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Chemistry/math pedagogical question about unit conversions/dimensional analysis


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In an attempt to ease the first few weeks of chemistry I’ve been reviewing scientific notation and unit conversion/dimensional analysis problems with my rising 10th/11th graders to whom I will be teaching both Algebra 2 and Chemistry. My rising 11th grade has always struggled with math.  I started with an example of converting 6.5 m to yards.

 

It did not make any sense to her. What she wanted to do,and this does make sense to her, is multiply 6.5 by 100 to convert to cm. Then divide by 2.54 to convert to inches. Then divide by 12 to convert to feet and then divide by 3 to convert to yards.

 

Which of course is correct. I tried to show her that what I was teaching her was the same calculations.  I told her that when she is working with units that are less familiar to her and less intuitive to her that knowing the process would make things easier and that the process is faster.  I am looking ahead to working with moles and atomic mass,etc. Sometimes in chemistry I figure out what to do based on where to place things so the units cancel or end up in the right place. Maybe that is a crutch and her way is better. 

 

Talk to me about this from a pedagogical standpoint.  Probably eventually she might understand or at least remember it enough to do it, but it will be a long time coming if the past is any indicator. Since there is a way she understands it, at least with familiar units, should I let her do it her way? Should I let her do the problems her way and then redo them with her the other way?

 

BTW, none of this is “attitudeâ€. She really just has some mental glitches when it comes to mathematics and just does not see things the way most kids do. She also has difficulty remembering processes/concepts. Lots of reteaching and repetition and mixed practice is the only way I’ve found thus far to make slow/steady progress in math. 

 

Thanks,

Kendall 

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If she can do them reliably her way, I would let her.  That's how my son does it (and how I did it until someone showed me the "proper" way--years after I had gotten a degree in biochemistry).  She does need to see that the units work her way, so I would have her show the units (if you're not already) and have her do the math on them as well.  Once she gets there, if she understands what moles and atomic weights are, she should have no problem with those calculations either.

 

I am currently tutoring an adult student who is taking the pre-nursing chemistry sequence at the local CC.  Her instructor insists that they do it using dimensional analysis.  In fact, the instructor seems to be very vocabulary and procedure oriented and not at all concept oriented.  My student can rip through a complex unit conversion using dimensional analysis but underneath it all, she has no idea what she's doing.  And she won't let me explain because she's afraid to learn anything that is different from what the instructor says.

 

Your daughter is showing that she knows the relative sizes of meters, cm, in, yards, etc., and how to move from one to another, which in my mind is important.  This is how the Teaching Company high school chemistry course guy does it--you know, the guy that is so good at explaining how to do the math.  He says that calculations in chemistry are easy if you really understand what's going on in the problem and what the numbers represent physically.  This is what your daughter is doing.

 

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Some places teach dimensional analysis by having students write out the conversions, and they multiply everything on top and then divide by everything on bottom. It looks like a line of numbers divided into boxes. If that's what she's not getting, I wouldn't worry about it. She will probably need to be able to show units cancelling, though. The only thing that I would suggest is to see if she understands that the reason to multiply by 100 is because 100 cm is the same as 1 m. She divides by 2.54 because that's how many cm = an inch, etc. So, write the numbers as fractions with 1 as either the numerator or denominator. 6.5 m x 100cm/1m x 1in/2.54 cm, etc. She could even set up her problem the way that she understands, and then just put the 1 part of the unit in afterwards - if she understands that dividing by 2.54 is the same as multiplying by 1/2.54 then it shouldn't be a problem.

 

I don't know if this will help - basically, I just told you to write dimensional analysis in a different way than what you might have been trying, but sometimes saying it a different way helps. While I'm not a big fan of blindly following a formula, it can sometimes be really hard to track down students' mistakes when you can't see all of the units and where they cancel. In my co-op class with 25 students, I'll put in the effort. With a TA grading 200 chemistry tests in college, they may not.

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Step one: Look up the meaning of the word "pedagogical". O.K. "Having to do with a teacher."

 

At the risk of "preaching to the choir" here is my response:

In an attempt to ease the first few weeks of chemistry I’ve been reviewing scientific notation and unit conversion/dimensional analysis problems with my rising 10th/11th graders to whom I will be teaching both Algebra 2 and Chemistry. My rising 11th grade has always struggled with math. I started with an example of converting 6.5 m to yards.

 

It did not make any sense to her. What she wanted to do,and this does make sense to her, is multiply 6.5 by 100 to convert to cm. Then divide by 2.54 to convert to inches. Then divide by 12 to convert to feet and then divide by 3 to convert to yards.

