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The answers provided here so far give lots of good tips but I think they're not addressing a key part of the question, which is "why do we need to count two events (50,52) and (52,50), instead of one event (50,52)?" The answer is that you can do it either way. The textbook way is counting P(X=50 \cap Y=52) + P(X=52 \cap Y=50) + P(X=51 \cap Y=51) = P(X=...

12

I struggled with this as a new lecturer, and I found a few ways to manage the process of using a big blackboard: Observe people lecturing When I first started lecturing, I visited the lectures of other staff to see how they did it. I realised that when I was a student I had seen many lectures, but I had never thought about the actions of the lecturers at ...

9

One thing to watch out for is learning to understand what the student's problems with the subject are. Once you identify that, fixing is typically easy. Take care when designing exams and homework. Ask yourself what you want to accomplish (what should be measured, what you want them to learn). It is important to think of grading when designing exams, don't ...

7

This is a very good question. The issue comes up frequently. I explain this using a toy model: throw two regular six-sided die. What is the probability that the sum is 3? With some physical modeling, you can become convinced that the answer is 2/36=5.5%, This corresponds to the two possibilities 1+2 and 2+1. But why are these two possibilities distinct? We ...

7

IBL is a really wide umbrella term, nowadays at least. I strongly suggest browsing the Academy of Inquiry Based Learning website. They have videos for many of the talks at the R.L. Moore conferences, such as this one that seems to be about "larger" classes. The canonical text on "Moore method" may have some ideas for your larger classroom situation, but I ...

7

After teaching college courses (full disclosure - I'm a biologist) for about 15 years, my main teaching discovery is this: Teachers overestimate how much students learn from a lecture. This occurs because a) teachers themselves were part of the select group that learned great from lectures, or they wouldn't have survived the flaming hoops of academia, and ...

7

I recommend the book How to Teach Mathematics by Steven G. Krantz. He covers almost all of what you are asking from low level courses, advanced courses, large and small. The second edition also contains many essays on reflections on teaching that I found enlightening.

6

Because you used the education-research and reference-request tags, I will give my usual recommendation of Powerful Learning: What We Know About Teaching for Understanding for the reason that Schoenfeld's chapter is well-referenced and written for practitioners. The benefit here is that it is a good introduction to methods of "teaching for understanding" ...

6

I would like to mention this recent opinion article in the AMS Notices (vol.66, no.7; PDF download) by Colin Adams (author of The Knot Book). His main point is that we should try "to impart a love of mathematics." He ends with this anecdote: I remember seeing a lecture by a professor who had won a variety of teaching awards. As I always am, I was ...

5

I read in an article the other day that said rather cynically that a sign of a good classroom is one where students are allowed to say more than one sentence about mathematics without being interrupted by the instructor. I've been reading a lot of blogs written by university-level mathematicians who are interested in teaching, and the latest buzzword ...

5

A simple search give you plenty of reference materials. Here is The Academy of Inquiry Based Learning. And here is The Journal of Inquiry-Based Learning in Mathematics. Enjoy exploring and experiencing.

5

I will tell you as a previous teacher my hope that I wish I didn't have was that I was going to teach my students math and that they would understand because I was going to teach them the "how". This is an unrealistic goal (particularly for a beginning teacher). This pessimistic view will probably be met with down votes, but you need to hear it as a ...

5

I spent two days sitting in on math classes at Phillips Exeter Academy. It is an outstanding example of inquiry/problem-based mathematics education in practice. Consider their curriculum. Their format is simple: students do 4-10 homework problems per night (all of these are posted online here), come in to class every day and each put their solution to one ...

4

Here are references directed at teachers more than policy-makers, that analyze teaching from the teacher’s point of view, including the decisions that teachers must make as the lesson unfolds, and how their decisions affect the students. Teaching Problems and the Problems of Teaching by Magdalene Lampert Connecting Mathematical Ideas: Middle School Video ...

4

This is only a brief answer, but I hope that it's a useful counterpoint to the others which present a very positive view of discovery learning. The one reference that I have found most useful for discovery- and inquiry-based learning is the paper Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, ...

4

I have loved the zen of teaching. Accept where a student is, and figure out how to help them move forward. I remember years ago being a bit bored teaching algebra for the nth time. And then it got less boring, because I was able to focus more on the students and less on the content. Or rather, my focus was on how the students were seeing the content. Don't ...

