This math review covers pretty basic things. The first thing we went over was how to do unit conversions such as converting feet to meters using a conversion factor. I think I do a pretty good job of this, showing exactly why you can multiply/divide by the factor and it's ok to do so (ie, because a conversion factor is essentially multiplying by one). I very carefully show how if you have the same units in the numerator as you do in the denominator, it cancels out, leaving you with the new units you need.

We discuss ratios as a means of comparing things and determining how many times larger one quantity is than another. The lab packet each student has even has a big bold statement saying "

**Notice that a ratio does not have any units!**"

We also cover a bit of scientific notation and how it relates to the metric system. Should be simple enough. After all, it's base ten. If you need to go from meters to centimeters, just move the decimal point two places to the right.

We discuss triangles. Not even in the depth a high school student had in which they have to learn how to do sine and cosine, but just the realization that there's a base, height, and some angles.

So the first week of lab, I go through all of this, taking what is a ridiculous amount of time doing so given that it's so basic. Every step along the way, I ask if there are questions, if it looks familiar, if they think they could replicate what I'm doing, etc... I don't continue until I see, at the very least, some general head nodding from the majority of the class.

All in all, if people can't do this, there's no reason they should ever have graduated high school. Yet last night in my lab, a sizeable amount of students demonstrated that they lack these skills as well as any sort of basic algebra (junior high type stuff).

Our lab last night was over angles and parallax. Since the angles involved are generally pretty small, we can skip the nasty trig functions and just use the small angle approximation (angle = 57.3ºBase/Height). In this equation, there are three unknows: angle, base, and height. Thus, if you know two, you can find the third.

Since the angle is the measurable quantity in all cases we study, this means that they will have to solve for the base and the height. Out of 21 students, at least three were completely unable to do this (possibly more if they worked with a partner that explained it to them). To be fair, one is a non-traditional student and admitted she hadn't used algebra in any recognizeable form in at least a decade.

In one section, they had to measure the height of a classmate, and knowing this, derive the distance after measuring the angle. For some reason, my students have no concept of what a "meter" is and were getting heights of 30m before I peered over their shoulders and asked if their partner was really as tall as the building we were working in.

We also measured the height of boxes having known distances and measured angles. The answer would come out in meters, but I gave them the true value, so they could calculate their error, in centimeters. It should be obvious that you're going to need to convert them all to the same units, otherwise the dimentions don't come out right. To make sure they would recognize this, I told them to write their units on everything. Some still didn't do so, and wondered why their errors came out to be several hundred percent. After figuring out that units were the problem, a good two or three still didn't know how to convert within the metric system.

One of the questions at the end of the lab, asked them to compare their observed angle for a certain measurement, to the largest oberved stellar parallax angle in astronomy (.76 arcseconds). The point of this question is to drive home the fact that stuff is so far away, that even the largest shift is still about 50,000 times smaller than what they were measuring in class. To make this comparison, the students need to put things in the same units. They measured in degrees, the given angle was in arcseconds. The conversion was given (1º = 3,600 arcseconds). A full

^{3}/

_{4}of the class was unable to answer this question without assistance. So much for them understanding what I'd said at the beginning of the semester about convserion factors. And apparently they missed the bolded statement about ratios not having units too.

Admittedly, this is a 100 level course, taken by primarily freshman wanting to get their lab science requirement out of the way so they never have to think about that awful "science" thing again. But admittedly, this is a 100 level course, taken by

*college*freshman. The skills we use here are ones that are taught to students(with the exception of those with developmental disabilities) in 10th grade or before.

And yet somehow, many seem to be making it through America's public school system, and accepted to a university without even a smidgeon of proficiency. I suppose this is the true meaning of "No Child Left Behind": We'll drag them kicking and screaming along, whether or not they understand anything.

## 12 comments:

Hi Jon,

I think the question you have to ask about the students centers around learning and not so much about teaching. You are correct, ALL of these students have worked these various kinds of problems before and yet they do not carry any knoweldge away from their experience.

Your lab section gives context and application to these mathmatical skills; which should improve the students learning. I would be willing to bet however, that if tested next year, fully 90% of your current students would not be able to do these problems.

While we all (society) tend to blame schools and teachers, I think that there is clearly something we do not understand about how humans learn.

I must admit to not being surprised -- as a T.A., I worked with students who couldn't understand how to read or plot a graph, as an example. My favorites, though, were the students who, upon seeing something like F=ma on the board, would exclaim "I don't do math, I'm a business major."

(Must wonder what they teach in business classes. There's apparently no numbers involved.)

