Not Sure Should Continue with AOPS---Need High School Math Advice

[ResearchMama2]

2 hours ago, lewelma said:

Sorry OP, this is answering Researchmama's request for high school sequence and texts, and does not answer your OP.

Researchmama2: So it looks like you are interested in the actually progression and materials used for advanced math students. This is what my older boy did in highschool. With this background, he was able to walk into graduate level math classes as a freshman at MIT. He is graduating in May with a double major in Math and Physics and has been accepted into the #1 ranked graduate school in his subfield of physics.   

Formatting is a bit off, but I bolded the courses so you could see which resources went with which courses.

Things to note: 1) he did Introductory Algebra over 3 years. Back then the preA book was not printed, and he did every. single. problem. in the entire Intro Algebra AoPS book. My biggest advice to you is to make Algebra rock solid. Three FULL years for this very advanced boy and it was sooooo worth it. 2) He did number theory and combinatorics over many years because he was studying for the competitions -- I simply split into two half courses and gave him a full credit each. 3) He was *very* underwhelmed by the university courses he took, which is one of the reasons he decided to leave NZ and go to the USA for his education. He earned 98% with a course median of 64%, and he was 5 years their junior. We were told that this level of math is what we would find throughout NZ universities.  4) He did Calculus twice, once theoretically and once practically. 

HTH

Ruth

                                                                       9th           10th       11th      12th

Also Multivariate Calc in summer after 12th grade so he could test out.

Algebra 1. 4th-7th (self study with text)
Textbook: Introduction to Algebra (Art of Problem Solving), by Richard Rusczyk

Geometry. 8th (Self study with text)

Textbook: Introduction to Geometry (Art of Problem Solving), by Richard Rusczyk

Algebra 2. 8th (self study with text)
Textbook: Intermediate Algebra (Art of Problem Solving), by Richard Rusczyk

Precalculus. 9th (AoPS online class, 1 credit)
Textbook: Precalculus (Art of Problem Solving), by Richard Rusczyk and Mathew Crawford

Olympiad Geometry. 9th (AoPS online class, 0.5 credits)
Textbook: material provided in the course

Combinatorics and Probability. 9th and 11th grade (Blended class: AoPS online class and self-study, 1.0 credit)

Textbooks: Introduction to Counting and Probability (Art of Problem Solving), by David Patrick
Intermediate Counting and Probability (Art of Problem Solving), by David Patrick
Problem Solving Tactics, by Angelo Di Pasquale et al.
Selected readings were provided for advanced topics

Calculus 1 – 10th (AoPS online class - 1 credit)
Textbooks: Calculus (Art of Problem Solving), by David Patrick
 

Calculus 2. 10th grade (1 credit) (blended class, AoPS online class and self study using Anton which has a practical physics/engineering approach)

Textbooks: Calculus (Art of Problem Solving), by David Patrick
Calculus, by Howard Anton and Irl Bivens

Number Theory. 9th and 10th grade (Blended course: AoPS online class and self-study, 1 credit)
Textbooks: Introduction to Number Theory (Art of Problem Solving), by David Patrick

Problem Solving Tactics, by Angelo Di Pasquale et al.

Selected readings were provided for intermediate and advanced topics

Linear Algebra. 11th grade (local university)
Textbook: Elementary Linear Algebra: Applications Version, by Howard Anton

Foundations of Algebra, Analysis, and Topology.  11th grade (local university)
Textbook: Materials produced by professor

Real Analysis. 12th grade (self-study -1 credit)
Textbook: Principles of Mathematical Analysis, by Walter Rudin
Lecture Series: Real Analysis, MIT OpenCourseWare

Multivariate Calculus 12th grade (self study )

MIT opencourseware

Thank you, thank you, thank you!! This is so incredibly helpful!! 
 

I can’t believe you are from NZ! My husband and I visited years ago and we absolutely loved it! We plan to visit again when our children are older. Such a beautiful country with the most wonderful people! 
 

Thank you again! 

[8filltheheart]

@Researchmama1 @ResearchMama2  What Ruth's ds accomplished demonstrates what homeschooled kids with a passion for something are capable of achieving.  It wasn't the math curriculum he used that got him into MIT.  It was who he was as a person with an intense drive to learn. In addition to math, he is an accomplished violinist, did in-depth science projects, read the Economist for pleasure reading, etc.  His list of math classes isn't a checklist of things to do to succeed in elite admissions.  If you go back and read Ruth's threads over the yrs, you'll see that math and math competitions are the air her ds breathed. His path isn't something that is replicable.  Using AoPS for math will not result in Ruth's ds's path.  AoPS is in the periphery. His path was far more than a list on a transcript.

