AOPS Pre-Algebra - Is my son writing too much text?

[Asker123]

My son 11 years and 4 months old is doing self paced AOPS Pre algebra from the book. Before that he did BA online level 3 to 5 ( we skipped level 2) in total 2 years.  From school perspective he completed level 5 and will be going to level 6 in Sep but you might be knowing school levels do not compare with his learning through beast academy and AOPS at all.

He has just completed chapter 3 of Pre-Algebra and he has not encountered any difficulty so far. All the Review Problems and Challenge Problems are also solved by him. I have to admit that I do not check his work. He self checks it from the solution manual. If the answer and the method of the solution manual is same as his own he gives himself a tick mark..

I somehow think that he is writing too much text and that makes his pace a bit slow for my liking ( I am comparing my own style in 80s and 90s and can be wrong. We did not used to write so much in Math until later in grade 11 or 12) . What do you guys think? Any feedback is welcome. I have attached some of his work for the last few Challenge Problems from chapter 3 - Number Theory  that he did last week.

Should I advise him to summarize his response rather than such elaboration? Or should I just leave him as is?

Thanks in advance....

 

 

Edited July 18, 2023 by Asker123

[daijobu]

Wow, that is a lot of writing.  But I'm reluctant to discourage any student from writing.  It's just such an unusual occurrence, I'm kind of in awe.  It's actually longer than the AoPS solutions.  

[maize]

I would be delighted if any of my kids were willing to elaborate on their responses like this! I would not discourage it if it is not bothering him. It's fantastic that he lays out his thinking so clearly and thoroughly.

 

Edited July 18, 2023 by maize

[daijobu]

Also, his solution is very similar to--but not quite the same as--the AoPS solution:

Solution: We’ll say that a student “touches” a locker if she either opens or shuts it. So, the first student touches every locker, the  student touches every  locker, the  student touches every  locker, and so on. Since the  student touches every  locker starting with locker 2, the  student touches every locker that is a multiple of 2. Similarly, the  student touches every locker that is a multiple of 3, the  student touches every locker that is a multiple of 4, and so on.

Therefore, locker number  is touched by every student whose number evenly divides . So, the number of times locker  gets touched equals the number of positive divisors of . If a locker is touched an even number of times total, then it is closed after the whole process, since each “opening” touch pairs with a “closing” touch. Similarly, if a locker is touched an odd number of times, the last touch will leave the locker open.

Therefore, the lockers that are open at the end are the ones whose numbers have an odd number of positive divisors. The only positive numbers with an odd number of positive divisors are the positive perfect squares, so we must count the number of positive perfect squares less than 1000. Since  is less than 1000 and  is greater than 1000, there are  positive perfect squares less than 1000.

 

I mean, even if he were copying the solution text, I would not necessarily discourage the practice.  I copy math equations by hand as a tool for my understanding.  But I might ask him about it.  Not in a negative accusing way, but just from a position of neutral curiosity.  

 

 

[Asker123]

No, he does not copy from the Solution Manual at all. I know that. He only opens the solution manual when he has completed his own solution and wants to verify.

Together we have an agreement that we will stick to the philosophy of AOPS that you learn by solving the problems (specially the hard ones)  yourself. Also that he has to read the solution manual completely even if he solved the problem himself.

I was only worried about the pace. Because he writes so much, he has taken full 2 months to complete 3 chapters of Pre algebra and filled 2 Legal pads completely.  I think not every chapter is long like chapter 3 so in that case, I will just let him work at his own style throughout summer. We believe he should have no issue till chapter 9 as all these topics are augmenting what is taught in BA . Chapter 10 is Angles so getting involved in Geometry will be good.

Thanks for your feedback so far. 

Edited July 18, 2023 by Asker123

[Ting Tang]

I really have no advice, but I just wanted to say this is amazing.  He might be moving at a slower pace, but I feel like this shows a love and appreciation for math.  

[daijobu]

My goal for mathy kids is to have algebra completed by 8th grade, because that gives students enough time to take calculus by senior year.  I would only intervene in another year or so if it looks like he might not make it.  He may pick up the pace as he matures.  

Those legal pads are going to be a fun keepsake for him!  

[Asker123]

2 hours ago, daijobu said:

My goal for mathy kids is to have algebra completed by 8th grade, because that gives students enough time to take calculus by senior year.  I would only intervene in another year or so if it looks like he might not make it.  He may pick up the pace as he matures.  

Those legal pads are going to be a fun keepsake for him!  

Introduction to Algebra AOPS book - is that correct? By 8th grade do you mean beginning of 8th grade or end of 8th grade? 

[daijobu]

One typical math sequence in the US by year is:

Algebra I

Geometry

Algebra II

PreCalculus

Calculus

Taking calculus in senior year, while somewhat arbitrary, is what naive application readers at highly rejective colleges look for in a STEM student.  And frankly, the strongest engineering students at state flagships are going to have already seen calculus in high school, so it's a good idea to keep up with them, though technically not required.   

If you are on this sequence, then you'll want to have completed algebra by the end of 8th grade so your student can take geometry as a freshman.   But if you check the older messages on WTM you'll find that AoPS Introduction to Algebra covers a typical algebra I class plus a lot more, including algebra II topics.  So if you think you'll be switching to another math curriculum in high school, you may not necessarily need to complete the entire AoPS Intro to Algebra text.  Check the archives or start a new thread if you need to drill down on this topic.  

 

[mathmarm]

Personally, I don't think it's too much written text. He's processing and explaining things as he's going and recording the thoughts and explanation well for his age.

So instead of asking myself if my son's writing too much, I'd ask if he's

• often including too much non-essential detail
• writing at a reasonable pace
• rambly or focused in this thought process
• writing logically sequenced solutions
• formatting his writing well for readability

One thing that I'd coach him on is improving the readability of his work and eventually teach him to use abbreviations/shorthand. Not that he needs to use them, but so that he is able to read them when reading others work.

 

[daijobu]

If your student is interested in stepping up his solution-writing skills, he might look into learning Latex and Overleaf.  In particular, critical equations should be on their own line and center justified.  More white space will make it more readable.  

But my sense is that he seems to be writing to himself, not for an audience.  There is just something about writing things out by hand, whether that's a solution or a graph or a diagram.  It slows the student down and causes then to really think about what they are doing.