Monday, August 16, 2010

Teaching For the Test

I've been slaving over this long enough—I'm just going to post it. More on this to come.

In my last post I outlined the trap in which American education reform is mired: our best hope for improving education—tying incentives to outcomes—has reduced inner-city education to a massive, year-round test-prep regimen. I want to talk about how we could improve the situation, but first we need to look more closely at the phenomenon of test-driven education.

The phrase "teaching to the test" gets thrown around a lot, and I don't want to write facilely about the ills of No Child Left Behind, because that leads to the kind of simplistic ideology I'm trying to avoid. I think that there are serious downsides to NCLB, and I'd like to discuss those in detail, but I'm hindered by the narrowness of my experience. I can talk with great specificity about the impact of NCLB on New York State middle-school mathematics education; but it's hard to be sure how generalizable that analysis would be. At some point, I'd like to conduct a qualitative study of state exams and curricula in a variety of subject-areas, at a variety of grade-levels, from a wide selection of states—but that study will require months of research. Anecdotal evidence from teachers in other states and subject areas, however, suggests that New York State middle-school math is not some wild aberration. What follows, therefore, in lieu of more definitive conclusions, is my analysis-by-extrapolation of the specific mechanisms by which NCLB is undermining education in America.

Outcomes and Assessments

First of all, let's talk about the relationship between outcomes and assessments. The principle insight behind high-stakes testing and No Child Left Behind is that we should hold educators accountable not for what they do with their time but for the results they produce. That's a good idea (see my last post). The problem comes from the difficulty of determining what educators have produced. To do that, we introduce assessments, i.e. state exams— but our assessments are lousy. They don't really tell us much about the depth of student understanding; they tell us only about students' surface knowledge of a subject: whether they can apply the Pythagorean theorem, find the indirect object of a sentence, name three causes of the First World War.

As has been well publicized, under NCLB many schools, especially those in poor districts or districts with high percentages of at-risk students[1], have come to align most or all of their curriculum and instruction to these same shallow state tests. Some probably do this because they really are motivated only by the exam[2], but most do it despite the best intentions of their administration and faculty and in contradiction to their own stated aims— because, when it comes down to it, they often have no choice.

When I talk about how shallow state tests are, people tend to assume that means they're easy—that they test only the simplest mathematical procedures and concepts. If that were true, it would easy for a dedicated teacher or school to go above and beyond test-prep, teaching students about the deep structures and beauties of mathematics, literature, etc. In fact, though, state tests—even at the primary school level—cover a daunting array of information, much of which will be unfamiliar to your average adult.

See, for example, Appendix A, a list of New York State's procedural standards for 6th grade math. Most adults, even those with a good understanding of mathematics, cannot tell the difference between a bar-graph and a histogram (standard 6.S.4) or "Determine the number of possible outcomes for a compound event by using the fundamental counting principle and use this to determine the probabilities of events when the outcomes have equal probability" (standard 6.S.11). Yet, under NCLB, a teacher with a classroom full of 6th graders who can't perform multi-digit subtraction or tell with any certainty whether 1/5 is larger or smaller than 1/6, will be forced, if they want to keep their jobs, to ignore such gaping conceptual holes and power through a unit on data representation and compound probability.

Advocates of high-stakes testing are often the same people who argue for "back to basics" education. That viewpoint critiques progressive educators for focusing on abstract ideas of curiosity, creativity, and deep understanding, when students lack the basic skills and knowledge necessary to write a complete sentence or perform long-division. That critique may be apt, but basic skills and conceptual depth are not natural enemies: a strong foundation in the former is the quickest route to the latter. The attempt to create rigorous curricula in the absence of conceptual depth has led to the very opposite: a system in which there's no time for the basics or concepts.

This all adds up to the Heisenberg Principle of Education: we create assessments to measure student outcomes; but the assessments come, instead, to determine the outcomes. Viewed in the most extreme terms, this alters the entire project of education.

Why We Educate

Before I got involved in inner-city education, my vision of the ideal classroom was one in which students and teacher were deeply intellectually engaged with subject matter, and the primary source of student motivation was not grades or any other extrinsic element, but curiosity. Under the constraints of NCLB, such a model is a fantasy, and—considering the furious pace that must be maintained in order to meet state standards in many schools—a dangerous one.

