In high school math class, there’s pretty much a complete separation of form and content—we learn about algebra, geometry, functions, and calculus with very little connection to any real-world problems, it’s pretty much all form and no content, or we could say all theory and no applications. But in high school English class, there’s no separation at all: we read a bunch of books and learn how to write, but the writing assignments are particular things we have to do. Yes, there are some tricks like the “5 paragraph essay,” but writing is not taught in the abstract way that math is taught. Even when we’re asked to write a 5-paragraph essay, we’re asked to write about a particular topic.
I’m thinking is there are problems with both math and English teaching. In math teaching, I like the separation of form and content; my only problem with the standard math sequence (algebra/geometry/functions/calculus) is that it’s all form and no content at all. I’d like there to be some content in addition to the form. In English teaching, I’d like form and content to be more separated, so that students can learn how to write without having to write boring essays about the books they’ve just read. Reading literature in English class is great; I’d just separate that from the writing lessons.
The next question is how this applies to statistics classes. Statistics already has a pretty clean separation of form and content—that is, methods and applications—and we tend to teach them together. So I think that, for all the problems with statistics teaching, we do well on this dimension.
I’m curious what Basbøll thinks about all this.
Using concrete vs abstract examples is a heated topic in cognition and education research.
Here is a review article in favor of abstract education for math:
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.709.6843&rep=rep1&type=pdf
And I’ve just seen this recent article that suggests otherwise:
https://onlinelibrary.wiley.com/doi/full/10.1111/cogs.12851
We’ll have a clearer picture after this question has more time to get hashed out at conferences and in the literature.
It is more than a question of concrete vs. abstract, but when and how should different types of examples be deployed in order to support the development of particular kinds of knowledge.
For example, “concreteness fading” (a horrible term, but a good idea) involves presenting concepts initially using concrete examples and gradually “fading” them out in favor of more abstract ones (https://link.springer.com/content/pdf/10.1007/s10648-014-9249-3.pdf).
Moreover, it is not just a matter of what kinds of examples are used, but what the student is supposed to do with them. For example, making explicit analogies between examples can facilitate learning abstract concepts, even if the examples themselves are concrete (https://loewenstein.web.illinois.edu/papers/Gentneretal%20JEP03.pdf).
And this is to say nothing of differences between individual learners, since chances are the same set of examples/tasks will not be equally effective for everyone.
If you are waiting until learning scientists “hash out” these issues, let me know how things look at the heat death of the universe…
I’ll look at your article, but it’s hard for me to understand how that’s a heated topic.
Well, it shouldn’t be “concrete vs. abstract” as there isn’t some kind of dichotomy there. Perhaps there’s a question as to whether there’s a sequence of concrete to abstract for most learners, but again it’s hard for me to imagine how that would be controversial.
I was trained in Montessori education. As part of the math curriculum, trainees worked with concrete materials to learn about squaring or cubing binomials (or trinomials). It was amazing to watch people who could never understand a binomial squared or cubed, or remember the algorithms, work with those materials and then be able to UNDERSTAND what they were doing well-enough so that they would never forget the algorithms once they worked on transitioning from the concrete materials to the abstracted algorithm.
Of course, math geeks like the readers here probably never needed such tools as they easily worked in the abstract early on. One interesting factor is that sometimes it’s very intellectually curious students who have the hardest time jumping to the abstract because they always want to ask WHY an algorithm works, and the explanation is actually quite complicated or the teacher doesn’t know the answer.
As a result, many times those students think “I can’t do math” because they wanted to know those answers and then felt that they aren’t “good at math” because they couldn’t find the answer to their question whereas students who never asked WHY just moved on and accepted the algorithms and did better on tests. This problem reflects that much of our educational paradigm hinges on evaluating students only by judging them in comparison to the performance of other students.
