My middle school math department is having a debate on how to teach solving equations.

I have a strong opinion on this, but am going to withhold it for the sake of an unbiased conversation. Our main differences are on whether teaching students to solve equations vertically or horizontally will lead to deeper understanding and accuracy. We’d like to teach them consistently across all 3 grade levels and are particularly interested to hear from teachers who teach algebra and higher level high school classes. Is one method more powerful for students than the other? Does having an entire math department teach one consistent method lead to more students understanding equation solving in more complex applications?

One new aspect which I plan to adopt is what a new teacher to our department taught us (I should mention that she’s new to our school, but taught 15 years in Oakland before coming to Berkeley). She uses a vertical line through all the equal signs (see below) to help students ‘see’ that equality is always maintained.

With no further adieu, I present vertical and horizontal: (and by all means, feel free to critique both notations)

And I need to make a public thank you to Donors Choose who got Chevron to give a whole lot of money to fund California teacher’s grants. I just got a $2000 tablet PC laptop and Fluid Math thanks to them.

And finally, this is no longer very timely, but 2 posts ago there was a really great conversation about how to best help students see and correct their errors.

What I tried:

- I made a spreadsheet of what % of students answered each answer choice for each question (it was an all multiple choice exam).
- I analyzed the data and chose the 6 most common errors. I used examples of students’ work and handwrote a worksheet where there were 10 problems on these 6 topics. I told students that 6 were wrong and 4 were right. I broke them up randomly into pairs and they had to analyze the student work and find and correct the 6 errors. We then discussed some of the problems whole class.
- Finally, they had 20 minutes to look at their exams and correct any mistakes they found. It was a test taking time (i.e. no talking, notes, etc). I DID NOT tell them which ones they had gotten wrong, I just wrote at the top of their exam how many problems they had wrong and how many errors they needed to correct to move 1 grade higher on the exam.

What happened?

In brief, there was very little change in exam scores. A few students went up by 1 or 2 questions. No one went up by more than 2 questions. And a few students went down by 1 or 2 questions. In terms of overall grades, it was a total wash. And most students were too scared of going down to even try to find their mistakes. This surprised me as I thought that after seeing common errors, spotting their own would have been easy. Their lack of confidence in finding their own errors (when it really counted…scores could go up or down on the exam) was enlightening (as well as disconcerting).

Next steps? Not sure. Lots of ideas…Stay tuned.

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Hi it’s me again, I had been waiting to hear how the errors session had gone. You said something really important in closing in your prior post on this subject. You said these students are used to not being able to do something. In my opinion, your strategy is aimed at students who already have some selfconfidence in maths. Your students do not fit this profile. They have no selfconfidence in maths. The foundation needed for succes is not there yet.

So my suggestion is to first focus on building up their selfconfidence to the point where you can have a conversation about mistakes. Building up selfconfidence is something I do know something about. It starts with focussing strongly and almost solely on what is going right. I expect that if you asked your students to tell you what is going right for any given maths problem, they will look at you like they see water burning. (Is that an expression in English? I sometimes translate literally from my first language and then don’t realise)

Anyway the thinking is this: get them used to the idea that they are doing stuff right. Have them notice that when they learn something, they do more stuff right. And then introduce using mistakes to learn from. If you make a mistake, purposefully, on the board, and then take them through questions on ‘ok, what went right?’ they may find the mistake or you can point it out and discuss how finding a mistake made you feel when you were their age (hooking your audience – and give them something suitably similar to what they seem to feel), and maybe they can say how they feel when they made a mistake. one or two should be brave enough, and others may raise their hands if asked if they ever felt bad about making a mistake in maths. This should have you all on the same page and starting point and then you can have a discussion of how you could turn this around by looking at a mistake as an opportunity, a chance to learn something new and add to the list of stuff they are doing right. After this step you can practise all of them finding your mistake on the board. maybe as an opener each class for a while. And only when they are celebrating the mistake found and the opportunity for learning it presents can you start them on finding each others or their own mistakes, as their feeling of confidence in finding the mistakes has been built up, as well as their selfconfidence in handling the mistake. these positive feelings will balance the negative feeling they will still have in finding a mistake they made themselves, and allow them to use the process they have been practicing and do some useful learning with it.

After all that, you can start up the strategy you tried again, and probably find it succesful as it now has a foundation to stand on.

Honestly, I can’t see a big difference between the two solution methods. It took me a few passes through the two examples to even notice what you meant by “vertical” and “horizontal”. While it is probably true that it will be better for students to see the same method across the math classes, I don’t know if this is one of those big deal things. I think the more powerful tool that you are using is color-coding.

I have a personal preference for the vertical method, because I like to draw zeros and ones over the terms that “cancel” so that students can see that we are making zero or making one as we go along.

