Thursday, October 9, 2014

Review for Chapter 4 Test (Day 45)

Today is Day 45, which marks the end of the first quarter on this blog. Because of this, I'm posting another test.

The quarter mark is a good time for me to discuss my philosophy regarding grades. In particular, a question that often comes up among teachers is, should teachers let students retake tests?

A grade represents a percentage, a scale from 0 to 100. One of the most commonly used grading scales in schools is something like this:

90-100 = A
80-90  = B
70-80  = C
60-70 = D
0-60   = F

Now let me give two more examples of percentages used in the world of sports. In baseball, we have the concept of batting average. Although batting averages are calculated to three decimals rather than two, they are still essentially still percentages.

It's often said that baseball is the only place where one can fail seven out of ten times and still be considered successful. This is because .300 is an excellent batting average. A player with a batting average of .300 will be one of the top hitters in the league. On the other hand, a .200 batting average -- the dreaded Mendoza line -- is a terrible batting average. A .200 hitter is likely to be demoted to the minors or cut from the team. The highest ever batting average for a full season is slightly over .400 (Ted Williams around 70 years ago). For the recently completed season, the average hitter was batting around .250, maybe slightly above this.

Now suppose a batter gets a hit in his first at-bat, then makes out the second time up.(Let's keep it simple and say that he only has two at-bats in the game -- maybe he walked, or sacrificed, or was pinch-hit for after his second at-bat.) So, has his average gone up, down, or remained the same after this game?

To answer this, notice that the batter went 1 for 2 in this game -- so he hit .500 for the game. So the correct answer is that his new batting average is closer to .500 than his average before the game. If his average began at over .500 then it dropped, if it was at .500 then it remained the same, and if it was below .500 then it rose. And so, since realistic batting averages are well below .500, the true answer is that going 1 for 2 raises a player's batting average.

In other words, a single hit raises a batting average more than a single out lowers it. Even though batting averages range from .000 to 1.000, half the scale -- .500 and above -- is irrelevant because it refers to unrealistic averages (except for a player who has only one at-bat the entire season, gets a hit, and finishes the year hitting 1.000).

Now let's move from batting averages to team winning percentages. Now unlike batting averages where .300 is excellent, a winning percentage of .300 is terrible. This is because every game has one winner and one loser. So by definition, the average winning percentage in all of Major League Baseball is .500 (unless there were rain-outs that aren't made up or 163rd tiebreaker games, and these only change the average barely noticeably from .500).

A winning percentage of .600 is excellent. Only one team had a winning percentage that high (namely the L.A. Angels). A winning percentage of .400 is terrible. Only one team had a winning percentage that low (the Arizona Diamondbacks). Half of the scale -- above .750 and below .250 -- is irrelevant since it's extremely rare that a team would have such an extreme record after 162 games. (Notice that in football, where only 16 games are played, records such as 16-0 or 0-16 occur from time to time.)

A team who wins a game and loses the next will find that its record is closer to .500 after those two games than before them. In particular, a first place team -- presumably with a record above .500 -- will find that its winning percentage has dropped. This often shows up when two teams battling for first place have played different numbers of games -- one team having played two more games, with one more win and one more loss, than the other. The team having played more games will find that its record is closer to .500 -- and therefore lower -- than the other team's. That team is said to trail the other team "by percentage points."

Now let's return to school grades. For batting averages .300 is excellent, but in school 30% is surely a failing grade. For winning percentages .600 is one of the best teams in the majors, but in school 60% is the lowest passing grade. All of the grade boundaries -- from B to A, C to B, and so on -- occur in the upper half of the scale. The lower half of the scale -- from 0 to 50 percent -- is irrelevant as far as determining letter grades is concerned.

To the extent that C is average, the average student is earning between 70% and 80%. So we expect the average score in the class to be around the midpoint of this range -- 75%. So the average batter gets one hit every four at-bats, the average team wins two out of every four games, and the average student gets three out of four questions right.