Your comment about math geeks made me think of this example from the blog just a couple of days ago. The subject was about conformal prediction. There were a number of posts and I exchange some private emails about the subject, and I’ve come to the conclusion that all the people working with this method are speaking geek. Admittedly, my math skills have deteriorated over time (and may not have been that great to begin with), but virtually all the literature (including the “gentle introduction” seem far more complicated that they need to be. I think the concept is actually quite simple:
When you see how a new observation stands in relation to the existing class in terms of probability, that provides a measure of the probability it will belong to that class. Since this is done for each class, some classification probabilities will be lopsided enough to assign the observation to a single class (subject to your chosen threshold), while others may pass that threshold for multiple classes (and some others may not pass the threshold for any classes).
The mathematical descriptions just don’t say much to me; if anything, they make the concept seem complex. What I came up with is 3 possibilities:
1. I really don’t understand conformal prediction, so my simple understanding is just wrong.
2. Dale is from Venus and Geeks are from Mars: all the people explaining conformal prediction in mathematical terms understand each other while I just speak a different language.
3. The relatively straightforward concept has become obsfucated by unnecessary mathematics. Since I am an economist, I am well aware of the possibility that the pursuit of rigor can stand in the way of insight.
If #3 is correct, that might explain why the concept is so little known and used, despite its obvious importance for machine learning models, and despite the methodology having been around for almost 20 years.
I was impacted reading this. I was informally homeschooled for the first ten years of my life and then I was left to my own devices, spending all of my time on my computer, online. I’m 26 now and I’ve just entered university and the formal education system for the first time. From what others have told me, it seems like most high-school students who are successful have been successful because they adopted a perspective that viewed the goal of school, the winning conditions of the educational system, to be something other than the comprehension of the educational materials. So they don’t study to understand, they study to figure out what they need to do or say to win, to get a certain grade, or to simply pass. I haven’t experienced that environment, and my only goal has been to learn as much as I can and to do things ‘right’, rather than half-assed. This has presented me with challenges that many others have difficulty relating to / empathising with. For example, my first psychology lab report gave me immense stress because I couldn’t understand what the real significance of our study was. Even though no one else would ever actually see the paper, something felt wrong about the idea of submitting the report, condoning the study we did without actually understanding it. I feel like there’s a clash of values at many times. Exams are another good example. Scared the shit out of me my first time. I was freaking out, “What is with this boot camp set up and the atmosphere of suspicion and distrust towards the students?” When I asked my friends about it later they just said that that was normal. My time hasn’t been too bad overall, thankfully, and most of my unit coordinators have actually encouraged comprehension as a value and goal, structuring their units to facilitate that.
It is not all that new, though likely much more intense now.
When I was an undergraduate (1970s) I decided to focus primarily on understanding but not overlooking the need to play the game enough to ensure grades were adequate for what I wanted to do – B+. Same in my MBA program and then in my MSC in Biostats I eventual abandoned the degree when I felt it was getting in my way of understanding (at the time I was working in teaching hospitals and publishing my early papers on meta-analysis).
Then one of my kids was in undergrad and aiming for all A+s and I tried to discourage them. They graduated with the highest grade average in their program. Then a couple years later they mentioned that they had wished they had listened to me.
I think most would agree after a couples years out as long as they got do what they wanted. For instance, if you need all A+s to get into med school – you have no choice but to play the game (then work like the devil trying to understand things as an intern)…
“my only problem with the standard math sequence (algebra/geometry/functions/calculus) is that it’s all form and no content at all”
I asked the young woman behind the deli counter for 2/3rds of a pound of mortadella.
She went over to the slicer, came back with a stack of slices, and asked “is this good?”
I asked “is it 2/3rds of pound?”
She said “I don’t know how to put that in decimals.”
Is this a failing of the school system? Was she really never shown that you can just divide 2 by 3 to get the decimal equivalent? It seems like year after year in grade school, we started the year by hammering away at basic fractions. There has to be more going on here than bad curriculum. Was she just not smart enough? My guess is that she was plenty smart, but just could not foresee working at a deli counter and needing to do fractions; it was outside her worldview.
Matt –
> It seems like year after year in grade school, we started the year by hammering away at basic fractions. There has to be more going on here than bad curriculum.
