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17

Why do they think this? I can think of a few intertwined reasons why the students think that $-x$ is negative. Firstly, psychologically the $-$ sign has a very strong pull towards thinking of subtraction and negative numbers. When they were small, your students had more than five years to get used to the idea that $-$ meant subtraction and that subtraction ...


10

The following "definition" given in the Glossary of the Common Core State Standards for Mathematics (2010) give you an idea why students think this: Integer. A number expressible in the form a or – a for some whole number a. The problem is not about algebra alone, rather it is about disconnection of teaching of algebra and negative numbers in schools. It ...


7

...what are efficient structures for a revision lecture? Things I've actually tried, sorted by increasing student engagement: Merely provide list of topics for students to study on their own time. This was simple and fast, and it made it clear that they should expect questions from the entire course's material. It also put the review entirely on them to ...


6

I think it's appropriate to lead the student(s) through answering their own objections with some leading questions - perhaps questions like "When is $-x$ negative?" In particular it helps reinforce to students that this is the sort of question they should be asking themselves. Why do students think this? I think it's because they're applying a "rule" ...


5

I spend one entire class period with the students working in groups on a massive stack of problems. More problems than is really reasonable to complete in the time allotted. In this "review mode," certain kinds of problems are more effective than usual. One type: where the problem is easy once you identify the correct strategy, but the correct strategy is ...


5

In my calculus class, I give a practice test that covers (almost) the same material as the test. I give some class time to do the practice test but they largely do it in their own time. In the class before the test I discuss the practice test. The students know this is the last time for them to ask any questions. This system isn't perfect, but what is? It ...


5

Even for math grad students, I'd forcefully review much more than many traditions seem to indicate. That is, I would not presume perfect recall of the standard curriculum, especially either in detail or in "big picture". Further, in my experience, even very smart people with unusually good memories greatly benefit from repetition. It's not "one and done", ......


5

I'd conjecture that classes have not given sufficient time to the specific translation skill between the English and Algebra languages, especially the meaning of relation symbols like equals (=). Two days ago I had to spend a few minutes clarifying this in my College Algebra class. Q: "Algebraically, how would one say that $c$ is a negative number?" A: "$-c$...


5

Instead of thinking about $-x$, consider the expression $(0-x)$. Having (hopefully) being exposed to expressions of the form $(a-b)$ for a long time, where $a$ and $b$ can be arbitrary reals, there's no reason to assume that the value of the expression is positive (or negative) without further information. Therefore, when they see $\sqrt{-x}$, they can now ...


4

I tutor many students whose teachers have been telling them to pronounce -x not as "negative x" but instead as "the opposite of x." I assume that this semantic change is aimed at instilling in students the idea that -x need not be a negative number. You can ask questions such as "what is the opposite of negative 3" and "what is the opposite of 4" to get ...


2

I recommend devoting class time to helping the students review last year's content. I wouldn't recommend just telling them stuff, but getting them to identify things that they need to know themselves. First I would get them to try to write down things they learned last year. Give them say 15 mins to write down everything they could think of individually ...


2

I think some sort of examination program like qualifier, but for undergrads, would be the simplest way to drive for what you want. Perhaps like the Tripos, albiet not as hard. And with motivation being scoring on it (but hopefully not just in ranking, but in absolute sense). I think you have to consider the disadvantage of spending more time on basics ...


1

In my experience, the videos can be quite shallow, but then the textbook I use for A-level (16-18 year olds) has a very prescriptive approach. I can assist with the understanding, making-links aspect of their learning. I think providing links is fine and probably a good idea. I would expect students to soon realise that attending classes, and particularly ...


1

This may be unpopular, but what exactly is the purpose of giving a final? On my tests, I do sections that spiral prior knowledge so students have to still study those throughout the semester. So by the time the semester is over, I've assessed them multiple times on skills and conceptual understanding. I believe final exams cause unnecessary stress not ...


1

I will often give two review sessions. In the first session, I will reorganise the content of a particular section of the course into an order more useful for review. I tell the students that often we learn things in a certain order because it's the way our brain has to learn them, or because that's how the topic is built up axiomatically. However it's ...


1

Presumably you have access to a class list over the summer? I would send the students a series of review problems in the mail once a week that form some sort of scavenger hunt. You can include a stamped postcard so they can send the answers back to you. The answers to the review problems could correspond to letters of the alphabet (so a problem with the ...


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