Hi Grant,

I was itching to reply to your blog immediately but had a crazy week.

Remembering your definition of feedback – if your intention was to provoke as well as to advocate, then yes, I got quite excited about the article and postponed reading it until I could spend time doing a “closer” reading. And yes, it made me want to continue thinking about the counterparts for Math of the PGC KIAs (or the task frames you introduced in Mexico).

Regarding the references to conceptual understanding (or lack thereof) of the CCSS:

You started your blog by talking about how the Math CCSS encourages (or does not do enough to encourage) conceptual understanding. Thanks to my work with you for the PGC project, I had to read and re-read and re-read the Math Practice Standards, and I feel that the MPS (1) brings to the fore the need to address conceptual understanding and (2) describes ways that this can be done. The CCSS even describes how this can be done in relation to specific Content Standards in the Progressions documents of Bill McCallum at http://commoncoretools.me/author/wgmccallum/ (unless I am just reading so much more into the CCSS documents than is actually there ).

For example:

MPSportion of the MPS that

addresses conceptual understanding

a few examples in Progressions

documentssome personal thoughts

MP2: Reason abstractly and quantitatively

MP6: Attend to precision.

“[Mathematically proficient students] bring to bear… the ability to decontextualize--to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem…; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.”

They state the meaning of the symbols they choose…”

Gr. 6 Expressions and Equations:“A firm grasp on variables as numbers helps students extend their work with the properties of operations from arithmetic to algebra.” MP2

MP6 is critical because “the language of math” is such an important part of conceptual understanding. At the heart of Algebra is “generalization” or giving/using a general rule for doing many problems of a certain type (made possible by the language of Algebra, MP6), and according to MP2, the students have to know how to work with both the general rule (symbolic representation and the underlying properties of real numbers) as well as the calculations in context, and how to move fluidly between both worlds. (Thus, proficiency in Math will inevitably include elegance.)

I think of MP2 a little bit as helping students move between the worlds of applied math and pure math. Or, using the analogy of an appliance or machine, it is like moving between the worlds of “how to operate the machine” (where knowing only the algorithms, without knowing why, often resides) and “why/how it works.” Maybe because I was a Math major in undergrad, rather than an Educ major, I really believe that we can exist in the world of just knowing how to operate an appliance, but understanding, transfer, extensions, and experiencing the joy of doing pure Math, etc will not be possible. Example, echoing your blog, if students don’t get that the definitions, undefined terms, and postulates are important because the rest of the math are just the implications of these (all residing in the world of

pure Math), then they will really have to rely on just doing the algorithm by rote memory.

This will also help students with the “abstraction” needed for conceptual understanding, e.g., exploring why geometry as abstractions of real-world objects is exact but all measurements are approximate. (I remember The Education of T.C. Mits!)

The thinkboard of the Singapore Math framework will also be helpful in teaching conceptual understanding. See my crude snapshot below (this is anyway just a private correspondence so I hope its not breaking copyright laws). Their bar model is fantastic for developing conceptual understanding. (Just an aside – I see a lot of Singapore Math elements in the CCSS progression documents. They might have used it as one of their bases?)

MP7: Look for and make use of structureMP8: Look for and express regularity in repeated reasoning.

“[Mathematically proficient students] look closely to discern a pattern or structure.”“…notice if calculations are repeated…” and look for the generalization (e.g., abstract the equation by studying the rise over run of the line)

Gr. 6 Expressions and Equations:“[In Grade 6, students should] describe the structure of an expression, seeing 2(8+7) for example as a product of two factors, the second of which, (8+7), can be viewed both as a single entity and a sum of two terms.” (MP7)

We neglect doing Make Meaning activities about structure, e.g., more attention needs to be paid to the properties of real numbers (especially the Closure property).Example:- Which properties hold (and don’t hold) for

Naturals, Wholes, Integers, Rationals, Reals?- Simplify the expression 2x+3+5(x-1), giving

the property that “allows” you to do each step.

Other than equality, which you mention in the blog, the meaning of variable is neglected (but is now addressed in the CCSS Progressions documents).

MP3:Construct viable arguments …MP1: Make sense of problems…MP4: Model with math.

As you say in your blog, the CCSS says that student show conceptual understanding when they can justify the procedures used (MP3).

“Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.”

“…They are able to identify important quantities in a

Gr. 6 Geometry“Students develop area formulas for parallelograms, then triangles. They learn how to address three different cases for triangles: a height that is a side of a right angle, a height that is a side of a right angle, a height that ‘lies over the base’ and a height that

I think teachers have students do too little of mathematical INVESTIGATIONs in the “pure math” mode. I refer to investigations like the example of triangles in Gr. 6 Geometry, or asking them to just use the definition of exponents (xn = x·x·x …x·x )

to come up with the laws of exponents, including the different cases involved. These investigations will involve MP1, MP3, MP2, MP4, MP6, MP7 and possibly MP8.

Proving helped me a lot in gaining conceptual understanding. Is this done a lot in HS Geom in the US?

practical situation and map their relationships…They can analyze those relationships mathematically to draw conclusions.”

is outside the triangle.”Meeting MP1 means that students must “try special cases and simpler forms of the original problem in order to gain insight into the solution.”

Yee, Peng Lee (Ed.). 2008. Teaching Secondary School Mathematics: A Resource Book (2nd ed). Singapore: McGraw Hill.

Regarding the paradox in education

I agree with what you said about the paradox in education – novices needing clear instruction and simplified/scaffolded learning (just as I did when you said, in one AE SI, that “no thinking is possible without ambiguity/provocation…“ but the irony is that teachers think their job is to make everything clear to the student – will never forget that).

Now though, I think it is also true that their job is indeed to “make everything clear.” However, it is scaffolding of bits of knowledge in isolation that is the problem, instead of scaffolding (and revisiting, rethinking, refining) big ideas or, for example, simplifying and scaffolding by following the sequence in the picture above. Many teachers do start with concrete (manipulatives) but still too quickly jump to the algorithm, instead of investing time in meaning making to make generalizations about the concepts. The problem might be difficulty in coming up with the big ideas –many many Math EQs online are “Why is studying (Math topic) important, or EUs like “(Math topic) can help us in many real-life situations.” It did take me a long time to think of good better EQs and EUs in Math.

Grant, an Arias (new ASCD format I saw last March?) book from you and Jay of EQs in Math and/or “unpacking” the Math Practice Standards would be helpful !

Thanks for this wonderful blog!

Rita

PS – Having “defended” the CCSS Math Practice Standards, I just remembered the well-publicized father/engineer who ranted against the way the teacher taught subtraction to his son, using the CCSS standards. Don’t know much about it, but we do have to make distinctions among: needing to teach conceptual understanding, the way it should be taught, how it should be taught (the extent and developmental appropriateness for each level, etc.), why it might be harmful to teach conceptual understanding if teachers don’t really have conceptual understanding themselves, etc.