11

I found this an effective teaching technique. I take a topic they know, and find a Wikipedia article discussing that topic. If you are specifically focused on proofs, as opposed to more generic descriptions, you can find many proofs in Wikipedia. E.g., of Sperner's Lemma, or Euclid's proof of $\infty$ # primes. Then I project the text in class, and have ...


11

I think the given example is highly appropriate. You cannot cover every possible combination of ideas in class. Students display understanding of a concept (rather than "recipe following") by showing the ability to adapt at least a little bit to novel conditions. I think the problem you gave is a great homework problem. I personally like homework to be a ...


10

You may be interested in Academically Adrift: Limited Learning on College Campuses, Arum and Roksa, 2011. Also summarized in http://www.newyorker.com/arts/critics/atlarge/2011/06/06/110606crat_atlarge_menand . They have a lot of discussion of something called the Collegiate Learning Assessment, which is a standardized test of critical thinking. They find ...


10

I would frame this issue a little differently than you have. I think it's unreasonable, at least in the context of courses which aren't well into a math major, to ask students to do something they have not been taught to do. That is, the problems on an assessment should be the same as problems they've seen already. The catch is that "the same" is actually ...


9

I know I shouldn't add another answer, but I think my other answer went off on a tangent that didn't really address your question. I did not initially read it carefully enough to realize you were tutoring students, not teaching a class. I really don't think the label of "Teaching Critical Thinking Skills" is either relevant or pertinent--teaching is ...


9

When I first started asking students if they "checked their answers" on a test, a number of them asked me what I meant. This was specific to algebra, and I told them they should take the answer, say, the X intercepts, and put those back into the equation to see if it resulted in Y being zero. Many had never heard of this, and thought that checking simply ...


8

My advice is to minimize the amount of such synthesis required. Don't make it a large fraction of your tests, if at all. Teach the students the methods you expect them to display on the exam. Not something requiring some spark of creativity. Program for success. Creativity is tough in general and even tougher under test conditions. If you push too ...


8

I was also a double major in mathematics and philosophy as an undergraduate (at a state university in the U.S.). I think that both are incredibly important, and I'm happy to have both under my belt. Actually: philosophy was my primary major, mathematics secondary. (This confused a lot of my instructors in higher-level math courses looking at the course ...


8

Teach the students to perform a sanity check at the end of every problem, and a check-by-substitution when practical at the end the problem. If either check fails, they can use a technique for finding errors, such as: Rory's binary search for the mistaken step. Keeping a "top 10 list" of most common mistakes, such as some of the following: ** (not) ...


7

One definition of "abstract" is " disassociated from any specific instance". In mathematics, we "abstract" by finding properties which underlie a class of examples. For instance, the concept of a "group" is an abstraction of many different concrete instances of groups which were important to mathematicians: composing symmetries of spaces (such as the ...


7

There is quite a bit of evidence that critical thinking can be taught (which I found surprising). College students' gains in critical thinking skills over the course of their education are correlated with high standards set by instructors and greater time spent studying. So I think the basic answer is that you have to assign students tasks that require ...


6

I teach high-school calculus, and many questions can be checked by a graphing calculator. So my first strategy is to teach the use of the calculator and using it to check answers when possible. However, often a student will see that his final answer disagrees with the calculator but he does not know which of his steps introduced an error. I have found that ...


6

One possibility to encourage sanity checks is to practise sanity-check-type questions and include them in tests. An integration-related example could be: A student has calculated the area bounded by $y=x^2$, the $x$-axis, $x = 0$ and $x = 4$ to be 128 square units. Without using an integral, explain why 128 square units is too large to be a correct answer....


5

Not sure how to answer the question apart from suggesting a book about mathematical proof that can be found free online, and that is on the easier side of understanding. It's actually been adopted by a lot of universities and so I'm certain it is a quality textbook. https://www.people.vcu.edu/~rhammack/BookOfProof/ I don't know if it's for kids or grade ...


5

There are quite a few textbooks that have critiquing sample proofs as exercises. Here are three I know of: A Transition to Advanced Mathematics by Smith, Eggen, and St. Andre The Foundations of Mathematics by Thomas Q Sibley How to Read and Do Proofs: An Introduction to Mathematical Thought Processes by Daniel Solow This past year I used the Solow text ...


