I have Opinions. 😜😂 I think in some areas, I am more of a traditional/classical homeschooler, like with history. However, when it comes to math, I am full Poppy. Phrases like "rigorous daily practice" and a "need for continuous practice to ensure fluent mastery" make me itchy. I'm curious, does anyone know if Egan enjoyed math? Was he, personally, enchanted with numbers? I feel like if he were the sort of person who did math for fun, he wouldn't have said those words.
Regarding the practices, I LOVE the concept of ladders. The math that I made for my own kids uses this concept, except rather than going through a ladder in a day, for example, the ladders are tackled simultaneously and spread out over the year. Essentially, there's one ladder for each area of math (decimals, fractions, geometry, probability, time, graphs, rounding, etc.) each divided into 36 steps. The kids just do one step on each ladder each week, going back to the previous week or two if necessary. It's like built-in spaced-repetition and, in my opinion, significantly reduces the need for "continous, rigorous practice" which frees up times for math games and other ways of making developing math fluency enjoyable. However they are used, ladders are so effective, because they build confidence in a nonstressful way while letting the student do all the actual mental effort.
I think the "making friends with numbers" and the "origin stories of math" practices are also brilliant. I haven't ever seen those suggestions before, but I love them and wish my kids were younger so it would make more sense for me to geek out on them. I don't know where it fits, but I feel like connected to these practices somehow should also be math metaphors and stories. I'm not sure, but I think they are pretty popular in Waldorf. I don't have a good reference for them unfortunately, but whenever I'm tutoring, I always try to come up with *something* because it makes whatever concept we're working on more friendly and memorable. Here are some examples I can think of off the top of my head:
- Long division - A certain number of people (the divisor) go to the house of Count Divide and hand him their objects that need to be divided among them (the dividend). He takes the objects inside his house, does his calculation, then writes the answer on his roof for how many each of them should get (the quotient). -- This was the first math story I ever saw. I believe it was in Oak Meadow's first grade curriculum that I used over a decade ago. I read it to my then first grader and was like, "this is genius, we'll just do this for everything in math."
- Fractions - obviously, you're going to a pizza party
- Borrowing/Carrying - the different place values are actually apartments for different types of creatures. There are doors between the apartments and they can visit their neighbors, but they have to first transform into the other creature type according to rules, like 10 goobers (single units) makes a doodle (a ten).
- Negatives/Positives - with younger kids I use a squirrel holding an acorn in a tree (positive) or burying it (negative). with older kids, I use money. earn money (positive), borrow money (negative).
- Perimeter - you're a farmer walking around your sheep pen checking for holes in the fence.
- Functions - are factories. You dump in your raw material, X, and some magic happens on the inside and it spits out Y on the other end.
- Solving for the Unknown in Algebra - A number commited a crime and doesn't want you to know who he is so he's wearing an X costume. You have to be a detective and use the clues in the rest of the problem to figure out his real identity.
- Mutivariable equations - You're throwing a party and your best friends get awesome gift bag X while everyone else gets mediocre gift bag Y.
Anyway, Egan was all about stories. I feel like it would be a shame for an Egan math curriculum to not incorporate them somehow when they can be so helpful!
"The kids just do one step on each ladder each week, going back to the previous week or two if necessary. It's like built-in spaced-repetition and, in my opinion, significantly reduces the need for "continous, rigorous practice" which frees up times for math games and other ways of making developing math fluency enjoyable."
This sounds like continuous rigorous practice though! Continuous rigorous practice doesn't have to be miserable, it just has to be frequent and thoughtfully chosen.
