UHP
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Posts posted by UHP
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On 7/28/2023 at 1:01 AM, Gil said:
As we look back, we noticed that the things that we made a concerted effort to know throughout their K-8 years are still known pretty darn well relative to subjects in which we made no concerted effort to memorize a body of information. What's more, is that they know more about those memorized topics/areas relative to a topic in which they learned about things only at a surface level.
18 hours ago, caffeineandbooks said:I agree with your observation, @Gil: the subjects where we took the time to "hang memory pegs", to borrow a WTM phrase, are the ones where my kids seem to have the most capacity to hang new knowledge.
Would you two share some examples of things your kids have memorized?
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@mathmarm I think you're familiar with some of Saxon's books, do you have an opinion about his calculus book? Until now I assumed I would be using it when the time came.
21 hours ago, mathmarm said:I'll be teaching from and assigning problems from the book, its not for the student to use to study independently,
I am about halfway through Saxon Algebra 1 with my 8-year-old, and I'm really impressed. At 8 she could never use it independently, but the skills are broken down and sequenced very thoughtfully, and it's easy enough to adapt the text to worksheets that can hold her attention and that she can understand. The problem sets give the right amount of practice. I've been hoping the sequels are just as good.
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6 hours ago, Mom_to3 said:
He is finishing up AoPS Calc this summer (we postponed/delayed Calculus for as long as possible, as we didn't want to scare off private high schools).
I haven't heard the concern in parentheses before. Do I understand, some private schools see a liability in applicants who know calculus?
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39 minutes ago, Not_a_Number said:
I'm a firm believer in the discovery method and really only teach using it, but I don't like the way AoPS does it.
Over the years you've told something here about your approach. What's different about the way AOPS does it?
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1 hour ago, Cake and Pi said:
I'll say that Saxon is absolutely fantastic for teaching math computations, but only meh for problem solving and application. I taught Saxon Algebra 1 to high schoolers at a small charter school one year, and the problems it were significantly easier than what I'd seen in AoPS Prealgebra. I'd encourage you to look into AoPS (maybe even back up and do some Beast Academy) and EMF. Those would be my top choices for a very accelerated younger math student.
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Oh! I like Direct Instruction! A couple of my kids learned to read with 100 EZ Lessons, and I used CMC with my developmentally disabled DS#4 for a year before we moved him back into public school. Kiddo did better with CMC than any of the other curricula we tried (Right Start, Math U See, Ronit Bird, local public school math). He never finished the kindergarten level, though. I'm out of littles to teach, but I'm very curious about the upper levels. However, if it's similar to Saxon, I guess I'd worry that it would be too straightforward and easy (as in, not enough true problem solving or out-of-the-box applications) for a very math gifted/accelerated kid.
My first exposure to Saxon was one of the grade-school books, maybe "Saxon 2" meant for second graders, years ago when I was still deciding whether to homeschool. I was not impressed at all, and more recently was surprised when a DI professional told me (when I asked what to do after CMC) to check out Saxon Algebra 1. The explanation or accusation I heard is that the grade school program was written after John Saxon had died, possibly for kind of cynical reasons. Similarly I was warned that after Saxon died the publisher "ruined" the Algebra program with their 4th edition. I don't know how accurate this is, I took the advice without checking it out for myself.
I can imagine the DI professional and CMC fan I talked to scowling at the idea that the later levels of CMC are comparable to the grade school texts published by Saxon the company (which he did not respect) but not written by Saxon the author (whom he respected a lot).
But I can see in the Saxon problems something in common with the CMC problems. They are like you say easy, *in a way*. In Saxon, every problem on every problem set (anyway, so far) is of the same type taught or at least modeled earlier in the book. If it is the first problem of its kind, it was introduced in the very same lesson. If it was introduced many lessons ago, there is a little parenthetical annotation that reminds you exactly where. CMC is structured pretty differently in the details, but it too has a tremendous amount of scaffolding (often really ingenious scaffolding) and the students are never presented with a problem that the problem's author thought they would have trouble solving.
This kind of handholding approach has lots of pitfalls, if the course designer had a superficial grasp of either the material or of the learning process. Maybe the kids will be taught how to add and multiply fractions, without ever realizing that fractions can be compared to whole numbers and indeed that some fractions are equal to whole numbers. Maybe they will be taught some procedure for column addition and subtraction, but get easily confused about which operation is called for in simple word problems. I grew to trust CMC a lot for having anticipated and guarded against these kinds of pitfalls, partly from experience, partly from reading some of the theory, and partly from reading the memoirs of the program's creator.