4

Bloom's taxonomy is not very helpful for maths, as many of the verbs don't make much sense, and ones that do have meanings occur in the wrong order. There are a couple of people (at least) who have written versions more adapted to maths, with more understandable verbs. I don't have the references to hand though. Edit to add some references: Anderson & ...

4

There are numerous little things you can do to instill an IBL atmosphere in your class; most of them can be found on the AIBL website mentioned earlier. I would also recommend watching this TED talk: http://www.ted.com/talks/dan_meyer_math_curriculum_makeover?language=en I love his examples of getting the students to be the ones to generate the questions....

4

I don't think I can improve on juod's excellent answer, but there are a few points worth elaborating. First, physical objects are always distinguishable. (I am here ignoring the phenomenon of identical particles in quantum mechanics, the issue there being that we really have to rethink the concept of physical object when talking about the electrons in a ...

3

Many teachers use Twitter to share and discuss teaching. One large community includes teachers from early childhood to university level, mostly from North America but also from other countries. It calls itself the "Math(s) Twitter-Blog-o-Sphere" and uses the hashtag #MTBoS to identify tweets where they want to share or discuss their teaching ideas. You can ...

3

For an absolute free-for-all with no restrictions on extended discussions (unlike here) or anything else for that matter, you could try sci.math at Google Groups. WARNING: It is currently infested with cranks and trolls, but you can just ignore them. They can't block your postings. I wouldn't recommend it for students. The trolls there seem to delight in ...

3

The succinct answer is that there are two ways to choose $50$ and $52$ - if two boxes are chosen one could have $50$ and other $52$ or vice-versa - but this will be more apparent to a student if you write out the space of all $25$ possible outcomes as a list of ordered pairs. "Indistinguishable" does not mean "identical", just as "isomorphic" does not mean "...

3

It was quite a shock to me when I found out I had the expert's curse (can't understand the problems newbies have because at their root is something so familiar that you can't imagine somebody doesn't know it). Now I know I'm cursed, dunno if it has done my victims much good...

2

Study the evidence. People believe all kinds of things about teaching, but in many cases their beliefs are contradicted by evidence. A classic, important paper in physics is: Hake, "Interactive Engagement Versus Traditional Methods: a Six-Thousand Student Survey of Mechanics Test Data for Introductory Physics Courses," Am. J. of Phys, 66 (1997) 64 These ...

2

There are a couple of Facebook groups that might be useful: If you are interested in discussing research in mathematics education (including "action research" in your own classroom, which sounds like what you are interested in), try the Math Education Researchers group (of which I am a moderator). If you want to discuss curriculum and best practices with ...

2

The Mathematics Teaching Community would seem to be a good place, but unfortunately it seems to be unavailable. (It has been unavailable for a few years now, I think.) From the University of Georgia website: The Mathematics Teaching Community is an online community for those of us who want mathematics teaching to be a vigorous, vibrant profession. It's ...

2

Let me start with mentioning this now well-known experiment: Improved Learning in a Large-Enrollment Physics Class. I highlight part of which that is related to your question. During week 12, we studied two sections whose instructors agreed to participate. For the 11 weeks preceding the study, both sections were taught in a similar manner by two ...

2

I don't know from what principles you're teaching probability, but the answer I would give is the following: When calculating probabilities, it is important to remember the basic principles. We have a set of possible outcomes $S$ and an event $E \subseteq S$. The probability that an outcome in our event is chosen is precisely $\frac{E}{S}$, *given that ...

2

You didn't disclose the level you teach. When I first ran into this issue (i.e. the need to explain this), I went to the example... You have 2 coins. There are 3 possible outcomes, 2 heads, One Head One Tail, 2 Tails. Are they each equally likely, each 1 in 3? I then walk them through the physical experiment. Even though we flip two coins at once (as ...

1

Challenge him with the following: let‘s think it through with a soccer match. In how many ways can the match end, such that the sum of the goals is 3? According to his logic only two possible ways (1, 2) or (0, 3) - I‘m pretty sure the fans won‘t agree. Same goes for the colored dice - how many ways can they sum up to 4? Thus what’s the probability for ...

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