You’ve hit upon a pet peeve of mine. Or more specifically, the way that division of fractions is taught year after year to students before they (hopefully) catch on. And it’s rare that people ask why the same concept has to be taught over and over.
It’s easier to see with division of fractions than with more basic work with fractions.
Division of fractions is quite abstract. Ask people who have memorized an algorithm of “invert and multiply” why that works, or really even what it means to divide fractions, or when in real life they have ever divided fractions, and you’ll just get a blank stare.
The problem arises when you ask learners to do something in the abstract, by memorizing an algorithm, when they don’t have a concrete understanding of what they’re doing. That happens because people assume they understand a logical scope of sequence of how we learn – in this case first you teach addition and subtraction of fractions and then you move on to multiplication, and then division – when actually there are HUGE assumptions being made there without supporting evidence.
You can see a similar problem when teaching English to a non-native speaker. Teaching the use of the definite versus indefinite article comes early on in the scope and sequence. Yet you can find VERY sophisticated speakers of English as a 2nd language, who can engage with very complicated topics, and still get the use of the definite and indefinite article wrong. Try asking a native English speaker to draw a flow chart of when to use the definite and indefinite article, and you’ll get a blank piece of paper.
I think it’s a mistake to focus on the individual student or teacher or school. It’s more fundamental, in that we THINK we know the proper scope and sequence of what to teach, basically, just because we’ve arbitrarily designed one and then think that’s how people actually learn.
Next time try 11/17 of a pound, maybe you’ll have better luck.
Matt: even stock market info is reported on the news in points instead of percent.
I think the problem with English teaching is not so much that we need seperation of form and content but that they aare simply not using relevant content.
There is too much of literature and not enough of functional, contemporary English.
And I don’t mean contemporary literature but the English that we use in real world, applied contexts. Wish they did more comprehension and drafting using (say) journal articles, newspapers, legal briefs, county codes etc.
Rahul –
> I think the problem with English teaching is not so much that we need seperation of form and content but that they aare simply not using relevant content.
I think that’s a critique that certainly isn’t limited to English teaching.
Speaking of form vs. content, Dart + Flutter is this new language + package that is big these days for people writing UI applications on phones and whatnot.
The docs for R functions go description, technical definition, and then examples at the bottom: https://stat.ethz.ch/R-manual/R-devel/library/base/html/rep.html
The docs for Flutter go description, examples, and then technical definitions later: https://api.flutter.dev/flutter/material/OutlinedButton-class.html
This stuck out to me cause a lot of Flutter stuff is by example. What was happening to me was I would get an example and most things kinda made sense, but then I wanted to double check the meaning of some specific arguments, so I search for a thing, and then the docs start with more examples! EZ to work around cause I can just scroll down or look at the sidebar, but it stuck out.
This is an interesting thread for me as a former stats and econ teacher. Like some of the others who follow this blog, I received training in teaching “reform calculus” back in the late 80s when that was a thing. At its heart, the movement was about finding a better balance between inductive and deductive reasoning, which is closely related to application vs theory. I was also trained in case methods, which can be used in lots of settings beyond the applied professions.
Over time, however, I began to see a different distinction emerging, between exemplary applications and messy ones. This first appeared as a difference between textbook-type examples, which I was busy formulating for my courses, and student-driven questions that arose in project work. The latter rarely fitted neatly into the just-so role that applications were supposed to perform. Eventually I realized this was a feature, not a bug. Ultimately, learning how to use a tool needs to be accompanied by learning its limits: what it does and doesn’t do, what kind of applications it works best in, and how it can also lead you astray. Messy problems are great for this. I suppose this is what critical thinking is really about in technical fields.
Once I got to this point I became thoroughly disenchanted with existing textbooks in both econ and stats. I sometimes continued to use them with misgivings, sometimes not. I spent more time divising applications and projects that raised interesting, open-ended question about methods, without (of course) cutting back too much on teaching the methods themselves. I guess I went from a two-way division of methods and applications to a three-way between methods, neat applications and messy ones.