@ Eva: I so appreciate your thoughtful response to my original post and to this one. Where/what do you teach? Your comments here are quite right..I tried to have students correct common errors in others’ work and then immediately have the chance to find their own errors….

In past years, I have done a ton of error correction similar to this http://www.teachingchannel.org/videos/my-favorite-no?fd=0

Leah, from this video, is a colleague of mine in Berkeley and I have learned a lot from her in terms of doing error correction with students. For whatever reason, I have used warm ups for other things this year and hadn’t done a lot of error correction prior to that week.

I agree with you…if we spent a bit more time as a whole class doing error correction then students would have a lot more success when given the opportunity to find their own errors on an exam.

@The Space Between the Numbers: It was refreshing to hear that you don’t see a huge difference between the vertical and horizontal solution methods. The math department at my school has had long, vociferous debates about this notation. Maybe we (the department) should agree to disagree!

Hi Allison,

I am training to be a maths teacher. It is a second career for me, I wanted to become a teacher and maths seemed like a subject that is interesting and also, there are more jobs teaching secondary school maths than teaching English, for example. (I live in the Netherlands) For me, the subject is only the medium I will use for teaching.

The curriculum used to train teachers here focuses mainly on young people just starting out in life and in teaching, which is frustrating in my position. So naturally I gravitated towards reading blogs to get involved in conversations on teaching at adult levels. It is a great addition and motivator.

I teach in a high school, and looking at the “successful” students’ work, I can say I don’t really see a relationship of one being better than the other. I will say this – to encourage my students’ understanding of concepts as concepts instead of an algorithm with a ‘right’ way, I show them as many ways as I know of to get a solution. I also say – there may be other ways to organize your ideas. But the key is be organized. If you can’t tell me WHY you’re doing a step, the don’t do it.

Apologies for the long, convoluted, and somewhat tangential comment but…hey, why not?

Mr. T: Thanks for your comments. It’s especially good to hear from high school students. Both from comments on this blog and speaking to colleagues at other schools, the consensus seems to be that the notation, whether vertical or horizontal isn’t the most important aspect. But teaching for understanding is where it’s at. I know that part…but our department had a big debate about trying to be more consistent in notation from grades 6-8. It’s nice to hear the feedback as it can hopefully bring us to a deeper conversation about student understanding and misconceptions instead of being bogged down in trying to find consensus on notation.

I also don’t think there is much difference in the methods but I do think it’s important for your department to teach a method consistently. As a high school teacher, I learned that the middle school teacher had quirky names (i.e. ‘break out of jail’ for distributive property) for mathematical processes. After I learned what they were, it was a lot easier to remind students of things they had already learned by saying the funny names. Anyway, at the least, the teachers should be aware of the way teachers before them have taught in order to create a flow in their learning and to stand on the shoulders of those before them instead of recreating the wheel.

In my current ninth grade Algebra class, all the students solved problems vertically except one. Eventually, when she would work problems on the board she would write vertically for her classmates but on her paper she would work them horizontally. Which proves to me, that when a student truly understands the concept, they can easily maneuver between either method.

For my test corrections, I have students fold a piece of paper in half hot dog style and on the left they rework the problem and on the right, they write an explanation of what they did wrong. It was helpful for some to see that they mainly made arithmetic errors; they realized they did know what they were doing, they just made little mistakes.

Also, there is an interesting idea here http://noschese180.posterous.com/day-22-quiz-day#more that you could try where students grade themselves first and give feedback and then you give feedback on their feedback.

I teach 7th-12th grade math and personally, I much prefer for students to learn the vertical method. That may just be me, or my school, but many of my students obsess over details. They want things to look right- even if those details don’t much matter. With the horizontal method, I can see my high school students forgetting to leave spaces for where to place the stuff that needs to be subtracted and having to erase each time they forgot to leave spaces. This may seem like a small problem, but it can make a relatively short problem set take much longer and could really annoy perfectionist students. Also, the horizontal method is not very good at all for division. I spend a lot of time in my high school classes urging students to rewrite all division with fraction bars instead of division signs because fraction bars are (a) much more versatile, and (b) help when the students start canceling. If students get in the habit of getting rid of stuff vertically, doing division vertically will come more naturally to them. I agree with some of the other comments in that yes, it’s just a small detail, but a lot of students find it comforting to know how they’re expected to lay things out. Also, once they’re taught one way, it’s almost impossible to reteach it in a different way because they’re set in their ways now and vertical is much more versatile in later math classes.

Lizzy: Thanks for your detailed thoughts. I have been sharing the feedback with my colleagues. We’re still divided, so having opinions from math teachers outside our school has been nice.

You meant to say “with no further ado”

Not “no further adieu” That expression has nothing to do with french for goodbye!