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5

I am going to try the following activity as a first introduction to Mathematical Induction on Monday next week. I will let you know how it goes. The implication $P(k) \implies P(k+1)$ let's you "hop around" the natural numbers, deciding the proof of new statements using your knowledge of the truth value of old statements. However, it is a bit too ...


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Re (1), you should lower your standards. As previously discussed, the population for Math55 in high school is tiny. (Add onto that the difficulty of finding them and it just makes little sense to try to develop this.) In addition, you really lack the math knowledge OR the practical pedagogical experience to develop Math55 for high school (even the time, ...


4

Here's another use-case that came up in my college remedial algebra class tonight (and again, this boils down to translations to mixed numbers): Finding fractions in a graph. So the specific example that presented itself tonight was a book exercise: "Graph the equation: $y = - \frac 5 3$". At that point, my students could tell me that this would be ...


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For (2), I'd suggest creating the lessons first as blog posts or youtube videos and then sharing them. I think the math reddit might be a good place, but I'm not active there. These can then in turn serve as advertisements for any active curricula you'd like to implement. For (3): Start with an overall structure of the course. Make a list of course-level ...


5

Do people hate math, or do people hate that they have to do math in school ? I suppose every so often we meet a true hater, someone who reviles math in every avenue. But, I think that is anomalous. In reality, most people hate math because it is harder than their other subjects by in large. Moreover, that is not a product of math teachers being "mean&...


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The poet Samuel Taylor Coleridge had an answer (namely, that imagination is underutilized). You can read his full critique of mathematics education here: https://allpoetry.com/A-Mathematical-Problem


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For me it is simply a difference between diving 1 unit vs. dividing a group(more than 1 unit). I teach very small kids and sometimes I explain them a difficult topic for example fractions and try to find out WHAT EXACTLY are they not understanding, why they don't understand. So I have found that difference of diving/splitting 1 vs dividing/splitting group is ...


2

The appearance of "mixed numbers" is inherent to the Euclidean algorithm. For example, $$\eqalign{\scriptsize{\frac{355}{113}}\rightarrow& 335=\color{red}{3}\cdot113+16,\ \scriptsize{\frac{355}{113}}=3\scriptsize{\frac{16}{113}}\\ \scriptsize{\frac{113}{16}}\rightarrow&113=\color{red}{7}\cdot16+1,\phantom{xx}\ \scriptsize{\frac{113}{16}}=7\...


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I think you are doing a good job, engaging with the kid and moving things along. There's a lot we don't know about the brain and about pedagogy, even for normal learners. Then for syndromes like ADHD, there's a lot unknown about what exactly that means. Surely not the same thing for each person so diagnosed. And then there are confounding variables to ...


2

Disclaimer: I'm not a psychologist and I'm not trained to deal with "complicated" students such as this. It's perfectly possible the method I used was counterproductive, I wouldn't know. However, we eventually were successful and the student managed to master the concepts and passed the exam comfortably. Method used I provided the student with a ...


3

I did some research, which you can follow here, that explains the why of the nomenclature we use. Basically, something is proper if it is contained in something else, and improper otherwise. For example, the proper divisors of 12 are 1, 2, 3, 4, and 6. This set excludes 12, because that represents the entire portion, e.g. no fraction occurs. Similarly, in ...


6

Interestingly enough, we do not have such a distinction in France. From the French Wikipedia article about fractions (emphasis mine): Dans l'enseignement français depuis la fin du xixe siècle, la fraction est définie comme le quotient de deux nombres entiers sans contrainte sur la taille du numérateur et du dénominateur (...) In French education, since the ...


6

The use of rulers with fractional inches is the first thing that springs to mind. Like this: The four keys have a width of $2 \frac{11}{16}$ inches at the tops of the key caps. If I calculated a length, and got $\frac{43}{16}$ inches, I'd have to convert it to $2 \frac{11}{16}$ to actually measure it. The "improper fraction" $\frac{43}{16}$ is ...


7

Student are introduced to fractions as part of a whole. They are then taught that improper fractions can be more than a whole - this is not ideal terminology or helpful for understanding. Improper fraction is a terrible name since it implies that there is something wrong with the fraction. Once student start to do calculations with fractions greater than one,...


8

I do not know of any relevant research. Here are my own not-research-informed ideas. Most people refer to fractions as parts of a whole. If someone says "I lost a fraction of a pound on my diet", you can be fairly certain that they didn't lose $\frac{23}{1}$ pounds. Since the common usage of the word and the mathematical usage differ, it is useful ...


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added Oct 6 The reason mixed numbers are found in US education is that mixed numbers are found outside of school in the US, so the children need to learn to understand them. Mixed numbers are found in road signs, cooking recipes, length measurements, and so on. (Denis Nardin commented that mixed numbers are never seen in Italy. Meters, centimeters, and ...


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