Monday, September 26, 2016

SMP Scavenger Hunt

    Find evidence of each Standard for Mathematical Practice.  In one week, with at least one example from every grade level.  No cheating by looking at the teacher's manual.  On your mark, get set, GO!

SMP 1- Make sense of problems and persevere in solving them
Grade 2, Friday, September 16

Even after his teacher told the class to put their journals away, this student continued to work on a self-created broken calculator  problem.  I was impressed with the way he challenged himself, his determination, and the willpower he showed not to use the broken key.

SMP 2- Reason abstractly and quantitatively
Grade 4, Thursday, September 15

The fourth graders were presented with the above problem.  Because they were in the middle of a unit on estimation and rounding, all students obediently rounded each of the numbers to the nearest hundred.  They added 500 + 200 + 300, got 1,000, and answered yes.
  All, that is, except for this student:

As he later explained, there was no need to round the numbers to the nearest hundred.  He front-ended them, added 400 + 200 + 300, got 900, saw the 63 left from 463, and immediately knew they would have more than 950.  Done.  He was also able to calculate mentally how far past 950 they would be, without even being asked by the teacher. 

SMP 3- Construct viable arguments and critique the reasoning of others
Grade 1, Wednesday, September 21

Is there an easier way to incorporate this practice than Which One Doesn't Belong?  I've used this prompt countless times, and never once has a student explained that the basketball didn't belong because, "It's the only one that bounces.  If you try to bounce the other ones they'll get smushed."  Gotta love those first graders!

SMP 4- Model with mathematics
Grade 3, Tuesday, September 20

Multiplication and addition equations, pictures and number lines.  Same problem, many paths, many models:

SMP 5- Use appropriate tools strategically
Grade 5, Monday, September 19

Volume time again.  I described this project here.  Calculators and rulers are the tools of choice.

SMP 6- Attend to precision
Grade 5, Monday, September 19

Estimation on a number line.  Too low, too high, and just right.  Which student attended to precision?

SMP 7- Look for and make use of structure
Grade 3, Wednesday, September 14

This student is using the structure of the hundreds grid and what he knows about place value and patterns to help him fill in missing cells.

SMP 8- Look for and express regularity in repeated reasoning.
Grade 5, Wednesday, September 21

Deriving the formula for finding the volume of a rectangular prism.

Some reflections:
  • I'm fortunate to work in a position that gives me access to a wide range of grade levels.  The fact that I was not able to get into a kindergarten classroom doesn't mean that there isn't great math work and thinking happening there, because there is.  
  • I realized there were two practice standards I didn't fully comprehend.  My friend Graham Fletcher helped me with SMP 8, which he describes as "algebrafying."  The other was SMP 2, and I'm still not totally sure my example fits. 
  • I found many activities and tasks with overlapping standards.  For example, the volume project that I used to illustrate SMP 5 could also fit with SMPs 1, 3, 4, 6, and 8.
     Most days I find myself lost in the content standards: what they really mean, how they fit together and progress across grade levels, whether or not our curriculum really does align, how we can do a better job engaging kids and hitting them with meaning.  Pulling back for a week and viewing the math in my school through a practice standard lens was refreshing, and made me realize that I was missing the forest for the trees.  Lesson: don't miss the forest for the trees.

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.