In the most successful post-NCLB inner-city classrooms, teachers engage students through games, competition, challenges, and fast-paced drills, in order to push them through a curriculum that they don't well understand and for which they often lack the foundational skills and knowledge. A good teacher, in this context, can provide students who never really understood division with tricks and mnemonics that will get them through a unit on fractions. The resulting sense of competence and success will obscure students' underlying sense of confusion—indeed, these students will have come to see mathematics as an arbitrary language with no discernable logic, and will not even recognize that they do not comprehend it—after all, they passed the exam, didn't they?

I don't want to exaggerate the situation. At its best, the inner-city classroom described above can be an exciting, safe, positive space for students. Issues of comprehension are less severe in subjects other than mathematics and in schools with wealthy or ability-selected students. Still, the primary aim of these classrooms is not that students fully comprehend what they're learning, nor that they engage deeply with their subject matter; the primary aim is that they be able to perform a set of discrete tasks, in the exact contexts and formats that are presented on the state exam.

This can be seen most clearly in teachers' attitudes towards conceptual understanding. Progressive teachers tend to view deep understanding of subject-matter as their ultimate goal; they teach specific procedures, facts, and rules only as stepping stones towards this type of comprehension. Teachers in inner-city classrooms talk about comprehension a lot, too. To some degree, they talk about it as a goal in and of itself, but usually it's justified, or even brought up in the first place, as a way to improve retention: the better kids comprehend material, the better they'll be able to remember the things they learn. Higher retention means less class-time devoted to review and, ultimately, higher scores on the state exam. Thus, for progressive private-school teachers, procedures are a tool to achieve conceptual understanding, whereas for those subject to the constraints of NCLB, conceptual understanding is a tool to achieve better retention of procedures.

[1] I'd like to know just how many teachers that is, but I've had a surprisingly difficult time figuring out what percentage of teachers or schools serve poor or high-risk students. The department of ed has a nice little digest of education statistics, which I've referenced in a couple past posts, but it doesn't have data on number of schools or students by income level of district. More surprisingly, it doesn't have data on number of schools designated as underperforming according to NCLB; this seems like an intentional omission, but who knows. It does have data on number of schools and students by type of local: about 23% of public school students are in cities with populations over a hundred thousand; I think we can safely assume that most of these attend schools where passing state test is of paramount concern. Another 19% go to rural schools, many of which, I suspect, are poor and in danger of failing to make state standards under NCLB. My data here is very limited, unfortunately.>.

[2] For the most jaded, burnt-out, or lazy educators, this is only natural: their motivations were already self-interested and economic. For most others, and to varying degrees depending on the individual, the material rewards and punishments tied to the exam have probably displaced the intrinsic motivations that originally drove them to educate. There's plenty of psychological and economic research on the capacity of extrinsic motivators to drown out intrinsic ones and for economic incentives to eliminate considerations of personal responsibility or community-mindedness.

Appendix A: New York State 6th Grade Procedural Math Standards (back to main text.)

What follows is a list of the mathematical procedures that New York State 6th graders are expected to be able to carry out. I'm glazing over a big issue here: the list of NYS 6th grade math standards is actually much longer than this, but most of them are conceptual and not tested on the state exams. I have a lot to say about that, but I'm not going to say it now. The list below is broken into various "strands," under which are various sub- and sub-sub-headings. Each strand represents a different area of mathematics, and you may wonder why there are so many different mathematical fields covered in a single year; in fact, all of these same strands are covered year after year, so that students learn a little more about each one each year; this is called a spiraled curriculum. Again, I have a lot to say about that, but I'm not going to get into it right now.

Number Sense and Operations Strand

Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.

