For those of you who are new to the scene, please allow me to introduce you to Loree and Mahesh...
Loree Silvis is a former primary-grade teacher who co-led the development of the PNOA and has consulted on math instruction and assessment with elementary teachers in WCSU in the past. 
Mahesh Sharma is a former Professor of Mathematics who is known for his work on math learning problems and has an ongoing relationship providing professional development for math instruction for all grade levels in WCSU.
Loree and Mahesh have a lot of common ground, such as:
- Focusing instruction and assessment on developmental learning progressions starting with developing number concept and flexible number sense to build numeracy.
- Emphasizing automaticity and fluency (which Loree defines as: flexibility, accuracy, efficiency and appropriateness).
- Emphasizing the importance of using concrete and visual models to build conceptual understanding.
- Defining conceptual mastery in terms of having the ability to apply and connect the concept flexibly, and communicate understanding of the concept using multiple representations.
- Insisting that "I don't Know" is not an acceptable "out."
- Believing that students must move from: sequential counting to one-to-one counting to subitizing (recognizing groups without counting) before being able to develop additive reasoning.
They also have a few key differences relating to pedagogy. The key conflict that I can see in their approaches stems from differing beliefs about how and when to move from one to one counting and subitizing to additive reasoning.
Mahesh believes that students need to move from seeing number as discrete (for example, a collection of 6 objects) to seeing number as a length (for example, this dark green rod is 6 units long) to developing fluency with additive sight facts. As a result, he sees Cuisenaire rods as the appropriate, efficient tool for developing that concept and fluency. He discourages discrete counting (counting on) to develop sight facts.
Loree believes that students are not developmentally ready to see number as a length until second grade. As a result, she sees Cuisenaire rods as developmentally inappropriate. Instead, she relies on discrete objects (chips, pennies, pattern blocks, etc.) and ten frames to develop sight facts.
To illustrate this difference, take 9 + 1. Mahesh would approach it this way:
In contrast, Loree would approach it this way:
You may be wondering how to reconcile the two approaches? I was wondering that myself, so I talked to Loree and did bunch of research on developmental math cognition before talking to Jen. Here's what we came up with:
We feel that it is essential that once students develop one to one counting mastery at the applications and communication level (outlined in this document), we must push the focus away from counting to developing conservation of length (finding lengths that combine to equal another length) and the commutative property of addition. Over reliance on counting takes away from the development of deep number sense. This is why we see high school students counting on their fingers to solve math problems.
That said, we can (and should!) be checking to be sure that students understand that how a cluster of 9 discrete objects is represented by the blue rod. This video illustrates: https://www.youtube.com/watch?v=qR_6vEOu5No
Student who don't see quantity will build the Cuisenaire "staircase" to show you that "blue is nine because it's the ninth rod from the left." In other words, they see number as simply cardinal ("the ninth rod") rather than quantity and length. We want them to develop the flexibility to see it all of these ways,and the ability to consider the contexts and level of efficiency inherent in each way.
Colleagues administering the PNOA may be thinking that this poses some problems since there are ten frame items on the PNOA. That is a valid point that is addressed in the Untangling Assessments post.
Thank you for addressing this issue head-on!
ReplyDeleteMary