The greatest gift of homeschooling is being able to help our own children find for themselves what their passion is.  What is it that feeds their internal motivation to want to learn more?  For example, the student I posted about earlier who is in Harvard's MD-PhD program, in high school she "Investigated vertebral, liver, and splenic iron storage in pediatric patients with sickle cell disease and thalassemia through MATLAB analysis of MRI studies" under the supervision of an MD-PhD at a children's hospital.  Her passion for medical research is inspiring.  What is it that inspires them to become who they want to be?

Edited April 14, 2022 by 8filltheheart

[ResearchMama2]

On 4/12/2022 at 2:44 PM, daijobu said:

What a funny coincidence that you mentioned Princeton math specifically.  One of my favorite slides from Richard Rusczyk was from a talk he presented at Math Prize for Girls.  (One of my dd's attended for a few years before the pandemic.)  I've added RR's comments below.  It seems like Princeton is in the habit of admitted unprepared students.  

And why would anyone tell this person not to apply to medical school just because of one lousy math grade?  That's just so... weird.  Heck, I got a C in chemistry the first time around and I was admitted to UCSF.  Not exactly Ivy League, but still.  

2014 MPG video

 

 

Right after I started Art of Problem Solving I received an email from someone who attended Princeton who attended right around the time I did.  

 

I want you think for a minute what this student’s middle school and high school teachers thought when he went off to Princeton.  They thought, “We succeeded.  He went off to Princeton; we’re awesome.”  They never saw this.  I’m sure he didn’t go back to his middle school teachers and say, “Yeah what’s up?!?  You didn’t prepare me for this.”

 

So they didn’t get this feedback, and this happens a lot.  I saw this a lot at Princeton, this happens a lot now.  Kids go through school, some very good schools, they get perfect scores on everything, and then they come to place like MIT, a place like Princeton, they walk into that first year math class, and they see something they’ve never seen before: problems they don’t know how to solve.  And they completely freak out.  And that’s a bad time to have these first experiences.  Having to overcome initial failure. 

I just listened to this entire presentation and it was outstanding. I very highly recommend it to anyone else who might be interested. Thank you again for sharing!

[lewelma]

9 hours ago, 8filltheheart said:

@Researchmama1 @ResearchMama2  What Ruth's ds accomplished demonstrates what homeschooled kids with a passion for something are capable of achieving.  It wasn't the math curriculum he used that got him into MIT.  It was who he was as a person with an intense drive to learn. In addition to math, he is an accomplished violinist, did in-depth science projects, read the Economist for pleasure reading, etc.  His list of math classes isn't a checklist of things to do to succeed in elite admissions.  If you go back and read Ruth's threads over the yrs, you'll see that math and math competitions are the air her ds breathed. His path isn't something that is replicable.  Using AoPS for math will not result in Ruth's ds's path.  AoPS is in the periphery. His path was far more than a list on a transcript.

 

Thanks for adding this, 8. I completely agree. I know for a fact that his math did not get him into MIT. There were very few international students from the competitions that MIT accepted, and the vast majority (who did way better than he did) were rejected.  These types of schools are lottery schools, otherwise why would my son have gotten into MIT but rejected from Harvard, Princeton, and Stanford?  There is no logic. And to be clear, we never ever planned for my son to attend university in the USA, let alone an elite. His entire homeschooling career was about doing what he was passionate about and he never spared a thought to what might 'look good'. Not even one thought, because university entrance here in NZ is automatic if you just pass the national exams. So his goal was just to follow his passion for math and music and great literature. So hopefully Researchmama can use my course list and resources to see what can be done with a kid who is passionate and capable, but not see the list as a way to gain admission to an 'Ivy.'

[daijobu]

11 hours ago, ResearchMama2 said:

I just listened to this entire presentation and it was outstanding. I very highly recommend it to anyone else who might be interested. Thank you again for sharing!

I'm just glad you could read the slides.  I'm having trouble getting my images to appear on my WTM posts.  

[daijobu]

11 hours ago, ResearchMama2 said:

I just listened to this entire presentation and it was outstanding. I very highly recommend it to anyone else who might be interested. Thank you again for sharing!

Another favorite quote:  "They don't know Polish.  I don't know Polish.  And they will all get 5's [on AP calculus]."  

So true!  

[Flatlander]

In reply to the OP, my kiddo did AoPS PreAlgebra, then Ch 1-14 of AoPS Intro to Algebra.  Kiddo then tried out the LPS, where it turns out "Algebra II" is largely a repeat of material covered just in AoPS Chapters 1-14.  Kiddo is likely to return to homeschooling next year and, despite not being 'mathy', is looking forward to returning to AoPS because it explains why things work not just how to do it.

Last year when we were frustrated with our slowing pace through Intro to Algebra and with the LPS math placement test looming, I was grateful when someone pointed out the sequencing the the AoPS material.  In their own class sequence, they repeat Intro to Algebra chapters 10-13 in two courses.  Those chapters are really dense!

