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I'm teaching first semester algebra, the majority of the students share the rest of the classes. Sometimes I want to take examples from other courses so they can see that everything intertwines, but it has happend too many time that whenever I try to do that they say that they haven't study that with the corresponding teacher for whatever reason. To give a for instance, yesterday I wanted to show them that sequences could be seen as functions, but they told me that they haven't study anything about them! which is a little frightening because at this point in the semester they are suppose to know something.

When that happened at the beggining I opted to just go over what I wanted to show them in particular very quickly, but as we progress I can tell that they are falling behing way too much and fast that is also affecting my own course. At first I thought that they were exaggerating, they even told me that they geometry teacher hates them, but now the good students are failing as well. What should I do? or more over, what can be done?

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    $\begingroup$ Does first semester algebra mean group theory, or does it mean high-school algebra? Does geometry mean high-school geometry, or does it mean a higher-level course? I don't know your country's educational system, but if the context is high school, then I wouldn't have expected students in the US at this level to have heard of functions or sequences. $\endgroup$ – Ben Crowell Oct 11 '14 at 20:10
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If you want to rely on things convered in other course, one thing I think you should try to do is to get information from the teachers of these other courses what they have discussed; if possible, in detail, so that you know exact terminology and notation they use.

In my experience (both as teacher and student!), students can be (for various reasons) quite unreliable in conveying what they have seen in another course and (should) know, especially when asked in the middle of a large lecture, possibly using slightly different terminology.

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  • $\begingroup$ I think that the option to check with the other teacher is a little bit difficult, but I'll try, I know anybody could get counfused just by changing notation. I know I shouldn't rely on what they say, however I have use some examples that are pretty generic, I mean, shouldn't they know what a sequence is? May be they are falling behind because of other reasons? They still have issues that have impact on the class $\endgroup$ – Ana Galois Oct 8 '14 at 17:30
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    $\begingroup$ When you try to check what was covered, I would (at least at first) avoid all direct or even only indirect mention that you believe there is some problem. Just explain in a polite way that you ask so that you can reference results that they have convered. Regarding your speific question if they should know what a sequence is: I do not know the curriculum where you are and when the term started. So I cannot tell. To understand the situation better: do you feel this year is much worse than others or is it your first year teaching this there? $\endgroup$ – quid Oct 8 '14 at 18:19
  • $\begingroup$ Thanks for the follow up. I'll be careful when consulting the other teachers. I know that the students should already know what a sequence is, I've been a calculus tutor before in my school, and I'm confident that they should know that by now. It is my first year with a huge group like this, but I have some experience helping smaller groups and I've seen that in general when they have had "good teachers" at least they know the generic examples that I've been using in my class now. I just want to know if there is something I could do to help them? $\endgroup$ – Ana Galois Oct 8 '14 at 18:33
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    $\begingroup$ Regarding the "should know": I think you should also take into account the possibility that, yes, they should know, but they do not (or at least not to the extent you think) yet it is not the other teachers fault. Of course, it could be their fault, but it is not clear. Regarding how to help: I think mainly you should try your best to teach a good course yourself. And, if you notice that the examples from other courses causes problems perhaps do not do them as much (even if abstractly they are a good idea). There is a considerable risk in trying to convey too much. $\endgroup$ – quid Oct 9 '14 at 15:33
  • $\begingroup$ Now, if you get really convinced there is a major problem with another course you should likely do something about it. Try to talk about it with an experienced colleague you trust and/or with the person responsible for the organisation of teaching. Details could vary depending on context and this part of the question might be more suitable for academia.SE $\endgroup$ – quid Oct 9 '14 at 15:37
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Here is an example from an algebra class I teach.

We cover the quadratic formula, completing the square, the multiple ways to find the vertex, and try to connect it all together with the effects on the graph of the functions. In practice problems from the online homework, the students see things like

Solve $x^2 - 9x + 18 = 0$.

However, when I ask them to do the following:

Solve $x^2 - 9x = -18$.

They invariably respond that they have not seen that kind of problem before. This is a very robust and repeatable experiment.


What I am saying is that your students have probably seen what you are talking about before, but you may be surprised how few connections they are making between material. Instead many math students (and some teachers) think that mathematics is all about formula-plugging and pattern-matching.

Asking the other teacher for the problem sets the students have seen on sequences will probably give you a lot of information.


As a further example:

1a) Find the y-intercept of the following line: $2x + 5y = 10$

1b) Find the x-intercept of the following line: $2x + 5y = 10$

College algebra students will almost always think that 1a is easy and 1b is confusing. If you are wondering if they understand intercepts and you give them only (1b), you would probably think their other teacher had never even mentioned intercepts, when actually the problem is a lack of conceptual understanding and sense-making.

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  • $\begingroup$ "...but you may be surprised how few connections they are making between material." That's exactly why I do this examples using geometry and calculus, so they see that the connections are there and that they are useful. $\endgroup$ – Ana Galois Oct 9 '14 at 18:27
  • $\begingroup$ I +1ed your answer, but find it frightening. For your second example, a student had to graph this exact equation. I watched the manipulation to y=-.4x+2. When I offered "set X=0 to get Y intercept and Y=0 for X intercept, I got the blank stare. Really? $\endgroup$ – JTP - Apologise to Monica Oct 9 '14 at 18:28
  • $\begingroup$ Almost all of the students I talk to give me blank stares. And most of them faced with 1b will manipulate to $y=-\frac25 x + 2$, set y=0 and then manipulate back to x=5. Or they will manipulate to $y=\frac25 x + 2$, graph it with their graphics calculator, and then use the graph to find the solution. $\endgroup$ – DavidButlerUofA Oct 9 '14 at 18:52
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    $\begingroup$ In fact, this answer touches upon a very deep and widespread issue in teaching mathematics. It is so much easier to have decent grades by asking students to reproduce step by step a few model exercises, that the fundamental question of showing them how symbolic notation carries meaning is very, very often lost. It might be that this phenomenon is indeed interfering with the perception of @AnaGalois. $\endgroup$ – Benoît Kloeckner Oct 9 '14 at 19:51
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    $\begingroup$ > "...think that mathematics is all about formula-plugging and pattern-matching". I just wanted to remark that from my experience, much of mathematical skill at all levels is based on pattern-matching. I'd even venture to say that the main way in which one improves in mathematical thinking is by gaining more diverse and higher level tools of pattern matching (and manipulation). I agree that basic blind pattern-matching isn't enough, but we shouldn't disparage what may be the most refined element in our cognitive toolkit. $\endgroup$ – yoniLavi Oct 10 '14 at 20:26

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