Now that our family health crisis has abated (my daughter is doing well), I guess it's time to jump back in with both feet. A math program leader in a district with which I am consulting, asked for my opinion on an important issue of curriculum and instruction.
How much time should middle school teachers spend on the traditional vertical algorithm for adding and subtracting mixed numerals vs. converting to improper fractions immediately?
I assumed that both methods are still commonly taught with about equal time given to each, but I wasn't all that sure about how that was across the country. This is where I need the help of my informed readers.
First, my thoughts. From a practical perspective of those who utilize fractions in their occupation, I would guess that the mixed numeral form is most commonly employed. Whether it's the carpenter taking measurements to see how many board feet of wood must be ordered (or for precise measurement to the nearest sixteenth of an inch) or someone following a recipe in the kitchen, I can't imagine that converting to improper fractions would be their first choice. On the other hand when I personally need to add fractions in a math problem, I usually use improper. I took an informal survey of one of the groups I'm working with and the majority stated they were taught both methods and some preferred working with mixed numerals and others said it's more complicated that way.
Are the number of steps roughly the same?
Mixed Numerals Algorithm for Subtraction:
(1) Convert the proper fractions to common denominator form.
(2) If needed, regroup, i.e., "borrow" 1 from the whole number part of the larger mixed number, convert the 1 into common denominator form and combine this with the other fraction (of course students are shown short-cuts for this which they blithely and mechanically follow without much thought).
(3) Subtract the whole numbers and the proper fractions.
(4) If the resulting fraction is improper convert it and add the whole number part to the previous result.
Improper Fraction Algorithm:
(1) Convert each mixed numeral to an improper fraction by the traditional algorithm (again blithely and mechanically without much thought).
(2) Determine a common denominator (or the lcd) and convert each fraction.
(3) Subtract the fractions.
(4) Convert the answer to mixed numeral form by the traditional division algorithm.
Now I may have combined steps or there are oversights but essentially they appear to be roughly the same number of steps. However, the difficulty or complexity level of the steps
may not be equivalent.
I also feel that the mixed numeral form requires somewhat more conceptual understanding even if the child does it routinely. It may also prepare the youngster for working with algebraic expressions like A + B/C, but that's debatable. Further there seems to me to be a strong connection between the Mixed Numerals Algorithm and adding and subtracting denominate numbers. For example:
Subtract
15 hr 37 min
9 hr 46 min
I doubt that we would encourage students to convert both to minutes first, subtract, then convert back to hrs and min. I could be wrong there!
I feel there are arguments on both sides here. My instinct is that both need to be taught but it's not clear to me how much time should be spent on each method. Certainly some youngsters could handle both with facility while some would struggle mightily with at least one of these methods.
Further, I suspect there are some youngsters who convert mixed numerals to improper fractions procedurally without full conceptual understanding that a mixed numeral is an addition problem!
Your experiences and thoughts...
Saturday, October 18, 2008
Adding and Subtracting Mixed Numerals - A Survey of Methods Taught
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13 comments:
I'm sorry but I can't tell you any answer to your question.
I just wanted to point out that there are several ways of doing a subtraction in the mixed numeral form, once the fractional parts have the same denominator.
For example:
3 1/5 - 1 3/4
= 3 4/20 - 1 15/20
Here's one way: subtract in parts.
3 4/20 - 1 15/20
= 3 4/20 - 1 - 4/20 - 11/20
= 2 - 11/20 = 1 9/20
Another way, like you said, convert 1 whole into 20/20:
3 4/20 - 1 15/20
= 2 24/20 - 1 15/20
= 1 9/20
And yet another, allowing a negative fraction in between the steps (this is similar to my first one):
3 4/20 - 1 15/20
= (3 - 1) + (4/20 - 15/20)
= 2 + (-11/20) = 1 9/20.
Would this "flexibility" in methods perhaps make this method (I mean where we just convert the fractional parts to like fractions) superior in the sense that more kids will be able to grasp at least one way of doing it?
Then again maybe it just adds to the confusion. But perhaps you could first show them the first step:
3 1/5 - 1 3/4
= 3 4/20 - 1 15/20
and then let the students figure it out by themselves... see what they come up with.
Thank you, Maria. Some interesting alternate methods there that deepen understanding.
I'm hoping to hear from teachers currently following a middle school curriculum. However, I am interested in others' views on the relative merits of these methods. If some teachers feel that improper fraction methods are easier for students, the other methods may be deemphasized. Do you think this is ok?
As students get older they reach for the fraction keys on their calculator or change everything to decimals and convert back to fractions at the end. Of course graphing calculators do not allow entry of mixed numerals.
I've not taught this yet, but one thing that occured to me is you have to manipulate much bigger numbers with the top-heavy fractions method... so weak students will be put off by that. Another reason to go mixed.
Here's a middle school perspective.
I agree with your comment that a higher conceptual level is required for the mixed numerals method. But most
students don't have that, and I think it is a dangerous method to teach as it leads to so many conceptual errors
later.