I should add that I was influenced along the way by a physicist colleague who argued for the importance of problems that had no general solution but only approximate local ones that depended on priorities and context. This for him was what design was about. I came to think of a continuum between contingent design and more deductive, general “solutions”.
I’ll second your experience and preferences. Just wondering – did your teaching evaluations take a hit when you moved away from textbook examples towards real messy project work? (mine did, and it took a lot of work to stem the slide)
Not really, although I taught in a highly nonstandard environment (Evergreen State College), so my experience may not generalize. Above all, experiential learning is a calling card at Evergreen, so my approach didn’t stand out too much. Also, since there are no requirements or majors, students self-select more or less on the basis of the kind of education they’re looking for. I would be very interested in whether teachers at more conventional schools face disincentives I was able to dodge.
I tried to skip as much school as I could to attend lectures at surrounding universities. I was more interested in attending the lecture series held in Cambridge, Ma. I got to hear Joseph Campbell a couple of times. His wife was a dancer, as I recall. So that was a point of interest. I also heard many authors at various venues. I would actually jog to Harvard Square Bookstore as a teen.
I found high school pretty boring.
The insight of your physicist colleague has always interested me. In economics, so much work goes into setting up the assumptions of the problem so that the answer to the problem has either (a) a closed-form solution; or (b) a form that is amenable to econometric estimation. We then get students to memorize the closed form or the specific econometric technique. I always thought, from grad school on, that this was at least slightly (and sometimes hugely) backwards: start with a minimum of assumptions and use computers to do the hard part of coming up with an approximate answer. Experience with a number of different problems, or sensitivity analyses on a single problem, will graph out the solution… no closed form required. Toy problems have closed forms… very little else does.
Fractions are intrinsically hard. I’ve watched teachers and kids struggle with them in all the elementary grades. They have plenty of (mostly fake, some almost real) applications, and drill on the algorithms. They still struggle. I don’t think you can solve this problem by properly adjusting the curriculum.
From https://math.stackexchange.com/questions/1127483/how-to-make-sense-of-fractions/1127776#1127776
>In English teaching, I’d like form and content to be more separated, so that students can learn how to write without having to write boring essays about the books they’ve just read.
I’m not sure I understand exactly what you mean, Andrew. The way to separate form and content in English class (as I agree you should) is to fix the content (the books students are asked to read) and assign the form (a prompted n-paragraph or n-page or n-word essay). That turns it into something like a math problem.
In fact, I often fantasize about being an actual English teacher and assigning a five-paragraph esssay with the prompt, “What happens in the first act of Hamlet?” I’ve written about this at great length on my blog, but the point is that something (and not just any old thing) does in fact happen in that act and it’s a real skill to be able to break it down into exactly three major events organized into an arc the can be introduced and concluded. (Or the student may come up with another way of using five paragraphs.)
Are math or statistics problems “boring” simply because they have right answers, or because solid craftsmanship in the student is easily recognized by the teacher?
Thomas:
Here’s how I see it:
– In English class, the “form” is how to write, and the “content” is what is being read and what is being written about. I feel like there’s not enough separation here. Ideally I’d like to see form taught separately from content, or maybe I should say I’d like to see content taught separately from form. I guess there’s some form-without-content in English class—I’m thinking of the notorious five-paragraph essay—but not enough. Much of the time it seems to me that high school students are given assignments to write essays about the books that they’ve read, so that they’re simultaneously being asked to master both form and content. It seems to me that they should be able to write about the books they’ve read without having to use this difficult essay form.
– In math class, the “form” is what we call mathematics (algebra, geometry, functions, calculus), and the “content” are applications of mathematics. High school math seems pretty much 100% form and 0% content. I like the separation of form and content but I don’t like that there’s no content.
In response to your comment: Yes, I agree that requiring an essay on the topic of what actually happens in the play would be great (except maybe they’d just copy it from wikipedia, but that’s another problem). My impression is that the usual assignments in English class won’t just ask for an essay on the plot, or the characters, or the theme, or the staging, or whatever; rather, they’ll require that the student make a case for some argument. I don’t like that. To me, the “content” of Hamlet is the plot, characters, theme, staging, etc., not this artificial thing where you have to make a case.
