Addition
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| The original problem was 72 + 47. |
Subtraction
Multiplication
Some noticing and wondering:
- These students have underlying issues with place value.
- All these students have seen and used base-10 blocks. They've been taught to add, subtract, and multiply using partial sums, differences, and products. For some reason they default back to a traditional algorithm, even though many of them haven't even encountered the traditional algorithm yet. So they must be picking it up somewhere. Maybe on the bus?
- Are these examples of what I've heard called over-reliance on the traditional algorithm?
- All of the answers are wrong, but some egregiously so, like 1,000- 500 = 1,500. Or 37 x 35 = 245.
- Is there a difference in the way kids do these kinds of calculations when they are presented vertically vs. horizontally? Has anyone done any research on that?
- These are not necessarily representative; many kids can add, subtract, and multiply multi-digit numbers just fine. But do we confuse some kids by attempting to teach them multiple ways to do these multi-digit calculations? Are some kids better off just learning one way? Or at least one way at a time? This is something I hear quite often from teachers. Are they right?
- Graham Fletcher, in his addition and subtraction and multiplication progression videos, urges us not to rush students through conceptual stages of understanding. My guess is that what we're seeing here is the result of such rushing. It's also likely that there are students who are calculating correctly by following the traditional algorithm, but have little or no understanding of the underlying concept. Masked by correct answers, their misconceptions go undetected, and that is just as troubling as what we see above.
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| And there's always this. |
Fantastic selection. Will you share this with your students?
ReplyDeleteThanks John. Absolutely. I've already seen some of them. One thing I neglected to mention in the post, is that there are some students who are reluctant to use anything other than the standard algorithm, even though they employ it incorrectly. I think they see other algorithms (like the partial products, for example) as something like less "grown up". This resistance isn't always easy to overcome.
ReplyDeleteI think it's interesting you don't mention family at all. Don't you figure that's where the standard algorithm attempts are coming from?
ReplyDeleteI also notice that you assume the first subtraction problem is a result of the kid trying to use an algorithm without understanding and getting it wrong, while in similar circumstances with sixth graders, I've tended to assume they just misread the problem as addition. I'll be checking that assumption next time it happens.
Good call! Absolutely Julie, family is a likely source of standard algorithm attempts. As for that first subtraction problem, I know that the students did not misread it as addition because I was right there when it happened. But that being said, I do agree that some students arrive at incorrect answers because they have not attended to the operation.
DeleteSeems like a case study in Math Mistakes. Could be fun to do this with kids and ask what the student was doing right and where he or she lost her way.
ReplyDeleteThanks Robert. Those kinds of error analysis questions are ones we've seen appearing with increasing frequency on our program's unit assessments. Those are contrived by the test makers, where these are actual examples and might be more powerful for the kids. I like your focus first on what the student was doing right before heading to where he or she lost their way.
DeleteThese are all situations where the popular instruction "Check your working" is not going to make any difference. "Check your answer" would be much more useful, if they had some ideas on how to do this. Working back, or undoing the calculation, is the best. Stuff about checking the units digit, about magnitudes, about diagrams with a number line. None of them guarantee the correctness of the "answer" but they provide good evidence. We shouldn't be giving kids things to to if they are unable to assess the result and have to rely on "teacher".
ReplyDeleteI'm glad you've brought this up. Often in the journal work kids do they are asked to make an estimate first. Some have pretty sketchy ideas about how to do this, or even why they're being asked to do it. Few even go back and compare their estimate to their actual answer. They are asked questions like, "Does your answer make sense?" But they don't know what to base their response on. They don't know whether or not it makes sense.
ReplyDeleteIt's both the brilliant thing and the problem about teaching arithmetic - it's coming from everywhere, not just the teacher - from families and friends, from older year groups. That it's currency, and that it's obviously useful currency like this means it's not questioned (Whatever they say about algebra, not many people would say" I left school a long time ago now, and I've still not had the slightest use for addition"). It also means that students are coming to class with the advantage of having all sorts of people them show them ways and algorithms. The disadvantage is that however neat, sequential and logical the teacher or school's progression is (like Graham's!) someone's going to scramble it, maybe by introducing a way that seems more grown-up to the students, maybe by teaching it without understanding, maybe by skipping a few steps that would, a little slower, build a real solid understanding.
ReplyDeleteIt's one of the reasons I really enjoy teaching maths that isn't arithmetic: coming with the class to something that's fresh, like new snow, where they have to rely on their own perceptions and resources.
I wonder if it all goes back to Fibonacci and his book, Liber Abaci? He was showing adults and learner merchants how good the new Hindu-Arabic numerals could be - a wonderful and powerful invention. But of course elementary schools weren't big back in the middle ages, and the idea of children building up to something through intermediate stages isn't there at all in the book - "let's cut to the chase and I'll show you the algorithms". It's a meme that still has a lot of momentum!
