## Comments on the Common Core mathematics 1st draft (K-5)

The common mathematics standard for 48 states and 2 territories of the United States is up. Here are some random thoughts:

Kindergarten: Know the decade words to ninety and recite them in order (ten, twenty, thirty, …).

This is the first I’ve heard the term “decade words” (it isn’t on Wikipedia either), but I guess we’re supposed to know it now.

Kindergarten: Analyze and compare a variety of two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, component parts (e.g., number of sides and vertices) and other attributes (e.g., having sides of equal length).

Emphasis mine; I don’t think vertices have ever been mentioned much at this level.

a. Addition is commutative. For example, if 3 cups are added to a stack of 8 cups, then the total number of cups is the same as when 8 cups are added to a stack of 3 cups; that is, 8 + 3 = 3 + 8.
b. Addition is associative. For example, 4 + 3 + 2 can be found by first adding 4 + 3 = 7 then adding 7 + 2 = 9, or by first adding 3 + 2 = 5 then adding 4 + 5 = 9.
c. 0 is the additive identity.

I have a niece in first grade who made a poster which mentioned these properties by name. Yet, some people never have learned them: here’s a story about adults who never realized addition is commutative.

Grade 1: Solve word problems involving addition and subtraction within 20, e.g., by using objects, drawings and equations to represent the problem. Students should work with all of the addition and subtraction situations shown in the Glossary, Table 1, solving problems with unknowns in all positions, and representing these situations with equations that use a symbol for the unknown (e.g., a question mark or a small square). Grade 1 students need not master the more difficult problem types.

I sensed the standards were going to introduce algebraic concepts early, and here they are at grade 1, making explicit use of the terms “equation” and “unknown”, with only the concession that a blank square is used rather than a letter like x.

Grade 1: Understand that the length of an object can be expressed numerically by using another object as a length unit (such as a paper-clip, yardstick, or inch length on a ruler). The object to be measured is partitioned into as many equal parts as possible with the same length as the length unit. The length measurement of the object is the number of length units that span it with no gaps or overlaps. For example, “I can put four paperclips end to end along the pencil, so the pencil is four paperclips long.”

I like this one. I find it genuinely useful for students to invent their own units of measurement, and even (at a later grade level) their own subunits within the units.

Grade 2: Solve word problems involving dollar bills, quarters, dimes, nickels and pennies. Do not include dollars and cents in the same problem.

Grade 2 is where money is introduced. The emphasized caveat is curious, if understandable; I suspect it is research-based.

Grade 3: Understand that multiplication of whole numbers is repeated addition. For example, 5 x 7 means 7 added to itself 5 times. Products can be represented by rectangular arrays, with one factor the number of rows and the other the number of columns.

Grade 3: Understand that a unit fraction corresponds to a point on a number line. For example, 1/3 represents the point obtained by decomposing the interval from 0 to 1 into three equal parts and taking the right-hand endpoint of the first part. In Grade 3, all number lines begin with zero.
2. Understand that fractions are built from unit fractions. For example, 5/4 represents the point on a number line obtained by marking off five lengths of ¼ to the right of 0.

Well, a little comment. It appears the terminology in the draft is coming from this paper mentioned in the bibliography of the standards, also mentioned by a supporter of multiplication-as-repeated-addition in one of my posts. Therefore, before anyone starts throwing flaming rocks, I’d recommend reading the source paper.

Grade 3: Find an unknown length of a side in a polygon given the perimeter and all other side lengths; represent these problems with equations involving a letter for the unknown quantity.

Here’s more evidence the standards are thinking of algebraic concepts early.

Incidentally, grade 3 is also where area is introduced. A study by Patricia Clark Kenschaft mentioned in this Peter Grey essay claims “In one visit to a K-6 elementary school in New Jersey she discovered that not a single teacher, out of the fifty that she met with, knew how to find the area of a rectangle . . . They taught multiplication, but none of them knew that multiplication is used to find the area of a rectangle.”

Grade 4: Solve word problems that involve multiplication of fractions by whole numbers; represent multiplication of fractions by whole numbers using tape diagrams and area models that explain numerical results.

Tape diagrams are standard in Japanese classrooms and should also be familiar to Singapore experts. Here’s an example: Grade 4: Explain why multiplication and division strategies and algorithms work, using place value and the properties of operations. Include explanations supported by drawings, equations, or both. A range of reasonably efficient algorithms may be covered, not only the standard algorithms.

Aka “lattice multiplication if you want to”, or any of the other methods. I am quite interested in what sort of drawing would explain the algorithm for long multiplication; does anyone have one?

