A brief review/lecture at the end to tie together all of that practice intoa coherent story then wraps it all up.
https://scholarscompass.vcu.edu/cgi/viewcontent.cgi?article=...
For each class, a text lesson is assigned. This assignment includes a reading and specific problems associated with the reading material. Each cadet is expected to "work the problems." (Note: Prior to 2000 these problems were called "drill problems"; the current terminology is "suggested problems.")
• "One learns mathematics by doing mathematics." Cadets are encouraged to be active learners and to "do" mathematics. Group work is encouraged and expected. Special projects are a major portion of each core mathematics course-work on these projects is done in teams of two or three.
• Cadets are required to study the concepts of each lesson in such a way as to be ready to use them in three ways: 1. To express them fluently in words and symbols 2. To use them in proof and analysis 3. To apply them to the solution of original problems
• The instructor's goal during each lesson is to cause the maximum number of cadets to actively participate in the day's lesson. One of the instructor's roles is to facilitate the learning activity in the classroom. This may take the form of a question or a remark to clarify a point.
• Class begins with the instructor's questions on the assigned text lesson. Cadets are asked if there are questions on the assignment. Example problems are worked and discussed. Cadets are sent to the boards to work in groups of two or three on specific problems that are provided (so called "board problems"). These board problems may be similar to the problems assigned with the text lesson or they may be "original."
• Cadets are selected to recite on the problems they work. Questions are encouraged.
• The instructor spends a few minutes to discuss the next lesson. This practice is commonly called the "pre-teach."
First, I'm a slow reader, so I always feel penalized when it takes me twice as long to get through a text as classmates.
Second, math/engineering/science lessons typically build upon understanding the first example. If you don't understand or have questions about the earlier parts of the lesson you will have a hard time completing the lesson.
Third, most text books I have encountered are terrible. Grade and High schools typically are trying to get the cheapest books so their dollar stretches further. In college, too many Profs/Departments push certain books because of kickbacks.
Finally, too often enough people don't complete the readings, so you end up covering the material in class anyway. Or worse, not at all. I had several profs who's assigned reading was never to be discussed in class but was prominently featured in tests.
I much prefer the typical lecture that allows for questions and discussions during the class. That way I can quickly address the issues I have with the material when I encounter it instead of having to wait till the next class hoping I don't fall too far behind.
Newer learning systems like Duo Lingo, ALEKS, and Brilliant do an excellent job of constantly, actively engaging students with the lesson, tightening the feedback loop between teaching the student something and checking whether they actually learned it to seconds rather than days.
After experiencing such systems for myself I'm blown away that they aren't already the norm.
I’m currently nearing a 2000-day streak and have repeatedly gold-starred the German course as they add more content, and Duolingo isn’t the only app I’m using.
Despite this, while my vocabulary is OK, I don’t conjugate even close to correctly, my grammar in general sucks, and I can only comprehend real-life spoken German if the speaker talks very slowly and clearly and uses a sufficiently short sentences — from experience, the sort of conversation you’d find in an interview in a general interest magazine in the waiting room of a Hausarzt.
I’m also trying to learn Arabic on Duolingo. Over a year into that course, I still can’t even read the entire Arabic alphabet.
Their app has to be "fun" or in flow rather than in that difficult challenging place to actually help you grow.
In learning both German and Old Norse the most helpful thing for me was to translate texts, read them aloud to a fluent speaker and get feedback. Which is hard to scale.
... Do Old Norse graded readers exist?
Also, when you make a common mistake during a lesson, Duo Lingo will often interrupt the lesson with an impromptu tip showing you what your mistake was, why it was a mistake (what grammatical rules it broke, etc), and how to avoid that mistake in the future. Those have been pretty helpful in my (admittedly limited) experience, though I suppose it's possible the prevalence of those tips depends on the course.
Unfortunately the interstitial hints do vary by course. Spanish course has stuff like that very frequently in the early lessons (I have not done the later Spanish lessons); the German course barely has them at all anywhere, possibly not at all (if I had perfect memory it would be much easier to learn the languages).
For language specifically, the only way you're ever going to get anywhere close to the level of a native speaker is by actually conversing with native speakers. I'm still just starting out with Duolingo, but my plan is to finish the course I'm in (or at least get a decent way into it), then switch to Tandem or some other service that lets you trade lessons with native speakers of another language.
If you can recommend any novels for mid-skill non-native speaking adults, I’d be interested.
0. Homework is never collected or graded, but don't be fooled into thinking it's not required -- that is, if you don't do the homework, you are extremely unlikely to pass the exams/course. Essentially, this is not knowledge we were learning -- it is skills that require practice. Homework is an opportunity to practice and hone skills.
1. Each lecture introduces a concept and/or technique, and works through a few demonstrative problems to show what it means or how it is done. Homework is assigned from textbook problems that involve the same techniques with progressive difficulty or complexity. The textbook used that pattern where odd-numbered problems included solutions, and assignments usually involved the ones with solutions.
2. The last one-quarter to one-third of every class period was dedicated to review and questions about the homework assigned for the previous class. Because we had the correct solutions in the text, we knew what to ask about (i.e. the ones we couldn't get to come out right). This particular instructor was fantastic at thinking on his feet and working problems on the fly, correctly and without preparation, so usually he'd just work the problem on the board and we could stop him to ask for a more detailed explanation if necessary.
Granted, this model didn't work as well for his linear algebra class. Since many of those problems involve long slogs through tedious and error-prone matrix operations before/while you were really dealing with the concept or technique being introduced, he couldn't as easily demo entire solutions during the question/review periods. I suppose that difficulty would apply to several other higher-math topics, as well, but even so, later in my education I often found myself wishing this or that professor would follow the pattern of my humble calculus teacher.
My math teacher in high school gave us a 15 minute test every week and would just randomly pick 4 students who must hand it in of which one presents their solution giving the teacher time to grade the tests. Doing something every week makes it less stressful.
Most students do their math homework begrudgingly because they get in trouble otherwise; this does not mean that doing the assigned problems does not help them learn...
While it's been decades since I went to college, I'm surprised this is no longer how it's done. That was pretty much the routine when I was in school.
At the end of the class, the professor would say, "Next week, we'll be doing X, Y, and Z. It's chapters A, B, and C in the book." You'd prepare for it over the weekend. The following week, we'd have a mixture of lecture, discussion, and quizzes.
Is it the other way around now? Lecture first, then the books and papers?
This seems to be the continual heart of opposition to restructuring math curricula. Whether it's my parents generation recalling how their parents couldn't make heads or tails of new math or parents slightly older than me struggling to comprehend the Common Core math they're supposed to guide their children through, the essence of the complaint is the same: "how can I teach my child what I was never taught myself?"