What grades should kids get? Notes on evaluation for the mathematically-challenged.

Here is a part of a post from Ben’s.  A teacher– let’s call him Mr John Speaking– who uses T.P.R.S. in their language class writes:

“I was told by a Defartment Chair a few weeks ago that my grades were too high across the board (all 90s/100s) and that I needed more of a range for each assessment. Two weeks later I had not fixed this “problem” and this same Defartment Chair pulled me out of class and proceeded to tell me, referencing gradebook printouts for all my classes, that these high grades “tell me there is not enough rigor in your class, or that you’re not really grading these assessments.” After this accusation, this Defartment Chair told me I was “brought on board [for a maternity leave replacement] in the hopes of being able to keep me, but that based on what he’d seen the past few weeks, I’m honestly not tenure track material.”

Obviously, Mr John Speaking’s Defartment Chair is an idiot, but, as idiots do, he does us a favour:  he brings up things worth thinking about.

There are two issues here:

a) Should– or do— student scores follow any predictable distribution?  I.e., should there be– or are there–a set percentage of kids in a class who get As, Bs, Cs, Ds and Fs?

b) How do you know when scores are “too low” or “too high”?

Today’s question: what grades should students get?

First, a simple, math idiot’s detour into grading systems and stats.  The math idiot is me.  Hate stats?  Bad at math? Read on!  If I can get it, anyone can get it!

It is important to note that there are basically two kinds of grading systems. We have criterion-referenced grading and curved (norm-referenced) grading.

First, we have criterion-referenced grading.  This is, we have a standard– to get an A, a student does X.  To get a B, a student does Y, etc.  For example, we want to see what our Samoyed Dogs’ fetching skills are and assign them fetching marks. Here is our Stick Fetching Rubric:

A:  the dog runs directly and quickly to the thrown stick, picks it up, brings it back to its owner, and drops it at owner’s feet.

B: the dog dawdles on its way to the stick, plays with it, dawdles on the way back, and doesn’t drop it until asked.

C: the dog takes seemingly forever to find the stick, bring it back, and refuses to drop it.

So we take our pack of five Samoyed Dogs, and we test them on their retrieval skills.  Max, who is a total idiot, can’t find the stick forever, then visits everyone else in the park, then poos, then brings the stick an hour later but won’t drop it because, hell, wrestling with owner is more fun.  Samba dutifully retrieves and drops.  Rorie is a total diva and prances around the park before bringing the stick back.  Arabella is like her mother, Rorie, but won’t drop the stick.  Sky, who is so old he can remember when dinosaurs walked the Earth, goes straight there, gets the stick, and slowly trudges back.  So we have one A, one B, one C, one C- (Max– we mercy passed him) and one A- (Sky, cos he’s good and focused, but slow).

Here are our Samoyeds:

Now note–

1. Under this scheme, we could theoretically get five As (if all the Dogs were like Samba), or five Fs (if everybody was as dumb and lovable as Max).  We could actually get pretty much any set of grades at all.

2.  The Samoyed is a notoriously hard-to-train Dog.  These results are from untrained Samoyeds.  But suppose we trained them?  We used food, praise, hand signals etc etc to get them to fetch better and we did lots of practice.  Now, Sky is faster, Rorie and Arabella don’t prance around the park, and even silly Max can find the stick and bring it.  In other words, all the scores went up, and because there is an upper limit– what Samba does– and nobody is as bad as Max was at fetching, the scores are now clustered closer together.

The new scores, post-training, are:

Sky and Samba: A

Rorie, Max and Arabella: B

Variation, in other words, has been reduced.

3.  Suppose we wanted– for whatever reason– to lower their scores.  So, we play fetch, but we coat the sticks in a nasty mix of chocolate and chili powder, so that whenever the Dogs get near them, they get itchy noses, and very sick if they eat them.  The Dogs stop wanting to fetch our sticks.  Some of them will dutifully do it (e.g. Samba), but they aren’t idiots, and so most of them will decide to forget or ignore their training.