I don't have the same trust in John Saxon, who doesn't lay out his process or his thinking in very great detail anywhere that I have found. But 37 lessons in, the program is growing on me.
When the pitfalls are addresssed, there are virtues in "this kind of handholding approach." For me, the big one is that when a kid gets one of these easy problems *wrong*, it is because she has misunderstood something *small.* You, the tutor, are very well situated to remedy the misunderstanding.
AOPS advertises that its kids are constantly being challenged with tricky problems. What leaves me nonplussed about this, is what the tutor can learn from the pupil's mistakes. If she gets a tricky problem wrong or gets stuck, it's not because she is misunderstanding or just missing one thing. It could be many diffuse things, and what is the right course of action to take as the teacher? The answer I've heard, "Well, she just wasn't ready for that problem. We don't expect kids to be able to solve all or even most of the problems in our book," doesn't satisfy me.
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38 minutes ago, Malam said:
Which levels of CMC and Saxon did you go through? Wouldn't going through the daily lessons take one year per year? How did you go through k-6 by 2nd grade?
Here's the story so far:
When she was 5.5, I started the Give Your Child a Superior Mind curriculum, more or less the math that Engelmann and Bereiter were teaching at their Urbana preschool in 1964-1966. "Superior Mind" is hard to find, but I discovered much later that essentially the same thing is outlined in Chapter 10 of this book, which someone has scanned and put online. We finished right around the time she turned 6.
At the time, I thought that "Superior Mind" and "Teach Your Child to Read in 100 Easy Lessons" were the only Engelmann products that could be used at home. For nine months after finishing it, I improvised math lessons. At first, largely cribbing from Beast Academy, but I grew to like that less and less over those months.
In month 6 or 7 of that period, I found this blog, where a UK teacher went into some detail about how CMC works. I was able to adapt the fraction lessons she outlines there, and was really impressed. I got optimistic that the whole program would work at home after all. I also learned that @mathmarm was using the Engelmann classroom products at home, very successfully. I took her advice and searched for the programs on ebay. There is something tricky about this: each program has several components and it is hard to find information online about which components are necessary.
At the beginning of 1st grade, we started CMC Level E (which might be 4th grade math, if Level A is kindergarten). This wasn't a very thoughtful placement: if I did it again I would start further back, maybe way further back. But Level E was the first program I found all pieces of. I tried out Lesson 1 as an experiment and it seemed to work well, so we just kept going. We went through one lesson every day for about 50 days, then (as the lessons got more time consuming) slowed down to one lesson every two days for the rest of the year. Altogether the program took us 200 days, we finished a little less than a year ago.
She learned all kinds of really sophisticated stuff from Level E, but after finishing it her "math facts" were still very weak. She was still skip-counting to find 6x4 and so on, did nothing from memory. It slowed her down a lot. Because of this, and also partly as a nod to "summer vacation," we didn't start Level F right away but spent the summer doing "Correct Math Multiplication" and "Corrective Math Division," two other Engelmann products that are supposed to be remedial. For both programs, we started with Lesson 27, not Lesson 1, finished in September of 2022. Then Level F took us until the middle of February.
1 hour ago, wendyroo said:I also like the Math circle lessons and problems from the University of Waterloo as a way to go broad.
New to me and looks very interesting!
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2 hours ago, Malam said:
Which daily lessons are you using?
For 1.5 years, I used a program called Connecting Math Concepts, which really did have daily scripted lessons. We finished them in February and started Saxon, which is less transparent and requires a lot more daily preparation for me.
Saxon was recommended to me by another fan of CMC, in fact very highly recommended. But it came with a warning to only use the third edition.
49 minutes ago, Not_a_Number said:I’d go broad if I were you.
What does it mean, go broad?
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3 hours ago, Cake and Pi said:
He's 5th grade by age and just took his calc 2 final at the university today.
Could you tell something about what he was doing in 3rd and 4th grade? I'm very curious.
My 2nd grader isn't crazy about math, doesn't study it or play with it on her own outside of our daily lessons. But by now those lessons have brought us through the usual K-6 curriculum and 30% of the way through Saxon's Algebra 1. It's conceivable we'll be finished with Algebra 1 at the end of the summer. I've been having a hard time picturing the next few years.
