how we're using AoPS with a young child (8yo)

[serendipitous journey]

We are far enough along with this that I think it is worth sharing, and I'd love to see how others are working this out. 

 

A. began AoPS Algebra during the fall, roughly 45" a day though not necessarily all at once.  We began algebra when he was failing to engage with pre-algebra (he was antsy and bored-seeming, though he had NOT mastered all the material) and Singapore's Discovering Math also failed to engage.  I provided a lot of active teaching support and focused (thanks to help from the board, esp. Ruth) on teaching him to write his work in an organized way and helping him learn to read the math and work toward self-teaching. 

 

We started on the chalkboard and now A. uses 1/2" graph paper for his work.  We pull out PreAlgebra for necessary review (only happened with fractions so far, but now we have a strategy!)

 

In chapter 4 or so, by which time he had a handle on linear eq'ns, I started him on the Intro. Counting/Probability and Intro. Number Theory books.  There were two goals here: 1) leavening the algebra using a structured program (A. does not do math voluntarily, though he requires it for his well-being, and so we do best with a structured program that I can use with a child who is being temperamental and difficult) and 2) spreading the Intro topic books out a bit -- I had gathered from Ruth's comments that this is helpful, particularly with a young learner, so that the child retains more. 

 

We didn't start Intro. Geometry because I was out of charter-school funds.  :)   I plan to start geometry by the fall. 

 

We are doing one section at a time (the teaching and the problems of a section) or a review.  Then we do another book.  Here's our cycle:

 

Intro. Number Theory

Algebra

Intro. Counting/Probability

Algebra

 

and so on.  I'm not sure how Geometry will fit in, since it might space some of our learning out further than ideal.  But so far this is going very well.   Esp. for a child who doesn't "like" math -- he gets interested and excited despite himself. 

 

ETA:  We have NOT been doing challenge problems, because he was too discouraged and overwhelmed and I didn't think the pedagogical payoff was worth the price.  Now that we're about to wrap up chapter 5 in Algebra, I think I'll have him do selected problems from chapter 2's Challenges.  And so on -- so that the challenge problems do not involve "new concepts" so much as give him practice in solving challenging problems.  Not sure what we'll do with the Number Theory/Probability; either revisit the challenges after we finish the books, or do them some chapters behind our learning, or I suppose we may skip them but I really don't like that plan! 

 

oh -- we do drills still, though not every day.  CalcuLadder or Math Sprints. 

[Gil]

Listening in! Please link me any posts that helped you figure this out, I'm desperate!

[serendipitous journey]

Listening in! Please link me any posts that helped you figure this out, I'm desperate!

 

oh, :grouphug: , it's not simple or easy. 

 

Here's my cry-for-help thread, which got me started ... I'll be offline most of the day and can't follow closely, I am sure others will post their helps too...

[EndOfOrdinary]

We use the PreA and then Counting Prob books with one chapter in one, then switch. Geometry starts in the fall. Ds is asking to skip Algebra for the time being due to how much more visual he is. I told him we could try it. He already had a handle on most linear equations, though not quadratics.

 

I have found a whiteboard ultimately the best for him to work at. We discuss the initial content and first four problems together. Walking around and really moving through the problems (he is kinesthetic and it does wonders). These are normally the leading problems which try to get you to develop a process. Most all discussion is Socratic, but I have found it helps him to have a sounding board so he does not jump too far and get lost when his brain starts connecting the dots.

 

He then writes down the steps to his process. These are attempted on the next problem. If it works, great! If it doesn't we step back and he tries again. This helps him work through the math in almost a scientific method way. Slowly, I am pulling myself out of this part. I hope by a month or so into next year he will have this down without me. Ds is not a risk taker so my presence is a feeling of "back up" for the time being.

 

Once he feels secure in his process he is on his own with the solving and reading till the exercises.

 

Exercises are done on day 2. He writes out his process and does the exercises at the whiteboard. I am easily accessible for check in if he is confused in exactly what the problem is asking. When he says he has done all the problems, I come back and pick three. He has to show and teach me how to do those three problems.

 

This has eliminated forcing him to write down every problem in a notebook. That was a major issue for him with the level of waste. I have informed him in years to come it might mean stacks of notebooks, but as he gets older the writing is getting better and more will fit on the page. It also speeds the process along a bit and does not create erasing frustration.

