Monday, July 17, 2017

How Do You Get to School?

"The standards do encourage that students have access to multiple methods as they learn to add, subtract, multiply, and divide.  But this does not mean that you have to solve every problem in multiple ways.  Having different methods available is like having different means of transportation available to get to work; flexibility is good, but it doesn't mean you have to go to school by car, then by bus, then walk, then bike--every single day!"
Bill McCallum

     Last month I stopped by a second grade classroom where the teacher was administering an end-of-year math assessment.  I paused by the desk of a student who, with a look of frustration on her face, was puzzling over this question:

     "What's the matter?" I asked.
     "I forgot how to use an open number line," she responded, head down, staring at the blank page.
     "Do you know another way to solve the problem?"
      She looked up at me.  "Partial sums?"
     "Could you show me how you would do that?"
      Here's what she produced on a piece of scrap paper:

And wrote the answer, 79, in the space provided.

Question: Do you mark this wrong because she couldn't show her thinking on an open number line?

     Continuing to make my way around the room, I came upon this response:

     "Tell me what happened here," I asked.
     "I got confused about using the number line."
     "Do you know another way to solve the problem?"
     "I could draw base-10 blocks."
     "Could you show me how you would do that?"

Question: Do you mark this wrong because he couldn't show his thinking on an open number line?

     I spent the next several days looking through other end-of-year assessments for examples of questions where students were being commanded to solve problems using specific representations and methods.  Here's a sample from grades 1-4:
  • Use the break-apart strategy to solve each problem.
  • Use the turn-around rule to solve.
  • Explain two different ways you could use doubling to solve 6 x 8.
  • Explain how you can think addition to solve 14-7.
  • How can you find the sum using a number grid?
  • Explain how you can use the near-doubles strategy to find the answer.
  • Use base-10 shorthand.
  • Use an open number line.
  • Solve using partial-sums addition.
  • Solve using U.S. traditional addition.
  • Use partial products or the lattice method to solve.
  • Use U.S. traditional subtraction (this for 38,000 - 23,177.)

     As I recall, in my math classes growing up there were no multiple methods or representations.  You memorized your facts and used the traditional, standard algorithm.   I'm sure I had classmates clever enough to devise alternate strategies on their own.  As for me, I was out of luck.  That's too bad.  I wish I had the exposure to the multiple methods and representations that are now considered essential components of math education today.  If I had, maybe this wouldn't have happened.
     But in the leap from standards, to curriculum, to assessment (especially assessment), something has gone awry.  We want to expose kids to multiple representations and methods, and encourage them to experiment with, explore, connect, and analyze them.  But do we want to force kids to use them on summative assessments? For a grade?  The two students wrestling with question 16 above each had their own way of thinking about 43 + 36.  But the directions to the problem, which instructed them to show their thinking on an open number line, only served to shut their thinking down.  How did it make them feel?  And how will they feel when they get their test back and see that a problem that they can find the answer to is marked wrong because the way they want to show their thinking is not what the test maker wants?
     Providing access to and connecting different models, methods, and representations for students as they find their way to computational fluency is very important.  But I think that in forcing the issue we run the risk of doing more harm than good.  How kids get to school is dependent on many variables, none which are under their control.  The ultimate decision rests with us adults.   How about we let the kids decide for a change?    


  1. Great post Joe! Really interesting to think about "multiple representations" and the decisions we make in the name of representations.

    Personally, I think that some representations might be more productive than others, and I agree that being able to represent in more than one way is important (that's relational understanding), however, I also agree that we need to be very purposeful when we ask students to use a specific representation and when we ask students to choose their own representation.
    Might this decision depend on whether this is a learning experience, or whether it is a demonstration of learning? For example, I might purposefully design a problem with a context that relates to a number line.
    You've got me thinking for sure!

    Thanks for sharing!

    1. Thanks for the comment Mark. We can find out how comfortable students are with various methods or representations (what's the essential difference anyway?) in settings that are less threatening and emotionally charged than during summative assessments. I believe it's one of the reasons that the CC has gotten a bad name in certain circles.

  2. I loved your post. I heard Jason Zimba speak last year and he says the same thing you do. Have the kids use and learn different methods to solve the problems so they can find what makes sense to them, and what's more efficient for the problem they're solving. But on a test just ask the question. Leave HOW they solve the question up to the child. He said Asking them to solve the problem specific ways on a test are NOT part of the standards we should be testing! So, great post, I can't wait to share it with others!

    1. Thanks for sharing Jason Zimba's comments. Bill McCallum also agrees. Now all we need is Phil Daro. Maybe we can get him on the record!

  3. Another great post, Joe! There are many reasons to teach students various strategies for solving problems. I remember distinctly when I learned you could solve a problem in more than one way and sadly this was when I was an adult! So true for many of us. As I toyed with multiple strategies in my teaching, I slowly became aware of how important it was to be intentional not only with the teaching of the strategies, but how I was helping students to use them and see their value. Thanks for shedding some light on something that I think we all need to talk about more.