Maybe it would help to call the form of composition, “prose” (and to me that means essential the arrangement of “paragraphs” into “essays”). Maybe there should be a “prose class”, like a math class, separate from English class. Then we’d get into that 0% content situation, where student would be asked to tell stories about how they get dressed in the morning or write descriptions of the tree in the schoolyard. But these exercises all have many things to recommend them.
Then, in English class, they’d be asked to use their prose skills on Hamlet or Beloved, just like students are asked to use their math skills in an economics or political science class. (A quick aside: in How to Draw Hands, Oliver Senior at one point tells you plainly that if you can’t draw a foreshortened cylinder you should put down his book and work on that problem first.)
Some of this is just going to have to be oldfashioned drilling — boring work that makes you better at something through repetition.
I’m just thinking out loud here. I like the question. And I do wish we were as good (precise) at distinguishing between good and bad prose (by which I don’t just mean grammar) as we are at recognizing good and bad math (by which I don’t just mean arithmetic).
I wonder if courses in logic would help to distinguish between form and content in writing?
This thought was inspired by my first exposure to Lewis Carroll, which was within an introductory logic course in high school (this was years ago, I doubt many places offer such a course today). The point of the Carroll examples was to illustrate how he captured the form of a logical argument, but with silly premises that led to equally silly conclusions. The “content” (premises and conclusion) was clearly distinct from the “form” (the rules linking the premises and conclusion together).
More generally, getting some instruction in logic seems like a useful way to distinguish between what (rhetorical) writing is supposed to accomplish and how it does so. I guess this is more like doing a sentence diagram than an outline. Outlines are often unconnected bullet points, whereas a logical argument connects those dots into a meaningful structure. Then “writing” can focus on how to use the tools of language to convey that structure.
Assignments would provide a simple logical argument, and the job of the student would be to figure out how to best realize that argument in language. I think this is a bit more engaging and general than writing about getting dressed or the plot of Hamlet. A logical argument at least has a “point”, no matter how silly it may be.
Interesting, I was wondering about distinguishing learning about abstract probability models versus how to connect those to the empirical world. That is how well does the model represent important features for the purpose of a given analysis and what to transport what was learned about repeatedly happens in that abstract model to what reality happened to produce the observations.
The first being “form” and the second being “content” (suitability of premises and conclusion)…
My high school English teacher allowed us to use whatever medium to express our thoughts. I was quite enthusiastic in choosing. For many English assignments, I would express myself in two or three mediums: art/painting, poem, and script. The other classmates chose essay format.
I found the essay assignments tedious. I became a bit more comfortable with essay writing after reading Joseph Cambpell’s and Northrop Fry’s books. Then I could analyze the content and had more to add.
Inspiring students to write well is a rare talent.
It seems to me that through high school there was always some content along with the form: the famous “word problems” that are dreaded by many. Stuff like: Someone wants to paint the walls and ceiling of a room that is 24 feet by 19 feet and is 8 feet high; one quart covers 100 square feet; how many quarts do they need. A lot of these seemed completely artificial, which might be what you’re complaining about — a search for word problems online found “Minimum profits for company x, whose profit function is f(t) = 100t2 – 50t + 9” — but I think quite a few of them were at least moderately realistic. Maybe they don’t teach that way anymore, I dunno.
But once I got to college, the math classes often had no applications at all. You learn how to do, say, integration by parts, without reference to any real-world applications in which that ability is handy. That didn’t bother me at all, I already knew that I would need to be able to do integrals (I was a physics major) and I didn’t really need specific examples as motivation.