[wapiti]

Listening in! Please link me any posts that helped you figure this out, I'm desperate!

 

If you're up for some reading, here are some random threads that might provide bits of food for thought:

 

http://forums.welltrainedmind.com/topic/313660-those-using-aops-with-young-kidsmy-ramblings/

 

http://forums.welltrainedmind.com/topic/318251-very-early-aops-users/

 

http://forums.welltrainedmind.com/topic/422132-aops-perspectives-please/

 

http://forums.welltrainedmind.com/topic/503363-jacobs-or-aops-algebra/

 

http://forums.welltrainedmind.com/topic/381150-for-those-using-aops-intro-to-algebra/

 

http://forums.welltrainedmind.com/topic/444635-aops-alternative-usage-question/

 

http://forums.welltrainedmind.com/topic/339565-i-know-the-general-consensus-is-that-its-okay-to-start-aops-after-sm5what-about-mm/

 

If I correctly remember your specific situation, my guess is that you could seamlessly (well, except for the workbook aspect) move from finishing MM6 to Jacobs Algebra.  However, I also love what AoPS Prealgebra brings to the table, though whether that's a possibility for your specific situation depends in part on how you intend to use your resources and how much stretching your students are ready for.  If you're up for teaching socratic style, on a white board, the AoPS lesson problems set that up perfectly, IMO.

[serendipitous journey]

...

 

I have found a whiteboard ultimately the best for him to work at. We discuss the initial content and first four problems together. Walking around and really moving through the problems (he is kinesthetic and it does wonders). These are normally the leading problems which try to get you to develop a process. Most all discussion is Socratic, but I have found it helps him to have a sounding board so he does not jump too far and get lost when his brain starts connecting the dots.

 

...

 

I'm glad to see you mentioned this!  When things get challenging, we walk around too.  Sometimes toss a ball back and forth.  I understand this is an esp. powerful tool for helping boys to think (getting them moving was a key point in a book on boys' education I read a while back). 

 

Thank you for writing out your process.  I'm esp. eying the way you have your child teach a subset of problems back to you, that sounds really excellent from a teaching & a working-together point of view. 

 

 If you're up for teaching socratic style, on a white board, the AoPS lesson problems set that up perfectly, IMO.

 

oh I agree so strongly!  Taking it to the board, and thinking things through together, has been key here also. 

 

wapiti, I hope things are working out well at your DS' school. So glad that there are test scores to motivate the principal; and hoping the principal is being motivated in the most useful direction!!

[Runningmom80]

Can anyone link to resources that show how to teach "socratically?" I took a theory of lit class based on Socrates, but admittedly my eyes glazed over half the time and I BS'ed my way to a B.

[acurtis75]

I'm certainly not an expert but here is one link:

 

http://www.stanford.edu/dept/CTL/Newsletter/socratic_method.pdf

 

It is a rather technical explanation. Google had several other resources. This is probably a gross over-simplification but attempting to apply the principle with dd in math is "asking leading questions to make the child think through the problem logically and get the answer". The idea that the article references of a "guide on the side".

 

I haven't had any formal training but the basic goal I'm shooting for is to prompt her to think logically through the problem. I've been intending to read and study more about the method as we move in to literature and other more formal studies but for math I think it is a little more straight forward.

[sunnyday]

My understanding is that "Socratic discussion" is an actual thing with structure and form and stuff but it's done in a large group. But informally the phrase "Socratic questioning" usually means a humble questioner (Socrates was known for his stance that "I only know that I know nothing") who assumes that the answer to the questionee's problem is in him, so that in the questioner's great but understated wisdom he elicits a certain train of thought that leads the questionee to that answer deep within.

 

Here's a reference I've looked at before that gives you ideas of *types* of questions to ask: http://www.umich.edu/~elements/probsolv/strategy/cthinking.htm

 

In practice, in math, I think it would involve a lot of long pauses. Occasional "why do you think *this* information was included in the problem?" Or "Hm, do you think that the challenge will be organizing our work in order to be sure we've answered the correct question?" Or "Great answer, I think I understand what you are saying, but can you say it so it's true for all numbers by using properties of those numbers?" But always with a genuine sense of humility. As MCT says in his Classics book, you don't want to be "fishing" for one right answer. You want to be honoring the thought process that the child in front of you is using.

