Teaching to the Standards

Here are two course descriptions from the course catalog for George Mason U's Graduate School of Education (Fairfax VA). They are courses in teaching math in elementary and secondary school classrooms. Forget about teaching to the test. It's about teaching to the standards. And in case you didn't know there were "national" standards, I would guess that term means NCTM's standards:

633 Advanced Mathematics Methods for the Elementary Classroom (3:3:0) Prerequisites: completion of elementary education (PK–6) licensure and EDCI 631. Focuses on teaching all children, including those from nonmainstreamed populations. Emphasizes teaching problem solving and higher-order thinking skills promoted by the National Council of Teachers of Mathematics and the Virginia Mathematics Standards of Learning. Uses techniques and materials to develop specific problem-solving strategies in a hands-on, activity, and workshop-oriented experience. Explores the teaching of problem-solving, reasoning, communications, and connections in PK–6 mathematics by working with manipulatives and technologies. Field experience in public schools required.

672 Advanced Methods of Teaching Mathematics in the Secondary School (3:3:0) Prerequisites: EDUC 522 and EDCI 572. Focuses on learning processes fundamental to the development of mathematical thinking from a problem-solving perspective. Introduces students to national and state standards regarding content and methodologies appropriate for teaching school mathematics. Examines a variety of instructional strategies and materials, and relates them to the broad scope of mathematical content in secondary curriculum. Field experience required for those seeking initial teacher certification.

-- BarryGarelick - 25 Oct 2005

Comments

In our state, which uses Standards-Based Education, the standards are not standards, they are frameworks. Even when you get down to a school's curriculim document, the requirements are vague. There are a lot of conceptual understandings, spiraling and allowance for social promotion. When someone talks about NCTM's standards, what do they mean? Are these real grade-by-grade standards or just guidelines.

This means that our schools can only base decisions on the results of the state's standardized tests (NSRE - New Standards Reference Exam). I have been in those meetings. A teacher-parent group evaluates the statistical reports and looks at the relative changes in scores over the years. The math section is broken into sub-sections like Problem Solving. If the scores are going down, then they will try to do something to fix it. These are small relative changes and the fixes are small and relative.

I found it quite incredible that nobody looked at the questions and the absolute difficulty of the test. Not even any of the parents. I did. There was a sample 4th grade math test booklet containing about 5 or 6 questions. These weren't simple questions like multiplying two-digit numbers. One fourth grade question displayed a bar graph with 4 bars. It showed the results of a class election and how many votes each child received. The heights or amounts of the bars were less then 20. The first question was: "How many total votes were cast?" The second question was: "Who won the election?" Alloted time: 15 minutes. The NSRE is a fuzzy type of test that uses fewer real world problems where the student has to do more explaining. From these fewer, real world questions, the graders have to break-out scores for things like Problem Solving and Number Skills using a rubric.

Another 4th grade math question showed a window divided into a rectangular grid of panes: 43 in one direction and 44 in the other. The student had to determine how many more panes they needed if the window had one more row and one more column. Alloted time: 15 minutes. I would like to see the grading rubric for this question.

Two problems. The first is that the tests are incredibly easy, but nobody looks at that. This has to do with fundamental assumptions and they are not going to change those. What is incredible is that none of the parents looked at the problems. Perhaps they saw that the scores were really bad and couldn't imagine that the tests were so easy. I have also found that a lot of parents don't have a good idea of what kind of education they want. They may not like some of the homework that comes home, but that's about it. Many parents seem more interested in whether a particular teacher is good or bad. Perhaps they see that a good teacher can make up for a bad curriculum - at least partially.

Second, the state uses a fuzzy testing standard to match the way classes are taught. Everything is relative. Nothing is related to any outside or world-based criteria. They greatly influence the curriculum and teaching methods, they select the type of state test, they decide on the grading scale, and they select the consequences, but still schools complain. At least in high school, there are the externally-defined AP standards.

It's a great scam. National teachers' organizations and state public school administrations define the fuzzy standards and the local schools simply deflect criticism by referring to national standards. There are no other choices because there are no other choices.

So I want to know exactly what these national standards are? Is there a document somewhere? Does it specify grade-by-grade knowledge and skills? I suppose I should know after all of these years, but I don't.

-- SteveH - 25 Oct 2005

There are no national standards. When people say national standards, they mean NCTM's "standards" which as you know, really are not. Whenever ANYONE tells you they are abiding by national standards, call them on it immediately. If they say NCTM, then ask "Are those required by law?" Answer: No. I've caught a few fuzzies off guard this way. It's fun. They get rattled. They know the game they're playing. They then are quick to say that states by and large base their standards on NCTM's. But you're right that the frameworks are really where the rubber meets the road.

As Catherine pointed out elsewhere (I think on the NCTM online chat discussion thread), NCTM is planning to revamp its standards. Perhaps it will be grade by grade. NCTM has published its standards in a document called PSSM, which you can view online at NCTM's website, or purchase the thing. You can't view ALL of it online, however, forcing you to purchase it.

-- BarryGarelick - 25 Oct 2005

"NCTM has published its standards in a document called PSSM, "

Of course. How could I forget? Perhaps I was hoping for something more specific.

-- SteveH - 25 Oct 2005

NCTM Table of Standards and Expectations

http://standards.nctm.org/document/appendix/numb.htm

The standards are fuzzy to match the teaching methodology.