Grade 5: Explain and justify the properties of operations with fractions, e.g., by using equations, number line representations, area models, and story contexts.

How many 5th grade teachers can do this, let alone their students? I am hoping one result of the standards is an effort to improve teacher background knowledge (as opposed to just testing the students more).

Grade 5: Graph points in the first quadrant of the coordinate plane, and identify the coordinates of graphed points. Where ordered pairs arise in a problem situation, interpret the coordinate values in the context of the situation.

The pre-draft gave a hint that coordinates were going to be introduced early, but I was curious how early. 5th grade is the answer. I do like spending time with just the 1st quadrant, and think it will be a useful scaffolding (rather than a topic that might as well be repeated from scratch the next grade on).

[I will put my comments up for 6-8 and high school level up later this week.]

### 8 Responses

1. Thanks for posting this Jason.

I haven’t been willing to comment on these, because it’s just too big for me to wrap my head around. And I’m convinced that different kids need to do their learning on different timetables.

But I like what you’ve given me here.

2. I agree with your hope that this will increase teacher background knowledge. Teachers at the elementary level (at least many, not all) significantly lack background knowledge in math and it shows. We tend to use textbooks as a crutch. It’s a serious problem.

3. Thanks for the summary! Like Sue, I haven’t managed to wrap my mind around the proposed standards, but I can take in these bit-sized pieces.

“Understand that a unit fraction corresponds to a point on a number line.” This is a great improvement over the earlier draft of Wu’s paper, which said that a fraction “is” a point on the number line. That grated on my mental ears like fingernails on a chalkboard.

Never heard the term “tape diagrams” before. But the drawings are quite useful, whatever one calls them.

In answer to your question, a rectangle drawing does very well for showing the partial products of the multiplication algorithm. Say you want to diagram 25 x 37 = ?: Mark off 20 and 5 more on one side of the rectangle, and mark 30 and 7 more on the other side. Draw lines to cut the rectangle into four partials: 20×30, 20×7, 5×30, and 5×7. The total area is the sum of the four smaller block, and this method easily extends to larger numbers (and to algebra) if desired.

I’m concerned about requiring *explanations* from elementary students. My experience with my own kids is that they can understand math very well (as indicated by easily working multi-step word problems) and still not be able to put an explanation into words. It would be great, however, if elementary *teachers* were able to explain these things!

My 5th-grader would heartily endorse a study of graphing. She had one worksheet in her Math Mammoth book and has been asking for more ever since, but I haven’t gotten around to finding projects for her…

• In answer to your question, a rectangle drawing does very well for showing the partial products of the multiplication algorithm. . .

I see how this matches neatly with the lattice method, but do students understand how it matches with the “standard” method?

I’m concerned about requiring *explanations* from elementary students. My experience with my own kids is that they can understand math very well (as indicated by easily working multi-step word problems) and still not be able to put an explanation into words. It would be great, however, if elementary *teachers* were able to explain these things!

I don’t have the exact quote offhand, but at one point they emphasize “grade-appropriate” explanations, so I imagine they’re not asking for it exactly like a teacher would understand it.

• In answer to your question, a rectangle drawing does very well for showing the partial products of the multiplication algorithm. . .

I see how this matches neatly with the lattice method, but do students understand how it matches with the “standard” method?

And after a lot of thought, I came up with a way to draw both matching with the standard algorithm and include carrying, but it’ll take a while to finish (and I’m unsure if a student could pull it off).

4. I have been following the common core standard process just a little. Here in WA State, we are just adjusting to new, more rigorous standards this year. The common core standards, when implemented, will represent another level of rigor.

The important piece that we shouldn’t lose track of is that these standards are meant for ALL kids. We can no longer continue to say “well these kids over here just can’t do it.” We have to provide a wide array of learning opportunities that are inclusive of all regular ed kids, we have to institute data systems so we can determine gaps in learning early, and we have to hold elementary, middle, and high school teachers accountable for much deeper content knowledge.

5. I’m with Sue here that it’s folly to link standards to grade levels, but that’s really a problem with school in general rather than our common core standards.

• I’m not totally antithetical to the idea, especially because it seems like all geometry is currently pushed to higher levels (when it is much better integrated in most countries). If even teachers can’t tell how to find the area of a rectangle (see quote in post above) then we’ve got a major problem.

Also, the idea (although I don’t think they fully pulled it off) is that the standards also give advice on what _not_ to teach; that is, some topics are done currently by teachers yet are one too many.