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Wonderful question, unpleasant to contemplate.
I also have an 8-year-old. One thought I've had so far (not with high confidence), is that nine tenths of the opportunities that could open up for someone who learns to speak French or Chinese as a kid, might in 10 years be just as open to any one-tongued person.
I think (or only hope) that the three Rs (to a high human standard: reading a lot, writing well, lots of math) will pay off even as computers work more and more miracles in those areas. They will give our beautiful human kids a little bit of good judgement to face whatever bewildering things are coming.
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The first chapter (or more) of "the 8th day of creation", by Judson, might be good to read aloud for a 6th grader. It might be a little grown-up (I don't mean R-rated), it's adapted from some old New Yorker articles. But it gives a lot of interesting context to the "molecular revolution" and it's not too technical.
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3 minutes ago, mathmarm said:
I was told that it would cost several hundreds because they'd have to stop production, set the machine specifically to label land features only but no countries or borders and then run off a small batch of globes (can't make just one) and then reset the machine to resume normal production.
Wow. You could ask if they give tours!
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There's a famous recording of Tennyson reading "Charge of the Light Brigade."
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Does anyone know a place to buy a globe that has no country names or borders drawn on it?
I would like one for discussing ancient history.
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Sometimes doing statistics, the question of interest about the mode is not "what is the mode?" It's "how many modes are there?" For instance if your bar graph has two different highest bars, and those bars are far apart from each other, it might be beacuse there are two interesting populations in the pool of samples that your data came from, with different averages.
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Eager to hear answers, it's a great question.
My sense is that median and mode are introduced later than mean. But, the procedures for computing medians and modes are easier than the procedure for computing means. (Well, for homework-sized sets of data). I'd like to know what conclusions a kid can be taught to make after those computations. ("Since the median is so much larger than the mean, ___" Fill in the blank, in lots of examples.)
Engelmann's Connecting Math Concepts introduces the mean in Level E and reviews it in Level F. It does not introduce median or mode. The explanation provided for the mean (simply called "average" in the student materials) is something like this:
"The average rain for the week tells the amount of rain there would be each day, if every day that week it rained the same amount."
"The average exam score for the class tells the score each kid would have got, if they had all got the same score."
Some of this language is accompanied by diagrams that show a pair of bar graphs, one with bars of varying heights and one with bars all of the same height. I think it's a fair explanation, as long as your kid doesn't get too distracted by the metaphysics of it ("what do you mean, 'if it had rained the same amount each day?' What if it hadn't rained at all that week?")
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1 hour ago, Kela said:
She can't seem to tell the difference between 12 and 21, 13 or 30.
Does she also have this problem with say 64 and 46, or is it always a teens number that trips her up?
For early math learners, there are four categories of two-digit numbers:
Category 1: "Easy -ty" numbers. They include the numbers in the 40s, 60s, 70s, 80s, and 90s that do not end in a zero. Kids catch on right away: to read something like "64", simply say the first digit, then say "ty", then say the second digit.
Category 2: "Hard -ty" numbers. These are numbers that do not end in a zero in the 20s, 30s, and 50s. Kids have to learn that when reading such a number we start by saying "fif", "thir", and "twen" instead of "five", "three", and "two"
Category 3: "Decade" numbers. These are the numbers 20, 30, 40, 50, 60, 70, 80, and 90. Kids have to learn that they don't say e.g. "seventy-zero."
Category 4: Teen numbers, the numbers 10-19. This is a minefield for little kids.
I feel like the most effective and fastest procedure for teaching kids to read two-digit numbers, would be to teach category 1 to mastery, then category 2 to mastery, then category 3 to mastery, then category 4 to mastery. But this isn't always expedient because usually kids are using the teens numbers to count before they are reading and writing numerals.
Here are two exercises for introducing category 4, that I learned from "Connecting Math Concepts":
Quotea. Write on a sheet of paper, in pencil: 7 4 9 6
Have the kid read them: point at 7 and say "what number?", point at 4 and say "what number?" ...
b. Say "I'm going to change these numbers into teens numbers.
