V---: “Mr. AB, this is boring.”
Me: “Well, V---, I’m glad it’s that easy for you.”
V---: “No, Mr. AB, it’s not that it’s easy, it’s really hard. It’s also just, you know, really boring.”
Me:
---
Dear Readers:
Be honest.
If you were required to add two mixed numbers, say five and one-fourth and three and three-fifths, could you do it? Fluently?
I’m going to guess that unless you teach math, the answer is no. Not because I have low expectations of my readership but because I suspect the last time you did such a problem was when you yourself were in school. It doesn’t matter who you are. As far as I know, not even professional math-people, inclusive of engineers and architects, compute fractions anymore.
Can we even envision a concrete articulation of such a problem? I'll try...
Tom goes to the butcher and buys a three and a third pound roast chicken. His roommate Scott, with inhuman precision, eats two and four ninths pounds. How much is left for Tom, who for inexplicable reasons must know this quantity to the immeasurable and irrelevant accuracy afforded only by unconverted fractions.
Let’s understand why this is worthy of such gripe: in the process of completing this problem, our ten year olds must complete ten constituent steps. This includes deriving the least common denominator, performing two simple and two double-digit multiplication problems, three addition problems, a double-digit long division problem with a remainder and then the isolation of the final answer from that division problem. Depending on the problem, they might also have to do two more division problems to reduce the fraction to lowest terms, bringing us to a full dozen. All must be performed perfectly, as we only give partial credit in high school on up.
Not only is this skill obsolete, it’s not even a prerequisite for ensuing and more important ones. Certainly, I recognize that doing basic operations with fractions is helpful for algebra, but mixed numbers? As far as I know, this skill is at the pinnacle of irrelevance. My students don’t even really need this for sixth grade, let alone later life. Yet I am forced to spend three weeks on it now, because someone, somewhere, decided that understanding how to operate mixed numbers with unlike denominators constitutes “proficiency” in fifth grade math.
To me, the saddest part is this: I can, and I will, teach my fifth graders to master this skill. But why, why, why?!? The list of other, infinitely more valuable aspects of math that I have to leave out (logic, probability, measurement, statistics, just to start…) to make time for this irrelevant crap would make any one who does use math in a meaningful way cry. Imagine the service to humanity if I taught all my fifth graders the difference between mean and median, instead? How many misleading and pointlessly dumb statistics might we be spared? But no…
Now readers, try one and a third subtracted from four and seven-eights… You won't, will you? Because you know it's a waste of your time!
If I am wrong and you are in (or know of) a career where these skills are vital, please correct me. Even with a dream class of budding young mathematicians, teaching this unit is awful. I'd love to provide a reason why it should be more than excruciating tedium. In other words, help me fill in that blank I was stuck in today.
Wednesday, February 13, 2008
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19 comments:
There are a number of professions where you nominally have to deal with mixed numbers: how many 1 1/3 foot crossbraces can you cut out of an 8 foot stud?
However, those are generally simple problems, and the people who have to do them don't rely on what they learned in 5th grade to do them.
Ideally, the answer to your question would be that operations with mixed numbers would lead to a better understanding of fractions, which would in turn result in grater success when they get to algebra and eventually calculus.
Practically, when I have to teach my remedial 8th graders to deal with mixed numbers, the only thing I teach them is to convert to and from improper fractions: everything in between is just review of stuff they already know.
Making up real world word problems, on the other hand, is often a doomed effort.
I'm still an undergraduate student, and have only done one summer of real-world research, but I think that mathsy people do need the skill of adding fractions with unlike denominators. When we are lucky enough to work with actual numbers, most of my professors leave things in fractional form until it becomes absurd. The fractional form will often show a relation between answers for related inputs, and then we can devise a general formula from it. We skip out on mixed numbers completely, but that is only because we can mentally go from (whole number + fractional part) to (fraction with a larger numerator than denominator). Somewhere down the line, we needed to learn that lesson really well.
I would also not discount the importance of algebra. Physics students (and I think engineering and optics) here are taught to leave numbers out of their equations for as long as possible, because working purely with symbols often simplifies the problem and allows greater understanding of what is going on. Knowing how to add whole numbers and fractions together is vital to them.
I'm 36. I've infrequently used mixed fractions when I cook. But still - it's pretty important when you're in the middle of dinner and can't find the right measuring tool.
Of course, now Google Math is a big help.
I deal with mixed numbers in my job in the Navy. Even with sophisticated computerised combat systems, you still need to be able to solve a large number of mathematical problems quickly in your head. A lot of these problems invovle fractions and ratios, and the initial form of the data may be mixed numbers.
We certianly emphasise the ability of quick mental calcualations, since even the most expenisve computer systems are "garbage-in, garbage-out" - If the input data is wrong, or the underlying assumptions are incorrect, you will get a dud answer. Besides, a person working out a mental solution can usually generate at least a ballpark answer quicker than a computer, and we always attempt to verify our computer based results with mental caluclation.