Barely related: About five years ago I finally got rid of my old grad school copies of Gradshtey and Ryzhik ( https://en.wikipedia.org/wiki/Gradshteyn_and_Ryzhik ), which we used to call “Gradschool in Physics” and Abramowitz and Stegun (https://en.wikipedia.org/wiki/Abramowitz_and_Stegun ). I realized I would now look stuff up online rather than use the books. I considered hanging onto them, but then thought what am I gonna do, pull down A&S and look at the Bessel function chapter someday just for fun? Anyway these are purely mathematical but their existence was motivated by their use for doing physics and engineering. It’s kind of a pity that they aren’t needed anymore.
Phil:
I was thinking about word problems when writing my post, but I don’t think these are real “content”; I think they’re just a direct application of “form.” Shakespeare is content; part of the point of English class is for students to learn to appreciate literature. But I don’t think any part of the point of math class is for students to learn how many quarts of paint to use. So I think these are different.
Then what would be an example of mathematical content?
Phil:
One example I can think of offhand is the growth of an epidemic. One common math “word problem” that does have real-world application is compound interest. But somehow I have the impression that when compound interest is discussed in a math class, it’s more for the math than the content. In contrast, when we read Shakespeare in high school, it was for the Shakespeare itself.
I may have been out of high school far too long, very likely, or we do not use it in the Canadian (Ontario) system but this is the first time I have ever heard of the “5 paragraph essay”.
It strikes me as the rough equivalent of encouraging poetry writing but insisting that it must be in Shakespearean sonnet format. I would have thought one would have tried for coherence/logical presentation and then asked for form though the reason for 5 paragraphs leaves me bewildered.
Back in my misspent youth, I studied Latin and an “interesting” exercise was to reduce a multi-sentence English sentence ta a Latin rolling periodic sentence. English does not easily allow something like this but Latin–trained English writers could come close. Quite seriously it could be possible to reduce a 5 paragraph essay to 5 or 6 sentences. This reminds me of the old grammar–checker that used to warn that a sentence had more than 12 words.
Why 5? Why not more or less as the subject demands?
>the “5 paragraph essay” … strikes me as the rough equivalent of encouraging poetry writing but insisting that it must be in Shakespearean sonnet format.
That’s exactly right. It’s like encouraging mathematical thinking and assigning polynomial equations to solve. It’s not the only thing the student should ever do in the math classroom. But it is one very teachable (and learnable) form.
There’s nothing wrong with telling a student to write three quartrains and a couplet. It teaches them a great deal about what poetry is. But you obviously do not tell them at a “poem” is always and only three quartrains and a couplet.
>the reason for 5 paragraphs leaves me bewildered … Why 5? Why not more or less as the subject demands?
All assignments have to have more or less arbitrary constraints or the students wouldn’t know when they’re done. They get, e.g., one week to write three pages about two texts that have been studied in class, sometimes answering a specific question. When I assign 5 paragraphs it means they can write at most 1000 words (5 x 200 words) and at least 30 sentences (5 x 6 sentences). I expect them to spend about 3 hours doing it (3 x 30 minutes plus some editing). I tell them they want to develop an ability to compose an argument efficiently and that, like writing a sonnet, this is one way to train this ability.
I don’t tell them that any topic is ideally treated using five paragraphs. But I do give them a problem that can be solved in five paragraphs. This means they have to come up with a major thesis that can be established by supporting, elaborating, or defending three claims. And they have to find a way to frame or motivate the thesis so that it becomes interesting to a peer (an intelligent, engaged classmate).
>Quite seriously it could be possible to reduce a 5 paragraph essay to 5 or 6 sentences.
Again, you are right, but this is a strength not a weakness of the form. You are telling students to elaborate an argument that could be summarized in 5 sentences (the key sentences of each paragraph). The summary will not be persuasive (it will lack detail and evidence) but it will give a good sense of what you would come to believe if you were persuaded. Looking at those sentences is a good test of whether your essay is interesting.
Somehow people who easily accept this sort of thing in math instruction think nothing like it makes sense when teaching writing. I think the mistake here is to think that writing is just an application of sincerity to the page. Writing is in fact a very artificial thing. It’s a craft. It’s hard. So to teach it you break it down into simple problems for the student, and you tell them where they’re doing it right and where they’re doing it wrong (where there is a better way). That’s just like teaching math.
