Wednesday, September 14, 2016

It's Never Too Late to Learn

Some teachers wonder then what their role in the classroom could be if it is not focused on demonstrating and explaining strategies to students and monitoring students' progress in using these strategies.

-Susan B. Empson and Linda Levi

      I'm late to the math party.  I came to my current position as a K-5 specialist from a general elementary education background.  I taught all subjects during the 23 years I spent in my grades 2 and 3 classrooms, and math didn't stand out in any significant way.  I knew who Marilyn Burns was, but if I had ever met the luminaries that my colleagues in the MTBoS speak of so highly, people like Van de Walle and Fosnot and Kamii and Richardson, for example, it was long ago in teacher school, and I'd certainly forgotten who they were.  So I play catch up.
    Cognitively Guided Instruction (CGI) is something that, upon hitting the MTBoS, I heard referred to again and again.  Glowingly.  Reverently.  And so this summer I decided to find out what the fuss was all about.  I wanted to explore Extending Children's Mathematics: Fractions and Decimals, and put together a small, very informal book study PLC composed of our two grade 4 teachers and my former partner and fellow specialist Theresa.  We met twice in July and once in August to discuss our reading assignments and share the work we had done.

Kudos to my colleagues who gave up some of their summer evenings to participate in this project.

    It was an incredible experience, and we learned so much from the book and from each other.  We increased our own content knowledge, and are now armed with some tools that I hope we can employ to better analyze student work.  
     A few weeks ago I came across a set of photos I had taken late last school year of responses to an item on a grade 5 unit assessment.  The problem seemed pretty simple, and I was struck by the many different ways students had attempted a solution.  At the time I was ignorant of CGI, but looking at them now through a CGI lens I can better understand and describe what's happening.

The student multiplied 4 x 2 and 4 x 5/8 and added the two products together.  This was a good illustration of the distributive property of multiplication over addition.  This student was also able to use relational thinking to express 20/8 as 2 1/2.  He is able to decompose 20/8 into 8/8 + 8/8 + 4/8.  He understands that 8/8 equals 1 whole, and that 4/8 is equal to 1/2.

    The students who got it right were almost uniform in implementing the above strategy, although there were some outliers:

Convoluted, but it worked.  This student is also able to use relational thinking to express 2 5/8 as 21/8 and 4 as 32/8.   Why did he feel it necessary to have common denominators?

   But what had really caused me to stop and take some pictures was the many and varied ways that students had managed to be wrong.

This student converted 2 5/8 into an improper fraction.   Did he really understand why 2 5/8 was equal to 21/8 or was he just following a procedure: multiply the denominator by the whole number and then add the numerator?   The multiplication across the numerators is incorrect.  Careless mistake or confusion about the identity property of multiplication?

No question here.  Issues with the identity property of multiplication.

This was a common mistake: multiplying 4 x 2 then just adding the 5/8.  This student has used an equal sign to separate the expressions, rendering an incorrect equation. 4 x 2 does not equal 8 + 5/8.

This is diagrammed as if the student were multiplying 42 x 5/8.  When multiplying 5/8 x 4 the student flipped the 4 into 1/4.  Perhaps he had heard something about "invert and multiply"?  However he did not do this when multiplying 5/8 x 2.  In the top left corner he is adding his two products, but how he got 10/628 is a mystery.

With no actual work, I deduce that this student multiplied 4 x 2 to get 8, then, in the mixed number, multiplied 2 x 5 and 2 x 8 to get 10/16.  He finished by adding 8 and 10/16.

At first look it appears as if this student multiplied 4 by every digit he saw in the mixed number.  How did he get 160?  By multiplying 8 x 20?