    1. Thanks for sharing your experience, which is identical to mine! I agree that we need to help students see the value of the different strategies. But as for requiring specific ones on a test...that's clearly not in the spirit of the standards.

  4. Joe,
    Thanks for the interesting post. As a classroom teacher, and now an Elementary Math Specialist for my district I have thought a lot about this issue of ensuring certain strategies for all students, and have concluded that that we should, at times, teach and assess certain strategies.

    In fact the standards that Bill and Jason helped to write suggest as much; specifically calling out that students should be able to use the traditional US algorithms, and, in this case, to be able to represent addition and subtraction under 100 on a number line (2.MD.6).

    I think there are a couple of places where I find disagreement with your position above. The first being that we need to consider that this is one task among many. Presumably, there would be other tasks on this same assessment that would encourage a student too solve this type of problem using any strategy. This task appears to be designed to specifically measure the students' ability to use a number line. As with any assessment task, there is room for criticism, which I will get to below, but if we want to determine whether a student has achieved the goal of CC standard 2.MD.6 we need to write questions specifically to that purpose.

    The other place where I feel a disconnect with your post and the reality of most classrooms is where you say, "How kids get to school is dependent on many variables, none which are under their control." Of course kids come from many places and situations, however in our classrooms we share experiences daily. The task you use in your illustration is embedded in a set of curricular materials. Those materials have very likely spent a series of lessons on how to represent addition on a number line using similar problems. Hopefully, they have even spent years on this, starting in kindergarten. Given that learning this skill (and related concepts) is a goal of the lessons, isn't it valid and reasonable to assess to see if the students have achieved that objective.

    Whether learning to use a number line in this way is not at issue here. Every set of standards, curriculum writer, and teacher needs to make decisions about what to teach. If they believe that a doubles +1 strategy, traditional algorithms, number bonds, etc. are important enough to expect all students to learn, then those ideas are also likely important to assess.

    I too am not delighted with how the number line task was written. When writing tasks to get students to represent addition and subtraction on a number line I prefer to give the answer, so it isn't even about solving the problem. For example, how would you represent 32 - 29 = 3 on a number line? Is is one long jump back to 3, or a small jump up from 29 to 32? Or is it 29 tiny jumps back, or two jumps up from 29 to 30 and from 30 to 32? Or is the student completely unable to make a sensible representation?

    Curriculum designers and teachers need to utilize common representations, strategies, terminology, symbols, etc.for many reasons. Knowing that students have a functional understanding of those can be important, especially within the context of a well articulated curricular program. In this case it is the use of a number line. CC can get a bad name when things are taken out of context. The purpose of this task is specific, and taken in isolation that can look restrictive, however within the context of a set of curricular materials with many objectives it makes more sense.

    One last point. It is important to recognize that state and other large scale assessments cannot make the same assumptions that instructional materials can. If this were a task on one of those types of tests I would agree with you wholeheartedly.


    1. Thanks for your very thoughtful (and respectful!) pushback. It is true that the number line question is just one among many on that assessment, and that taking it out of the context of the entire assessment and from its embedded place in the set of curricular materials may make it seem worse than it actually is. Maybe I used it as a straw man. I like your rewrite, where the answer is given, which deemphasizes the solving part and puts the focus on using the representation. I never thought of how a change like that could impact the nature of the response, and now I’m really curious to know how those students would have responded to the question: How would you represent 43 + 36 = 79 on a number line?
      It’s important for us to know what it is our students know, and what kind of progress they’re making. Without that, we can’t help move then forward. We need to assess, but it’s how we do it and what we do with the results that I want to think about. Reasonable people can disagree about the standards themselves, but they’re a fait accompli. At the classroom level, it’s not the teachers making the decisions about what to teach, or what to assess, at least in my district. The state adopts the standards, the district adopts a curriculum (hopefully) aligned to the standards, and the teacher is expected to implement the curriculum. The unit assessments that go along with the curriculum assess what’s been taught. (And the large-scale tests like the PARCC supposedly assess the standards too.) Some of the questions are good and some are not so good.
      I will readily admit to being overly sensitive to the effect these assessments and their results have on the psyches of the kids, and how they impact their emotional disposition towards math, which to me is more important than any content standard. I come by it honestly. I have a well-documented case of math anxiety, about which I still have a tremendous amount of resentment. I’ve been in many classrooms, grades 1-5, over the past 8 years and watched countless kids take countless assessments where they are asked to solve problems in very specific ways, ways that, yes, are indicated for mastery in the CC and that have been taught as part of the curriculum, but ways that still don’t make sense to some of their developing minds. Saying that, “You must be able to use (insert method here) to solve (insert problem here) or else you aren’t measuring up”, communicates a message. That message is damaging to some students. I don’t know how to assess the damage, but it’s real. So we worship at the altar of accountability and data, but are never ourselves held to account for the damage that’s done in its name. Upon reflection, that’s what’s lying underneath this post.