Introduction to Algebra A - Introduction to Algebra, Chapters 1-13

Introduction to Counting & Probability - all of Introduction to Counting & Probability

Introduction to Number Theory - all of Introduction to Number Theory

Introduction to Algebra B - Introduction to Algebra, Chapters 10-22

Introduction to Geometry, then Intermediate Algebra, etc

 

 

 

[Malam]

On 4/9/2022 at 8:20 AM, 8filltheheart said:

(My entire homeschool philosophy is about nurturing internal motivation and ownership over their learning and futures.)

How do you, as a parent who is external to your child, encourage internal motivation and ownership, especially from a young age?

[Malam]

On 4/13/2022 at 7:38 PM, lewelma said:

.Algebra 1. 4th-7th (self study with text)
Textbook: Introduction to Algebra (Art of Problem Solving), by Richard Rusczyk

What was his math background in grades prek - 3 like to prepare him for AoPS algebra in 4th grade?

[lewelma]

3 hours ago, Malam said:

What was his math background in grades prek - 3 like to prepare him for AoPS algebra in 4th grade?

My ds is very gifted in math so not a particularly good model for anyone. From age 4 to 6.5, we played shop or played card games. We estimated numbers and made up funny word problems for each other. I took a completely unschooled approach to math. But at the age of 6, he *invented* algebra, which obviously took me by surprise, the mother who was happily playing shop with my little boy and who had never shown him *written* mathematics. I had no idea at the time that he was special at math. He was just a boy who liked to ride his bike and play with numbers in his head. At the age of 6.5, I finally bought him his first workbook - Singapore math grade 3. And by the age of 7, he became completely self-taught in math, refusing all help from me or any other adult. He also refused to read any textbook explanations. So somehow, and I am not sure how, he invented all of elementary school math (fractions, decimals, ratios, etc.) by himself and then I guess he checked himself with the answers. On about his 9th birthday, he had finished Singapore Math and started independently learning AoPS Intro Algebra. That book slowed him waaaay down. It took him a full 3 years to do every single problem in the book, and to do the entire book without any help from anyone. It was just who he was and the book was written for a boy just like him, a boy who absolutely thrived on the discovery method. 

So to answer your question about his preK to 3 background, he was unschooled with a mathy mom until age 6.5 and then the rest of primary school math was self taught through his own personalized discovery method.  

Edited May 29, 2022 by lewelma

[8filltheheart]

11 hours ago, Malam said:

How do you, as a parent who is external to your child, encourage internal motivation and ownership, especially from a young age?

Not really the right thread for this topic, but in short, our homeschool does not look like a traditional classroom.  It is not a list of textbooks completed over the course of a yr.  I work with my kids to find topics that they want to learn more about and give them the opportunity to spend time doing exactly that.  Assignments are across curricula.  So if they are in 3rd grade wanting to learn about bees, science would be reading about bees/watching documentaries about bees.  Their writing assignments would be connected to what they are learning about bees.  We'd look at where else in nature you find hexagonal constructions like in beehives, maybe build some cells with Zometools.  These sort of things tend to take on a life of their own and when they are fascinated it could lead to looking at other designs in nature, etc.   (For example, I had a ds who at that age wanted to read everything he could find on bees and ants.  He did for months.  I didn't tell him he had to move on to a different subject and tell him we needed to check of a list of science topics.  He was fascinated, so I let him run with his fascination.  By 8th grade, his fascination shifted to physics.  During high school, he self-designed a dark matter/black hole study.  He had stacks of thought-experiment books.  His freshman yr of college he took 400 level electromagnetic wave theory.)  

I want to instill in them the desire to learn and have found letting them pick topics of interest and following natural trails from those interests leads to the highest levels of learning.  They often have questions that I don't know the answer to, so I simply respond, "I don't know.  Let's see what we can find out."  Turns out that is a life skill that I didn't realize at the time I was providing them.  When they have gone on to college, so many of their peers wait for someone to explain everything to them.  If those peers don't understand something in class, they struggle.  My kids take a different approach.  If they don't understand something, they go searching for explanations until they do.  It is what we have always done.  😉 

[Malam]

4 hours ago, lewelma said:

My ds is very gifted in math so not a particularly good model for anyone. From age 4 to 6.5, we played shop or played card games. We estimated numbers and made up funny word problems for each other. I took a completely unschooled approach to math. But at the age of 6, he *invented* algebra, which obviously took me by surprise, the mother who was happily playing shop with my little boy and who had never shown him *written* mathematics. I had no idea at the time that he was special at math. He was just a boy who liked to ride his bike and play with numbers in his head. At the age of 6.5, I finally bought him his first workbook - Singapore math grade 3. And by the age of 7, he became completely self-taught in math, refusing all help from me or any other adult. He also refused to read any textbook explanations. So somehow, and I am not sure how, he invented all of elementary school math (fractions, decimals, ratios, etc.) by himself and then I guess he checked himself with the answers. On about his 9th birthday, he had finished Singapore Math and started independently learning AoPS Intro Algebra. That book slowed him waaaay down. It took him a full 3 years to do every single problem in the book, and to do the entire book without any help from anyone. It was just who he was and the book was written for a boy just like him, a boy who absolutely thrived on the discovery method. 

