is a major problem that no one is ignoring at our CC.
However, many students fail because they never attend class, or don't attend frequently enough to engage with the instructor. With any instructor, no matter how talented ze might be. I've written about that in an old bit of wishful thinking about new student orientation. Coming straight out of HS, they believe they were taught pre-calculus or Algebra II, so they just don't believe they need to go to class and actually learn math. Older students, out of school for years or decades, know they don't remember anything from school so they take it seriously and often do quite well. An age-based breakdown of performance in Developmental classes might be worth looking at, Dean Dad.
Or Dean Dad might only need to walk by a classroom or three on a regular basis and take a sort of visual attendance. Is the room still full after 4 weeks? Maybe that, rather than calculators, is the real problem.
This cartoon from last week really captured my view of "modern" calculators. 
Click on the image to see the entire cartoon from XKCD, including the highly relevant punch line.
I was originally going to riff off of this cartoon to discuss "modern technology" in the classroom, but then Dean Dad's article came along. Just for perspective, the current model (TI-83 Plus) shows up priced between $89.99 (on sale at Staples for the new school year) and just under $100 (at Walmart and Amazon). For comparison, the CPI says $110 in 1996 will buy about $150 of normal goods today, but computer prices have been going down even as performance increases. For many decades.
Further, the cartoon is not exaggerating the connection to 1996. Today's TI-83 Plus is still running on a 6 MHz Zilog Z80 microprocessor, an 8-bit cpu that dates to the mid 1970s (as an upgrade to the legendary Intel 8080 chip). The Z80 was used in such memorable machines as the Kaypro II (running CP/M), the TRS-80, and the Sinclair and Timex notebook-sized computers. [The Kaypro, like the Osborne, was a "luggable" computer that would have to be sent in checked baggage today. I still remember using both of those.]
Not exactly MODERN technology, particularly when you consider the limitations of the 96x64 screen compared to, say, a (much smaller) iPhone. This has practical effects in that the calculator has great trouble graphing certain kinds of functions and the interface for "tracing" to a zero is really crude. More importantly, for whatever reason, I see no improvement in algebra skills associated with the month or more of time spent specifically on using this technology. Students do not use the graphs to check their answers, but that is a topic for my other postings on this topic.
Other observations:
The Plus indicates it has 512 kB of flash memory rather than 32 kB of RAM on the original model. The Silver Edition has a 15 MHz cpu and even more memory, but the added speed is one reason why you can clear its memory so much faster than on the Plus. AFAIK, the main difference is that you can clear uploaded programs on the 83 Plus but cannot clear the equivalent programs that are installed OEM on the Silver Edition. The main advantage for students is that you can connect any of these to a computer and download modestly sophisticated applications into Flash memory that are run with the Apps key.
Naive instructors believe that students have to laboriously type in crude crib sheets listing, say, trig identities or chemistry and physics formulas as fake programs. Many do this, but TI provides sophisticated, indexed crib cards - and similar tools are also available from others on the internets. Anyone who "limits" students to a note card of notes but allows a TI-83 without clearing it is laughably naive. Might as well let them bring in a notebook.
This is the first of three articles concerning calculators and mathematics triggered by a blogspot and IHE blog article by Dean Dad, a community college dean who appears to be writing from another part of the country yet has the same problems we have at our CC. The original article concerned calculator use in "developmental" math classes that typically cover fractions and 7th grade algebra. I have already commented on the blogspot version of this blog (more than once) and the two together have generated more than 80 comments. I added some more in my second article of this series.
I won't actually comment on this topic here. My purpose is solely to set the terms of the debate, as it were, because the wide-ranging discussions of this topic by Dean Dad and others are seldom clear about which of the four or more levels of "calculator" available to students are being discussed and/or which of the three or more levels of math classes (plus physics and chemistry) provides the context for the discussion.
The divisions I make are somewhat arbitrary and perhaps idiosyncratic, so I want to spell them out somewhere without cluttering up a discussion of the teaching and learning issues as I see them. That way I can link here for future discussions of this topic and not have to repeat myself.
Although I think three levels of "calculator" suffice for most classroom use, and hence for later discussion, I think I need to list at least five to make the definitions as sharp as possible.
The three in the middle, in all caps, are the ones I will refer to most often within the context of lower division classes taught at a community college.
For the record, I allow Scientific calculators in my introductory physics classes but do not allow formula sheets or cell phones or Graphing calculators to be used on exams. I encourage students to get one of the high-end Scientific calculators that can be used throughout their engineering career, including on licensing exams, so they become fluent in its use.