Point to 7 again, and say "What's this number?" After kid answers "seven," change it into a 17:
17 4 9 6
Say "Now it's not 7; its 17."
c. Erase the 1 or cover it up to show
7 4 9 6
Say "What is it now?" When the kid answers "seven", change it into a 17 again:
17 4 9 6
d. Say "What is it now?" and wait for the kid to answer "seventeen." If she doesn't answer 17, say "No, now it's seventeen" and go back to step c.
e. Point to the 4. "What's this number?" After the kid says 4, change it into a 14:
17 14 9 6
Say "What number is it now?" Hopefully the kid answers "fourteen".
f. Repeat step e with the 9 and the 6.
g. Now your paper says
17 14 19 16
Say "My turn to read these numbers." Touch each number and say "Seventeen, fourteen, nineteen, sixteen." Then say "Your turn to read these numbers" and see if the kid can do it.
After this, or the next day, give the kid a simple worksheet with 1-digit numbers on them:
_6 _7 _8 _9
and tell the kid to turn them into teen numberes. She should write a "1" in each blank space.
Another one:
QuoteWrite on a sheet of paper: 9 6 8 5 4
Have the kid read them: point at the 9 and say "what number?", point at the 6 and say "what number?" ...
Say "I'm going to change these numbers into teen numbers."
Point to 9. "What teen number can I change 9 into?" She should answer "nineteen." Point to 6. "What teen number can I change 6 into?" She should answer "sixteen." Repeat for 8, 5, and 4.
Point to 9 again. "What do I write in from of the 9 to make it 19?" Kid should answer "one." Then say "Tell me if I write one the right way." and change the 9 to 91:
91 6 8 5 4
Ask "is that the right way?" Kid should answer no. Say "The 1 is not in front of the 9."
Change it to show nineteen:
19 6 8 5 4
"Is that the right way?" Kid should answer yes. "What number did I write?" Kid should answer "nineteen."
Repeat with 6. Write 61 first, ask the kid if you wrote it the right way, then change it to 16, ask "What number did I write?"
For 8, do not change 8 to 81. Start directly with 18. "Did I write it the right way?" If kid answers "no" tell her she's wrong. You did write it the right way, the one is in front of the 8. "What number did I write?" Kid should answer eightteen.
For 5, again do not change 5 to 51.
For 4, go back to writing it the wrong way first.
By now your paper should have
19 16 18 15 14
written on it. Close the exercise by having the kid read "all these teen numbers."
If that's easy for her, try introducing 13 the next day, 11 and 12 the day after, and 10 the day after.
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1 hour ago, Not_a_Number said:
I cover the idea of an equals sign as meaning that two things are the same considerably before algebra, so I don't think of that as an essential algebra concept. I can believe it can be, though.
I can't resist the opportunity to write that Robert Recorde invented the equals sign in 1557. You can tell that it's a marvelous invention by reading the algebra that Omar Khayyam was doing 500 years earlier. Khayyam states problems like
QuoteWhat is the amount of a square so that when six dirhams are added to it, it becomes equal to five roots of that square?
and reasons about them in the same kind of prose. To me, the fact that our good notation makes these things accessible to little kids shouldn't take them out of the universe of "algebra." It reminds me of punctuation and spaces, invented in very late antiquity, that made literacy accessible to little kids. We can dream of future inventions that will make the secret knowledge of today's geniuses accessible to little kids.
Here is a little bit of Recorde 1557, "The Whetstone of Witte...", quoted in a footnote on wikipedia:
QuoteHowbeit, for easie alteration of equations. I will propounde a fewe examples, bicause the extraction of their rootes, maie the more aptly bee wroughte. And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in worke use, a paire of paralleles, or Gemowe lines of one lengthe, thus: = , bicause noe 2 thynges, can be moare equalle.
"Gemowe lines" means "twin lines."
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4 hours ago, Arcadia said:
The course framework 85 pages pdf
https://apcentral.collegeboard.org/courses/ap-precalculus/course-framework
Thank you for highlighting this!
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1 hour ago, Clarita said:
Beast Academy people. When your child gets stuck on a Beast Academy concept/problem what do you do or how do you help them?
What concept or problem is troubling them lately?
I found BA's introduction to fractions to be really dismal.
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1 hour ago, 8filltheheart said:
What you are describing is solving for a variable that is a constant. If a single variable is considered the key pt of algebra, than 1st grader filling in a blank, 1+ ___ =3 are essentially doing algebra since the ___ and a variable represent the same thing. What makes the statement true?
