Addition Numbers & Operations in Base 10

How Many More Miles Total?



The submitter of this mistake notes,

This mistake brings up the concept of teaching with keywords to me.  I asked the student to tell me how he got his answer; he pointed to the word “total” and said that he needed to add.  I’d like to know what ideas educators have to intervene when a student is already clearly looking for keywords and is not making sense of the problem.
What do we mean by “make sense of a problem”?
Are we imagining an all-math skill, tools that can be used to make sense of any math problem no matter the topic or age of a student? Something like “read the problem carefully!” or “draw a picture!”?
Or are we imagining a local skill, some way to make sense of this problem and problems like it? Something like…well, I’m not exactly sure what would help someone make sense of this problem. Maybe, “if you see names, you might be comparing!” or “if you see a lot of numbers in a problem, rewrite them in a list so you can focus on what the problem is asking”?
I suppose that I’m inclined to think of keyword mistakes as a lot like applying the distributive property where it doesn’t belong or other “over-extending” mistakes. Generalizations are smart things to do, and a keyword generalization is a smart thing to think also, and it’s usually correct. Knowing that this mistake exists, I might create a set of problems all that contain the word “total” with some being summing up problems and others being compare problems and so on. My reasoning is sort of simple: this kid thinks that “total” means add ’em all up, so let’s provide him with counterexamples and then urge him to make a new generalization.

20 replies on “How Many More Miles Total?”

I’m waiting for the student who points to the word “some” and tells me that it means “add.” :^)

Seriously, though, I think most experienced teachers could have called this before reading what the student said. And yet books and various “experts” continue to provide lists of key words and promote looking for them in math problems as part of the magic way to increase test scores. And therein lies a major part of the problem: as long as you teach math as the discipline in which getting higher test scores is the goal, kids are going to be inclined to look for and be deceived by simplistic reading of problems. There’s simply no substitute for decent comprehension skills coupled with a bit of common sense. Math isn’t the only place where being this simple-minded will lead to glaring errors, of course, but somehow it seems so sad to see kids reach absurd conclusions by using cheap tricks they’ve been encouraged to employ to AVOID mistakes.

I like to talk about making a “map” of the problem. It says, when we strip it down, how many more miles Denise ran than Jamal did. So we know we need to subtract SOMETHING. OK, what numbers are we subtracting? How do we get those numbers?

I also am really big on the use of “what kind of thing are we looking for?” A distance? An angle measure? A temperature? Math is a way of making sense of the world, and for our answers to make sense in the world, we need to know which kind of measurement a number means. (I.E., I’d also want students to find how many miles, and maybe have a distractor answer with a common answer number, but the wrong units.)

Sadly, keywords were the first thing I thought of when I looked at the student work. After many failed attempts (on my part) of asking “Does that make sense?” after they have arrived at the answer (right or wrong) and getting “Yep” I have started to think about this a lot when assigning questions.
The best I have come up with is holding Ss accountable for making an estimate or giving a range that makes sense before they jump in. It is my hopes that after practice with this process of estimating, then solving, then revisiting, it becomes metacognitive?
I find Ss dont like to do the reasonableness thinking after they have arrived at the answer, not that they shouldn’t but they dont. They see the work as done. By having them make a reasonable estimate or range, they are more inclined to check back with it after they finish.

They don’t like to do reasonableness because since they never did any sense-making, “reasonable” doesn’t even exist for them. So it seems like a dumb idea, or they don’t even know what it would look like. Back to the beginning – retell the story in your own words would be a good place to start here to get at, “So, what’s going on in the story? Can you tell me about that?”

This is so true Annie, it all starts at the beginning with sense-making. That really seems like the key to them really be invested in the problem to want to think about reasonableness in the end. In thinking about an estimate beforehand, I hope they are pushed to make sense of it before solving.

there’s a sort of “iatrogenesis” at work here: keyword-recognition is *actively* promoted and in trying to instill, say, better reading skills, we can confront the need for “unlearning”. and here’s what think is an under-noted twist. “first read through the whole problem”. this is a bug not a feature in my working world for real. i’ve found it useful to urge certain students to instead “begin by looking ahead at the *question*” (but not as useful as i’d like… mostly i tell them this because it’s what *i* do and mostly they steadfastly refuse to take me seriously; still, for me it’s pretty close to the *heart of the matter* when students can’t yet understand how to set up “word problems” [“define variables” (with units, of course)] and me sitting there listening to them read through a problem they really ought to have been thinking about for a while now doesn’t help *me* in any way…). i also agree with MPG that test-taking has quite a bit to do with this pathology.

I hope the writers of this test put in a note that everyone who selected a) probably needs a lesson in reading comprehension more than math.

well i think we could start by making actual understanding a priority over test scores.

