29

To start things off, some moments my students move around during class: I have given students tape measures and had them determine how much they would spend at the paint store if they wanted to paint the walls and ceiling of the lecture room (while projecting two images on the overhead: one of a paint can label, showing the number of square feet per gallon; ...


25

Perhaps emphasize to students the spirit of equivalence relations. They partitions sets into equivalence classes--cutting down the amount of cases necessary to prove something. To illustrate this, take a geometry example first. "Is similar to" is an equivalence relation on the set of regular polygons (I'll omit a proof here). Now say I need to ...


21

Sonnert and Sadler (2014) investigated what factors influence success in college calculus courses, including putting students into a pre-calculus course, which they described as "often a review of material students learned (or were supposed to learn) in high school." This may correspond to what you are calling "remedial." In their results, they found ...


18

When a student writes incorrect notation, ask them to read it out loud. I would say something like: Something here doesn't look right, but we can fix it. Could you read this work out loud? I think probably you are not super familiar with this topic, and that's okay, but this can help us fix it. I had success with this when dealing with a student who ...


16

In general, all applications are going to be more of the "here is what we have to check to make sure that our algorithm/theory/definitions work". We don't usually encounter practical problems where we're given a completely arbitrary relation and have to check if it's an equivalence relation. Here are some of the more common equivalence relations ...


15

I don't know about the research you ask for - supporting remediation. I am learning a lot lately about how problematic remediation is. There are two sorts of problems - placement and the remediation coursework. Placement tests can be harder than the tests students encounter at the end of their courses. I took the ACT/Compass placement test that our college ...


12

The Hungarian Quicksort Dance demonstrates a computer science algorithm with dance. It's pretty advanced, but the idea is to have your students physically act out algorithms. Perhaps something similar can be done with a number line. Line your students up and have one walk down the line to demonstrate addition and subtraction, or take big steps for ...


11

I've had a remarkable (remarkable to me) success in a different environment on a different topic, so I am not sure this will translate. But I'll mention in anyway. With one change, the same quiet "labs" turned into such lively conversations that I now have to raise my voice above the din to interrupt them with instructions. The change was from allowing ...


11

Equality vs. Identity Alexei touched on this topic by mentioning hash tables, but I would like to spell it out more explicitly, because this is a critical and fundamental topic in software engineering, and essential for every programmer to know and understand. Every high level programming language has a mechanism for comparing two values for "equality&...


10

This is an opinion, but perhaps worth sharing: When I assign that I expect students (particularly lower division students, such as those in calculus or precalculus classes) to do on their own time, I assume that they are going to collaborate, ask the Google, and otherwise use all of the resources that are available to them. Demanding that they do the work ...


9

Personally in my own math courses, I have found that the gold-standard is to have an advance cycle of short-answer responses, document the most common student responses, and then turn those into the multiple-choice options in the future. Disclaimer: In general I frown upon multiple-choice testing, since the math discipline is inherently about explaining/...


9

Start calling on students individually without waiting for any one to raise their hand. For a long time I never did this since I am conflict avoidant and since I didn’t want to put students on the spot, but then I witnessed another instructor masterfully call on every student throughout the course of one class period. The key point to add is that when a ...


8

Use them for what they're best at testing, like conceptual understanding questions. One way to do this is by having students evaluate statements they would never be expected to come up with on their own, but should be able to understand the truth or falsehood of. This can also be useful for questions that have many different way of getting to an answer that ...


8

This grew a bit long for a comment. (My first note is similar to Alexander Woo's remark about factoring polynomials; perhaps he intended "polynomials" to subsume the case here, in which we add constant functions...) Given $413 + 91$, it may not be clear that this can be re-written as $7 \times 59 + 7 \times 13 = 7(59+13)$. (Plenty of people seem to ...


8

Remediation is more about the politics of mathematics education than about learning mathematics. When students don't use their mathematics they often lose it quickly. My own experience was that it was much better for students to take the courses that their high school record said they had taken the prerequisite courses for than to have them take "remedial" ...


