Three weeks into the new academic year, it is clear that thirty to forty percent of my algebra 1 math lab students, all 9th graders, misunderstand subtraction, even with whole numbers. Some of them even took algebra 1 last year and received As, Bs, and Cs, while others received Fs, and yet others took algebra readiness or a similar course, also with grades ranging from Bs to Fs.

When faced with an expression like the following:

most students usually find the correct solution, which is 3. However, reverse the order as follows:

and half of my students write their answer as

which is incorrect, as opposed to the correct answer of:

even when the earlier example preceded this one.

When asked how they arrived at positive three instead of negative three, to a one they responded that they subtracted two from five. Why that is so was not as clear. It was difficult to get a clear explanation of why they started with the five in the expression. I asked if they saw the five first, since it was largest, and many nodded that was the reason. Whether that is the case or not is still unclear.

What is clear is that these students have not developed a sense of number, or more specifically a sense of operations with numbers, especially when denoted symbolically. In an attempt to explain various methods to evaluate such an expression, I reminded students to read from left to right, as they did any western-based writing, to include mathematics. While the order of operations dictates the order in which operations are usually carried out within any expression, students should still follow a left to right sequence through any expression to reduce errors with operations.

Additionally, I reminded them to think about the expression, and what it implies. Specifically, in the first expression above, they are asked to take away three (*items*) from five (*items*), which obviously leaves two (*items*). However, when asked to take away five (*items*) from two (*items*), they need to recognize that they only have two (*items*) to subtract from initially. Hence, they need to borrow three (*items*) to complete the operation. I used money as a context for this class of problem, which they seemed to understand. However, some may not have understood.

For a visual representation, I showed them how to denote two positive items in one color (blue) from which they are to add five negative items (red), which when the positive and negative numbers cancel each other out, leaves three negative items in red. I also used positive (“+”) and negative (“-“) sign symbols to denote the different numbers in a problem, which followed the same logic.

Additionally, I showed them a number line where they start from two and move left, the negative direction, by five arriving at negative three.

Students seemed to appreciate our discussion, and signaled they understood. However, in subsequent, similar expressions, they made the same mistake, likely a result of continued misunderstanding.

The following day, after reviewing the principles we discussed the prior day, students were asked to work on a double-sided worksheet with integer addition and subtraction on one side and multiplication and division on the other. Most exercises involved one and two digit integer operations, mostly with two numbers, sometimes three for addition and subtraction. The following table depicts student results for a sample of six expressions.

As can be seen, students still struggle with subtraction. Except for the the expression -2 + (-3), a sizable number of my 9th grade algebra 1 math lab students have not attained mastery of a grade 5 / grade 6 level standard. Without mastering this concept, and any associated procedure, they will continue to struggle with algebra. That is simply not acceptable to me. So, I will continue to work with these students to ensure they conquer operations with signed numbers.

These students missed the concept when they were first introduced to subtraction. A common issue I face with my students is the lack of effort on the part of the teacher responsible for teaching these concepts (very early on). I also encounter teachers who admit to “brushing over” difficult concepts because they are unsure about how to explain and teach the concepts. This obviously causes problems in algebra and higher level math topics. Best wishes!! Let me know how you get the concept across at such a late stage in the process!

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Agreed that they misunderstand subtraction. To what extent its the fault of an earlier teacher, versus a variety of other factors complicates how best to resolve the issue. There are degrees of understanding and abilities in all people, students and teachers alike, even with what seems like the simplest of things, in this case, subtraction.

The most learned of teachers may have difficulties imparting deep understanding to all students, especially when the time allocated to cover a list of concepts is not matched to the time it takes for any student to learn the concepts. I believe this is the fundamental problem in education today.

We expect anyone and everyone to learn at the same, or very nearly same, pace irrespective of individual differences. On its face, this is ridiculous, since adults are unable to master hardly anything (of challenge) at the same rate. Why in heavens sake do we expect young children and adolescents to do so? The only reason that makes sense, at least when taken in historical perspective, is the necessity to educate as many people as possible, as efficiently as possible. The resultant educational system satisfied that goal in some sense by rapidly increasing the number of citizens with high school educations. However, it was not designed to maximize learning for all, which at the time may not have even been something discernible as a need.

Today, with the advent of sophisticated technology, we have the means to craft learning solutions that can be highly individualized. However, it has significant upfront and ongoing costs. As the Tea Party, and other tax averse factions, rally against taxation, they prohibit additional investments necessary to evolve our education system. So, we are stuck with a nearly impossible task of educating an amazingly diverse student population, regardless of their preparedness, or even desire, to learn what society deems necessary, all the while those who could help improve the situation remain self-absorbed in their personal affairs, and not that of our great nation.

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Excellent and compelling points!!! I face the same issues and share similar arguments. I must pick and choose my battles. You seem very persistent and focused. I’m sure you’ll make the best decisions for your students, under these less than ideal conditions!!

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I do my best, which is all I can ask of my students. BTW, I’m about to post my reply to your comment, as writing it lit a fire in me. Thanks for the opportunity to express my thoughts on the matter.

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You’re welcome!! I look forward to reading future posts. You say the very things most of us only dare to think about!!

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The kids who can’t master subtraction are unlikely to ever stop struggling with algebra. On the other hand, it’s an overstatement to say “don’t get negatives, don’t get algebra.” They will have an imperfect understanding of both. I wouldn’t recommend beating the horse to death. SPend a month on it and two months later, they will still have forgotten it.

