# How actually are prime numbers taught in elementary school in United States and how easily do students learn what they're being taught about them?

I read the question https://math.stackexchange.com/questions/1593091/how-to-explain-why-study-prime-numbers-to-5th-graders and according to the body of the question, some students sigh. Also according to https://psychology.stackexchange.com/questions/20867/why-cant-we-learn-what-we-dont-like, it's sometimes hard for people to learn stuff that's not interesting for them. I don't know which education system the person who asked about teaching prime numbers was talking about, so I decided I don't care to determine which one it is and will just ask about the education system in United States because the fact that some education system has that problem is a good sign that the American one might have that problem as well whether or not it's the same education system.

I think some schools just expect students to figure out what the teacher means by what they're saying and some students just cannot figure out what the teacher means and really struggle with their attempt to. I would love to get an answer by a teacher who is trying to teach prime numbers to elementary school students about what's happening with their attempt to teach them prime numbers. I would like them to tell me what teaching style they're using. I would also like to know what they're having trouble teaching the students in a way that they can understand. Are they trying to get the students to come up with an explanation all on their own of what the fundamental theorem of arithmetic is to teach them how to think for themselves?

• This question seems too broad to be given a clear answer. Could you try to narrow down what, specifically, you want to know? Mar 2 '20 at 23:28
• I have had success teaching prime numbers but feel your question is very broad and I am not sure what you specifically are asking If your question is what is my trouble - I don't have one. If your question is -Are they trying to get the students to come up with an explanation all on their own of what the fundamental theorem of arithmetic is to teach them how to think for themselves? I can say no. Perhaps you can edit your question to make it clearer. Mar 3 '20 at 7:47
• @AmyB I think it's really hard to figure out how to make my question really clear. Maybe you could give a full detailed explanation on how you teach and how your students are discussing the topic to gain an understanding. Now that I realize it's unclear and I got an unsatisfactory answer, it came naturally for me to figure out how you could adapt to my lack of clarity, then then I realized that in the process, I figured out what my question might actually be. I don't know for sure. Maybe the answer will still be unclear and gradually I'll learn from experience what ways of asking get what Mar 4 '20 at 2:28
• types of answer. I can't promise that but maybe if I figure out what I want to ask, I'll ask Robbie_P_Math for permission to edit my question because it might invalidate their answer. Mar 4 '20 at 2:30
• @Timothy Editing your question for clarity won't invalidate an answer - please edit and I will be happy to answer. Mar 4 '20 at 3:52

I would love to get an answer by a teacher who is trying to teach prime numbers to elementary school students about what's happening with their attempt to teach them prime numbers. I would like them to tell me what teaching style they're using.

This may not be exactly what you're looking for but just the other day I introduced prime numbers to my first grade (7 year old) daughter. The setting was more relaxed and comfortable for her than a classroom but I don't see why the same technique wouldn't work in class.

What we did was to see how many rectangles we could make with a given amount of square Legos. I wrote the numbers 1 through 13 or so vertically and then we wrote the number of rectangles and the dimensions of the rectangles next to them. If we could only make one rectangle, eg $$1\times 7$$, we circled the number. I wanted to go up to at least 12 so she would see a case where three rectangles were possible. Now, the number 1 ended up being circled and I quickly mentioned that because 1 is so special we don't usually bother calling it a prime, but that's just a technicality that I don't think is very important now (or ever for that matter). With older students it's worth discussing though.

edit: This year she's using the Investigations 2017 Student Activity Book for Grade 2 and there's an activity called "Building Rectangles" (pp. 109-111). It's exactly what I described above (even ending on the number 12) except it doesn't make any mention of primes. Nevertheless, I think it's clear that that's what it's leading up to, perhaps in a later grade, so I'm pretty confident that primes are introduced this way in at least some schools.

As for the second part of your question, I'm sorry to say that I still can't be of much help there!

• It was worth an upvote but it still doesn't solve my problem. I believe there is a way to answer the question in such a way that it would solve my problem. It's just very hard to do. Maybe if a teacher wrote a long detailed answer on how they teach and students discuss the topic with them or wrote the specific thing they're trying to teach that the students are struggling with and how they're presenting it to the students, it may give me the answer I'm looking for and I might put a check mark beside it. Mar 6 '20 at 6:56
• I think that if you want, it might be fine for you to ask a question on the Meta site of this question about getting an answer to this question from a teacher. See meta.stackexchange.com/questions/343265/…. Mar 25 '20 at 22:58
• This doesn't really quite answer my question. The question was how they're actually getting taught in school, not how you think they should be taught. I know it's useful to say how they should be taught but schools aren't going to change it just based on this answer. Just because it's so obvious that they should teach it a certain way doesn't mean they are doing it. See matheducators.stackexchange.com/questions/13150/…. That looks like a very confusing set of rules. On the other hand, you could just simply demonstrate the example in class where Sep 16 '20 at 4:35
• you ask how to add $\frac{3}{4}$ and $\frac{5}{6}$. Then feed the idea of converting them into $\frac{9}{12}$ and $\frac{10}{12}$. When you don't accept what they cannot do of showing their work in that way, they learn less. Sep 16 '20 at 4:38

I relate prime numbers to prime factorizations of big numbers and usually ask the students, "which is easier to work with? Large numbers all at once, or small numbers one at a time?"

If it's elementary school, I'd think of arithmetic of fractions.

• Do operations with highly composite numbers in two ways: one without prime factorization, and one with. I think elementary school is mostly computation and so being able to do that in much more efficient ways will probably look very impressive and meaningful.

Another thing I might do is talk about the Fundamental Theorem of Arithmetic (FTAr) and why $$1$$ isn't considered a prime due to how it would mean different prime factorizations of the same number (or how you'd never be finished a factor tree).

And, to make the students feel they're learning something that has a lot of meaning, you could show them a section from a university-level algebra book that has the FTAr and say they're doing things that even the "big kids" and adults use.

I'd suppose that the biggest difficulty would be having students adopt Prime factorization (and cancellation) and not resort back to laborious multiplication and divisions.

• I still don't quite understand exactly what you're doing. Also, sometimes when a kid is taught something, it comes naturally for them to explain the reason what they were taught makes sense. If some of your students are doing that, I would love to know what their explanations are. Maybe if your explanation to them is clear and understandble enough. I'm not a teacher and don't know how kids learn but if I was a helper who could teach them the way I like, then for the students who aren't very enthusiastic about prime numbers, I might just state what a prime number is. Surely, that's Mar 3 '20 at 3:28
• understandable. Mar 3 '20 at 3:28
• I think kids shouldn't be asked what the prime decomposition of a number is. It might not be obvious to them that it's unique up to changing brackets and orders of terms. Also, to add to the confusion, they'd be getting taught that 1 is the empty product and question and ask what is the empty product. A lot of students are struggling in school. According to huffpost.com/entry/teaching-discovery-learning_b_856463, teaching is not learning. The education system won't take no for an answer and teach less material in more depth like they do in Finland. I think they shouldn't be taught Mar 25 '20 at 22:47
• to multiply fractions by prime decomposition. Giving students so much material because they might need some of that material later doesn't work. The material should be moved to the job where they need it. Also, if multiplication is defined by prime decomposition, then we can say that by definition, 14 × 15 = (2 × 7) × (3 × 5). Now what is 2 × 7? By definition, it is equal to 2 × 7. Now it's not defined at all. Mar 25 '20 at 22:54