So to answer your question about his preK to 3 background, he was unschooled with a mathy mom until age 6.5 and then the rest of primary school math was self taught through his own personalized discovery method.  

Awesome, thank you for explaining! By "card games" do you mean some kind of "educational" game like SET or more traditional ones like Go Fish?

If he didn't demonstrate any particular math talent, how long would you have continued the unschooling approach before adding official curriculum? 

[Malam]

1 hour ago, 8filltheheart said:

Not really the right thread for this topic, but in short, our homeschool does not look like a traditional classroom.  It is not a list of textbooks completed over the course of a yr.  I work with my kids to find topics that they want to learn more about and give them the opportunity to spend time doing exactly that.  Assignments are across curricula.  So if they are in 3rd grade wanting to learn about bees, science would be reading about bees/watching documentaries about bees.  Their writing assignments would be connected to what they are learning about bees.  We'd look at where else in nature you find hexagonal constructions like in beehives, maybe build some cells with Zometools.  These sort of things tend to take on a life of their own and when they are fascinated it could lead to looking at other designs in nature, etc.   (For example, I had a ds who at that age wanted to read everything he could find on bees and ants.  He did for months.  I didn't tell him he had to move on to a different subject and tell him we needed to check of a list of science topics.  He was fascinated, so I let him run with his fascination.  By 8th grade, his fascination shifted to physics.  During high school, he self-designed a dark matter/black hole study.  He had stacks of thought-experiment books.  His freshman yr of college he took 400 level electromagnetic wave theory.)  

I want to instill in them the desire to learn and have found letting them pick topics of interest and following natural trails from those interests leads to the highest levels of learning.  They often have questions that I don't know the answer to, so I simply respond, "I don't know.  Let's see what we can find out."  Turns out that is a life skill that I didn't realize at the time I was providing them.  When they have gone on to college, so many of their peers wait for someone to explain everything to them.  If those peers don't understand something in class, they struggle.  My kids take a different approach.  If they don't understand something, they go searching for explanations until they do.  It is what we have always done.  😉 

Any tips on finding age-appropriate curricula on the fly for whatever rabbit hole your kids fall down?

[lewelma]

10 hours ago, Malam said:

Awesome, thank you for explaining! By "card games" do you mean some kind of "educational" game like SET or more traditional ones like Go Fish?

If he didn't demonstrate any particular math talent, how long would you have continued the unschooling approach before adding official curriculum? 

Card games like addition or multiplication war. Pretty simple stuff. That was not really my focus, we just did them because they were fun. 

Being a mathy person, I could have unschooled him for a long time by creating a mathematically enrich environment. During k to 6th grade, we were doing quite a lot of science that required primary school math - like weighted averages, scientific notation, fractions, graphing etc. And I was getting ideas from books like Family Maths and Games for Math. So the key with my approach was lots of math just in life.

So some specific examples from ages 4  to 6.5 (beyond just card games, cooking, playing shop, etc):

When we saw a flock of birds, I would get my son to estimate the number and then count them to evaluate his estimate. I would show him how he could imagine them in 10 groups, count just the birds in one group and then multiply by 10.

When we would walk into town, we would take turns making up word problems for each other, so he had to not just answer mine, but had to create his own. And he always wanted to make them more and more complicated to see if he could trip me up since I had to do them in my head (but he had to also to check my answer).

When we would sit a the stop light, I would ask him to consider the rotations of the light, how much time each took, and suggest improvements based on what we knew about the traffic.

When we went to the beach, I would ask him to estimate the speed of the wave. Then figure out if there was a way to check his estimate. Would he need a watch and a ruler? Could he walk the speed to compare, or could he time the waves coming in? How actually, could he figure it out? What assumptions was he making? Where was the error in his estimates? 

We were doing a LOT of math, but without a textbook. The goal was engagement and ownership. And I think that this background was part of why when I finally bought him a program, that he had the confidence and desire to figure out stuff on his own and not use the math workbooks as written. He was used to using Math as a tool to solve questions in real life. He was not used to just doing drill in a math book where he just did whatever the book told him to do. So perhaps my early approach together with his innate talent is what directed him into the discovery method and self-learning path, both of which drove him forward all the way through university. 

Edited May 29, 2022 by lewelma

[ResearchMama2]

@MalamI sent you a DM.