I wonder if the problem is that we don't teach concepts in math -- we teach how to get answers. So the students are lost if they can't simply type something into their graphing calculators to get an answer.

Sigh. Now I understand why I'm worried about taking a teaching job -- not sure I can put up with this for the rest of my career :).

I went to an engineering school. I was astounded at how bright my class mates were. All of them. Yet, only 30% of us graduated. That was about 1980. I've asked around recently, and the new rate is 25%. You have to be more than bright. You have to be dedicated. No one is going to make you dedicated, because no one can.

I never had that bit about cancelling units until my undergrad work. I thought it was fantastic. It's so simple, and gives you confidence that the conversions you're doing might be right. I immediately applied it to odd things like e = m(c^2). Most people don't go out of their way for new applications (new problems). Most people do the minimum to get through the assignment. So, the assignment has to be enough work to give some chance that they'll gain competence. And, there's the "use it or lose it" issue. Most people probably don't do geometry often enough.

My son is in 4th grade. I recently heard that the USA is up there with the best countries in math and science up to the 4th grade. Then we fall behind. In 2nd grade, he was having trouble with math. That's addition and subtraction. I thought that was silly, as i really know this stuff, and as a parent, i could give him some real one-on-one help. And, in just two months, using an abacus, i got him through it. Since he doesn't bring an abacus to school, i showed him how to do it (with the same technique) on his fingers (the thumb is worth five, and each hand is a digit). Two more months. But how many parents can and will do this for their kids? Do you have time to do one-on-one with each of your students? I thought not. No teachers do. And, if parents would do the math, it would be obvious.

How to do finger math.

So, i'm one of those who

get it. I got through school and ended up with competence. But, i have to say, that most of my learning wasn't in school, per se. I'd take the homework home, read the book, decide if i got it, and reread if needed. I'd do the problems, check the answers with some sort of cross check, if available. But i'd decide if i really understood the problem types and how to approach them before moving on. The key, i think, is that by doing the work on my own, i'm doing it at my own pace. And, at the end, i wasn't interested in the grade so much as having gotten something out of the course. That may have been the attraction i had to go to a pass/fail school. Every now and then, i'd take a course where gaining competence got you a good grade. Apparently it's harder to set up a course where this is true than one might expect.So, how does one put together a lecture when those that

get itdon't learn that way? Is the lecture for those that don't? Perhaps there is something wrong with the whole lecture concept. One alternative lab/lecture scheme is limiting the lectures to ten minutes (as a rule of thumb - obviously, there are concepts that can't be done in under 20). So, the lectures are injected into the larger lab.steven case: Sadly I'm forced to agree that 90% of students will not be able to replicate their results from this, or any lab in a few weeks let alone a year. But ultimately, I wouldn't really expect them to. These labs are all fairly specific using equations that have almost no application outside of this class. Thus, there's no reason to remember them. The big issue that I have with what I've observed is that they are graduating high school, and being accepted into universities without having the basic tools thatareutilized throughoutmanydisciplines. You can't build a house if you don't know how to use a hammer.squawky: Your statements concerning students being lost if they can't type something into a calculator is all too familiar. Many in my class tried to simplify things by plugging everything into the equation they could in order to only leave the one variable, but many still had difficulty rearranging things from there.stephen: Sadly, I'm all too frequently forced to be one of those people that only does the minimum amount as are most upper level students in my department as I've found out. Unfortunately, classes become so loaded that there's just not enough time to really play around with equations to the full extent I'd like.That being said, at the lower levels, mostly freshman, there are more than enough time to take the time to get a better comprehension.

So here is the perspective of a science educator in training:

There are multiple aspects to the problem Jon described in science education ad what we know about learning. One is that, despite the calls and papers on science education reform, science is still taught like it was in the 1950s; treating all students like they are going to be future scientists. This may, or may not, be the case with the curriculum in Jon's case, but based on his description it's my hunch that it is.

A study shown that students in these 'traditional' settings will only retain 30% of what they learned that they did not know before they entered the class. This is independent of the quality of instruction. Interviews of literally thousands of students right after they leave class show that most students only remember what the overall lecture was about. Counterintuitive facts are forgotten by the majority of students in 15 minutes. There is also cognitive overload where so much information is given in a non-stop lecture that the efficiency of processing everything is significantly reduced. Students need processing time to assimilate and accommodate new information. Something is fundamentally wrong with how science is taught to non-majors. I have actual numbers and sources if someone is interested, I am just a bit on the lasyy side to get it out of my bag at the moment ;)

Another element is translation of skill sets. One of the key elements (and assessment) of learning is being able to translate a skill into a different subject or novel situation. Here is the part where many students are failing. In my experience math is never taught in the context of anything but itself. If you present the students with a simple math problem of "find x" then I think you might see a higher success rate. But asking them to apply the same skill set in a different environment illustrates how well they were taught math.

