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Ask the student to "talk through" their calculations Having a student verbalize their calculation may force them to pay more attention (or a different kind of attention) to their work that causes them to catch the errors as they make them. This feels very related to rubber duck debugging. At the very least, if you're working with a student one-on-...


14

I used to have this problem. What helps me more than anything is: Solve it two different ways if you can and make sure they agree If you are finding a general formula, test it on some examples If neither of the above are possible, re-read every step of your work with an attitude like it's trying to sell you a used car. Adopt the useful exaggeration that ...


9

The student who designed this problem wasn't thinking about the different wholes. IN your students problem, there are 3 different wholes. Anna's flowers - The whole is 5 flowers and $\frac{4}{5}$ are daffodils Beatrice's flowers The whole is 3 flowers and $\frac{2}{3}$ are daffodils The flowers of Anna and Beatrice combined. The whold is 8 flowers and $\...


8

The word you are looking for is mediant. The mediant of two fractions $\frac{a}{c}$ and $\frac{b}{d}$ is $\frac{a+b}{c+d}$. According to Wikipedia, It is sometimes called the freshman sum, as it is a common mistake in the early stages of learning about addition of fractions. Teachers often do this in grading papers. For example, a test has two parts: the ...


4

The student was conflating the sum $$\frac45+\frac23$$ and the weighted average (where the weights account for group-size differences) $$\left(\frac{\color{#00F}5}{\color{#180}{5+3}}\right)\frac45+\left(\frac{\color{#00F}3}{\color{#180}{5+3}}\right)\frac23\\=\frac{4+2}{5+3}$$ of the two given fractions/rates. Clearly, the sum of two positive fractions is ...


3

Teach methods for verifying results. Often once you have an answer, there are simple calculations to check its correctness. For example: Numerical calculations: round the results and compute backwards. Example: if you get 1234 * 56 / (789 - 987) = 349, compute e.g. 300 * (700 - 900) / 50 = -1200 which is not close to 1234, because of the dropped minus sign. ...


2

Three categories of error I often encounter are "Incorrect negation", "trouble with quantifiers" and "Confusing general parameters with specific objects". Everybody knows that students have trouble dealing with negating quantifiers in the context of, for example, an $\epsilon-\delta$ proof in an Analysis course. We also see ...


2

My friend, who is currently teaching an abstract algebra course, recently provided me with an example of an incorrect proof that I think showcases a few types of faulty reasoning that are common among students. I'm sharing it here with his permission. Proposition: Suppose $R$ is a ring and $I$ is a prime ideal in $R$. For all ideals $J$ and $K$ in $R$, if $...


2

Allow and promote using a calculator This is one of the recommendations listed for accommodating students with dyscalculia, but this could help whether or not a student has dyscalculia. It's not really coaching and it doesn't address the underlying source of errors, but at the very least it will keep those arithmetic errors from cropping up and getting in ...


2

What property of fractions or addition of fractions could they be misunderstanding, and how would you explain to the student where they have gone wrong so that they don't repeat this in the future? [Emphasis added.] Short Answer Fractions are numbers and they behave like numbers when we do operations on them. Many students never learn this. You (and the ...


1

As far as "just practice", you shouldn't trivialize this or think of it as so simple as yes, no. Your job is to train the trainee. Many errors are common and repeated. So? This is not so different from sports or music. Develop a set of drills, games, etc. Try different things with different people. The problem here is not so Boolean as math ...


1

I believe that the majority of students with this problem are doing what is called "end-gaining" in the Alexander Technique: https://www.hilaryking.net/glossary/end-gaining I'm not an expert, but I think that in this case, it means that they are overly focused on getting to the end of the exercise, and they have a mistaken belief that the means for ...


1

A common error I see when teaching function composition is students seeing it as multiplication. Many factors contribute to this, but examples with multiplication in them don't help: Q: Let f(x) = 2x and g(x) = x+1. What is f(g(x))? A: f(g(x)) = 2(x+1) Some students will see this and think that the answer somehow involves multiplying f(x) by g(x).


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