4.  Also note who we don’t have in our Dog Pool:  Labrador Retrievers (the genius of the fetching world), and three-legged Samoyeds.  There’s no Labs because they are three orders of magnitude better than Samoyeds at fetch, and we don’t have three-legged Samoyeds because, well, they can’t run.

In other words, we could reasonably get any mix of scores, and we could improve the scores, or we could– theoretically– lower them.  Also, we don’t have any Einstein-level retrievers or, uhh, “challenged” retreivers– there are no “outliers.”

Now, let’s look at “bell curve” (a.k.a. norm-referenced) grading.  In this case, we decide– in advance— how many of each score we want to assign.  We don’t want any random number of As or Fs or whatever– we want one A, one F, etc.  We want the scores to fit into a bell curve, which looks like this:

We are saying “we want a certain # of As, Bs, Cs, Ds and Fs.”  Now, we have a problem.  In our above stick fetching example, we got an A, an A-, a B, a C and a C-.  We have no Ds or Fs, because all of the Dogs could perform.  None of them were totally useless.  (After doing some training, we would get two As (Samba, Sky) and three Bs (Rorie, Max and Arabella).  But if we have decided to bell curve, or norm reference, our scores, we must “force” them to fit this distribution.

So Samba gets an A, Sky gets a B, Rorie gets a C, Arabella gets a D, and Max fails.

Now, why would anyone do this?  The answer is simple: norm referencing is only a way to sort students into ranks where the only thing that matters is where each person ranks in regard to others.  We are not interested in being able to say “in reference to criteria ____, Max ranks at C.”  All we want to do here is to say where everyone is on the marks ladder compared to everyone else.

Universities, law schools, etc sometimes do this, because they have to sort students into ranks for admissions purposes, get into the next level qualifiers, etc etc.  For example, law firm Homo Hic Ebrius Est goes to U.B.C. and has 100 students from which to hire their summer slav– err, articling students.  If they can see bell-curved scores, they can immediately decide to not interview the bottom ___ % of the group, etc.  Which U.B.C. engineers get into second year Engineering?  Why, the top 40% of first-year Engineering students, of course!

Now I am pretty sure you can see the problem with norm referencing:  when we norm reference (bell curve), we don’t necessarily say anything about what students actually know/can do.  In the engineering example, every student could theoretically fail…but the people with the highest marks (say between 40 and 45 per cent) would still be the top ones and get moved on.  In the law example, probably 95% of the students are doing very well, yet a lot of them won’t be considered for hire. Often, bell-curves generate absurd results.  For example, with the law students, you could have an overall mark of 75% (which is pretty good) but be ranked at the bottom of the class.

So where does the idea for norm referencing (“bell curving”) sudent scores come from?  Simple: the idea that scores should  disitribute along bell-curve line comes from a set of wrong assumptions about learning and about “nature.”  In Nature, lots of numbers are distributed along bell-curve lines.  For example, take the height of, say, adult men living in Vancouver.  There will be a massive cluster who within two inches of 5’11” (from 5’9″ to 6’1″).  There will be a smaller # who are 5’6″ to 5’8″ (and also who are 6’1.5″ to 6’3″).  There will be an even smaller number who are shorter than 5’6″ and taller than 6’3″.  Get it?  If you graphed their heights, you’d get a bell curve like this:

If you graphed adult women, you’d also get a bell curve, but it would be “lower” as women (as dating websites tell us) are generally shorter than men.

