Posts by ancoop
A few years ago I got turned on to Geogebra, probably the world’s most awesome and low-barrier-to-entry mathematical software. The best part of Geogebra is that you can do most everything through the web client or the mobile app. (I have the desktop client, but I never use it!)
Here are some doodads I’ve built with Geogebra.
A tool for visualizing how the Riemann sums obtained by taking a product partition of the rectangle approximate the volume under the graph. You can specify how many subintervals to partition and into, and change the function .
The same as the previous applet, but this one is over the region , .
A tool for visualizing the relationship between the level curves of a function , the graph of , and the intersection of the graph with horizontal planes.
You specify a curve, and this applet animates its Frenet-Serret frame.
Change the slider to alter the slope of the line; watch as the reflections of the points and — and the image of Satan from the Codex Gigas — move along with the line. The matrix representing the reflection is obtained from the coordinates of the reflected points.
This isn’t a Geogebra gadget itself, but it shows how awesome Geogebra is for doing exploratory assignments.
Abstract: Mathematical induction has some notoriety as a difficult mathematical proof technique, especially for beginning students. In this note, I describe a writing assignment in which students are asked to develop, describe in detail, critique, defend, and finally extend their own analogies for mathematical induction. By putting the work of explanation into the students’ hands, this assignment requires them to engage in detail with the necessary parts of an inductive proof. Students select their subject for the analogy, allowing them to connect abstract mathematics to their lived experiences. The process of peer review helps students recognize and remedy several of the most common errors in writing an inductive proof. All of this takes place in the context of a creative assignment, outside the work of writing formal inductive proofs.
what are grades?
Grades serve a few purposes:
- Grades give feedback. They allow you to know what you know and what you don’t know. This is how most professors think of grades most of the time.
- Grades allow evaluation and comparison of students. This is how most students think of grades most of the time.
- Grades are motivators. Hope for a good grade, and fear of a bad one, is the predominant way professors get their students to do the work necessary to learn.
These purposes hold for any assignment, and for overall course grades. But they are present in different mixtures. A grade on a homework assignment is mostly feedback-oriented; whereas an exam grade is about evaluation; whereas a final grade is a strong motivator.
Most courses are graded based on the accumulation of points. Each assignment is worth some number of points, and the percentages are averaged together (possibly with some weighting) to compute a final average, which is then converted to a letter grade. The primary advantage of this system is that it’s easy to administer and describe. However, it has a several major shortcomings:
- comparing unlike things: Averages ignore the fact that some things just aren’t comparable. Exams and homework play very different roles, and the meaning of grades is different in each one. So what sense does it make to add those scores together?
- penalizing failure: One of the best ways to learn is to try, fail, and try again. In an average-based system, any mistake a student makes hurts her grade. So a student may not want to take that risk — a risk that may be necessary to truly understand the material.
- eliding important differences: Under the points-accumulation system, a student who does poorly throughout the semester but by the end of the semester has learned the material thoroughly may get the same final grade as a student who muddled through the whole time. For example, the following three functions have the same average, but the one on the left is the ideal for a course and the other two are almost an anti-ideal.
- not incentivizing complete work: Let’s say you’re the general of an army fleeing a much-larger and better-equipped enemy force. You need to cross a large river. You task your engineers with building ten bridges across the river, but their skills and the available time and materials mean they cannot complete the task. Would you rather they built ten partial bridges (say each bridge is 80% complete) or eight full bridges? It seems obvious that you want as many full bridges as you can get. Even six complete bridges (“only 60% average”) would be preferable to ten 80%-complete bridges (“80% average”). A grading system that regularly gives Bs or even As to students who don’t get the main point is not doing its job.
- incentivizing the wrong things: Many students have told me over the years that their strategy for getting a good grade in one of my classes is: “bank” points early, then check out later in the semester. With an average/points based system, this is a reasonable strategy. In terms of learning, though, it’s just about dead wrong. In almost every course I teach, the last few weeks of material are the most interesting and most critical for future courses to build upon. So a student who banks enough points early doesn’t get the main point of the course. A grading system that regularly gives Bs or even As to students who don’t get the main point is not doing its job.
For these reasons, I have moved away from a points-average system to specifications grading.
how specifications grading works (in my courses)
For each non-failing letter grade (A, B, C, D), there is a list of specifications. If a student meets all the specifications for a particular grade, they receive that grade. They must achieve all the specifications for the desired grade.
For example, in my MA 225 course, some of the specifications are:
- For a C
- get at least 50% of the points on each of two exams
- submit a proof by contradiction
- For a B
- get 50% of the points on one exam, 70% on another, including 70% on the final exam
- submit a proof by complete induction
- For an A
- get 70% of the points on one exam, 85% on another, including 80% on the final exam
- submit a proof that something is unique
These specifications are cumulative: to get a B, a student must complete both the C and B specifications; to get an A, the student must complete the C, B, and A specifications.
pass/fail (with multiple tries) where appropriate, points where appropriate
The principle of binary specifications — you meet them, or you don’t — also holds at the assignment scale. In the math major courses I teach, the vast majority of the coursework consists of writing proofs. In a very basic philosophical sense, a given proof is either correct, or it is not correct. Writing “8/10” on a proof doesn’t really reflect that the proof is “80% correct”; instead (when I write it), this usually means something like “there were a number of mistakes, but they were minor and I’m willing to accept them”. Whether this constitutes useful feedback to the student is unclear. It is also difficult for points to distinguish between a proof that captures the content correctly, but is incorrect for formal reasons, and a proof that has good formal properties but is lacking on the content side.
Having adopted a specifications grading scheme (hence, being freed from the need to report a numerical grade for each assignment), there are other possibilities.
