“Going Over” Problems

One of the most common things that I do, and I figure most teachers do, is to ask students to try a few problems, and then address that work as a full class. For the purposes of this post, I’m going to narrow the focus: I’m less interested at the moment in a problem-solving or inquiry activity, but instead on humble practice that solidifies learning — problems similar to ones students have seen before, that most will solve successfully but some will struggle with, both as formative assessment and a teaching opportunity. To drill down even further, I’m not going to focus on times when the majority of students are making mistakes — I think these are more clear-cut, at least for where I am in my teaching at the moment. Instead, I’m thinking about times when the majority of students either got questions right or think that they got questions right.

My go-to moves come in three general styles:
Pick a high-leverage question or two and either question our way through it as a class, or analyze one or two examples of student work

Moving on:
Folks are doing well, and I feel like I can address misconceptions individually or in some other way — instead of taking the time to look at questions students were successful with, let’s just move on to another learning opportunity.

Giving answers:
Give students answers to the problems, have them check the ones they got to, then move on.

I don’t love any of these. Questioning is one I often use, but when the class as a whole has been successful, it is hard to engage the group in looking at questions they have already completed, especially when they can sense that most kids can do it. Many times there is a unique strategy or particularly interesting problem to look at, but when there isn’t this seems like a bit of a hollow activity, and can take a bunch of time away from students actually doing math. Moving on is my favorite, but I worry that I send the message that the questions I asked the class to do are unimportant if we don’t address their work. And while giving answers makes that work more meaningful, it seems to reinforce answer-getting habits, and I’m skeptical it does a whole lot to promote learning.

So I asked Twitter over the weekend, and got a huge number of responses from some really thoughtful folks. Here’s what I learned:

Moving on is totally fine
If students are getting problems right, I should just tell them that, and say hey, we aren’t going to linger here, let’s move on to something more challenging. If I message the purpose behind it, and I don’t make a habit of giving students a cakewalk of a problem set on a daily basis, I shouldn’t feel any problem with doing it.

Giving answers ahead of time
Got this one from @k8nowak, and it’s one I’ve actually never used.
Screenshot 2015-05-27 at 10.57.44 AM
Kate talked about how he makes it routine from the beginning of the year that she will often give the answers to a set of problems to students ahead of time, so that they can see how they are doing. This also focuses student attention on the process, rather than the product — it underscores the emphasis on quality of work because students get feedback, and frees up teacher attention to what the students are thinking and how they are approaching the math, rather than telling them whether they are right or wrong.

Pose a similar, but different, problem
This one was from @hpicciotto, and is my favorite.
Screenshot 2015-05-27 at 10.57.12 AM
Henri proposed that, after students practice a set of problems that they feel confident with, but I know there may be some lingering misconceptions, that I pose a similar, but different, problem on the board for students to try. All students are actually doing math, rather than reciting math they have already done. It sends the message that the work students had been doing was important. And it exposes student misconceptions without having to coerce students into participating on a problem they’ve already tried.

These are small things, but the little things can make a huge difference, especially the small things that I do day after day.

6 thoughts on ““Going Over” Problems

  1. Henri Picciotto

    I totally support Kate’s approach. I learned it years ago in the ground-breaking, way-ahead-of-its-time _Geometry: A Guided Inquiry_ by Chakerian, Crabill, and Stein. They routinely gave key answers right there in the margin.

    Here are some variations on it (not from that book):
    – If I didn’t give the answer ahead of time, I might write it on the board when some students are done with the problem and are forging ahead. This can be useful whether or not those ahead-of-the-pack kids have it right or not.
    – I might give an answer ahead of time, which I predict will be an incorrect student answer, and state that it’s wrong. That can jump-start the necessary confrontation with a misconception.

    Note that this whole “going over” discussion also applies to homework.

    1. dkane47 Post author

      I like highlighting the likely misconceptions ahead of time — it’s another great step that encourages process over product, and gets students getting metacognitive while doing math.

  2. katenerdypoo

    i think you have a really wrongheaded idea here that i want to address:

    “While giving answers makes that work more meaningful, it seems to reinforce answer-getting habits, and I’m skeptical it does a whole lot to promote learning.

    i think this is a weird american quirk to not have the answers (i can vaguely remember my old american texts only having the answers to the odd or even numbers) readily available for the students. in both the dutch and international textbooks i use, every single answer is given and i *require* students to always check their answers.

    not having the answers not only makes the work less meaningful, it is potentially detrimental! whatever students practice, they get good at. say you set ten problems and the student diligently does all ten…incorrectly. now they’ve solidified their mistake in their mind. they don’t even know they’ve done it wrong. what is the purpose of this practice? how does this help them develop as a thinker? isn’t it more beneficial for the student to check their work, discover they have the problems wrong, analyze the problems to see if they can figure out why they have it wrong, and if not bring questions to the teacher the next day?

    1. dkane47 Post author

      Thanks for this, Kate. I think you’re right on. For some context — the school I teach at writes our own curriculum, and doesn’t use textbooks — everything is produced by the teachers. Maybe in part because of this, that’s a part of my classroom culture that has been a given for me — if I’m not habitually giving answers for students to self-monitor, and only giving them at the end to pass judgment as yes/no and then move on, I help to create an answer-getting mindset. I’d love to commit to that approach from the start of the year and see how it changes student perceptions of practicing math.

      Thanks for your thoughts!

      1. Carmel Schettino (@SchettinoPBL)

        This is a great post Dylan – some very insightful questions. I, too, use my own curriculum and don’t use a textbook so right from the start I’ve always told the kids there’s no “answer key” – they get used to not looking or asking for the answers. I have them present what we call “partial solutions” every day. Whether they know it’s right or not – so the discussion hinges on students commenting/addressing their process and viewing the multiple perspectives that they all have – the answer is the least important part of the discussion. As a side note though, there is also no lecture – the problems are scaffolded to introduce new material that the kids learn by doing the problems – so I would agree that if the goal of the problems is simply to practice material that was taught by direct instruction the day before, it’s pretty important for students to know for sure if they understand the teachers’ process I guess. That gives “going over” problems a very different meaning.

  3. Jen spencer

    I think that there is only a need to call the class back to “go over” a problem (using whatever strategy you like), if you don’t feel like 1.) you have been able to give frequent, widespread individual feedback a la rigorous circulation habits and practices 2.) the feedback you are giving is the same for much of the class and you are tired of repeating yourself 3.) you want to use a particular IP question to sum up the concept. I think there is this idea that you NEED to go over problems in front of the class so you know everyone “got it” but it just might be better to circulate clearly and efficiently, having a 2 second touch and go with each kid, on each circulation, to have the kids and you know they are on the right track. I found that the single biggest leverage point in my teaching was to give frequent, predictable feedback (check/circle/pre-planned prompt) in a predictable circulation pattern.


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