I also studied engineering at university. I guess an understanding of mixed numbers is essential for higher-level skills as algerbra and calculus. Not because we work in mixed numbers, but because you need an implicit understanding of the concepts. You need to be able to automatically convert them into a usuable form to solve your problems.
I actually did solve the both of your problems, using different techniques. The first I converted into decimals (5.25 + 3.6 = 8.85 = 8 and 17/20) - The second I just treated the whole number and fractional parts as seperate calculations, a short cut since I could see that I was subtracting a smaller fractional part from a larger one (7/8 minus 1/3). Converting to a common denominater we have 21/24 - 8/24 = 13/24, with the whole number part 4 - 1 = 3 your anser is 3 & 13/24.
The point of this is not to show much smarter I am than your 10 year olds, it's to demonstate the need for a implict understanding of what a mixed number is, and how you can use a variety of methods to solve the problem.
If you want to make your subject more interesting, talk about the various methods you can use to solve the problem quickly, and how to look at a bunch of numbers and decide what is the quickest method to a solution.
Obviously, this is going to be different for different people. Every person has a different way of understanding mathematics and their own unique solution algorithms. Maybe you can use it to teach your kids about different thought processes, including their own.
I always wanted to make up real life fraction math problems for my students that go something like this?
"You go to your drug dealer to get an ounce of chronic. He usually charges you 300 dollars for an ounce. Your dealer sells you 4 eights and 1 half once sac. He charges you 400 dollars. Did he raise his prices?"
Granted, it might be a bit too sophisticated for 10 year olds, but my high school kids would find this very useful. Too bad I can't commit it to paper...I might get written up or something.
Mr. AB--
I'm a physics grad student. I just got my M.S. and I'll be leaving academia in June to become a math teacher (through TFA, like you). I thought this post was really interesting because I can relate to it from both my "physics grad student" side and my "future math teacher" side. Here are a couple of comments:
1) We add fractions all the time in physics grad school, but I suspect that we are the only people on the planet who do so. And although we all learned to add fractions in 5th grade, we only learned to do it well when we were in college (by simplifying physics problems over and over and over). I think that this is as it should be: we developed the skill when we needed it.
2) If physics grad students are (almost) the only people in the world who still add fractions all the time, I do have to wonder why we teach it. Like you, I have the same question ("Why, why, why?!?") about a huge part of the current math curriculum. In the future, I will probably spend a month teaching some group of kids about the quadratic formula. But why should we give the quadratic formula so much attention? The only "everyday" problem to which it applies is the physics of projectile motion, and even this application isn't stressed in most of the algebra textbooks I've seen. It's true that we use the quadratic formula all the time in physics grad school, but that doesn't justify teaching it to everyone else.
3) Whenever I complain about this to educators, they usually misunderstand why I might want to teach something different. It's not that I have anything against mixed fractions and the quadratic formula; it's that, like you, I think there are so many better things to be teaching! If I had a dollar for every time I see a news article, political speech, or advertisement that misuses statistics (with Simpson's Paradox, the correlation-causation error, or any number of other errors), then I would be a rich man. Since statistics are so ubiquitous in the modern world, wouldn't it make sense to ensure that our students understand the subject deeply? And this is just one of many, many examples of topics we've left out.
4) You wrote that there is "someone, somewhere" who decides what "proficiency" means in fifth-grade math. Do you have any idea who this person is? How do I get that job? It would be nice if, ten or twenty years down the line, I could start working some sense back into the curriculum. Let me know if you have any ideas about this. =)
Take care, and keep up the excellent blog.
I did both of the problems in my head (I suppose that means I'm fluent at it) and came up with the correct answers...mostly just to see if I could do it. But to be honest, the reason I could do it so easily is because I was helping a sixth grader with his homework a few weeks ago.
I have used this skill a few times in my life, but probably not enough to make it worth three weeks of my life learning it. On the other hand, I love statistics and use them all the time! Teach mean and median because it'll make the world a better (and less bored) place
I'm with you for the most part, Mr. A.B. Although estimating the nearest integers to an improper fraction is important, the skill at doing arithmetic on mixed numbers is not. Even for the sake of developing the skill of keeping track of where you are in a 10-step process!
Curriculum decisions are made by school districts, influenced heavily by state standards. Teachers can usually be involved at all levels of the decision making. In my experience, many are strongly influenced by tradition and by fear of being accused (by some parents or mathematicians) of "lowering standards" if a traditional topic is omitted.
I did them in my head too. I have to disclose that I'm doing fractions with my homeschooled 9 year old but still, I use fractions ALL the time. I'm a homemaker so I'd be a little lost when it came to cooking and baking without them.
Keep teaching them. Who knows how many furture homemakers you have in your classes! :)
Why are math teachers so obsessed with the idea of finding real-world applicability to their subject? So what if fractions aren't needed in real life?
Math is a series of processes and rules. Kids learn to solve problems with these processes and rules, and the problems have one and only one answer. That's the purpose of math, for all but the fraction of people who will be going on to careers in the subject.