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I have a hard time accepting that these kids cannot master subtraction. Call me dense, for now. After lengthy efforts, I might revisit my perspective. I did not mean to imply they cannot get algebra, or aspects of it, by not understanding subtraction. I simply meant they will continue to struggle with algebra, much like my calculus students may get the calculus concept, but struggle with the algebra or trigonometry procedures necessary to work the problems successfully. I’m still in the honeymoon phase as a new teacher, albeit one with lots of grey hair.

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Thank you. If we keep handing the students 500 pound weights and saying “well, some of you will never lift them anyway… but I’ll expose you to it…” we’re doing a disservice, but we do that in math pretty much all the time. Students get behind… so instead of filling in the gaps, we go back to where they fell off the train and … run through the curriculum a little faster, every year, because after all, t hey need to catch up. Then we wonder why they don’t ever get it…

You might want to see how many of ’em don’t really get *subtraction.” See how many of ’em can figure out what to do with a “what do I add to 33 to get to 80” question (or, for many of ’em, a concrete version of that). When they don’t know what to do, and I then ask them “well, what do I add to 99 to get to 100?” they know “one.” I ask, “how did they get that?” and they say: “I added.”

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I agree asking someone to do something they cannot makes no sense, which is why I’ve reduced the 500 lb. weight to 5 lb. ones so they can lift them. They may still feel like 500 lb. weights, so I spot them in case they falter. Seems like you are having a bad day based on your two comments.

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I’m wondering who’s having the bad day. (Wondering… not knowing.)

My intention was to thank you for *not* doing the 500 pound weight thing.

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My bad. I did not see it that way, but I read that comment after the one on my MOOC post so might have been predisposed…good to know it was a thanks. 🙂

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I suspect that you are not seeing a conceptual problem as much as a cognitive mapping problem. I can do all of these problem, but I’m aware that when I do the problem 2 – 5 = ?, that I have to map this problem into two pieces, a magnitude and a direction in order to get the correct answer. I have to think spatially, not just symbolically, to understand the answer. If I or one of your students gets that the magnitude of the difference is 3, we’ve got at least half of the answer and 90% of the concept. The trick is to get them to realize that all of the problems are more than symbolic.

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I get the magnitude and direction aspect from 2-D vector perspective, which I explained via a number line but not as vectors per se, just as a starting position then movement in the positive or negative direction. It did not click, or I suspect they preferred not to use that approach for whatever reason. Same thing with manipulatives, be they written as + or -, red or blue, or double-sided discs (red/yellow) where one side is positive and the other negative.

Not sure if they do not like the visual / manipulative approach, or are so conditioned to the algebraic / symbolic approach, albeit in a misconceived way.

Still trying to decode this puzzle…thanks for your thoughts though!

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If we take way ( subtract ) something good ( positive ), we are worse off ( more negative, or less positive) than we were previously. On the other hand, if we reduced ( subtract ) something bad ( negative ), then we are better off ( less negative, or more positive ) than we were before.

Maybe more philosophy will do it.

Dave

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Dave – I’ve seen this issue with negatives a lot at the middle school level. Initially I think a teacher explains the concepts, but then it boils down to series of rules for students to memorize and the concepts are forgotten. Some people have a harder time keeping track of all the rules. And the rules are different for add/subtraction and mult/div adding to the confusion. I think the crux of the difficulty is that some of the signs are hidden, making it more difficult to understand and to apply “the rules”. I notice that the problem, -2+(-3), where students did best, displayed all the signs. Fundamentally I think many students do not understand that a minus sign is the SAME THING as a negative sign, such that 5-2 = +5 + (-2) and also that -2+5 can be re-written as 5-2. Students need to understand that -4-15 (least successful problem) is the same thing as -4+(-15), a format they apparently understand well. Notation is important as well as concepts.

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Hi Anne. I’ve explained the minus sign (as the operation of subtraction) is the same as the addition (operation) of a negative signed number, and vice versa. Some got it, but very few. It might have been more helpful to learn this in earlier grades.

I agree that writing a problem such that the signs of the addends are the same helps greatly. However, not all exercises are posed that way, and unfortunately, students are not clear on how to rewrite them accordingly. A group of my students refer to a process they call “KCO,” or “Keep-Change-Oposite,” where they manipulate the terms in every expression into signed addends. It seems to work for them most of the time. Coincidentally, it translates the expression -4-15 into -4 + (-15), albeit they do not include the parentheses when they implement KCO. I need to look into that “method” more.

Thanks for your thoughts. And on notation, if only my calculus students understood its importance better! Working on them, too. 🙂

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Of course, if we tell ’em a minus thing is the same hting as a negative sign, then when they see 4 (-5) they subtract. I spend a fair amount of time emphasizing that no, minus and negative are not the same (but that they do share that they’re telling us to do something opposite, which also helps with the subtracting negative things becasue the opposite of the opposite in language often gets you back where you started).

I stuck some camtasia explorations of integers on line at http://parkland.libguides.com/mat094cas with a few other visual and metaphorical ideas.

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In my classes we called it Keep, Change, Change. Of course if they know the proper ” negative ” symbols to use on the calculator that will work.

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Maybe it was all that fluffy discovery based, half right is OK, reform math they did in elementary. Instead of getting drilled and killed every time they messed up.

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