And so, unlike the batter who gets a hit then makes an out and sees that overall his average has risen, the student who gets 100% on one test and 0% the next can never have a higher letter grade -- only a lower letter grade is possible. Therefore, a grade is much more likely to drop very rapidly than it is to rise very rapidly. Since the average grade is around 75%, a student would have to receive 150% on a test in order for the grade to rise as rapidly as a 0% drops it. (And all of this is assuming that the student isn't at one of those schools that has abolished D grades and makes 70% the lowest passing score -- that 10% difference means that a student would have to earn 160% on a test to have the same impact on the grade as a 0%!)

I've seen students come up to me and ask me why their grade has dropped so quickly -- especially when there doesn't seem to be anyone in the class whose grade has risen as rapidly as their grades have dropped. They think that I'm a mean teacher and a harsh grader -- when the true reason is the nature of the grading scale, where all of the letter grade boundaries are in the upper half.

Now it's the end of the quarter, and grades are about to come out. Let's assume for simplicity that there are equal numbers of points possible in the first and second quarters (in reality, the second quarter may have more points because that quarter has a final exam while the first quarter doesn't).

Let's say a student is earning 10% at the quarter. Then this student will surely fail the class -- even if 100% is earned the second quarter, the average grade is only 55%, an F. This is akin to a baseball team who is 11 games back with only 10 to play -- that team has been mathematically eliminated from winning the division.

A student earning 20% at the quarter can still pass the class -- but only if 100% is earned the second quarter, which isn't very likely. And a student who gets 30% the first quarter would still need to earn 90% (an A) the second quarter to pass the class. The type of student who would earn only 30% in a quarter is not the type of student who would earn an A the second quarter -- which typically covers more difficult material than the first quarter. This is akin to a baseball team who is nine games back with only 10 to play -- although the team hasn't be mathematically eliminated, they are, realistically, eliminated from the division. It would take a miracle for the team to come back and claim the division, and a similar miracle is needed for the student to get any letter grade other than F at the semester.

In fact, in some ways, any quarter grade below 50% would realistically eliminate a student from receiving a grade higher than F at the semester. A 50% quarter grade would require a second quarter of 70%, a C, to pass the class -- and this might be doable, if the student works a little harder during the second quarter.

(And that, of course, assumes that 60% is the lowest passing grade. If the 50% student is at a no-D school, then that student will need 90%, an A, in the second semester to get any letter other than F on the semester report card -- once again, realistic elimination.)

When I am teaching class, what I want is for as few students as possible to be mathematically or realistically eliminated from getting any semester grade other than F, when there are still plenty of weeks separating them from the end of the semester. I have no sympathy for a student who is realistically eliminated when there are only nine or ten days left in the semester, but nine or ten weeks, that's another matter.

A player on a baseball team that is realistically eliminated from the playoffs may be traded to a team that has a shot of winning. And even if the player isn't traded, he will still play out the rest of the season, no matter how little his heart is in it, because he's under contract to do so. Even a college student realistically eliminated from passing a class can drop out and take a W instead. But for high school students, there is no choice but for the student to take the F.

A student who might have worked harder if a second quarter 70% to raise the grade out of the F range will probably give up if 90% is needed to avoid the F. A student needing 90% and knowing that such a grade is unrealistic might choose to cheat instead to get the 90%. Or the student might just give up and accept the F -- and earn 0% the second quarter by doing absolutely no work at all -- and likely disturbing the class and becoming a discipline problem the entire quarter.

So we see that a teacher who's more lenient regarding grades -- to the extent that there are very few students realistically eliminated with weeks to go in the semester from passing -- is more likely to have students work harder as the semester goes on. It's not that I want to avoid giving out F grades and just pass everyone -- I just want to avoid realistic elimination until the final weeks.

And one way to avoid realistic elimination is to allow students to make up the tests. And this is why I would allow students to make up the tests, to get rid of the zeros, 10%, 20% grades that lower a student's grade so much. In general, the way to maximize student learning is to have as many students near the grade boundaries as possible.

Once again, I call this the "Chapter 4 Test" even though there is not much from the U of Chicago's Chapter 4 at all. Only the last two questions are actually Chapter 4 -- and even then, I changed one of them to refer to rotations rather than reflections. Rotations and the parallel line tests are the focus of this upcoming test.

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