Now– pay attention, this is where we gotta really focus– there are THREE THINGS WE HAVE TO REMEMBER ABOUT BELL CURVES

a)  Bell curve distributions only happen when we have an absolutely massive set of numbers.  If you looked at five men, they might all be the same height, short, tall, mixed, whatever (i.e. you could get any curveat all). But when you up your sampling to a thousand, a bell curve emerges.

b) Bell curve distributions only happen when the sample is completely random.  In other words, if you sampled only elderly Chinese-born Chinese men (who are generally shorter than their Caucasian counterparts), the curve would look flatter and the left end would be higher.  If you didn’t include elderly Chinese men, the curve would look “pointier” and the left end would be smaller. A bell curve emerges when we include all adult men in Vancouver.  If you “edit out” anyone, or any group, from the sample, the distribution skews.

c)  Bell curves raise one student’s mark at the expense of another’s.  When we trained our Samoyed Dogs, then marked them on the Stick Fetching Rubric, we got three As and two Bs.  When we convert this into a curve, however, what happens is, each point on the curve can only have one Dog on it.  Or, to put it another way, each Dog has a different mark, no matter how well they actually do.  So, our three As and two Bs become an A, a B, a C, a D and an F.  If Rorie gets a B, that automatically (for math-geek reasons) means that Max will get a different mark, even if they are actually equally skilled.

As you can see in (c), bell curves are absolutely the wrong thing to do with student marks.

And now we can address the issues that Mr John Speaking’s Defartment Head brings up.  Mr Defartment Head seems to think that there are too many high marks, and not enough variation within the marks.

First, there is no way one class– even of 35 kids– has enough members to form an adequate sample size for a bell-curve distribution.  If Mr Defartment Head thinks, “by golly, if that damned Mr John Speaking were teaching rigorously, we’d have only a few As, a few Ds, and far more Bs and Cs,” he’s got it dead wrong: there aren’t enough kids to make that distribution  possible.  Now, it could happen, but it certainly doesn’t have to happen.

Second, Mr John Speaking does not have a statistically random selection of kids in his class.  First, he probably doesn’t have any kids with special challenges (e.g. severe autism, super-low I.Q., deaf, etc etc).  BOOM!– there goes the left side of the bell curve and up go the scores.  He probably also doesn’t have Baby Einstein or Baby Curie in his class– those kids are in the gifted program, or they’ve dropped out and started hi-techs in Silicon Valley.  BOOM!– there goes the right side of your curve.  He’ll still have a distribution, and it could be vaguely bell-like, but it sure won’t be a classic bell curve.

Or he could have something totally different.  Let’s say in 4th block there are zero shop classes, and zero Advanced Placement calculus classes.  All of the kids who take A.P. calculus and shop– and who also take Spanish– therefore get put in Mr Speaking’s 4th block Spanish class.  So we now have fifteen totally non-academic kids, and fifteen college-bound egg-heads.  Mr Speaking, if he used poor methods, could get a double peaked curve:  a bunch of scores clustering in the C range, and another punch in the A, with fewer Bs and Ds.

Third, instruction can– and does– make a massive difference in scores. Remember what happened when we trained our Samoyeds to give them mad fetching skillz, yo? Every Dog got better. If Mr Speaking gave the kids a text, said “here, learn it yourself,” then put his feet up and did Sudoku on his phone or read the newspaper for a year (I have a T.O.C. who comes in and literally does this), his kids would basically suck at the language (our curve just sank down).  On the other hand, if he used excellent methods, his kids’ scores would rise (curve goes up).  Or, he is awesome, but gets sick, and misses half the year, and his substitute is useless, so his kids’ scores come out average.  Or, he sucks, gets sick, and for half the year his kids have Blaine Ray teaching them Spanish, so, again, his kids’ scores are average:  Blaine giveth, and Speaking taketh away.

“Fine,” says the learned Defartment Chair, “Mr John Speaking is a great teacher, and obviously his students’ scores are high as a result of his great teaching, but there should still be a greater range of scores in his class.”