I think it is important to discuss the language of Mathematics so students realize that total does not always mean addition. I work with my students to try to teach them to look at things in context to get the full meaning. Mathematics does have contexts as much as English.

Technically, there is addition in it. It’s just not the only thing we do.

(Plus, if I were writing this problem, I wouldn’t include the word “total.” For some reason, that word tends to throw students off who might otherwise know exactly what’s up.)

With my 8/9 yr olds this year we’ve looked at the way we make this kind of mistake. Earlier in the year I asked them “There are 25 chairs and 5 tables in a classroom; how old is the teacher?” and they all gave me numerical answers! We discussed why we do this, and looked at how other older students do too.

I was really pleased when one of my students reprised this recently:

I think seeing the humour in this kind of question might be one of our ways of trying to get understanding. We talked about how, like a magician, the question is distracting us. Perhaps because a question like this really tries hard to do the distracting, students enjoy uncovering its trickiness more.

I wonder what would happen if you had separated the sentences so that it was clearly talking about a comparison?

I might start with the question and ask the kiddo to pretend nothing else is in the problem and make up a problem of his/her own that would have information to answer that question.

No one has yet considered having the student create a picture of the problem a la Singapore Bar Modeling. Drawing a pictorial representation could go a long way toward helping the student see the problem as more than numbers. I have had considerable success with this method, especially when introduced in the younger grades.

I’m not familiar with the name of that method (yay, something cool to research!) but I agree that diagrams/pictures of a math situation can help to strip away all the weirdness and help students to conceptualize exactly what’s happening and why.

(Plus, I tell my college students, a lot of the sciences–especially physics–tend to have problems that are a lot easier to work if we draw a diagram. I tell them horror stories about times when I thought I knew what was going on, but because I hadn’t drawn a picture of what I was looking for, I missed a vital part of the problem!)

I have been teaching math in elementary school for over twenty years and in that time I have found only two “game changers” 1- Hands on Equations and 2- Bar Modeling. If you teach preservice elementary school teachers this is a must to help kids get a natural picture of the situation. Students create bars that represent the numbers and the situation to lead them to a proper numerical calculation. There are bar modeling apps for iPads and YouTube videos. The time you invest in understanding how useful bar modeling will be worth it. Good luck.

I think the word “more than” should be pointed out here. If one student ran more than the other, and we’re asking how much more, then in addition to the total (of what?), we’re also going to subtract something.

When that’s explained, it shouldn’t take too much more thought for them to get to, “we need to find the total for Denise, and the total for Jamal, then subtract.”

The student is right, total does mean add. He just got it wrong where the adding takes place. The question is “what is the difference between Denise’s miles total and Jamal’s miles total”. You draw their attention to the root of the question – “How many more” – this means you’re looking for a difference, and explain that “total” here means adding up each one’s individually, i.e. you’re not asking about the difference for each day, but the difference total accumulated over the entire week, which is the difference between the totals.

When I was scrolling through the math mistakes and came across this problem, I deemed it to be worth looking at. To adults, this may seem like a very simple addition and subtraction problem because all you have to do is add the miles for each person separately and then subtract the bigger one to the smaller one. This is another one of those problems that is phrased in a way that could mislead the student. When it uses the word ‘total’, many students are primed to think that it’s asking for literally the ‘total’, which means the whole number. However, it’s easy to misunderstand the question if not taken the other parts into context. This question does include the word “more” which can be seen as a subtraction problem, so maybe the student was confused by why there is a “total” and a “more”. I think in today’s elementary math, it’s important to teach these words to the students and show them the possibility that more than 1 can appear in one problem.

I personally think it’s smart to look for keywords when solving word problems- it’s easier to figure out what the problem is asking for. However, before looking at keywords, it’s vital that the student look over the whole question; It’s true that the word ‘total’ is included in the question. However, the phrase ‘more miles’ indicates that it will be a problem regarding comparing A to B. I taught my cousin how to solve math problems in english, and in these kinds of questions, I advised my cousin to look at who the problem is about. For the problem above, there are two characters: Jamal and Denise. Then it’s smart to look out for two different situations: Jamal’s situation and Denise’s situation. Likewise, in the problem above, we see how much Jamal ran and how much Denise ran.

The correct way to understand the problem is as shown:
Jamal= 10+5=15
Denise= 12+7 =19
Jamal ran a total of 15, and Denise ran a total of 19.

How many more miles total did Denise run than Jamal?
Denise-Jamal =19-15=4 miles.

The mistake that this student made in this problem is perfectly understandable; It’s reasonable to think that students get problems wrong because they have trouble understanding what the question is asking. It’s important for teachers to make sure that students have clear understanding of forming word problems into mathematical statements.

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