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Question
QuickView
Choices
Curve Question
Difficulty
Psg/Game/S
Difficulty
Explanation
PT35 S4 Q01
+LR
Except +Exc
Most strongly supported +MSS
A
7%
159
B
78%
165
C
2%
161
D
9%
163
E
4%
158
125
143
161
+Medium 144.86 +SubsectionEasier

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The statements above provide some support for each of the following EXCEPT:

We have a Most Strongly Supported EXCEPT question. That means the four wrong answers will be strongly supported, and the correct answer will not. To be more precise, the question stem actually indicates that the four wrong answers will have “some support.” So they might not even need to be strongly supported – as long as the answer has even some support, it’s wrong. The correct answer will not have any support. If you think back to the spectrum of support, the correct answer will be anywhere from “merely consistent with” the stimulus, to anti-supported.

The graphical illustrations mathematics teachers use enable students to learn geometry more easily by providing them with an intuitive understanding of geometric concepts, which makes it easier to acquire the ability to manipulate symbols for the purpose of calculation.

This tells us that the drawings math teachers use help students learn geometry more easily. How? It gives them an intuitive understanding of the concepts, which makes it easier to use symbols for calculations. So drawings are good for teaching geometry.

Illustrating algebraic concepts graphically would be equally effective pedagogically, even though the deepest mathematical understanding is abstract, not imagistic.

Drawings would be equally helpful for teaching algebraic concepts. (Equally effective as what? As teaching geometric concepts.) However, the deepest understanding doesn’t seem reachable by drawings (images).

The stimulus isn’t giving us too much to keep track of. Drawings are good for teaching geometry. They’re also good for teaching algebra. The drawings don’t give the deepest understanding, however.

Let’s look for 4 answers that have support – these are the wrong answers, since it’s an except question. The answer that does not have support is correct.

Answer Choice (A) Pictorial understanding is not the final stage of mathematical understanding.

This is supported, since the deepest understanding is not based on images. “Pictorial” understanding is another way to describe understanding based on images. “Final stage” of understanding is another way to say “deepest.”

If you’re uncomfortable with immediately making these connections, that’s OK. But keep that same level of skepticism for all other answers. If you picked this answer over the correct answer, you should ask yourself why you thought the other answer had more support than this one. In addition, keep in mind that the wrong answers just need some support from the stimulus – they don’t have to be proven true. The “deepest” stage of understanding is close enough to “the final stage.” If you’re not at the deepest level of understanding, would it make any sense to say that you’re already at the final stage of understanding? No. And if you understand something at the deepest level – there’s no further level to get to – would it make any sense to say that there’s some other stage of understanding that you still have yet to reach? No – you’re at the final stage.

Correct Answer Choice (B) People who are very good at manipulating symbols do not necessarily have any mathematical understanding.

This has multiple features that can help us recognize it’s not supported. First, it starts by talking about “people who are very good at manipulating symbols.” Do we know anything about these people? There’s nothing in the stimulus about those people. We do know that being good at manipulating symbols for calculation is connected to learning geometry more easily. So perhaps this would have been a better answer if it had said that people who are very good at manipulating symbols (for the purpose of calculation) have an easier time learning geometry than people who aren’t as good. But that’s not what it says.

Second, this answer says that a group of people does not have “any mathematical understanding.” Where in the stimulus does it suggest anything about who lacks understanding? All we have are relative statements about what helps understanding geometry and algebra, and about how drawings do not give the deepest understanding. But the idea of “no understanding at all” is an absolute claim that the stimulus does not support.

This answer is trying to tempt you by saying something that sounds very reasonable based on real world understanding. There are probably a lot of people who are good at manipulating symbols, but aren’t good at math. Lots of people play Tetris or Candy Crush, or are very good at drawing, but aren’t good at math. But the question is whether this answer choice has support from the stimulus. None of the statements in the stimulus either alone or in combination support this answer.

Answer Choice (C) Illustrating geometric concepts graphically is an effective teaching method.

This is basically a restatement of the first sentence. So it’s supported.

Answer Choice (D) Acquiring the ability to manipulate symbols is part of the process of learning geometry.

This is supported by the language, which makes it easier to acquire the ability to manipulate the symbols for the purpose of calculation. This part was presented as part of an explanation for how drawings help students learn geometry more easily. If the ability to manipulate symbols for calculating is how drawing helps students learn geometry more easily, that suggests such manipulation is part of the learning. If it wasn’t part of the learning, then how would getting better at it help students learn geometry?

Now you might be thrown off if you interpret the answer as saying that the ability to manipulate symbols is necessary to the process of learning geometry. You might be thinking, “Yeah, I know it can help you learn geometry. But what if there’s a way to learn geometry without manipulating symbols? Something can help you learn, but not be necessary, right?”

To that I have two responses.

First, is saying that manipulating symbols is a “part” of learning asserting that it’s necessary?

Shooting three pointers is a part of playing basketball. Is this saying that shooting three pointers is necessary to play basketball?

Television is a part of living in today’s society. Does that mean television is necessary in order to live in today’s society?

Learning Latin phrases such as “arguendo” and “a priori” is part of the process of studying law in law school. Does that mean you need to learn Latin phrases in order to study law in law school?

Is it possible that X can be a part of the process of learning Y, even though it’s not necessary to learn Y?

I’m not asking for a definitive answer, but suggesting that the issue is not as clear-cut as you might think. These examples should counsel against interpreting “a part of” as “necessary to.”

Second, ask yourself whether you think this answer has less support than the correct answer. Are you being as critical of the correct answer and whatever support you think it has as you are of this answer?

Answer Choice (E) There are strategies that can be effectively employed in the teaching both of algebra and of geometry.

We know that drawings can be effective in teaching geometry and algebra. So there are “strategies” that can effectively be used to teach those subjects – techniques involving drawings or illustrations. This is an example of reasoning by generalization. The stimulus provides a specific instance of a strategy (graphical illustrations) that effectively aids in teaching geometry and algebra. From this specific instance, (E) infers a broader generalization that there are effective strategies for teaching both these subjects.