To this, we say a few  things

a)  How do we know what the “right” variability of scores is?  The answer:  there is no way of knowing without doing various kinds of statistical comparisons.  This is because it’s possible that Mr Speaking has a bunch of geniuses in his class.  Or, wait, maybe they just love him (or Spanish) and so all work their butts off.  No, no, maybe they are all exactly the same in IQ?  No, that’s not it.  Perhaps the weak ones get extra tutoring to make up for their weakness. Unless you are prepared to do– and have the data for– something called regression squares analysis, you are not even going to have the faintest idea about what the scores “should” be.

b)  score variability has been reduced with effective teaching.  There are zillions of real-world examples of where appropriate, specific instruction reduces the variation in performance. Any kid speaks their native language quite well.  Sure, some kids have more vocab than others, but no two Bengali (or English-speaking) ten year olds are significantly different in their basic speaking skills.  95% of drivers are never going to have an accident worse than a minor parking-lot fender-bender.  U.S. studies show that an overwhelming majority of long-gun firearm owners store and handle guns properly (the rate is a bit lower for handgun owners). Teach them right, and– if they are paying attention– they will learn.

Think about this.  The top possible score is 100%, and good teaching by definition raises marks.  This means that all marks should rise, and because there is a top end, there will be less variation.

Most importantly,  good teaching works for all students. In the case of a comprehensible input class, all of the teaching is working through what Chomsky called the “universal grammar” mechanism.  It is also restricted in vocab, less (or not) restricted in grammar, and the teacher keeps everything comprehensible and focuses on input.  This is how everyone learns languages– by getting comprehensible input– so it ought to work well (tho not to exactly the same extent) on all learners.

Because there is an upper end of scores (100%), because we have no outliers, and because good teaching by definition reaches everyone, we will have reduced variation in scores in a comprehensible input class.

So, Mr Speaking’s response to his Defartment Head should be “low variation in scores is an indication of the quality of my work. If my work were done poorly, I would have greater variation, as well as lower marks.” High marks plus low variation = good teaching. How could it be otherwise?

In a grammar class, or a “communicative” class, you would expect much more variation in scores.  This is because the teaching– which focuses on grammar and or output, and downplays input– does not follow language acquisition brain rules.  How does this translate into greater score variation?

a) Some kids won’t get enough input– or the input won’t be comprehensible enough– and so they will pick up less.  Now you have more lower scores.

b) Some kids will be OK with that.  Some kids won’t, and they’ll do extra work to catch up.  Result: variation in acquisition.  Now, there will be a few high scores and more low ones.

c) Some kids will hate speaking and so will do poorly on the speaking assessments, which will increse variation.

d) Many kids don’t learn well from grammar teaching, so in a grammar-focused class, you’d expect one or two As, and a lot of lower marks.

e) if the teacher is into things like “self-reflection on one’s language skills and areas for growth” or such edubabble and the kids are supposed to go back and rework/redo assignments, things could go either way.  If, for example, they re-do a dialogue from the start of the course at the end, they might– if the vocab has been recycled all year– do better.  If, however, it’s the check your grammar stuff, you’d again expect variation: only a very few kids can do that, even if their language skills have grown during the year.

And, of course, there is the “grammar bandwidth” problem: any effort to focus on a specific aspect of grammar means that other areas suffer, because our conscious minds have limited capacity. A District colleague told me that, for Level 5 (grade 12) French, the kids self-edit portfolio work. They have an editing checklist– subject-verb agreement, adjective agreement, etc– and they are supposed to go and revise their work.

The problems with this, of course, are two: in their mad hunt for s-v errors, the kids will miss out on other stuff, and we know that little to no conscious learning makes it into long-term memory.

Some real-life examples of how good instruction narrows variation  in scores:

At Half-Baked School, in the Scurvy School District (names have been changed to protect the guilty), TPRS teacher Alicia Rodriguez has Beginning Spanish.  So does her Defartment Chair, Michelle Double-Barreled.  When, at the end of the semester, they have to decide on awards– who is the best Beginning Spanish student?– Alicia has 16 kids getting an A, another 12 getting a B, and two betting a C+.  None fail.  Michelle Double-Barreled has one kid getting an A, a bunch of Bs and Cs, a couple of Ds, and a few failures.

