19
I'd avoid giving problems like that to students first learning indefinite integrals (either by not asking it at all, or specifying the range x>1 in the question). It's a subtle algebraic trap, and if the goal is to teach students the mechanics of integration, it's going to be distracting rather than helpful.
It might be an interesting question in a more ...
18
What is $\frac 1 a$? It is the unique (real) number such that $a\cdot \frac 1 a=1$. Does there exist a real number that multiplied by $0$ gives $1$? No. Why is this? Because if $0\cdot b=0$ which ever is $b$. This is about not being defined. Still... why is $\frac 1 0=\infty$ not so completely wrong? Because they can see that the smaller is $a$ then the ...
13
The first lesson I ever taught was on Standard Deviation to a high school Intro to Engineering class. It went fabulously and I stuck to the lesson plan covering every point on my slide show and getting through every activity only having to shorten the pair and share activity due to time constraints. The students understood everything, responded to discussion ...
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 ...
12
This is a hard question, because students are so used to manipulation of this kind. I have found you are right that absolute values can cause the worst of these examples.
Here is an example I ran into recently, which I hope will help your thinking. Observe that there are two different limits here:
$$\lim_{x\to\pm\infty} \frac{x}{\sqrt{x^2+1}} = \pm 1$$
...
11
Undergraduate students should be aware of their academic advising offices. Essentially every school employs full-time academic advisers to answer questions precisely like this one.
If you go to a larger school, the math department (or at least the college of education) at your school will have a professional adviser just for that content area. You need to ...
11
You are right to be concerned that the students are "missing something", but IMO the real problem here is that the question is completely artificial.
In any application of this type of integral, most likely $x$ will be known to be either positive or negative, but not both, and only one part of the "either-or" answer would apply. And there had better be a ...
11
One of the charms of graph theory is that people of all ages often enjoy learning graph theory ideas and tools. One place one can read about these ideas is in the book called For All Practical Purposes (I am a co-author) which has gone through 10 editions. The book was designed for college liberal arts students who might have almost no proficiency with ...
10
Turning my comment into an answer as per request:
I think the graphical approach gets the idea across. You could represent $3\times 1$ by a row of three dots; likewise, $3\times 2$ is two rows of three dots; etc. If you asked them to guess what $3\times 0$ is, some would probably tell you you'd have no rows so no dots, so the answer is zero. Humans are very ...
9
tl;dr do a ton of research first before making any decisions
IMHO, you shouldn't just whimsically pursue teaching as a "backup plan" if your current plan doesn't work out. Teaching is A TON of work from planning, grading, teaching, tutoring, after-school activities, etc. and if you don't have a true passion for teaching and the discipline and persistence ...
9
When introducing functions to a student, I usually give thought to two main methods, each with its pros and cons.
Method 1:
Use the set definition of the function. This is what you're attempting to do at the moment. The set definition of the function states that a function
is a relation between a set of inputs and a set of permissible outputs
with the ...
8
To me, the key point here is that the integral runs over a singularity. If you naively calculates a definite form that runs over the singularity you get the wrong answer. This is something I have done enough so that I have taught myself to be careful in this case.
I am more a physicist than a mathematician, so what I care about is the connection to a ...
7
Well, of course, if your plan is not working, then you have to adjust it.
This is easier to do, of course, if you are in the happy situation where you find that the students are able to handle much more difficult material. One can simply cover the planned material more quickly and supplement with more sophisticated asides or extensions of that material. ...
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 ...
7
A simple but rigorous definition is that a function is a graph in the plane that passes the vertical line test: a vertical line never intersects more than one point on the graph. (A graph is simply a set of points. "The plane" means the Euclidean plane.)
This avoids the problems associated with defining a function as a rule. Defining it as a rule raises the ...
6
As JoeTaxpayer suggested, my comment is more of an answer than a comment, so I'm copying my comment here. Maybe this will get others to answer, although I realize that appearing to admit that something like this has ever happened can reflect poorly on one's teaching. Personally, this almost never happened to me, if at all. I had more of a problem with having ...
6
Try applying physics. If you throw a ball up in the air, there are functions to describe the balls position, speed, and acceleration - all as functions of time. You input the time, and the function returns the an output like the position, speed, or acceleration. If you have a graph, you input x in the function and it returns the output y.
6
It takes a few seconds to download Dragon Box 12+, then your job is just about finished. It will take her a few hours to go through, but it will teach her well. It is also a game, so it neatly sidesteps any maths phobia that your students has acquired.
I have used this program with students over the last few months. I have found that most of the Algebra ...
5
Part of the answer to your question is that the statement 'mathematics is considered good in the UK and USA' refers, I believe, to research mathematics, not to undergraduate teaching.
I don't have direct experience of education on the continent, although my colleagues do. There is a significant difference in standard. The culture and the education system ...
5
First off I would like to point out that at least one of the things you're comparing is a bad comparison. Mathematical Physics (or in my country just mathematics for Physics majors) will have a completely different breadth to any standard math course for mathematicians.
Physics majors don't have to care too much about proofs and theory but they need to get ...
5
Input-output machine is the most intuitive thing to first give, especially to a weaker student. Ideally, you eventually want to have multiple frames of reference for a function, but to start with...go with input/output machine. For God's sake, don't start with ordered pairs. That is so aphysical and theoretical. It is probably the most powerful and ...
4
When I teach some basic graph theory, I explicitly mention (and include in the notes) that certain definitions and terminology vary from author to author, in some cases giving the alternatives. In fact, the field is notorious for this. Of course, there is a risk that students will use one of the alternatives instead of my preferred definition, but my ...
4
My suggestions (which assume that you are a lecturer in a large university class):
Design lesson plans with hinge questions that check for important understandings. In a large lecture hall, you can use individual student response devices so that every student votes on their answer to the question, and you get to see the distribution of responses. This ...
4
While my data organizing problems are usually more related to research, they still often involve notes, papers, and sheets of data collection interview items that are much like handouts.
One solution that I have used (although I am still perfecting it) is to scan everything and get rid of the paper copies.
Once I have electronic copies of things, a number ...
4
A graduate degree does not necessarily correlate closely with knowledge of the subject. It's common for people with a master's in math education to be hired at a community college despite simply not knowing calculus. Something like half of people interviewing for a tenure-track position will flub a question that probes their knowledge of first-semester ...
4
A little late to the party, but I wanted to add my two cents.
The students should understand that they are looking at the value of $1/0$ as a limit. This is good but it's not the entire picture, and using that value in normal computation will break algebra.
See the following:
$$
0 = 0 \\
0\cdot{1} = 0\cdot{2} \\
\frac{0\cdot 1}{0} = \frac{0\cdot 2}{0} \\
1 ...
4
This sort of example is not artificial and needs more careful treatment than it often receives. The inherent difficulty of the example is compounded by the tendency to discuss primitives without discussing their domains.
The badly named "indefinite integral" of a function is really an incompletely defined primitive of a function. By "incompletely defined" I ...
4
Here are GeoGebra materials: Graph Theory for Kids,
inspired by Joel Hamkins' notes, to which @A.Goodier pointed.
Four-color challenge.
3
I'd say you should look also at the "Minimal Qualifications" section, which seems to be very very lenient toward the amount of graduate mathematics taken:
All full- and part-time mathematics instructors at two-year colleges should possess at least a master's degree in mathematics or in a related field with at least 18 semester hours (27 quarter hours) in ...
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