The four levels of mathematics classes are defined as follows:
The distinction between Developmental and Intermediate might seem unnecessary to some readers, because both levels are usually non-credit classes at a university. Indeed, some universities define college algebra as a remedial course. I make the distinction because our math department teaches classes at the Intermediate level and above, while the Developmental classes are taught by a separate department that specializes in teaching those skills. I know that smaller colleges do not make this distinction, but we are not a small college. (We have more t-t faculty in our Developmental math department than a private school like Union College has in its regular Math department.)
If I just say "Algebra", I mean College Algebra. I will say "Basic Algebra" or "Arithmetic" when I am talking about Developmental skills classes.
For the record, our Developmental classes use a Basic calculator for some things but some exams must be taken without any calculator. (The placement test and exit exam do not allow use of a calculator.) I believe they allow the use of any calculator up through a Graphing calculator when they allow a Basic one, but that might depend on the instructor. Our Intermediate classes all use calculators. Our Algebra classes require a specific Graphing calculator that is also required for statistics. Our calculus classes are a bit less picky about which Graphing calculator students can use, but ban Algebraic calculators and computers except in some special situations.
Today, June 16, is:
When looking for the official BP info about the status of the well in the gulf, I found the following comment on the Huffington Post's Social News prominently in the news stack on Google:
FREEDOM BELL(This was a comment on a Huffington Post article reporting the great news that the well had been "shut in".)
“Me either. When did Wells of BP issue email and comments during the past attempts. When did Obama ever go on TV during a past attempt?”
Equinator
85 days, 16 hours. Why was this not done the first day? All that planning to watch out for the walruses must not have helped much.
I had planned to post this a week ago (obviously), so by now Dr. Crazy has beaten me to the punch. Yes, it is that time of year, the time when you realize that next month is August! The month when classes begin. Lest we forget, the month when meetings begin! The month when there will be a number of things on the table (figuratively and literally) that you know you could have done, oh, in July. Or June.
As an inveterate procrastinator, I have used false deadlines for ages. In this case, I am now pretending we are approaching mid August rather than mid July. Those things that need to be printed for the first few labs, the ones that don't even need to have a date changed? They are going to get done this month. After all, my classes are all full (and one is overflowing) so I know what the number count is likely to be and any leftovers can be used next semester.
Syllabus for fall? Almost done, apart from one tweak and final check of the calendar and exam schedule. Syllabus for the spring? Next on my list. (Winter break is always too short.) Busy work I know I will need to do during the semester? I actually got the template updated so I only have to work on the worst part of it during the next month along with one task I have simply avoided doing for, oh, about two years.
And I plan to clean my office. Next week.
I think it was sometime in grad school, as I got to know more foreigners, that how odd it is that we celebrate "July 4th" as if a date could be the name of a holiday. (Do you have July 4th in your country? Of course, but it isn't a holiday.) Similarly, it is always celebrated as the "birth of the United States" even though it was almost 13 years later, in March 1789, that the United States government as we know it came into existence. But we don't celebrate a "constitution day" holiday like some countries do, nor do we celebrate the ultimate event that truly sealed our existence as a nation (victory in the War of 1812), although we could have celebrated two others of almost equal importance yesterday (victory at Gettysburg in 1863) or today (victory at Vicksburg, also in 1863).
So "Independence Day", or "July 4th", does multiple duty as holidays go. Including, of course, the opportunity to set off illegal fireworks while watching state-sanctioned fireworks, watching NASCAR fireworks (last night's wrecks were spectacular) and the start of the Tour de France (also featuring spectacular wrecks this morning) in HD, and dining on the least healthy food this country has to offer.
I didn't appreciate the length of the Revolutionary War or the huge gap between it and the formation of our nation until I took a middle school government class. I had a crazy radical teacher who thought we should know the real truths of the history that was behind the sound-bite myths of political speech. So I know that the Revolution War began in 1775, before we declared our independence. I remember that blew the minds of some of my classmates, but it made sense that they might have wanted to win a few skirmishes before putting it all on the line.
Ditto for the wonderful detail that George Washington wasn't the first President of the United States. There were something like a dozen of them (aha, Wiki has both the full list starting in 1774 and the ten who headed the government), each serving as the "President" of the single house of the US Congress that (weakly) governed the confederation that was the United States for 8 years, starting in March 1781 even before the Yorktown victory, negotiated the treaty of Paris in 1783 that actually granted us our independence from Britain, and developed a Constitution that would dissolve that government in favor of a stronger one.
One wonders if the United States would have been reconquered by Great Britain in 1812 if not for that stronger federal government. Ditto for surviving the unpleasantness that came along 50 years later. Would there be Spanish speaking nations of Texas and California to our west and Florida to our south if we had stuck with a Confederation that ended up a part of the UK (like Canada) or split in half across the Mason-Dixon line?
PS -
Our menu includes chili dogs, watermelon, and beer from Vermont. While you digest that, check out the great pair of videos that Unbalanced Reaction put up today. And Dr. Crazy got to watch fireworks from the porch of her new house!