Number Systems


Read and write whole numbers to trillions


Define and identify the commutative and associative properties of addition and multiplication


Define and identify the distributive property of multiplication over addition


Define and identify the identity and inverse properties of addition and multiplication


Define and identify the zero property of multiplication


Understand the concept of ratio


Express equivalent ratios as a proportion


Distinguish the difference between rate and ratio


Solve proportions using equivalent fractions


Verify the proportionality using the product of the means equals the product of the extremes


Read, write, and identify percents of a whole (0% to 100%)


Solve percent problems involving percent, rate, and base


Define absolute value and determine the absolute value of rational numbers (including positive and negative)


Locate rational numbers on a number line (including positive and negative)


Order rational numbers (including positive and negative)

Students will understand meanings of operations and procedures, and how they relate to one another.



Add and subtract fractions with unlike denominators


Multiply and divide fractions with unlike denominators.


Add, substract, multiply, and divide mixed numbers with unlike denominators


Identify the multiplicative inverse (reciprocal) of a number


Represent fractions as terminating or repeating decimals


Find multiple representations of rational numbers (fractions, decimals, and percents 0 to 100)


Evaluate numerical expressions using order of operations (may include exponents of two and three)


Represent repeated multiplication in exponential form


Represent exponential form as repeated multiplication


Evaluate expressions having exponents where the power is an exponent of one, two, or three

Students will compute accurately and make reasonable estimates.



Estimate a percent of quantity (0% to 100%)


Justify the reasonableness of answers using estimation (including rounding)

Algebra Strand

Students will represent and analyze algebraically a wide variety of problem solving situations.

Variables and Expressions


Translate two-step verbal expressions into algebraic expressions

Students will perform algebraic procedures accurately.

Variables and Expressions


Use substitution to evaluate algebraic expressions (may include exponents of one, two and three)

Equations and Inequalities


Translate two-step verbal sentences into algebraic equations


Solve and explain two-step equations involving whole numbers using inverse operations


Solve simple proportions within context


Evaluate formulas for given input values (circumference, area, volume, distance, temperature, interest, etc.)

Geometry Strand

Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.



Calculate the length of corresponding sides of similar triangles, using proportional reasoning


Determine the area of triangles and quadrilaterals (squares, rectangles, rhombi, and trapezoids) and develop formulas


Use a variety of strategies to find the area of regular and irregular polygons


Determine the volume of rectangular prisms by counting cubes and develop the formula


Identify radius, diameter, chords and central angles of a circle


Understand the relationship between the diameter and radius of a circle


Determine the area and circumference of a circle, using the appropriate formula


Calculate the area of a sector of a circle, given the measure of a central angle and the radius of the circle


Understand the relationship between the circumference and the diameter of a circle

Students will apply coordinate geometry to analyze problem solving situations.

Coordinate Geometry


Identify and plot points in all four quadrants


Calculate the area of basic polygons drawn on a coordinate plane (rectangles and shapes composed of rectangles having sides with integer lengths)

 Measurement Strand

Students will determine what can be measured and how, using appropriate methods and formulas.

Units of Measurement


Measure capacity and calculate volume of a rectangular prism


Identify customary units of capacity (cups, pints, quarts, and gallons)


Identify equivalent customary units of capacity (cups to pints, pints to quarts, and quarts to gallons)


Identify metric units of capacity (liter and milliliter)


Identify equivalent metric units of capacity (milliliter to liter and liter to milliliter)

Tools and Methods


Determine the tool and technique to measure with an appropriate level of precision: capacity

Students will develop strategies for estimating measurements.



Estimate volume, area, and circumference (see figures identified in geometry strand)


Justify the reasonableness of estimates


Determine personal references for capacity

Statistics and Probability Strand

Students will collect, organize, display, and analyze data.

Collection of Data


Develop the concept of sampling when collecting data from a population and decide the best method to collect data for a particular question

Organization and Display of Data


Record data in a frequency table


Construct Venn diagrams to sort data


Determine and justify the most appropriate graph to display a given set of data (pictograph, bar graph, line graph, histogram, or circle graph)

Analysis of Data


Determine the mean, mode and median for a given set of data


Determine the range for a given set of data


Read and interpret graphs

Students will make predictions that are based upon data analysis.

Predictions from Data


Justify predictions made from data

Students will understand and apply concepts of probability.



List possible outcomes for compound events


Determine the probability of dependent events


Determine the number of possible outcomes for a compound event by using the fundamental counting principle and use this to determine the probabilities of events when the outcomes have equal probability

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