Sunday, October 12, 2014

Thinking Through a Decision: Tom and Addition

The Class, The Kid, The Activity:

I'm going to share with you a situation from my 3rd Grade class and a decision that I made. Then, I'm going to ask you what you'd do in this situation.

This post comes with a question and a meta-question. The question is, "What's the right teaching decision?" The meta-question is, "How much context needs to be shared so that we can ask useful teaching questions?"

I'll attempt to provide context at three levels: about the class that I'm teaching, about the kid that I want to focus on, and about his work on a particular activity.*

* I'm unsure how important school context is to this. Briefly: I currently teach at Saint Ann's School, a private school with competitive admissions in New York City.

The Class: 3rd Grade Math

My 3rd Grade class has 11 students. My school has about eighty 3rd Grade students, and about half of them are in a "fast-paced" track and half of them are in a "regular-paced" track. This class is a regular-paced class. We meet less often that most math classes meet: 4 times a week, 35 minutes a day.

We've been studying addition for the first five weeks of school as part of our first major unit. (I have lots and lots of curricular freedom, but I've elected to mostly use the TERC Investigations activities so far.)

Here's a sample of some of the work that we've been doing in class over the past few weeks.

In addition to our main activities, we've often been starting our sessions with number talks aimed at helping students build a greater facility with efficient mental addition.

The Kid: Tom

Tom is a 3rd Grader. (Tom is not his real name, but privacy etc.) Tom has worked slower than other students in class on most problems that we've done this year. I randomly call on students fairly often, and when Tom's name gets picked he often struggles to articulate a line of reasoning.

On independent work, Tom works slowly but carefully. He often articulates his reasoning clearly using equations. Here's a sample of Tom's independent work.

Tom has had moments of great thoughtfulness over the first few weeks of school, and it's been fun to watch him make progress. In sharing his reasoning he sometimes makes mistakes ("65 cents and 25 cents makes a dollar") but when given a chance to slowly think it through he's able to improve his thinking ("No that can't be right because 65 and 20 make 85 and 85 and 5 make 10...")

Another nice moment from Tom: he solved 27 + 8 mentally by breaking 8 into 3 + 5 and "making a 10" with 27 and 3.

In short: Tom uses a variety of strategies quite well, but his addition is inefficient and error-prone.

The Activity: Formative Assessment for Addition 

After a few days of some rich and interesting number talks, I wanted to know what addition facts my students still struggled with so that I could properly steer things in the next week. I knew that a few of my kids, including Tom, still struggled with mental addition, but I didn't have a good understanding of the types of problems that they struggled with. To assess which addition facts my kids "just knew" and which they had to figure out, and also whether they could figure them out, I decided to give my class 33 addition questions and ask them to work on them alone.

Knowing that these sorts of activities sometimes cause anxiety in students, I was careful in how I framed the activity for kids. I didn't tell them that the activity was timed. I took care to make sure that there was another activity ready for students who chose to move quickly that they could silently access. I also made it clear that I expected there to be questions that they didn't "just know."

Here's a video of how I introduced the activity.

Here's Tom's work on this activity.

(About the stars: instructions to the class were to "start with the ones that you 'just know' and put a star next to any that you need to think more carefully about, and then work on those. By the time I noticed that Tom wasn't returning to the ones he skipped, I decided that this would be helpful information anyway and there was no need to have him work on these problems. That might have been a mistake.)

I noticed that Tom has very little problem with adding 9's. Knowing his sometimes-facility with certain addition strategies (and his lack of quick recall of addition facts in general) I think that Tom is reasoning relationally here, e.g. 9 + 7 = 10 + 7 - 1. This is nice.

I noticed that Tom skipped lots of questions that involve adding 8 or 7. I noticed that he didn't "just know" 3 + 7, which might have been a useful fact for figuring out 7 + 6 or a lot of the other 7's.

Tom's work wasn't typical of the class'. Most students slowly, but accurately, answered all the questions. Casie's work (below) was more typical of the class. (The rest of the class' work is here.)

The Question and a Decision:

I've had students like Tom in every class I've ever taught. How do I help Tom while still pushing the rest of the class forward?

After thinking things through, here's what I decided:
  • It seems to me that Tom could use some more strategies. In particular, there's a handful of additions that he could know very quickly if he started seeing 6+7 and 8+7 as "near doubles." 
  • Tom could also answer a lot of the problems that he missed using a "make 10" strategy. But in whole-class situations he's already used this strategy. Why didn't he use it on this assessment? He might just not have made the connection, or he might not "just know" the numbers that combine to make 10. (He missed 3+7 on the assessment.) I also think that "make 10" requires you to hold a bunch of numbers in your head, and that this strategy tends to be more cognitively demanding than other strategies. He might not be ready for it yet.
  • Tom needs a chance to rethink the questions that he didn't get and make it an explicit goal to improve on them. Here's how I'll do it: I'll do a number talk that makes some of the useful strategies explicit, then I'll return their assessment work and ask them to work on any of the questions that they skipped or found difficult. I'll suggest that they use some of the strategies that were just discussed. (An issue is that many kids won't have any problems to work on, and I don't want to draw a distinction between those who do and those who don't. I'll mitigate this by giving everyone their work to improve and a follow up assignment that they'll work on individually and silently.)
  • Over the next few weeks I need to give Tom an opportunity to practice and learn some of these additions that he struggled with. I'll do this mostly in the context of number talks, because these are easily differentiable. I can put four problems on the board that increase in difficulty, and ask students to mentally work on all four. This could give Tom a chance to practice "6+8" while giving Casie a chance to try "46+68" in her head.
I'll want to check back in a few weeks to see what sort of progress Tom (and others) have made.