I would rather say "solving for an unknown" than "solving for a variable that is a constant." I'll contradict you and say that I do think that solving for an unknown is a very typical algebra problem. I agree that even very young children can be taught to understand and solve a limited variety of such problems. My own has had a procedure for solving 1+__ = 3 for as long as she's had one for solving 1+2 = __. But that procedure is not the equations-as-balancing-scales, simplify-a-little-at-a-time approach that I was taught in middle school.
I fully agree that learning about variables, functions of them and relations between them is an algebra skill of prime importance and one that is largely separate from "solve for the unknown." The contrast between them is interesting. I've seen lots of ideas around for getting 5-year-olds used to solving for the unknown, no ideas for getting them used to dependent and independent variables.
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1 hour ago, 8filltheheart said:
I don't believe that this is something that kids learn in algebra. It is a basic elementary concept.
In your view or experience, what is a more prominent thing that kids learn in algebra?
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3 hours ago, Ellie said:
And that's why algebra never made sense to me: people say stuff like that, but I don't understand why it is so.
As an arithmetic problem, I know that I can subtract the answer from the number given, and *that* answer will be the missing number. Easy peasy.
I have an inkling now that this kind of exercise, of writing down all the intermediate steps between the question and the answer, perhaps could be called something more attractive than "showing your work." To me, the expression has a scolding tone to it: you hear it when you get points off of your homework even though the answer is right.
One famous thing that kids learn in algebra is to write a new equation that differs from an old equation in a simple way. Add to both sides the same quantity, multiply both sides by the same quantity, etc. They learn that if the old equation is true, the new equation is also true. And for some rarefied examples they learn that they can make these modifications repeatedly and judiciously until they reach an equation that simply tells the solution of the original equation.
This process isn't the only way a kid could get the right answer: maybe they drew a bar-model diagram, or another kind of diagram, or maybe they just had a hard-to-articulate inspiration. If that's the case, and you tell them they must now write lots of equations in between the question and the answer, then you aren't telling them to "show their work," because they didn't work the problem that way.
Maybe there's an opportunity to start fewer fights by calling it something else.
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1 hour ago, Not_a_Number said:
We've had a sort of circuitous path with it. DD10 was writing full-fledged proofs a few years ago and still knows how, but we used to butt heads over it, and now she's able to do it but feels nervous about it. So I'm currently not pushing her to do this, although we've had a lot of conversations about how she ought to work on it again.
I remember two years ago some of the impressive things she was doing, that you wrote about here. What is she up to now?
And you wrote also, a little bit, about butting heads. I also anticipate some amount of butting heads, that I wonder if I can mitigate with any kind of careful planning.
1 hour ago, Not_a_Number said:So I guess my suggestion will be to work with what your child feels is useful, @UHP, and try to run with that. Do you know whether she'd find this useful herself?
Right now, I wouldn't heed her own sense of what's useful and what's not useful. I usually think I know better. But I am a little bit uncertain of what the learning objectives of "high school algebra" should be. I took it in 8th grade, and have certainly mastered it, but don't remember how very well.
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30 minutes ago, Brittany1116 said:
My 11yo is in the last bit of Saxon Alg 1. The habit training is important at 10-A=9 because it keeps them on track when they get to really convoluted stuff down the line and they 1) try to do it all in their head, or 2) skip 3 steps mentally and write down some work at the end, AND 3) ultimately get the wrong answer. Ask me how I know. 😉
My plan, with reservations, is also to use Saxon. Are you happy with it?
Calculus Textbook for Younger Students
in Accelerated Learner Board
Posted
No, we used a scripted program called Connecting Math Concepts, which is really wonderful. When we were approaching the end of it, another fan of CMC recommended Saxon to me, but with the warning only to use the 3rd editions not 4th. He also thought poorly of the Saxon elementary math books (maybe they are just called Saxon Math 1, Saxon Math 2, and so on). I don't know from experience whether he's right, but I followed his advice.
Yes basically. Each problem set is matched with a lesson. Each is 30 problems, only 3 or 4 of which are on the new material. The bulk of the problems are practice from older lessons. I copy almost all these problems out onto printer paper in big handwriting (for some reason much easier to hold her attention this way, than when asking her to read the problems herself from the textbook), skipping at most two in any problem set.