What this means is, 16 of Alicia’s kids can

a) write 100 excellent words in Spanish in 5 min, on topics ranging from “describe yourself” to “describe [a picture].”

b) Write a 600-1000 word story in 45 min.

Both will have totally comprehensible, minor-errors-only Spanish.

Michelle Double-Barrelled, on the other hand, has one A.  Her “A” kid can

a) do grammar stuff

b) write a 100-word paragraph on one of the topics from the text (e.g. shopping, eating in restaurant, sports s/he plays, family).

This will be not-bad Spanish.

Now, who’s doing a better job?  Alicia has more kids doing more and better work.  Michelle has a classic bell-curve distribution.  According to Mr John Speaking’s Defartment Chair, Mrs Double-Barreled has a “normal” range of scores.  Yet Alicia is clearly getting her kids to kick major butt. Hmm…

The point is, with appropriate and effective instruction– good Dog training, or good Spanish teaching– we are going to get a cluster of generally higher scores.  Poor or no teaching might produce something like a bell curve.

So…what does T.P.R.S. and other comprehensible input teaching do for student outcomes?

In my class, T.P.R.S. did the following

a) all scores rose.

b) the difference between top and bottom scores (variation) decreased.

c) I.E.P. kids all passed.

d)  First-year kids in second-year classes did about 85% as well as second year kids, despite having missed a year of class.

e) In terms of what the kids could actually do, it was light-years ahead of the communicative grammar grind.  Kids at end of 2nd year were telling and writing 400-600 word stories in 3-5 verb tenses, in fluent and comprehensible (though not perfect) Spanish.  Oral output was greater in quality and quantity too.

f) Nobody failed.

My colleague Leanda Monro (3rd year French via T.P.R.S.) explains what T.P.R.S. did in her classes:

“[I saw a ] huge change in overall motivation. I attribute this to a variety of well-grounded theories including “emotion precedes cognition” (John Dewey), Krashen’s affective filter, and the possible power of the 9^th type of intelligence, drama and creativity. (Fels, Gardener).   There is a  general feeling of excitement, curiosity, eagerness to speak French, incorporation of new vocabulary, spontaneous speech.

All but one student has an A or a B. The one student in the C range has significant learning challenges , and despite excellent attendance in all courses is failing both math and English. No one is failing.

[There was] far less variation. Overall, far greater success for all students. My contribution to the “Your scores are too high” comment is this: As educators we need to pose an important question:  Are we trying to identify talent, or are we trying to nurture and foster talent?  T.P.R.S. works to nurture and foster.”

And here are Steve Bruno’s comments on the effect of T.P.R.S. on his kids’ scores:

“I now get more  As and Bs [than before]. A few C+s and very few Cs. Let’s put it this way, in the past I’ve had to send between 20 and 25 interims/I reports (total 7 classes); this year, so far, I’ve sent just THREE! Of these, two had poor attendance; the other one is an L.A.C. student who is taking a language for the first time (Gr. 9).

Marks are also closer together.  Anyone who has been teaching C.I. will understand why this is the case:  Students feel more confident, less stressed and simply love T.P.R.S. They don’t like doing drills, or memorizing grammar rules, etc.

Here’s anther example [of how comprehensible input has changed student behaviour].  Last year, I had a few students who on the day of an announced test (usually, with one week of warning) would suddenly become ill, and, or skip. Some of my L.A.C. students would have to write the test in a separate room. Others would show all sorts of anxiety as I handed out the tests. Many of these students would end up either failing the test or doing very poorly.

This year, I have the same students in my class, and the day of an unannounced test, they don’t have to go to another room, nobody complains and they just get down to it and, yes, they do quite well, thank you very much!“

OK, people…you want to report on how things are going with T.P.R.S.? Post some comments, or email.