5

Go to Mathematics Stack Exchange or MathOverflow. There are many questions there looking for proofs and there are many different answers, some good, some bad (some are even wrong). Ask your students to criticize the proofs (Are the proofs they consider good the ones that are highly upvoted?). The commentary that you want is sometimes there (as comments). ...


3

Give your students a multi-step problem and 5 different "solutions" written by fictitious students (you, really). Say: Here are five different solutions to this problem. First, rank them from best to worst. Second, discuss your ranking with the student next to you and determine appropriate criteria for assessing solutions. Third, list what you believe ...


3

Obviously it's hard to tell from a distance. Here are some explanations: She may be more motivated to think about the material in the philosophy course. This may be because she can relate to the topics that they analyze. Like one commenter suggests, if you do not know the basic examples or motivation for group theory, it can be hard to feel motivated to ...


3

You might engage them in billiards on an elliptical table (1st ref below), or on a circular table but aiming to hit a second ball (2nd ref below). Nice analytic geometry coupled with intuitive situations. See this Numberphile video.                     Fig.1 (detail) in Reznick et al.     &...


2

The idea that math-study imparts certain fundamental intellectual skills was taken for granted for most of the history of Western education, and reaches back at least to Plato, who (reportedly) refused to accept students who had not mastered geometry. "Let no one ignorant of geometry enter here" was supposedly inscribed over the entrance to his Academy. In ...


2

Well, this is not a systematic course proposal, but a list of useful references I've collected about the subject: A must read for your staff team (it brings arguments from neuroscience): Mind in Motion: How Action Shapes Thought by Barbara Tversky. Perspective Taking: building a neurocognitive framework for integrating the “social” and the “spatial” by ...


2

Have them work through a few chunks of Euclid's Elements. Then they are doing proof, rather than reading about logic in a theoretical way, which is probably too abstract for kids in the age group you're targeting. The theory of logic will make much more sense to them after they have had an opportunity to practice it, and the whole point of Euclid's text is ...


2

My advice: Look at the test itself. If possible get some statistics on common errors. If not, ask other teachers or at least use your intuition. I.e., make an intelligent guess, "Bayesian estimate". Then design something (lecture and practice together) to hit the common mistakes. Don't reteach theory, concepts, etc. in some sort of organic manner. ...


2

A compromise approach could be to give a problem (with parts), as you might on a guided worksheet. Such as: A function $f$ has domain $(2,4)$. We define $g$ by $g(x) = f(x-2)$. a) Is $g(3)$ defined? b) Is $g(5)$ defined? c) What is the domain of $g$? You might even give a follow up problem (perhaps as extra credit) along the lines of: A function $f$ ...


2

First, let me endorse Jared's point that checking calculations and debugging computer programs have much in common. Good programmers build checks into their code before it evidences a bug. Second, one general technique, which only works in circumstances where at least one variable is present, is: look at extreme values of the variables. An example from a ...


2

"Do like a sports commentator: just say what's happening. As a sport commentator is talking to the public, whatever they are looking to the live-event or listening at the radio, they will understand just because he is saying exactly what's happening and nothing else. I think if you explain to your students, you're probably telling them your interpretation of ...


2

This is something that I've been specifically grappling with in my college remedial algebra classes for the last few years. JoeTaxpayer's observations in his answer very much match my own (that many students have never heard of checking solutions to equations until I make a topic out of it -- in fact, I'm somewhat embarrassed by how many years I spent ...


2

One of the crucial points here is that verifying a result must be different than repeating the calculation, let me explain: Verify that 221/12 = 18, rest 5: Instead of doing the calculation all over, calculate 12x18+5, which indeed equals 221. Verify that -2 and 3 are solutions of x^2-x-6=0: Fill in the values: (-2)^2-(-2)-6 = 4+2-6 = 0, and 3^2-3-6 = 0 ...


1

A pair of fresh eyes may see goodies and errors alike, that you cannot see yourself after writing the assignment. Thus, I experiment with letting students peer-review (an anonmized fraction of) their upcoming assignment using the same rubrick as I intend to use when asessing their final submission. For this purpose, I have a class on Moodle (my highschool ...


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