     These fifth graders are now off to middle school.  But their work remains, and looking at it through the lens of what I've learned this summer is going to inform what I do, not only with the new batch of fifth graders, but with all math learners in our school.  If I read my CGI right, these kids are victims of an over-reliance on procedural memorization.  They have bits and pieces of algorithms that they can't put together because they have weak conceptual underpinnings.  They're easy to spot. However it's possible that many of the kids who did get the correct answer also have shaky conceptual foundations, but are just better at memorizing a procedure.  How will we find them?
     CGI argues for a decrease in the amount of time teachers spend, "demonstrating and explaining computation and problem-solving strategies to students."  Instead, teachers are encouraged to allow their students' intuitive strategies to emerge first.  "Children have some conceptually sound understanding of fractions, even before instruction," Empson and Levi write, "(but they) can learn to ignore this understanding in favor of models introduced in school that portray fractions in narrow ways."
     This is a tremendous shift away from the traditional "I do, we do, you do" model, and entails teachers taking on a different role.  "This new role," the authors of CGI explain, "Centers on helping students communicate strategies to other students, directing questions to specific students to help them draw connections between these strategies and more basic strategies, introducing equations to represent students' strategies, and highlighting the fundamental properties of operations and equality that that underpin these strategies."
     That's a lot to ask.  But I've read the book and I'm a believer.  Let's roll up our sleeves and get to work.


  1. Is CGI a pedagogy, or a theory of how children think?

    From your post, and the fractions book:

    "The new role centers on helping students communicate strategies to other students, directing questions to specific students to help them draw connections between these strategies and more basic strategies, introducing equations to represent students' strategies, and highlighting the fundamental properties of operations and equality that that underpin these strategies."

    From the 1st Edition of Children's Arithmetic:
    "During this conversation, I realized how serious [the researchers] were about respecting teachers' judgments on particular issues. Since they had little evidence about representing these situations, they would see how teachers and children handled it. As they worked with teachers, sharing their research knowledge about students' learning of addition and subtraction, they would continue to learn from teachers and children."

    So there's something cool here, a shift in how CGI has presented itself over time. It's gone from resisting pedagogical prescriptions to offering them freely.

    (The new edition of Children's Mathematics is far less cautious. To me, that's something lost.)

    1. It seems that CGI is prescribing a pedagogical approach based on their theory of how children think. I do have the new edition of Children's Mathematics, but haven't read it yet. Do you mean it's less cautious in offering pedagogical prescriptions than the first edition? Is the something lost the teacher's autonomy to deviate from the CGI prescription?

  2. Still my favorite math book I've ever read.

    1. Thanks Kent. I'd be curious to know how you feel it's informed your practice.

    2. For one, the book (and CGI in general) taught me that kids do have ideas about topics that they haven't been formally instructed in, and it's better to figure out those ideas and try to build off of them instead of ignoring all the thoughts that kids bring with them into the classroom.

      Another thing I love from this book is the way they show how formal fraction notation, like 1/3, can hamper long-term understanding if it's introduced too early. I LOVE the way they introduce fractions using equal-sharing problems and then start talking about halves, thirds, and fourths as units, the way we would talk about inches and feet and miles. Obviously you can't just add a third and a fourth, in the same way you can't add an inch and a foot, without converting them in some way.

      I would say that I now backload my units with vocab and notation in a way that I hadn't in the past. For example, in my first unit of Algebra, I ask kids to describe the domain and range of functions, but I don't require interval notation. I just want them to say "everything less than -3" or "from 4 through 11" in words. Then once they have that big idea down, I can introduce interval notation (probably in unit 2, when we solve and interpret inequalities explicitly.)

      Lastly, the book very delicately tells teachers that they may be completely screwing up the way that we introduce fractions to students. It has made me wonder what other topics in math are just sequenced completely incorrectly.

      For example, I don't think that we need to ever teach one-step equations in middle school. To me, the idea of reversing the order of operations makes MUCH more sense to kids if we start with multiple operations. So I don't even touch one-step equations as a topic anymore. I start with two- and three-step equations, and I think kids get it a lot better. One-step just fails the reasonableness test for most kids. "I know that x is 4 when 3x=12. Why do I have to divide both sides by 3 to prove it? This is stupid." But if I say "Joe picked a number, multiplied it by 3, added 5, and then subtracted 7. He ended with 28. What did he start with?" The whole problem just makes much more sense to kids, and working backwards seems natural.

  3. Thanks Kent! I'm just a beginner here, but the GGI book has influenced my practice in similar ways. I had many "a-ha" moments, and, like you, I love the way they introduce fractions with equal sharing problems beginning with mixed numbers! Looking back in my book I wrote "Wow!" and "Duh!" in the margin right there on pg. 6. It makes so much sense. In terms of formal fraction notation, perhaps that's why the CC holds off on that until grade 3. I also agree that backloading the vocabulary is the way to go. Interesting: a common thread is that these practices are counter to the way things have traditionally been done.