Teaching shortcuts only really works when students have sufficient grasp of the basic concept to appreciate what it
is that the shortcut is doing. Otherwise they lose track of the whole idea, their learning becomes
compartmentalized (if I get one looking like that I do it that way, if I get one looking like this I do it this
way......) and they can't see how ideas link together.
I think it is best to teach a straightforward method which can always be applied in any situation. That's the general addition method. Then they already know how to convert between mixed and improper, and vice versa, so it's just combining two ideas they already have - nothing new. That's plenty complicated enough for the vast majority of students.
The smarter ones will start to see shortcuts of their own, and then, maybe, we can teach them the mixed numerals method.
I think you have to be very careful bringing in carpenters and cooks here - i.e. applications for mixed numerals. Many students won't be carpenters, and many won't cook - so what's the point. In addition - most won't have any appreciation for the application, even those who may, eventually, end up as carpenters or cooks (even if only as a hobby), until much, much later. (and the world's going metric anyway!) If you are going to have your fractions lessons out in the carpentry shop or in the kitchen baking muffins, then perhaps yes, it is appropriate, but not generally.
It's very dangerous linking maths to the real world. It is true that it can be done, but it has to be a context which resonates with students sufficiently well for it to stick in their minds. So we need to pick methods that can consistently be relied upon to work, and only later, as their appreciation of the concepts develops, do we show shortcuts and clever little tricks and oh-look-we-can-do-this kind of things.
Sorry about this - I think I've gone off on a bit of a rant (and you should see the bit I saved in a file for later use). Fractions is one of those topics I've spent a LOT of time teaching and thinking about. Often I think that they are one of THE most useful things we teach, they are part of that doorway into advanced mathematical ideas that so many find it hard to get through. I've always been able to DO fractions, but to be honest, I don't think I ever really UNDERSTOOD them until I began teaching them. They still blow me away. They're so simple, yet so complex.
We have to be really, really careful when teaching them, feeling our way through our students thoughts to make sure everything is set up right. If that sounds like someone playing with a ticking bomb then I agree - because if we don't get things set up right we make SUCH a mess of things for later!
I don't think enough people appreciate this.
Oh, and the 'improper fractions means bigger numbers' thing? Come on! They have to learn to deal with that, that's our job as teachers. I can assure you, any kid who's scared of big numbers is going to run a MILE when faced with any fraction, let alone a mixed one.
Alex and David--
Thanks for your thoughts...
David, I was hoping for the kind of passionate response you supplied from someone who has lived with this issue for some time and reflects deeply on his(her0 profession. Your arguments are swaying me, but I need to hear from others. Actually, it would be so beneficial to students if teachers had the opportunity to regularly meet and discuss issues of content and pedagogy.
I still believe there are some advantages to working with mixed numerals. A sense about the magnitude of these numbers can be lost when converting to improper and the final result may be way off without the student realizing it. Mixed numerals allow for quicker estimating at the beginning of the problem and, perhaps, some exercises in estimation should be assigned for that purpose. Further, while we may not all become carpenters or chefs, we will all have occasion to measure objects (metric is still a ways off here!).
Perhaps, the mixed numerals algorithm for adding/subtracting should be taught and practiced only with relatively simple fractions. The fact that the improper fraction method is then taught and most students might prefer it does not mean the other method was a waste of time. Analogously, in algebra, we commonly teach 3 methods for solving a system of linear equations: graphing, substitution and elimination (linear combinations). Students may prefer one method over another, but I've seen students make choices when needed depending on the form of the given system. Nothing wrong with that. Yes, these students are older and more mature, but most (not all) middle schoolers can handle more than we currently expect IMO, at least here in the US. I suspect, David, you have much more to say on this issue - I welcome that and so will my readers!
I'm not really seeing much need to add or subtract mixed numbers even in carpentry or cooking. I had occasion recently to want to make 3/4 or a recipe that already contained lots of mixed number measurements (so it would fit in my bread machine) but that was multiplying, and for that I definitely converted to improper fractions.
My intuition is to stick with converting to improper fractions. To me adding and subtracting with mixed numbers is more of a curiosity, similar to adding and subtracting times.
Oh, and Dave, if your students ever realize when their result is way off, you're doing something right. :) IME "sanity checking" is a very hard practice to drive home.
I think I would agree that maturity has a lot to do with it. Consider the age at which we deal with these topics - systems of equations come much later than fractions. Consider also the processes involved. Systems of equations involve an understanding of general methods - the fact that each problem is basically a variation of a previous problem. Follow the algorithm and you'll get the right answer.
But fractions, which do involve an understanding of general methods, are taught way, way earlier than systems of equations are, and often at that stage we haven't really started talking about the general nature of solutions and methods. I think perhaps we should. Anything that involves an algorithm is an example of a general solution.
For example, I often throw huge, enormous numbers into problems. The kids freak out, but my response is 'what? what's the fuss? So it's a big number? Who cares, is it REALLY any different from the previous problem?' I'm trying to get them to look at the PROBLEM itself, not the numbers. Once kids start doing that, once they start looking at the STRUCTURE of what's going on, you're winning. Once they start to realise that each problem is just a variation of another, THEN they're learning. Then they're thinking mathematically.