>Quite seriously it could be possible to reduce a 5 paragraph essay to 5 or 6 sentences.
Again, you are right, but this is a strength not a weakness of the form.
Actually I was saying that the entire paragraph can be one, well-crafted, sentence, not that the sentence would be a summary. It is not something one sees today as it seems to require a fairly high-level exposure to Latin composition. I do not think we have been seeing much such writing since the 19th Century. Just a minor point.
Ah, I see. That’s interesting. It may account for the idea, shared by many late-19th and early-20th century rhetoricians, that a paragraph is just an expanded sentence. They did know their Latin.
There are better examples from Marcus Tullius Cicero but here is one of his sentences
From he beginning of the Second Catilinarian:
Tandem aliquando, Quirites, L. Catilinam furentem audacia, scelus anhelantem, pestem patriae nefarie molientem, vobis atque huic urbi ferrum flammamque minitantem, ex urbe vel eiecimus, vel emisimus, vel ipsum egredientem verbis prosecuti sumus.
C. D. Yonge’s translation:
At length, O Romans, we have dismissed from the city, or driven out, or, when he was departing of his own accord, we have pursued with words, Lucius Catiline, mad with audacity, breathing wickedness, impiously planning mischief to his country, threatening fire and sword to you and to this city.
https://latin.stackexchange.com/questions/2121/descriptive-example-of-ciceros-style
“my only problem with the standard math sequence (algebra/geometry/functions/calculus) is that it’s all form and no content at all.”
I am skeptical that this statement is currently true. Certainly every present-day high school/college math textbook intended for a general audience (i.e. not for upper division math students / only math majors) has a fair number of examples, and the emphasis in college courses is to further increase this, hence the proliferation of courses like “calculus for business students,” “calculus for biologists,” etc. My older son just took the AP calculus exam, which (from the practice exams I glanced at) has several “real world” word problems. (These problems were long, awkward, and unrealistic, but that’s a separate issue.)
For writing, I’ll just point out that at least at my university, teaching grammar in writing courses is actively discouraged. The consequences are as you’d expect.
I got a little tired of the Mistress of Parenthesis who in a Linkedin forum, [like a cross school Mom[ would criticize everyone’s grammar and word usage. After a couple of weeks of bullying, I went for the jugular. I’m quite easy going. But I had it with her. She was a whole lot nicer after I gave her a piece of my mind. People take advantageous of my good nature.
I purposefully misspelled and used wrong grammar. Shucks, I was waiting for corrections.
“I was waiting for corrections.”
Got my jugular to think about.
>I’m curious what Basbøll thinks about all this.
I’ve been revisiting some old posts of mine while thinking about this. And I think “The Place of Form” provides a good statement of my thoughts on this. It’s halfway through a series that defends the five-paragraph essay against some critics (using a specific assignment as an example), starting here and ending here.
I realize that that’s a pretty longwinded answer. But for those who are curious it may be of interest.
As an English teacher, I may be too lenient. I would not penalize a student’s essay for lack of organization or structure. I would even accept an essay response such as, “I don’t have much to write”.
Fundamentally, I am curious as to how students reason. I would probably have a few after school sessions as a means of discovering what may prevent a student from writing an essay. Individual and small group learning is preferable. Often the teacher can’t devote much extra time to a student because of class size.
Writing is not only about form and content; but also about psychology and encouragement.
Are you / would you be as lenient about misleading graphs, inefficient code, bad math, and “I don’t really know how to do this regression”?
I think the trick to designing writing assignments is to make “I don’t have much to write” an answer that only someone who is actually failing the course (either form lack of ability or effort) would need to say. You then fail them on that assignment to let them know how they’re doing in your class.
I don’t think writing is any more psychologically interesting than the performance of any other competence. You have to make an effort and be willing to fail. Teachers have to be willing to tell you when you do.
Greetings Thomas,
Was I arguing for leniency about misleading graphs, inefficient code? Not in the least.
