Politician: The current crisis in mathematics education must be overcome if we are to remain competitive in the global economy. Alleviating this crisis requires the employment of successful teaching methods. No method of teaching a subject can succeed that does not get students to spend a significant amount of time outside of class studying that subject.
Summary
The stimulus can be diagrammed as follows:
Notable Valid Inferences
If we are to remain competitive in the global economy, we must get students to study math outside of class.
If we are to overcome the crisis in math education, we must get students to study math outside of class.
If students don’t spend a significant amount of time studying math outside of class, then we won’t remain competitive in the global economy.
A
If students spend a significant amount of time outside of class studying mathematics, the current crisis in mathematics education will be overcome.
This could be false. Spending a significant time outside of class studying math is a necessary condition, not a sufficient condition, of resolving the crisis.
B
The current crisis in mathematics education will not be overcome unless students spend a significant amount of time outside of class studying mathematics.
This must be true. As shown in the diagram, by chaining conditional claims, we see that studying math outside of class is a necessary condition of overcoming the crisis
C
Few subjects are as important as mathematics to the effort to remain competitive in the global economy.
This could be false. The stimulus does not talk about the relative importance of other subjects.
D
Only if we succeed in remaining competitive in the global economy will students spend a significant amount of time outside of class studying mathematics.
This could be false. Remaining competitive in the global economy is a sufficient condition, not a necessary condition, of spending a lot of time studying math outside of class.
E
Students’ spending a significant amount of time outside of class studying mathematics would help us to remain competitive in the global economy.
This could be false. We know that spending lots of time outside of class studying math is a necessary condition of remaining competitive. However, that’s not logically the same as “helping.”
"Surprising" Phenomenon
Why did the witnesses who gave less inaccurate details than usual at first end up giving more inaccurate details than usual later?
Objective
The correct answer will be a hypothesis that explains why witnesses who gave relatively accurate accounts when prompted to give inaccuracies ended up giving relatively inaccurate accounts when prompted to answer accurately. We don’t know whether these witnesses changed their testimonies at all the second time around, or if the rest of the group simply started amending their erroneous details. But a hypothesis that points to either of these explanations would help explain the situation.
A
These witnesses were more observant about details than were most of the other witnesses.
This would explain why the first group gave more accurate testimonies the first time around. But why would that change when they gave their testimonies a second time? We need something to reconcile that discrepancy.
B
These witnesses had better memories than did most of the other witnesses.
Like (A), this answer choice explains why the first group gave more accurate testimonies the first time around. We need to know why their testimonies were relatively inaccurate the second time.
C
These witnesses were less inclined than most of the other witnesses to be influenced in their testimony by the nature of the questioning.
These witnesses didn’t change much about their testimonies the second time around. So, while they weren’t led towards inaccuracies on the first go as the other witnesses were, they also weren’t led towards accurate details the second time.
D
These witnesses were unclear about the details at first but then began to remember more accurately as they answered questions.
This could be true, but it doesn’t reconcile the fact these witnesses were more accurate than others the first time around and less accurate the second. We need something comparative between the two groups.
E
These witnesses tended to give testimony containing more details than most of the other witnesses.
It doesn’t matter how detailed their testimonies were. We need to know why those details were relatively accurate on first examination and relatively inaccurate on second examination.
Summarize Argument
The meteorologist concludes that his station’s weather predictions are better than its competitors. This is because the majority of times they predicted rain, it rained. The station’s competitors have not been accurate most of the time.
Notable Assumptions
The meteorologist assumes that prediction accuracy is an effective indicator of usefulness and reliability. For example, it could be that the meteorologist's station has only predicted it would rain three times in a given year and it rained two out of those three times, while the competitors predicted it would rain 100 times and it rained 49 out of those 100 times. While the meteorologist’s station may be more accurate, it is not more useful or reliable. The author also assumes a conclusion about weather forecasts broadly using facts about rain—maybe people also care about whether it will snow.
A
The meteorologist’s station forecast rain more often than did the most popular news station in the area.
This strengthens the argument by addressing a potential weakness. It provides additional information that reinforces the meteorologist's assumption that the proportions he compares are comparable, as his station is more accurate even with a larger number of predictions.
B
The less popular of the competing stations does not employ any full-time meteorologists.
This does not affect the argument. How many meteorologists a station employs and their status of employment (i.e., whether they’re full-time, part-time, etc.) does not provide us with information about how reliable or useful their station is in predicting the weather.
C
The most popular news station in the area is popular because of its investigative news reports.
This does not affect the argument. A station being able to put on a good investigative news report doesn’t tell us anything about its weather prediction abilities.
D
The meteorologist’s station has a policy of not making weather forecasts more than three days in advance.
This does not affect the argument. We don’t know if this policy affects the station’s accuracy, and we have no reason to believe that all its competitors don’t share this policy.
E
On most of the occasions when the meteorologist’s station forecast that it would not rain, at least one of its competitors also forecast that it would not rain.
This does not affect the argument. It could be that on all the days the station predicted it would not rain its competitors predicted the same—but still, the station has a higher accuracy of predicting rain on the days it rained. This doesn’t change anything for the argument.
A
having exceptional lung capacity and an exceptionally powerful heart is an advantage in cycling
This possibility doesn’t undermine the argument’s reasoning. The author may agree that these qualities give one an advantage in cycling.
B
some winners of the Tour de France have neither exceptional lung capacity nor exceptionally powerful hearts
This possibility, if true, shows that last year’s winner doesn’t have to have an exceptional lung capacity. It’s possible that they had neither an exceptionally powerful heart nor exceptional lung capacity.
C
cyclists with normal lung capacity rarely have exceptionally powerful hearts
This possibility doesn’t undermine the argument, because it doesn’t relate to whether someone without an exceptionally powerful heart will have an exceptional lung capacity.
D
the exceptional lung capacity and exceptionally powerful hearts of Tour de France winners are due to training
This possibility doesn’t undermine the argument. The author could agree that these features are due to training.
E
the notions of exceptional lung capacity and exceptional heart function are relative to the physiology of most cyclists
The author concludes that last year’s winner has exceptional lung capacity. Whether this is relative to most cyclists doesn’t change the fact that the author still believes last year’s winner has exceptional lung capacity, because they don’t have an exceptionally powerful heart.
A
fails to consider the possibility that the trunks of other cars may pop open when those cars hit potholes
The argument concerns Cynthia’s car and what caused it to pop open. The author does not have to have any belief about whether other cars’ trunks can pop open due to potholes.
B
fails to consider the possibility that potholes can have negative effects on a car’s engine
The argument concerns the cause of Cynthia’s trunk popping open. Whether the engine was affected has no bearing on the cause of the trunk popping open.
C
presumes, without providing justification, that if one event causes another, it cannot also cause a third event
The author did not reason that the trunk couldn’t have popped open from a pothole because the pothole already caused something else.
D
fails to consider the possibility that one type of event can be caused in many different ways
The author overlooks that one type of event (Cynthia’s trunk popping open) can be caused in many different ways. This points out that even if the trunk popped open for other reasons on other occasions, it could have popped open on this occasion due to a pothole.
E
presumes the truth of the claim that it is trying to establish
The author did not assume the conclusion as part of the reasoning. The reasoning involves other occasions on which the pothole popped open. It doesn’t involve a restatement of the conclusion that on this occasion the trunk didn’t pop open from a pothole.