"work flexibly with fractions"

"develop meaning"

"understand the meaning and effects"

"select appropriate methods" - (but apparently not use them)

"represent, analyze, and generalize"

"relate and compare"

"identify and contrast"

"develop an initial conceptual understanding" (initial???)

"explore"

Does anyone have the full PSSM document? Do they get more specific than this? Has anyone looked at exactly how Saxon or Singapore Math couldn't meet this vague standard? Has anyplace supplemented either of these two programs to get it to meet PSSM requirements? What did they use?

Does NCTM specifically recommend child-centered discovery learning? Do they say anything about teaching methodologies? Is this separate from the PSSM? I know about their 1999 math curricula recommendations, but do they specifically tie teaching methodology with PSSM? Or, is this a separate issue or recommendation?

How about testing? Do they recommend one type of test over another? It's not clear how anyone can evaluate math curricula without seeing the tests and the grade-by-grade performance expectations. What does NCTM say about spiraling and letting kids move on to the next grade without mastering the material?

-- SteveH - 25 Oct 2005

Compare and contrast:

From the University of Oregon (Home of DI):

Mathematics Instruction Principles & Procedures:

In this course we will concentrate on three main tasks that are central to teachers’ work:

• Establishing a classroom culture that supports the development of students’ mathematical proficiency –– and, in particular, what this takes at the beginning of the year;

• Interpreting and developing students’ mathematical thinking;

• Designing, teaching, and improving actual mathematics lessons in classrooms

Being able to teach mathematics well is shaped by your knowledge of (a) what it means to be mathematically proficient and (b) the mathematics needed to do the work of teaching. To put this knowledge to good use, we will focus on honing your skills in developing students' mathematical proficiency and promoting equity in mathematics learning. Of central importance to all segments of the course is our commitment to preparing you to work effectively with diverse populations of children, ensure equal educational opportunity for all children, and use research to guide your educational practice.

Then there's some lip-service to the NCTM's "standards":

to explore, understand, and implement the National Council of Teachers of Mathematics (NCTM) Principles and Standards, state and local standards

And, finally here's a slection from the weekly schedule:

• Developing Understanding in Mathematics

• Teaching Through Problem-Solving

• Developing Early Number Concepts and Number Sense

• Whole-number Place-Value

• Strategies for Whole-Number Computation

• Fractions

• Geometry

This last part almost brought a tear to my jaded eye.

-- KDeRosa - 25 Oct 2005

I should get Temple (Grandin) in here to look at this page.

She spent an hour this weekend ranting to me about how all the people on Animal Welfare committees want to set standards like 'Handle chickens gently.'

I'm serious.

That is the formal standard these folks want to set.

Temple says, "I can't measure 'Handle chickens gently.' What does that mean? Give me something I can measure."

The really horrible thing is that Temple has data showing that when you set vague standards, animal welfare declines. A lot.

-- CatherineJohnson - 25 Oct 2005

The really horrible thing is that Temple has data showing that when you set vague standards, animal welfare declines. A lot.

Just like it does in math.

-- KDeRosa - 25 Oct 2005

"That is the formal standard these folks want to set."

To be fair, the more rigorously you set standards, the more it costs to enforce them (in time, money, good will, whatever). At some point the costs overwhelm the benefits. Down this road lies the $436 hammer.

Further, there is always something else you can do with the same money, so you need to balance the potential benefits of any policy with other possible uses of the same resources. For an example of the latter process in action, see the conference that Bjorn Lomborg set up to study the appropriate use of resources to improve the environment (short report: fix third-world water safety first) or the Oregon attempt to get cost-benefit ratios for various common medical procedures.

The trick is to set your specifications so the costs and benefits are in balance; this is non-trivial.

I do not mean this as an argument that our current math standards (for instance) are reasonable. Rather, I'd like to see the cost-benefit issue explicitly addressed. For example, what is the cost-benefit ratio of "My Favorite Number" essays?

-- DougSundseth - 25 Oct 2005

what is the cost-benefit ratio of "My Favorite Number" essays?

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-- KDeRosa - 25 Oct 2005

I took a look at the Michigan MEAP released questions for 4th grade.

There were only two released questions, and neither one of them actually involved doing any math.

The first one showed a grid of how much time a student had read every day for three weeks. The pattern was 30 one day, 20 the next for three weeks. But it had three whole missing pieces of data. The students had to fill it in and then fill in the entire 4th week. Then they had to explain how they solved the problem!!! I'm not kidding!!

How about: I could have solved this dumb problem in kindergarten where they taught me about patterns.

This was 4th grade and it was worth 4 points.

No wonder the scores are not declining: the students are not actually asked to do any math.

-- AnneDwyer - 25 Oct 2005

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It's a fine object of ridicule; so it has that benefit at least.

-- DougSundseth - 25 Oct 2005

New Zealand's curriculum organisation, called NZQA, introduced a new curriculum based on unit standards. So rather than externally marked exams as in the past, teachers mark their own kids work. These standards are meant to be comparable amongst schools so a kid from Porirua College with an Excellence in Algebra should be comparable with a kid from Auckland Grammar with an Excellence in Algebra (to pick two opposites).

I fortunately left school before this new system was introduced. But I know a lot of teachers. One source of complaints: under English, to get the top level (Excellence), the requirements state there should be "few" grammar mistakes per page. My old high school has defined this as "2 or less per page". They have no idea what other schools have defined it as, and the NZQA won't offer any help.

A number of schools are offering the International Bacculeaurate or the Cambridge Board Exam. For some reason this upsets the NZQA.

-- TracyW - 25 Oct 2005

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