NCTM Focal Points (Standards)

According to this bulletin from the National Council of Teachers of Mathematics, there are apparently three (3) focal points identified for each grade level, PreK-8.

Here are the focal points, followed by excerpts from NCTM's explanation.

• the use of the mathematics to solve problems;
• an application of logical reasoning to justify procedures and solutions; and
• an involvement in the design and analysis of multiple representations to learn, make connections among, and communicate about the ideas within and outside of mathematics.

"These curriculum focal points should be considered as major instructional goals and desirable learning expectations, not as a list of objectives for students to master."

". . .this set of curriculum focal points has been designed with the intention of providing a three-year middle school program that includes a full year of general mathematics in each of grades 6, 7, and 8."

Go here to read entire bulletin.

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My observations:

No matter how good these focal points (expectations) sound, don't kid yourself that they intend for your child to master them. These focal points are not to be considered "as a list of objectives for students to master." That isn't considered to be important to NCTM. Notice, however, that it is considered important that they learn to communicate about the ideas they are learning. And don't be surprised if that communication is written.

And . . .

If states and districts follow these suggested standards, your students will likely fall short of the goal of most families -- completion of pre-algebra and algebra by the end of 8th grade.

Geometry of Today is Not Geometry of Yesterday

I'm pulling up old stuff, I know, but you can't get better than Barry Garelick.

Barry posted this comment on the first Kitchen Table Math back on June 5, 2005.

"From NCTM's PSSM, here's what NCTM has to say about their geometry standard: 'Geometry: Geometry has long been regarded as the place in high school where students learn to prove geometric theorems. The Geometry Standard takes a broader view of the power of geometry by calling on students to analyze characteristics of geometric shapes and make mathematical arguments about the geometric relationship, as well as to use visualization, spatial reasoning, and geometric modeling to solve problems. Geometry is a natural area of mathematics for the development of stusdents' reasoning and justification skills.'"

Translation: High school geometry used to emphasize proofs. Now it just emphasizes shapes and formulae, with an occasional proof and in general is not much more advanced than the geometry presented in 7th grade, except for the fact that not much geometry is presented in 7th grade."

My observations and thoughts:

NCTM is the National Council of Teachers of Mathematics. They are a body of educationists (my word) who are responsible for writing the national math Standards which are supposed to define the expectations for students in each subject area at each grade level. I say "are supposed" because the expectations are so watered down and are so vague that no one can actually identify a specific expectation.

You have to hunt far and wide to find geometry in textbooks today. This "broader view" is part of that "1/8 inch deep and a mile wide" approach to teaching Math. The subject of Geometry is spread all through other textbooks and is no longer taught in a coherent fashion semester by semester.

A high school teacher commented to me 2-3 years ago how much he wished he could teach geometry as an isolated subject so he could concentrate the students' focus on geometry.

Your student is probably being robbed of the opportunity to learn to prove geometric theorems. It's no wonder our high schools students score so much lower than Asian students.

Expectations Need to Be Measureable and Concepts need Time

Standards are "expectations". They are the targets. They are what you are shooting for. They define what is expected of the students at specific grade levels in specific subject areas. (In the United States, curricular expectations are defined as "standards".) In order for standards to be effective, they must be specific.

Specific expectations are easy to measure. If your expectation is that your child know the capitals of all of the states, that is specific and measurable. You can easily discern if a student has met the standards. How? Have him demonstrate that he knows the capitals.

And that is why so many people are concerned about "standards". It is the vagueness of the standards that troubles us.

I came across an article by William H. Schmidt entitled "What's Missing from Math Standards?" which was published at the American Educator website in the Spring of 2008. He discussed the findings of the Third International Mathematics and Science Study (TIMSS).

According to Schmidt, TIMSS found that "student performance is directly related to the nature of the curricular expectations." He explains that he does not mean the instructional practices, but rather "the nature of what it is that children are to learn within schools."

"The TIMSS research has revealed that there are three aspects of math expectations, or standards, that are really important: focus, rigor, and coherence."

Here are Schmidt's comments about all three of these aspects of expectations.

Focus:

"Focus is the most straightforward. Standards need to focus on a small enough number of topics so that teachers can spend months, not days, on them. . . [i]n the early grades, top-achieving countries usually cover about four to six topics related to basic numeracy, measurement, and arithmetic operations . . . In contrast, in the U.S., state and district standards, as well as textbooks, often cram 20 topics into the first and second grades."

It is Schmidt' s opinion that this number of topics is far more than any primary grade student can absorb.

Rigor:

"Rigor is also pretty straightforward -- and we don't have enough of it.... .[I]n the middle grades, the rest of the world is teaching algebra and geometry. The U.S. is still, for most children, teaching arithmetic. . . [O]ther countries outperform us in the middle and upper grades because their curricular expectations are so much more demanding, so much more rigorous."

Coherence:

Coherence may not be as easy to grasp as focus and rigor, but according to Schmidt, "it is the most important element." He explains that there is a formal academic body of knowledge that has been parsed out and sequenced from kindergarten through 12th grade, and he describes how especially important this parsing and sequencing is in the subject of math.

"Topics in math really need to flow in a certain logical sequence in order to have coherent instruction. If you look at the math curriculum of top-achieving countries, you see a very logical sequence. The more advanced topics are not covered in the early grades. Now that seems obvious -- until you look at state and district standards in the U.S. Everything is covered everywhere. Far from coherent, typical math standards in the U.S. often appear arbitrary, like a laundry list of topics."

Some of you may want to consult the entire article to see why our country has such unfocused, undemanding, and incoherent math standards.

Two related articles can be found here and here.

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Some additional thoughts . . .

By spreading topics all through the curriculum, nothing is covered in depth. Teachers are expected to cover so much material during the year that they must fly quickly to the next topic, meaning that there isn't time for a student to grasp the coherence of one concept with another. There is rarely time for the feeling of "Aahhh! I get it!"

Depth is better than shallowness. Kids can get their thoughts around a concept and understand connections when a topic is covered deeply and thoroughly.

For some reason this picture is going through my mind right now: I'm thinking of trying to drink lemonade through a straw -- after we have spread it 1/8 inch thick all over the table or counter top. And we only have one minute to do it!

That might be how many of our children feel when they are trying to "get hold of a math concept". This produces frustration and a feeling of "there's something wrong with me -- I didn't get it."