I can't see why you need three weeks to teach the subject, though. Surely one week should be enough, with reviews to be sure they remember it a month and three months down the line.
Anyone who likes to cook uses fractions constantly to adjust for numbers of people served, and for sometimes not having correct measuring tools.
I learned last year that carpenters use them a lot, and the MAIN reason for dropping out of the carpentry profession is inability to do math calculations. I thought that wasvery interesting.
Before I was a teacher I was a stock broker, and I used fractions constantly. As you know, stocks are quoted in eights of a dollar, and brokers use fractional calculations for computing margin requirements. While it's true you get a computer print-out, you still have to understand it well and know how to calculate by hand for when the computers aren't working, or when planning a purchase with the client. (The equivalent would be for people to no longer need to know multiplication tables or even how to add or subtract, because of being able to use calculators. But of course you have to know, because what if you hit the wrong button, but can't recognize that your answer doesn't make sense?)
Personally, I always enjoyed working with mixed fractions. The problem is, as you say, WHO cares about the problems in math books? NO ONE.
I live in the Middle East and here are the kinds of problems my daughter came home with in the sixth grade:
Something on the order of this (I’m just making this up):
"A caterer is going to serve 87 people at a wedding. Each diner will need to eat 330 grams of chicken, x grams of this and that vegetable for each serving, and x amount of desert–for which each pie serves six diners. So then you are given the recipes (which are each given to serve different numbers of people, such as six, eight, or ten people) for the main courses, and deserts. Then, you are asked to calculate the entire shopping list, what quantities of each thing must be purchased for all the guests, and in addition, how much it will all cost (they give you prices in the store, per kilo, of things as part of the info in the problem)."
We find problems like this kind of fun to work on as a family–each of us sits at the table in a race to see who can do it faster, and get it right, then we compare answers (our daughter, my husband, and myself). When I was in school, I could never figure out why anyone would want to know about how fast a stick is floating in a river, or the kinds of problems they came up with. They were boring. We find big, complicated problems, like the one given above, interesting.
Eileen
Dedicated Elementary Teacher (in the Middle East)
elementaryteacher.wordpress.com
I can say that I never use mixed fractions in my daily life other than when I had to sharpen my math skills for the California Test of Basic Skills, which was required for getting a School Psychologist credential. Somehow, I don't think that will inspire your 5th graders much!
I do appreciate your point though. I find the same frustration when we are teaching 6th grade kids with learning disabilities "academic vocabulary" such as polytheism, which I'm pretty sure I have never used in a sentence even though I have my Ph.D.
8 and 17/20
My mom is a seamstress by profession and quilter by hobby. I grew up helping her and I can tell you that dealing with mixed numbers was a frequent occurance. However, sewing and cooking are the only times that I can remember using mixed numbers.
As for the pinnacle of irrelevance, when was the last time that you needed to know what your mitochondria were doing?
TFA Baltimore '98
Just add the integer parts (5+3=8) and then add the fractional parts (3/4 + 1/5 = (3*5+4*1)/(5*4) = 17/20). There's no need to convert to improper fractions etc.
OK, reading the comments above, I see SEWING, COOKING, and CARPENTRY as three areas that use them constantly. I don't know if Americans are not doing any of these three activities any more, but surely SOME people are. And there were several jobs above that mentioned use of mixed fractions: in the Navy, being a stock broker. Also, anyone who works with a lot of decimals or percentages will probably also work with mixed fractions, or anyone who has to measure things a lot in his work--like interior designers, measuring for custom curtains, measuring fabric to cover furniture.
Eileen
Dedicated Elementary Teacher Overseas (in the Middle East)
elementaryteacher.wordpress.com
I am glad that you are writing about this topic. Yes, if we think hard enough, we can find some places where manipulating mixed numbers is important, but the overarching issue seems to be that there is not enough time to teach everything, especially when kids come to us behind, so it would seem that we should focus more on the essentials (although who determines what is essential becomes the issue).
I got so frustrated every time we had to teach stem and leaf plots (and sometimes box and whisker - which are even more obscure). I loved them because I love math, but my 5th graders were still struggling with other basics that could have used extra time.
To build number sense, it would have be great to have spent more time on alternative algorithms like partial products (23x34=4*20 + 4*3 + 30*3 + 30*20 = 80+12+90+600=782), but they needed time with stem and leaf because it was on the state test. Sigh...
I'm a little saddened by your take on fractions.
You said you didn't have enough time to teach reasoning and logic, yet it takes both to really understand adding of fractions. It's much more than "making equivalent fractions with like denominators."
I will chime in! I teach 5th grade and my students are also whining and complaining about the very same topic. Which I also taught them in 4th grade! I think they remembered what I taught them because I a) reviewed skills throughout the year and b) demanded that they think back to last year.
Anyway, I was talking to my neighbor, who is a pastry chef. She said when she went to culinary school, she had to have someone teach her how to add fractions. And she was 25.
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