[EndOfOrdinary]

Socratic discussions are formal, with specific types of questions, done with a large group. My husband uses them in his alternative school. The level of planning can be a bit mammoth for these. At least, it was for him the first year or two he ran them. So if you are finding something that seems overwhelming, that might be the glitch.

 

Teaching Socratically or Socratic Method would probably be better search terms to avoid the issue.

 

Secondly, it takes some time for the kids to get used to. The first couple discussions my husband's tenth graders are just learning how to have a conversation where someone does not give them the answers and they passively listen. If it seems stilted and taking for.ev.er. That might just be your student adjusting to you handing the ball over rather than giving her the way to run. Do not give up on it initially.

 

My favorite phrases are, "Can you show me?" Giving him the marker to write.

"I think I gotcha. Tell me more."

"How's that workin' for you?" (Normally used when things are disintegrating rapidly into disaster)

"What do you think is happening here?"/ "Why did he do this?"

[serendipitous journey]

Just a quick thought (DH traveling, very little time here!) RE Socratic teaching. 

 

Okay, a few quick thoughts.  :)

 

1.  Socrates, in the dialogues, was definitely leading his audience toward a conclusion he had in mind.  Many uses of "Socratic" methods are of this sort -- you know where you want the child to get to and you guide them there.  There plenty of occasions to use this with children while working maths.  This is pretty doable, not like those crazy Socratic Discussions referenced above that require hours of prep. and, ideally, professional training (crazy for the homeschooling mama, that is; totally reasonable for the professional teacher with a class). 

 

2.  A variation is asking genuinely open-ended questions in order to explore a topic, without a predetermined conclusion.  The goal/purpose is, more or less, generating a variety of perspectives and probing them for consistency, insight, utility. 

 

It is this second method that has been the most powerful change in how I teach A. this year.  A good way to do this with math is to approach a problem (of sufficient complexity and interest, but one that is more-or-less within reach of the child) as an opportunity in problem-solving skills and cultivate the genuine belief that your child can have surprising, accurate insights into problem-solving.  When I am taking this approach deliberately (it is time-consuming and not always our modus operandi, just one formal instruction method) we:

 

First read the problem over generally, then re-read and make sure we agree on what it is saying.  I explain ambiguous or unfamiliar references (like what officers in a school club are).  

 

Then we each, separately (in our minds, not in separate rooms :) ), come up with a way to begin solving the problem.  Sometimes we might have a goal of each developing multiple strategies, depending on the problem.   The strategies are not limited to the branch of math being studied.  One could use geometrical strategies, algebraic, diagram-driven, whatever. 

 

Next we explain and compare strategies. 

 

Usually at this point A. picks one and proceeds.  In a very rare case, for some particular purpose, I might work out the problem with an alternative method and we can compare processes and results.   In PreAlgebra sometimes when A. chose a brute-force method I would work 1/2 the calculations for him. 

 

Repeat for each necessary step.  Or, develop an overall strategy and carry through for the whole problem. 

 

Related to this: we discuss, after doing problems that seem pointless, why the author assigned them.  We look over the section title &c and try to figure out the point being made.  I try to make sure A. knows why the math we are doing is relevant to anything, what folks find interesting about it, &c.  It helps to be using a good math program here; it's very easy with AoPS.

 

The general idea being to focus on the problem-solving we are learning, not getting to the end of the problem being solved per se.   I picked much of this up from the Stanford CourseEra class on teaching math offered last summer (from the part I actually completed, that is; the course frankly drove me bananas in many ways but this little bit was totally worth it).  I really like this approach because it keeps A. and I engaged together, and he feels like we are working together as opposed to me just shooting stuff at him.  He also has had some lovely insights!!! 

 

and, a bit of humor for when it's all too serious.  From Isaac Asimov's essay The Relativity of Wrong (which may be worth reading in its entirety in the context of this thread ...)

 

 

"First, let me dispose of Socrates because I am sick and tired of this pretense that knowing you know nothing is a mark of wisdom.

 

No one knows nothing. In a matter of days, babies learn to recognize their mothers.

 

Socrates would agree, of course, and explain that knowledge of trivia is not what he means. He means that in the great abstractions over which human beings debate, one should start without preconceived, unexamined notions, and that he alone knew this. (What an enormously arrogant claim!)