In Conclusion:
  1. What here would you do differently?
  2. Should I be giving more individual feedback to Tom? How would you give it?
  3. Why else might Tom be having trouble here, besides for the ideas that I suggested?
  4. Missing from all this is the social aspect -- how Tom interacts with other students in class. How do you think I could represent this?
  5. Oftentimes I'm unconvinced by a piece of writing about teaching because I worry that it's an unfaithful representation of the classroom. Do you feel convinced that this representation of my class is faithful?


  1. It's representative.

    He's not memorizing math facts.

    Here's the problem: some kids can memorize with more practice, others can't. Right now, you don't know which type Tom is, and that's worrisome because it can stress him out.

    What you want to do is figure out a way to let him memorize these, not "make 10". And then see if he can. Understanding that he might not be able to.

    And if he can't, then that's as far as he can go. I have high school kids who can't memorize. It doesn't mean they can't manage math.

    1. Strategies (like "make 10") are a helpful path towards committing addition facts to memory. I see this all the time in my elementary classes, and it's also a core result in math ed research (e.g. the CGI program).

  2. I don't disagree; there's nothing wrong with the "make ten" per se. But at a certain point (as you notice) most of the kids do memorize. So what do you do with the ones who don't? As you point out, the rest of them are ready to move on. I'm pretty sure there are countless teachers who thought well, I'll just let Tom work on this for a while longer. In some cases, Tom finally memorizes. In others, he doesn't--and now he's even further behind.

    I wrote about it here. For all the talk about the importance of memorization, most of the pushers ignore the fact that some chunk of kids won't ever memorize.

    I think it's generally irrelevant how he socializes with other kids, but what would be nice is recreations of conversations with you.

    1. Re conversations: agreed. I didn't want to reconstruct them from memory here, but now that I know that I'm writing about Tom, I'll take better notes on our interactions.

      In your post, you observe students who know math facts but flail at higher math, as well as students who don't know math facts but succeed at higher math. Sure, I'll agree with that.

      Thing is that I don't really see any reason why Tom can't learn these math facts. He's making progress. And even if it's not the end-all of math education, it's still very helpful knowledge.

      Even though I've had students like the ones you describe, I'm as-yet unconvinced that there's some portion of the population that is, for whatever reason, incapable of committing math facts to memory but otherwise capable of lots of other math. "Incapable" is going to be a tough claim to defend with evidence, no? How do you show that someone is incapable of something?

    2. I work with lots of students who are making progress... who I think *can* learn the facts. If it's not built in to their lives, though, they're not likely to. Mine have this problem: because they're behind in math, it takes them longer. That kinda means they don't have time to do extra things like ... learn their facts.
      I am watching 'em drill and drill and drill to get through their ALEKS problems, though, so I'm pretty sure that if that were included in their little "pies," they'd do it.

  3. I wonder if he might be a kiddo who struggles with multi-step directions (especially when spoken), and didn't get past "put a star next to teh ones you need to think about."
    In my experience kiddos who struggle to memorize *sometimes* do a whole lot better with one sensory channel than another (sorry, the "learning style is bunk" thinking is gross oversimplification)...
    he might also have trouble with what 7 and 8 are... friend of mine could not at age 40 add 7 and 4 to save her life ... but when we worked with understanding the quantities and broke it to four and 3 it was mildly life-changing.

  4. has some ideas I've found useful.

  5. Hi, Michael. You mentioned that Tom is having trouble seeing that 7+3 = 10, and you comment that the "make 10" requires kids to hold a lot in their heads. How about "make 5" as an intermediate step? For instance, to analyze 7+3, let's take the 3 and "make 5." To do that, we need 2. Now decompose 7 into 5+2, and we get this: 7+3 = (5+2)+3 = 5+(2+3) = 5+5 = 10. What do you think?

    Also, I was recently introduced to the idea of a "10-frame." Have you seen this strategy? If you put 7 red dots into a 10 frame (2 rows with 5 boxes each), you can instantly see that you need 3 blue dots to fill it. How about having having him do this a bunch of times: 7 + ? = 10...6+? = 10, etc. But make it visual with the 10-frame and some counters (or virtual counters).


  6. This is a wonderful post! It's interesting to hear your analysis and read the comments from other teachers. Everyone who has posted before me has made some great points. I especially agree with SiouxGeonz – I think Tom might have difficulty understanding multistep or verbal instructions. Did you mean for students to put a star in front of the problem or after the equal sign?

    Did you ever ask Tom if he noticed that he wrote a star by most problems involving 7 and/or 8? If you haven’t yet, it might be worth asking Tom how he decided which questions to star. A variation on the your activity might be helpful without being identical to the original: ask students to answer *or* guess an answer to each problem, then have them mark the questions where they "guessed." Some students will leave questions blank when they don’t feel 100% confident in their answer, even when they’d get a better score if they answered all the questions.

    Tom’s written answers have more variation in numeral size and placement (spaces between digits and placement within the answer box) than most of your other 3rd graders. This could be a sign that Tom needs extra help with executive functioning.

    It’s easy for me to imagine your classroom based on the narrative and the student work. Thanks for sharing your class with us!