As you say, this is correct. But, as you also point out, that approach won't get you anywhere when you are not inimately familiar with the units in question.

Which of course is correct. I tried to show her that what I was teaching her was the same calculations. I told her that when she is working with units that are less familiar to her and less intuitive to her that knowing the process would make things easier and that the process is faster. I am looking ahead to working with moles and atomic mass,etc. Sometimes in chemistry I figure out what to do based on where to place things so the units cancel or end up in the right place. Maybe that is a crutch and her way is better.

I'm not sure I agree that the process will always be faster. But it will take you to the correct answer in virtually every case. In other words, the process is more general.

Talk to me about this from a pedagogical standpoint. Probably eventually she might understand or at least remember it enough to do it, but it will be a long time coming if the past is any indicator. Since there is a way she understands it, at least with familiar units, should I let her do it her way? Should I let her do the problems her way and then redo them with her the other way?

As an engineer, I suggest she learn a "rigorous" way to convert units. This will allow her to correctly make the conversion regardless of whether the units are familiar or not. The best approach I know is to deal with all unit conversions as fractions. For instance, in the problem you listed above, HER approach would look like this (I'll use unit abbreviations):

Convert 6.5 meters to yards:
       m         100 cm          1 in           1 yd
6.5  -----  x  ----------  X  -----------  X  ---------  =  7.1 yd
       1           1 m          2.54 cm         36 in
In other words, I feel her approach is fine, but it should be done in a structured manner so that the conversions are not made in the wrong manner. You can explain to her that this approach is the same as hers and that allows you to correctly convert units when you don't know the direct conversion. You can also point out that IF you do happen to know the direct conversion, the process is faster, but NOT more correct:

Convert 6.5 meters to yards:
       m           1 yd
6.5  -----  x  ------------  =  7.1 yd
       1         0.9144 m
Finally, the key learning needs to come by helping her see that the "intuitive" approach sometimes falls down. The most challenging units I ever dealt with in my career came up when I was designing magnetic structures. You could give your daughter a taste of what you might come across by showing her the Wikipedia page for the SI unit for inductance: the Henry. Scan down the page into the "Definitions" section and notice the long list of equivalent ways that you can express inductance in SI units. That might be a bit of "shock-and-awe", but it helps to point out that units can and do become non-intuitive in real-life physics and chemistry situations.

BTW, none of this is “attitudeâ€. She really just has some mental glitches when it comes to mathematics and just does not see things the way most kids do. She also has difficulty remembering processes/concepts. Lots of reteaching and repetition and mixed practice is the only way I’ve found thus far to make slow/steady progress in math.

She sounds a lot like our DS17, whom I have discussed extensively in the thread on testing speed. I just did AP Chemistry with him last year. He did very well and got a 4 on the exam. He was very good with unit conversions, using a very structured approach of multiplying strings of fractions, as above. It's a bit slow for testing, but his accuracy was very good.

 

So I guess my experience tells me that I vote for the pedagogical approach, BUT I suggest that you teach her that using HER numbers so that it might become comfortable to her. Then try to bring her along from there to a more general understanding of why such an approach is often necessary.

 

Finally, I find it a shame that so much of math is done without units. I'm sorry, but Math does not exist for math's sake. It exists as a tool to solve real-world problems. Most of those problems cannot be solved without the proper handling of units. Just ask NASA about their Mars probe which crashed on the surface because they had not properly converted the units between two pieces of equipment which had to share information. :tongue_smilie:

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I think your DD's method is fine because it shows that she understands what it is she is doing. Many students who have been taught the "unit multiplier" method end up with wrong answers, because they have no clear understanding why they do what they are doing and invert what goes on top and on the bottom. Many don't carry the units, and so have no way to check that the units cancel properly.

I would meet her where she is and simply insist that she documents her process correctly by including units and not simply "multiply by 100" but "multiply by 100 cm/m", etc. But I much prefer a student to understand what they are doing and being a bit slower than a student who simply dumps unit multipliers into the equation because "that's the method". 

 

ETA: Through practice, she will gain familiarity with units. Even in chemistry. Many students struggle because they never truly understand what a "mole" actually it: a name for a specific number of particles, similar to a dozen. 

 

Sometimes in chemistry I figure out what to do based on where to place things so the units cancel or end up in the right place. Maybe that is a crutch and her way is better. 

 

Yes, that is a crutch (and a quick way to solve a problem if one doesn't want to spend time thinking about it.) Thinking about it is, IMO, preferable, at least until the learner has mastered the material.

Edited by regentrude
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It seems like you want her to go straight from meters to yards, using a conversion factor that she is unfamiliar with.

