If schools are going to be closed and teachers fired over state tests, the tests had better be good. So I downloaded the most recent 8th grade New York State math assessment to see how it was. Out of 45 questions, I had issues with at least ten of them.

In this question, there is a pretty big inaccuracy:

The problem with this question is that ‘the Pythagorean theorem’ is used for questions where you have a right triangle and the lengths of two sides are known and you use the relationship a^2 + b^2 = c^2 to find the other side. The Pythagorean theorem, which they are instructed to use for this question, is not technically, what you do. What they mean is that there is something called the ‘converse of the Pythagorean theorem’ which says that if a^2 + b^2 equals c^2 then the triangle is a right triangle and if a^2 + b^2 does not equal c^2, then the triangle is not a right triangle. I’m not saying that they should have said ‘Using the converse of the Pythagorean theorem’ either. They should have just said to determine if this is a right triangle or not. Their ‘hint’ actually makes the question inaccurate and can confuse kids

What really bothered me is that 11 out of the 45 questions, just about 25% of the test was based on one concept in Geometry: recognizing vertical angles, complimentary angles, supplementary angles, and related angles with parallel lines. In 8th grade, apparently, students must memorize that in the configuration below in question 14 where x and y are parallel lines crossed by line z, there are 8 angles formed. Angles 1, 3, 5 and 7 are all less than 90 degrees and equal to one another while angles 2, 4, 6, and 8 are all greater than 90 degrees and equal to one another. Also, the smaller and larger angles are ‘supplementary’ meaning they have a sum of 180 degrees. So if angle 1 measures 40 degrees (as does angles 3, 5, and 7) then angle 2 (and also angles 4, 6, and 8) measures 140 degrees.

I say ‘memorize’ rather than ‘learn’ since the actual proofs of these relationships does not happen until they are in 10th grade Geometry. At this point it is merely something they they learn to recognize. The over emphasis on this concept bothers me because this isn’t really a very important concept, mathematically. It is not particularly interesting, nor does it really ‘go anywhere’ for a few years. It is the kind of mindless thing that makes students think that math is not relevant, fun, or worth learning. For the amount of value given to these questions, it would be worthwhile for an 8th grade teacher to spend 25% of the school year drilling on these mindless problems.

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You might not find it interesting, but it’s a huge part of almost any standardized test you can think of (SAT, ACT, GRE, GMAT), and kids struggle with it a great deal in geometry. That’s probably why they teach it early.

What I’m curious about is this: on a question like #38, if a kid didn’t use the phrase “vertical angles”, but rather pointed out that since CD is a line APC + APD will equal 180 degrees and since AB is a line BPD + APD will equal 180 degrees, thus APC will be congruent to BPD (in other words, basically supplied the proof), would they get marked down? Sadly, my guess is that by many scorers they would.

Actually, the student would receive full credit for that reasoning.

Cal, I agree that these problems are important for the college entrance exams. But why do we let the commercial test-makers dictate our entire curriculum?

I’m just contributing this response from a friend of mine who is a math Ph.D. high school teacher here. If anyone would like to respond:

Hi Lee,

While I disagree that stating use the Pythagorean Theorem will confuse students I do agree that that hint should not have been given. In giving the hint the student is being asked to plug and chug rather than knowing how to determine if a triangle is a right triangle. The parallel line problem is also pointless from a real world application viewpoint but then that is not what standardized tests test.

Our EOC tests do a better job of having students think then simple rote memorization not on every problem but often enough. However ACT often has similar questions so it is important for students to at least be able to apply the concepts.

EOC (End of Course) testing is done to ensure that all students have a BASIC understanding of the math concepts they learned not always higher level critical thinking. Many of the same types of problems are given on the ACT because colleges want students who have a minimal working basic knowledge. Our EOC like the ACT is predominantly made up of application problems at a rudimentary level with about 1/3 of the problems at the higher cognitive level. It allows students to reach the BASIC level with just a rudimentary understanding of how to do mathematics but in order to reach MASTERY and ADVANCED the students must be able to answer the higher cognitive questions as well.

Every standardized test no matter how well written should not be a complete picture of student understanding. However every student should be able to demonstrate a BASIC understanding of the four fundamental operations (add, subtract, multiply and divide) with all real numbers (fractions and percents included) as well as some basic geometry (area and perimeter) that will be needed to function in the real world. Unfortunately this is not what is emphasized because we concentrate so much on Algebra 1 and Geometry concepts that most people will never use. Why we feel all students need to be able to do Algebra 1 or Geometery in order to graduate is beyond my understanding. Now getting into a university is a different thing, for that you have to know Algebra 1 and Geometry.

Sorry to be so long winded but we need to develop tests that provide a picture of what students know and can do so that they along with their parents and teachers will have a better understanding of how to prepare that student for the world outside of graduation not to keep students from graduating.

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Our new La. State Superintendent is planning to substitute our current standardized testing and EOC testing with the ACT test in high school and the soon to come PAARC test (Common Core consortium to which La. belongs) wil replace our state LEAP/iLEAP tests. No details given yet.

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The Pythagorean theorem is a perfectly reasonable and valid method for telling if this is a right triangle. Its the converse of the Pythagorean theorem.

The Pythagorean theorem and the converse of the Pythagorean theorem are closely related, but to say ‘use the pythagorean theorem’ to test if this is a right triangle is not accurate.