In MA 225 (starting Fall 2015) and 425 (starting Fall 2018), I give each proof a mark of S (“satisfactory”) if it is nearly perfect. If not completely correct, the S proof has at most minor flaws of formatting or language. S proofs are the kind of proof that nearly every professor would recognize as correct.Only S proofs count toward the final grade.
As with any pass/fail scheme, this might seem unduly harsh (if not impossible for a student to succeed in). To reward incremental progress, I have another possible mark: P (“progressing”), which indicates that the proof is incorrect, but can be salvaged. P proofs can be resubmitted without penalty.
For proofs that have no hope of being corrected, I mark U (“unsatisfactory”) — these are not counted against the student, except as missed opportunities.
One of the specifications for the course is then that the student must get a certain number of S marks during the semester. In MA 225, the number of Ss required for a course grade of A is around half of the total proof opportunities — which may seem low if you’re used to thinking in terms of averages. But because getting an S mark requires the proof to be nearly perfect,
The S/P/U system is not appropriate for all assignments, though: quizzes and exams are graded using points.
Since implementing the specifications and S/P/U systems in MA 225, I have seen a dramatic rise in students who complete the course successfully. This isn’t because the scheme makes getting an A easier; subjectively the quality of student work seems to have jumped. Because even a C student has to get some near-perfect proofs, this increase in the quality of student work is perhaps most dramatic among students who wind up getting a C.
I learned about specifications grading through a reading circle offered by the Office of Faculty Development; we read the book Specifications Grading by Linda Nilson. Robert Talbert’s thoughts have also been very useful to me.
As part of NCSU’s TH!NK Quality Enhancement Program, I have spent a lot of time learning and thinking about how to encourage critical and creative thinking in mathematics courses. The course I targeted as TH!NK Faculty is MA 225 (an introduction or transition course for mathematics majors).
mathematical thinking is weird, but not that weird
It’s true that mathematicians view the world in somewhat a different way. We have our jargon (which we use outside its technical scope — I can’t tell you how many times I use the word “modulo” in everyday conversation!), and depending on our specialization we have our pet obsessions.
I believe very strongly that, though the reasoning one does in mathematics may be rarefied, it is a rarefied form of ordinary reasoning, by which I mean the kind of reasoning it takes to successfully navigate the physical, social, and emotional world we live in. To train students in mathematics means to help them learn to harness their physical-collision-avoiding, social-bond-forming, managing-tears-and-laughter apparatus and bring it to bear in certain ways on certain kinds of problems.
Learning to think involves taking control over one’s own thinking process. One of the best ways to do this is to engage students in reflection:
- students should reflect on the content of what they are learning (“How are the topics we’ve discussed this week related to the topics discussed earlier in the semester?”)
- students should reflect on their process of working in mathematics (“When you first approached this problem, what were some questions you had?” “What was the hardest part about this problem?”)
- students should reflect on their work (“What went wrong?” “What went right?”)
- students should reflect on their thinking (“What has changed in how you think about [topic] after doing this assignment?”)
the language of critical and creative thinking
One way to make the connection between mathematics and the rest of life is applications (such as the application of ODE to population biology). But I am interested in connecting the practice of mathematics to the practice of (for example) biology. One of the innovations of the TH!NK initiative is the use of a common set of terms and descriptions for aspects of critical and creative thinking. By using the same language to discuss and reflect on thinking processes in the mathematics classroom as in the biology classroom, we emphasize that it’s the same process, just being applied to different tasks.
the standards, and some ways they apply to undergraduate mathematics
The 13 TH!NK standards are:
- for critical thinking, clarity, accuracy, precision, relevance, depth, breadth, logic, significance, fairness;
- for creative thinking, originality, adaptability, appropriateness, contribution to the domain.
As an assignment in my MA 225 class, I ask students to think of some ways that each of the standards might apply to the process of writing proofs. Students also indicated which standards might apply to the formatting, style, content, and process of a proof. Below are my condensation of their ideas (along with some of my own).
|Clarity||Everything in a proof has to be written clearly. The logical steps should be stated explicitly. Simple formatting elements, such as writing “we will show. . .”, and consistently using the word Proof at the beginning of each proof, help improve clarity.||formatting, style, content, process|
|Accuracy||In a proof (except possibly in a proof by contradiction), all the statements in the proof should be true. Definitions and statements of quoted theorems should be stated correctly.||style, content, process|
|Precision||Precisely stating what you’re doing in a proof is essential. Ambiguity and fuzziness about what a statement or term means makes it difficult to be accurate and clear.||formatting, style, content, process|
|Relevance||Not every fact is relevant to every problem; sometimes problems contain lots of information that doesn’t go into the final proof. Determining what’s relevant and what’s not is an essential skill in the proof-writing process.||content, process|
|Breadth||Sometimes a proof requires facts and techniques that aren’t obvious at first — allowing yourself to think broadly about what tools could be used is essential.||content, process|
|Logic||Mathematical proofs are based on logic: you need to understand the logical structure of the claim you’re proving, and you need to make sure that statements in the proof are arranged in a valid logical order.||style, content, process|
|Fairness||It is very easy to fall into the trap of motivated reasoning when writing a proof. When you know what the goal is, you may overlook a logical step, or make an unsupported (or false!) statement that would get you your goal.
When you are trying to prove an “obvious” fact, it is important to remind yourself to be fair to the statement–to treat as worthy of a proof.
|Adaptability||Often the first (and second, and third) approaches to a problem don’t work. They’re good attempts, but they don’t work. It’s important to be adaptable in your thinking, so you don’t get locked into one approach.||process|
|Contribution to the Domain||content|