 

She wants to from meters to centimeters to inches to feet to yards, using conversion factors that she is familiar with.

 

I would let her do it her way.  She's got the concept down and just taking a longer path.  Sometimes our math strugglers need the longer path first (you would not believe how I had to teach my math struggler to multiply multi-digit numbers at first.  She now uses the "regular" way but it took a couple years for her to decide that she liked it better.)

 

It actually is very important for her to make these "connections to prior knowledge" in order to learn this new concept.  She'll probably get to where she likes dimensional analysis better because it's shorter.  But it may be awhile and I wouldn't worry about it.  Whatever works for her.

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Thank you so much everyone, I will read more carefully this afternoon and post again, but I did want to quickly say that this below is what I was showing her to do it and this was confusing to her. She wanted to do them one at a time and now that I think about it, she wasn't writing anything down. So I will correct that at least.  More later...

 

 

Convert 6.5 meters to yards:
       m         100 cm          1 in           1 yd
6.5  -----  x  ----------  X  -----------  X  ---------  =  7.1 yd
       1           1 m          2.54 cm         36 in

 

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She wanted to do them one at a time...

O.K.  I see.  It's not terrible, but I agree that would be slow.

 

...and now that I think about it, she wasn't writing anything down.

Ack!  :scared:   The unpardonable sin!  (Until it comes time for test prep.  Then I find myself saying "Don't write THAT down!")

 

Oh, yeah, our kids will be so well-adjusted! :lol:

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Thank you so much everyone, I will read more carefully this afternoon and post again, but I did want to quickly say that this below is what I was showing her to do it and this was confusing to her. She wanted to do them one at a time and now that I think about it, she wasn't writing anything down. So I will correct that at least.  More later...

 

I help teach a high school Chemistry class. I think it's fine to do it one step at a time. It is slower but a lot of kids seem to need to do it that way at first and then eventually realize they can combine steps. I would be really vigilant about making sure she writes it down and WRITES DOWN THE UNITS. I see a lot of kids who don't  write the units but they end up making mistakes later because of this. When I grade labs I will return all labs without units as needing to be redone. Yes, I am mean. :) 

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Many don't carry the units, and so have no way to check that the units cancel properly.

 

This was a problem for my tutoring student.  She didn't get from the instruction that you actually do math on the units too, and it was a revelation to her when I pointed it out.

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In an attempt to ease the first few weeks of chemistry I’ve been reviewing scientific notation and unit conversion/dimensional analysis problems with my rising 10th/11th graders to whom I will be teaching both Algebra 2 and Chemistry. My rising 11th grade has always struggled with math.  I started with an example of converting 6.5 m to yards.

 

It did not make any sense to her. What she wanted to do,and this does make sense to her, is multiply 6.5 by 100 to convert to cm. Then divide by 2.54 to convert to inches. Then divide by 12 to convert to feet and then divide by 3 to convert to yards.

 

Which of course is correct. I tried to show her that what I was teaching her was the same calculations.  I told her that when she is working with units that are less familiar to her and less intuitive to her that knowing the process would make things easier and that the process is faster.  I am looking ahead to working with moles and atomic mass,etc. Sometimes in chemistry I figure out what to do based on where to place things so the units cancel or end up in the right place. Maybe that is a crutch and her way is better. 

 

Talk to me about this from a pedagogical standpoint.  Probably eventually she might understand or at least remember it enough to do it, but it will be a long time coming if the past is any indicator. Since there is a way she understands it, at least with familiar units, should I let her do it her way? Should I let her do the problems her way and then redo them with her the other way?

 

BTW, none of this is “attitudeâ€. She really just has some mental glitches when it comes to mathematics and just does not see things the way most kids do. She also has difficulty remembering processes/concepts. Lots of reteaching and repetition and mixed practice is the only way I’ve found thus far to make slow/steady progress in math. 

 

Thanks,

Kendall 

 

You've gotten lots of great replies, Kendall. :)  I'm another one who feels that if you're dd understands what she's doing without using DA, then that's just fine.  Many, MANY of the students that come to me for chemistry at the local community college just do not get DA.  I explain it, the math teacher explains it, the physics teacher explains it - it just doesn't click.  I wasn't taught to use it when I went through university so I suppose I can see where the kids are coming from.  When I teach the chem unit on moles and mole calculations, I show the students the DA method of solving problems (which, don't get me wrong, is great if it makes sense to them - it's a great method for self-checking) but I also have a bit of a "tool" that I call the mole highway to help students keep track of the calculations involved in multi-step problems.  Your dd sounds like she might really like the mole highway - if you're interested, PM me and I can send you a copy and an explanation of how I use it. :)

 

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