7

After writing an email, I received a response from Ernie Danforth, the NE regional vice president of AMATYC which answer my question. Keep in mind these are GUIDELINES, they are not hard and fast rules. No mathematics education courses do not count as mathematics preparation, although they are still considered valuable. You are correct that under the ...


7

I see the same mistake in my Calculus students' work a lot. And from my observations, I think there's a very simple reason for that — for typical students (here in the U.S.) this is multiplication, and in multiplication order doesn't matter. In fact, for typical students (…) almost everything is multiplication, including function notation; and ...


6

Accelerated developmental education includes a variety of mechanisms to shorten time students spend in remediation. It can include: Batching together multiple courses of remediation in one semester (say: 6 or 9 credits in one semester), likely with "supports" such as tutors and learning communities; batching together remediation in simultaneous credit-...


6

In general, I'd think that anything that helps students to break out of a tendency to passivity is good. Initiative should be rewarded, looking ahead is good, and so on. Sure, if there is a pervasive assumption that everything is "curved" in an invidious way, there are even larger problems... but the possibility of "curving" unfairly (don't do it) is not a ...


6

The question is broad, so my answer will be as well. Students who have been in school for a sufficient length of time have probably learned highly effective strategies for accomplishing their scholastic goals -- whatever those may be. You (it sounds to me) have identified that the learning strategies many of your students have adopted (and likely adopted ...


6

Here are some ideas: Try using shared vertical surfaces for doing problems. For example use whiteboards/windows. According to Peter Liljedahl http://www.peterliljedahl.com/publications/building-thinking-classrooms, having to share the writing space can encourage discussion. He suggests vertical non-permanent surfaces, but any shared space is better than ...


6

I teach in the Universidad Politécnica de Madrid, which is a fairly large public engineering school with research objectives. The students are comparable to those I have taught in engineering degrees at places like Georgia Tech or the University of Washington, although they enter the univesity with better preparation. The teaching of calculus in the UPM ...


6

If you have looked at modular arithmetic, then one possibility is: Give/recall some example of an algebraic argument in modular arithmetic; then point out that the argument is implicitly relying on the fact that congruence is an equivalence relation. So then you can explain: equivalence relations are designed to axiomatise what’s needed for these kinds of ...


6

The notion of equivalence relation is one of the basic building blocks out of which all mathematical thought is constructed. (Paul Halmos) What's the point of learning equivalence relations? The concept of equivalence relation is a generalization of the concept of equality. Why is it good to know that $a$ is equal to $b$? Because, in this case, all we ...


5

If I understand you, I do often think this way when solving a problem. For example, see this answer of mine at Mathematics Stack Exchange. However, limits involving infinity (as the independent or dependent variable) seem to be losing importance in textbooks. For example, I teach from Calculus: Graphical, Numerical, Algebraic by Ross L. Finney et al., and ...


5

Generalizing from your personal experience with a small, local group of students to trends in "mathematics education in the US" is a tremendous leap. The evidence you have for this "trend" could just as well be evidence that over the last 10 years you have become better at identifying students' weaknesses in arithmetic.


5

Am I just asking or expecting too much from this group? I empathize with your goals completely, and they are honorable. Bear in mind that sometimes, due to a conflation of factors, a particular class climate may be such that this just never gels no matter what you do. In my master teaching folder I have a half-dozen post-it notes for (hopefully very rare) ...


5

Dictionary/hash table relies on equivalence to bucketize items. So knowing that one would never try to build a hashtable by a distance between cities (objects on a plane): distance is not transitive. In real programming there is another common way to violate equivalence which is sort of implied in pure math/CS: "a == b hence in 5 minutes a == b too"...


4

I'd recommend not being intimidated by textbook trends! "In real life" (as opposed to textbook-life, for sure, and often opposed to required-curriculum "school-math"), _of_course_ the two things you mention, "the limit", and "how it is approached", matter a great deal. Do not be intimidated by silly books (written by non-mathematicians, almost entirely) to ...


4

I've recently switched to using OpenStax open education resources in my college algebra courses. That is, they are: Free of cost, free to redistribute, free to edit if desired. For many years I thought the quality of OER materials was unacceptably low, but in the last year or two they've crossed the threshold of usability for me. They're digital in format (...


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