Note, this is NOT the same as getting them to do umpteen variations of the same problem over and over again. That's drill and kill. No, I'm talking about discussion, challenge, forcing them to LOOK at what's happening, to compare and contrast problems - to apply higher order thinking skills to what they are doing rather than just zoning out. It's not easy, for some kids it's hell being required to actually think about this stuff, but it's worth it in the end.
I think as teachers we really need to have some appreciation of why mathematics even exists in the first place. It came from our inbuilt need to find order in the world around us. Now, that's a desire to generalise, it's not a desire to come up with itty bitty separate results all the time. So we need to work with those ideas with students. The numbers are important, but they're the building blocks, it's the structure and generalities that really matter.
Back to fractions. Again, I think it's important to work on general solutions. To develop an understanding of why they work. What exactly are we doing? What on earth is going on here? Once there's more understanding of that, we can start looking at shortcuts and alternative methods.
I do agree that mixed numbers methods could be great for developing estimation skills - I agree with that wholeheartedly. But that's different - I mean, why not just ignore the fractional bit and go with the whole numbers. Then it's just addition or subtraction of whole numbers.
Oh, I just remembered - that was one of my points - if we teach mixed number methods then later they try to apply them to multiplication and division. Now those are ideas which are complicated enough - you want to try working on the distributive property with them as well? (Have you ever really tried to convert a fraction division problem into plain language that kids understand? It's HARD!) Why not just teach them that to work with mixed numbers in maths they need to be converted so they're all one type of number, and then we can work with that. (Think about it - most of the time when we have various quantities involved in a problem, we convert to just the one).
And I agree about the discussion of content and pedagogy. It's professionally and personally enriching, and it makes us think. The more we share ideas, the better teachers we become.
You folks are certainly making me think!
Now I'm wondering where this topic resides in the K-8 Achieve, CA or Mass Math Standards. I don't remember seeing this explicitly stated. If anyone knows how it's worded in their state's standards, I'd be interested. Usually, this level of specificity is missing.
At this point, I'm still not convinced that students should not be exposed to both methods.
Dave,
I'm guessing that for word problems mixed numbers are preferred and for "math" without application improper fractions are preferred.
Time would always fall in the former category.
Subtract
15 hr 37 min
9 hr 46 min
?
6 hours with a shortfall of 9 minutes, but that's me.
Doubling back, I actually teach my high school students to do the word problem = mixed, math = improper business. The improper fractions are easier to work with, but you can get some pretty angry stares for asking for five-quarters of a pound of something or other - and you may not even get it.
Jonathan
As a newcomer to these math blogs, first I want to say these blogs are great. Thanks Dave for running this blog. I think it is very important for teachers to have discussions, just as it is important for kids to have discussions in class about concepts and strategies.
David, your posts seem contradictory to me. In the first post you say most kids can’t conceptualize a mixed number solution, so let’s make it easier for them. In a later post you say you don’t want them freaking out over the numbers; you want them to focus on the structure of the problem and solution. I agree strongly with what you say here: “The numbers are important, but they're the building blocks, it's the structure and generalities that really matter.”
My question then is: isn’t a mixed number solution with borrowing the same in structure as subtraction with whole numbers with borrowing? The main difference is that the numbers are different. The rate of decomposing the borrowed whole number is not 1 to 10, unless the fraction is tenths, but the concept is the same. It seems to me that this is an opportunity to help the students better understand what fractions are. If decomposing the whole number and combining it with the fraction is the problem, then that can be addressed separately before doing the subtraction with mixed numbers. In other words, they are not ready for this kind of mixed number problem if they do not understand that 1 + 4/20 = 20/20 + 4/20 = 24/20. If this is not where the difficulty is, then please tell me where the difficulty is.
Burt,
After re-reading all the comments I believe a balanced approach is still the best. You made a critical point in your argument. Identify those parts of the algorithm that seem to create the most difficulty and reinforce those ideas prior to teaching the algorithm. Analyzing and anticipating what steps cause the problems is so fundamental to effective instruction. This comes with experience and reflection.
On the other hand, many youngsters also have trouble with the synthesis aspect. That is, they struggle with multi-step processes and, particularly those with learning disabilities, need the recipe in front of them for some time if not always.
I still feel that students do need to develop better "fraction sense" than they currently have -- I guess that's obvious to our readers here! Expressing 1 as 2/2 or 5/5 or 13/13 is very important. I don't think we can simply deemphasize this without thinking carefully about the ramifications down the road. I also believe many middle schoolers are capable of more sophisticated thinking than we give them credit for. I see how well they can manipulate their parents -- surely they can figure out how to manipulate a mixed numeral!
BTW, I forgot that you were addressing the other 'David'in this thread and I was initially trying to figure out where I contradicted myself!
I also believe David had some excellent arguments and his sentiments are shared by many teachers with whom I've spoken.
Burt, thanks for your thoughtful comment.
Dave
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