A little anecdote on this: In one of my ed classes we got into teams and did 50 minute lessons plans where we had to teach, had some kind of lab or inquiry experience, and assessment. My team did the water cycle. For the lab element we had people pretend they were mater molecules and individually go around to different station (ocean, glacier, stream, etc) pick up a card saying what happened to them next (condensated and precipitated as rain into the ocean). Instead of a quiz where they regurgitate definitions we had them write a short story about what happened the them as a water molecule to see if they are really able to use these words in context. It also gives the creative sort an opportunity to shine as they are often marginalized in science classes. Everyone loved it and in general quite successful, but the only complaints from the peer review were from those with science and engineering backgrounds saying that creative writing in science was nothing more than busy work. If it was a class for science majors then maybe they would be right, but when they tried to teach their unit (on astronomy) they did it in a more traditional way and their lessons was a disaster.

When I was in High school, my Chemistry teacher who had just moved here from America said that the lack of imperial to metric conversions was his favourite thing about teaching european students. After reading this, I can see why.

No kidding. Every now and then I go an entire day just using metric units. Living in the states I get some very odd looks when I am asked how tall I am or how much I weight and I respond in centimeters and kilograms!

Being from Australia we hate ever having to convert something from metric to imperial units (sometimes american cable manufacturers only provide resistance per foot, or something equally useless).

Anyway, the problem of retaining basic information is not unique to mathsy/sciencey subjects. In Grade 5 we learned about homonyms and homophones and all that jazz. So we learned the correct their/there/they're to use, and the correct its/it's, and the difference between girls and girl's. But how many people still knew these things at the end of primary school? And how many still didn't know at the end of high school? I still have to correct other people in my engineering class when they do stuff like this. It's not too hard when English is your first and only language, but for some people it is. I think it's more about attitude towards learning, the teachers who taught this stuff were great teachers.

steven case: Sadly I'm forced to agree that 90% of students will not be able to replicate their results from this, or any lab in a few weeks let alone a year. But ultimately, I wouldn't really expect them to. These labs are all fairly specific using equations that have almost no application outside of this class. Thus, there's no reason to remember them. The big issue that I have with what I've observed is that they are graduating high school, and being accepted into universities without having the basic tools thatareutilized throughoutmanydisciplines. You can't build a house if you don't know how to use a hammer.squawky: Your statements concerning students being lost if they can't type something into a calculator is all too familiar. Many in my class tried to simplify things by plugging everything into the equation they could in order to only leave the one variable, but many still had difficulty rearranging things from there.stephen: Sadly, I'm all too frequently forced to be one of those people that only does the minimum amount as are most upper level students in my department as I've found out. Unfortunately, classes become so loaded that there's just not enough time to really play around with equations to the full extent I'd like.That being said, at the lower levels, mostly freshman, there are more than enough time to take the time to get a better comprehension.

So, i'm one of those who

get it. I got through school and ended up with competence. But, i have to say, that most of my learning wasn't in school, per se. I'd take the homework home, read the book, decide if i got it, and reread if needed. I'd do the problems, check the answers with some sort of cross check, if available. But i'd decide if i really understood the problem types and how to approach them before moving on. The key, i think, is that by doing the work on my own, i'm doing it at my own pace. And, at the end, i wasn't interested in the grade so much as having gotten something out of the course. That may have been the attraction i had to go to a pass/fail school. Every now and then, i'd take a course where gaining competence got you a good grade. Apparently it's harder to set up a course where this is true than one might expect.So, how does one put together a lecture when those that

get itdon't learn that way? Is the lecture for those that don't? Perhaps there is something wrong with the whole lecture concept. One alternative lab/lecture scheme is limiting the lectures to ten minutes (as a rule of thumb - obviously, there are concepts that can't be done in under 20). So, the lectures are injected into the larger lab.I must admit to not being surprised -- as a T.A., I worked with students who couldn't understand how to read or plot a graph, as an example. My favorites, though, were the students who, upon seeing something like F=ma on the board, would exclaim "I don't do math, I'm a business major."

(Must wonder what they teach in business classes. There's apparently no numbers involved.)

I wonder if the problem is that we don't teach concepts in math -- we teach how to get answers. So the students are lost if they can't simply type something into their graphing calculators to get an answer.

Sigh. Now I understand why I'm worried about taking a teaching job -- not sure I can put up with this for the rest of my career :).

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