 

In his discussions of such matters as "What is justice?" or "What is virtue?" he took the attitude that he knew nothing and had to be instructed by others. (This is called "Socratic irony," for Socrates knew very well that he knew a great deal more than the poor souls he was picking on.) By pretending ignorance, Socrates lured others into propounding their views on such abstractions. Socrates then, by a series of ignorant-sounding questions, forced the others into such a mélange of self-contradictions that they would finally break down and admit they didn't know what they were talking about.

 

It is the mark of the marvelous toleration of the Athenians that they let this continue for decades and that it wasn't till Socrates turned seventy that they broke down and forced him to drink poison."

 

[Runningmom80]

 

 

and, a bit of humor for when it's all too serious.  From Isaac Asimov's essay The Relativity of Wrong (which may be worth reading in its entirety in the context of this thread ...)

 

 

"First, let me dispose of Socrates because I am sick and tired of this pretense that knowing you know nothing is a mark of wisdom.

 

No one knows nothing. In a matter of days, babies learn to recognize their mothers.

 

Socrates would agree, of course, and explain that knowledge of trivia is not what he means. He means that in the great abstractions over which human beings debate, one should start without preconceived, unexamined notions, and that he alone knew this. (What an enormously arrogant claim!)

 

In his discussions of such matters as "What is justice?" or "What is virtue?" he took the attitude that he knew nothing and had to be instructed by others. (This is called "Socratic irony," for Socrates knew very well that he knew a great deal more than the poor souls he was picking on.) By pretending ignorance, Socrates lured others into propounding their views on such abstractions. Socrates then, by a series of ignorant-sounding questions, forced the others into such a mélange of self-contradictions that they would finally break down and admit they didn't know what they were talking about.

 

It is the mark of the marvelous toleration of the Athenians that they let this continue for decades and that it wasn't till Socrates turned seventy that they broke down and forced him to drink poison."

 

 

HA!  I wonder how long it will take before my kids are slipping something in my drink.  Maybe I'll go all Princess Bride on them and start developing my tolerence now.

[La Texican]

It is the mark of the marvelous toleration of the Athenians that they let this continue for decades and that it wasn't till Socrates turned seventy that they broke down and forced him to drink poison.

 

 

 

 

 

 

ROFLOL

[serendipitous journey]

ROFLOL

 

me too!!!  I read the whole essay during those Dark Homeschooling Moments. 

[anmom]

I know I'm butting in here, but I have been trying to plan my ds next step in math. We are using MM right now. I'm trying to decide if level 6 is needed. I am also considering maybe jousting armadillos then maybe aops pre-algebra, just to get him ready for algebra and stretch it out because he is young. What do you all think?

[wapiti]

I know I'm butting in here, but I have been trying to plan my ds next step in math. We are using MM right now. I'm trying to decide if level 6 is needed. I am also considering maybe jousting armadillos then maybe aops pre-algebra, just to get him ready for algebra and stretch it out because he is young. What do you all think?

 

You might want to start a new thread.  How young is he?

[anmom]

He is 8 now and will finish MM 5 within the next 6 months or so. I can start a new thread!

[serendipitous journey]

He is 8 now and will finish MM 5 within the next 6 months or so. I can start a new thread!

 

good idea -- you will get many more responses!  I am happy to have this conversation here, too, but many folks won't see it. 

 

I myself haven't any good advice -- we haven't used MM. 

[serendipitous journey]

I just realized there wasn't anything on this thread about the challenge sections.  I've edited the OP to address this as follows: 

 

We have NOT been doing challenge problems, because he was too discouraged and overwhelmed and I didn't think the pedagogical payoff was worth the price.  Now that we're about to wrap up chapter 5 in Algebra, I think I'll have him do selected problems from chapter 2's Challenges.  And so on -- so that the challenge problems do not involve "new concepts" so much as give him practice in solving challenging problems.  Not sure what we'll do with the Number Theory/Probability; either revisit the challenges after we finish the books, or do them some chapters behind our learning, or I suppose we may skip them but I really don't like that plan! 

 

ETA -- see also this post by kohlby -- the child picks 6-8 challenge problems to solve, which for this student resulted in better focus.  The child sounds like A., needing challenge + a lot of play time, so I might give this a go. 

 

Also, we don't do all the starred problems.