Thanks, Simon, for your insight and perspective. I suppose the "scrambling" is just something we as teachers have to deal with on a student by student basis. Some come to us with no previous knowledge or experience, others with vague, confused, or misguided notions, and still others with strong, solid foundations. So we need to sort everyone out, try to meet them where they are, and then bring them forward. I think one issue we run into here is the fact that we have a group of kids at different stages of cognitive development all together in one class and expected to move in step together through a prescribed curriculum. Hence the rushing. If we're trying to slow something down that's been gaining momentum since Fibonacci's day, we've got our work cut out for us!
ReplyDeleteOne of my favorite instructional moves (and an easy one to insert when coaching) is to ask the student, "Can you prove your answer using another strategy?" This works two-fold. Firstly it reinforces the use of multiple strategies, de-emphasizing the algorithm. Secondly it puts the 'rightness' in the mathematical proof and off of the teacher, aka "bearer of all things correct." When they try using a second strategy and get a different answer, now they have to try a 3rd strategy, perhaps even going back to using concrete manipulatives! ;)
ReplyDeleteThanks Nova, using multiple strategies is a great idea. I actually tried this the other day with a student. She solved a multi-digit multiplication problem using the traditional algorithm. Then I had her try the same problem using partial quotients, trying to show her how the two were related. This also reminds me of Howard's idea of "undoing" the calculation. I tried this today with another student who did a division problem with a 3-digit dividend and a 1-digit divisor, then had her undo it by multiplying the quotient by the divisor and seeing if she got the dividend back.
ReplyDeleteWho knows what else they learn on the bus. Thanks for sharing these. I am going to use these with my 4th and 5th grade students to see if they can find the errors.
ReplyDeleteThanks Matt. The bus can provide kids with quite an education. If your students figure out what happened with the second subtraction problem, let me know!
ReplyDeleteThanks, Joe, for inspiring my thinking. I began writing a response but it got way too long. So . . . I've just published a blog that is a response to your wonderful post: http://marilynburnsmathblog.com/wordpress/a-reponse-to-joe-schwarts-blog-about-algorithms/.
ReplyDeleteThanks Joe! I'm a fourth-grade teacher. It is good to know that I'm not struggling alone. I get so frustrated when I see these sorts of mistakes show up in my students' work especially when I know that they are, in some ways, above such mistakes. Like you, I feel like we've done the work necessary to build conceptual understanding and we've analyzed and practiced procedures for finding sums, differences, products, and quotients that tap into the students' understanding of place value. Still, this struggle is real! If there is one thing I wish, it is that students would be more generous with the amount of think time they give themselves. I'm working so hard to encourage my students to slow down and visualize the problems. I know that taking the time to analyze what's going on in a problem and thinking about what a reasonable answer will look like will benefit my students. I love Marilyn Burns' suggestion in her blog/response to your blog. Giving students specific parameters to consider will help to both focus my students' thinking and slow them down. I'm eager to see the impact that such parameters might have on my students' success. Thanks for helping me to think about this challenge more deeply.
ReplyDeleteThanks Marie. I agree that it's good to know we don't struggle alone. One of the greatest gifts of the MTBoS is the way it connects us across schools, states, and even countries.
DeleteI think Marilyn has summed things up in her response, there's really not much i can add. It can be frustrating though. Just today a student was working out the problem 245 - 5. He lined the numbers up vertically (at least he had the place values aligned), but thought he needed to regroup, and somehow wound up with 235. Why wouldn't he just do that mentally? I saw a student the other day actually add 16 + 1 vertically and get 27 because she misaligned the place values! She didn't blink an eye. We'll keep on plugging away!
Hi Joe. So, about confusing kids with multiple methods. I have seen some curricula that treat methods like partial products or partial sums as "something to teach", and I have also seen curricula that treat those methods (and also the standard algorithm) as "strategies to be used when it makes sense to use them". I wonder if the former creates situations for parents in which they don't see the need to (in their minds) make things more complicated so they pedal the standard algorithm, which to them is much simpler. It does seem simpler, so students default to it because the other methods are seemingly more complicated and they probably don't have enough experience with them to have made them their own.
ReplyDeleteI think the question of just teaching one way has to be examined in light of our goals. If our goal is to create little compute-ers, then memorizing basic facts and parlaying those into the standard algorithm are definitely the way to go. But if our goal is to create little thinkers and inventors, then we have to find ways to encourage flexible and strategic thinking that begins with the child's thinking and goes from there. Difficult to do in a whole class setting, as you point out, but the only goal (for me) that makes sense in the long run.
Thanks for your comments Aaron. I want little thinkers and inventors! What's ironic is that the standard algorithm may appear simpler, but in many cases the computation can just be done mentally. But you need to be comfortable decomposing numbers into pieces that make them easy to combine or take apart. Knowing how to make a ten, and how to add and subtract by multiples of 10 are important foundation pieces. That all encompassing term "number sense" comes into play here as well. I think of Andrew Stadel's example of giving one of his students a stack of 100 sheets of copy paper and asking him to count out 80. Do you need to count out 80? Or will 20 do?
DeleteIt is difficult in a whole class setting, but here is where I suspect that regular doses of number talks would be a good antidote. That's where the flexible thinking can be brought to the surface and the different strategies discussed and analyzed.