You have to admit, though, that stellar math high school students have gone on to conduct less than stellar graphs, bad math, regression, etc. Such a hypothesis has been implied on this very blog by quite erudite academicians. Just go ask Sander Greenland, as one critic of current statistics methods and analytics.
Additionally, a student responds to an essay, with “I don’t have much to write”, I’m not about to fail the student either. As I noted in my last post, I would have meet with the student after class hours and identity, to the extent feasible, why the student responded that way. In my experience there are specific reasons for lack of effort and ability. Each student learns differently in different settings. I have read essays that may lack structure, but they have contained unique insights. I believe too many students are stigmatized by the grading system.
Lastly, I did not claim that writing was more interesting than any performance of any other competence. But I do think that encouragement is one key strategy toward helping students to achieve their potential.
Glad tidings, Sameera!
Do we not agree that after a few weeks in a math course, the student should be able to solve a particular set of problems, some very easily and some with a little effort. Given an appropriate assignment, when the student says, “I don’t know how to proceed,” it’s a sign they’re not learning what we’re trying to teach them. Surely, that means they should get a grade (on the assignment) that reflects this lack of progress?
Why should it be different when we assign an essay about materials that have been assigned as reading and covered in class?
I’m arguing that encouragement plays the same role in mathematics and literary instruction. Sure, we have to encourage our students to learn the material and perform the related tasks (calculating, writing). But when they do it badly, or tell us they don’t even know how to begin, we’re not dealing with different “psychological” issues in the math and writing classroom. We’re just dealing with a student who is having a hard time mastering the subject.
There will always be such students in a class, some who lack the requisite ability and some who don’t make the requisite effort. The first step is to give them a low grade so they are aware they have a problem. Then, yes, of course, talk to them as time permits and reason allows.
Hi Thomas,
As an instructor, I would give students the option of revising their essay after I met them one on one: three revisions of their essay. The third would constitute the final submission.
Individual attention often results in improvements. I base that on my own experience as a tutor. The instructor’s attitude and versatility in educational approaches are key to bringing out student talents. Such instructional latitude is not so feasible in many school systems. Being able to discern what makes the student tick is a great skill. I am not sure how much individual attention is available, given the large class sizes. on average 18-21 students.
So you might find that I’m not so enthusiastic about grading per se b/c even more fundamentally, I am a proponent of a major pedagogical shift, one that, from my own observations, would be hard to achieve. That’s another subject.
What we view differently, it seems, is the value of assigning a grade? I suppose it depends also on your pedagogical & incentive preferences for students.
—–
>What we view differently, it seems, is the value of assigning a grade?
It’s not grading as such that I’m talking about but whether there’s a different rationale at work in grading/feedback for math and English.
If you’d let a student redo a math problem three times after giving them some one-on-one feedback then grading isn’t the issue.
I guess it’s sufficient that you tell the student that their first attempt would be worth revising before handing in a final version. I.e., you’re telling them it isn’t good enough. That’s all I’m after, really.
As an English teacher, I may be too lenient. I would not penalize a student’s essay for lack of organization or structure.
It depends on the level but I would do the opposite however I am not an English teacher. The 5 paragraph form seem weird but I need a relatively coherent, organized essay if it is to be of use.
I once got a report from a very capable consultant who could not write. He, really, was close to incoherent in text. In discussions I could point to his report and show that he had dealt with the issue superbly but I had not even seen it when reading it. It was hard to sell the report to the rest of the team.
For someone who knew the issue and the background it was a an excellent documest. To an outside reader (i.e. Treasury Board) it was gibberish.
Sorry, here’s the direct link to “The Place of Form”.
Anyone who has read Zen and the Art of Motorcycle Maintenance will recognize form vs. content as the main dichotomy Pirsig sets up in that book.
interesting post.
The point of integrating form and content is to enhance understanding.
The Stats – English symbiosis is in the verbal representation of findings
Wrote about this abundantly. The latest in https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3862006
See also https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3035070