Fair! I completely agree, Becky. 😊 I just think traditional math educators and I have different ideas about what constitutes "frequent." I'm pretty sure if I told my school's administrators that my kids worked on math once a week for about 30 minutes they would have something to say about the rigor and frequency of my kids' math education. 😂
I think Egan (and Brandon) are exactly right when they say that math can and should be meaningful, magical, and enchanting. What I'm pushing back against is they both seem to then add a caveat to ensure they are taken seriously acknowledging that OF COURSE not ALL math can be enjoyable and there will need to be some time spent on tedious, regular practice in a rigorous math education. Maybe I'm wrong and this isn't what they are saying. I admit I'm reading between the lines. Or maybe it is what they're saying, and I respect that, but since I don't have the goal of trying to get people to take me seriously, I have the freedom to politely, but adamantly, disagree. 😂 I do not think that any part of a quality math education, particularly in elementary school, needs to feel tedious to the student. I think Egan had it right with the first sentence of his quote and should have just stopped talking. 😜
>> "they both seem to then add a caveat to ensure they are taken seriously acknowledging that OF COURSE not ALL math can be enjoyable and there will need to be some time spent on tedious, regular practice in a rigorous math education. Maybe I'm wrong and this isn't what they are saying."
Ooh, this is good! I'm not sure if this is what I'm saying, or not. It's certainly close — and insofar as it is what I'm saying, I'm happy to defend that. But I feel like there's something we're missing, in this. Wanna record a conversation, and post it for paid subscribers?
Interesting! I think I'm interpreting the words tedious, frequent, and rigorous slightly differently. I think math inherently has tedious parts, but that people enjoy lots of things that objectively contain tedium. Kids often like coloring and repetitive craft projects after all! Both mine went through a phase of delighting in prime factorization. I have fond memories of row reduction. All of these can be described as hard work, but that hard work doesn't necessarily equate to distress. My own belief is that if kids see the kids and adults around them delighting in the joy mixed with the work and the tedium they will more easily be able to do the same. And of course I think the Eganization helps too. As for frequent, I interpret that as frequent enough that kids aren't becoming demotivated by having to relearn what they've forgotten rather than any particular frequency. And as for rigorous, that we've charted a coherent course through the world of math rather than declaring that baking counts as math and we're done here. But I absolutely agree that mathematical misery is counterproductive!
I think we are in near perfect agreement, Becky. I am fully on board with your points regarding rigor, frequency, and the value of an inspired teacher. There is definitely joy to be found by many in a challenge and satisfaction that comes from hard work. Also, I too can appreciate the beauty of a clean Gaussian elimination. I'm right there with you. I think the word tedious is where I diverge with you just a tiny bit, because tedium is in the eyes of the beholder. I completely agree that tasks many find tedious and monotonous can be sources of pleasure and satisfaction for some. When I was in grad school, one of my favorite ways to decompress after a long day was to clean my skateboard bearings. I've been a knitter since I was a teenager. I'm very familiar with finding bliss in mindless repetition where others find tedium.
The part that does not quite sit right with me is that there are a lot of people who WILL find repetitive mathematical exercises tedious. No matter how inspired of a job you do at presenting prime factorization, there are those who would still rather go outside and play than *practice* these skills. (Not that I'm one of them. I'm just saying they COULD theoretically exist. 😜) Where I'm pushing back is on the idea that repetitive mathematical exercises are necessary even when a child finds them tedious. I think you hit the nail on the head when you said that "mathematical misery is counterproductive." As homeschool parents, we can put the worksheets away when our kids hit a wall and I have a feeling you and I are probably more alike as homeschool parents than we are different. We know to stop long before misery sets in. I am just putting my stake in the ground (I *think* alongside you) saying that if the child is NOT a willing participant in planned mathematical activities, enforcing a plan of "rigorous daily practice" to ensure "fluent mastery" has the potential to do more harm than good. Actually, I think that language is not strong enough. I believe it WILL cause actual harm AND is unnecessary in achieving your goals. (You know, provided you have a more coherent plan than declaring your math curriculum to be solely baking. 🤪)
Yes, I think we do agree. I once had a conversation with someone in which I mentioned my younger one's amusing habit of asking me every morning if he had to do Beast Academy and then delightedly doing it as soon as I said that of course he didn't have to. "My son is older, so I just don't think it's enough for him to do math every day because he chooses to. I think he needs to know that it's required," she answered. I wanted to ask her if, as her kids ate broccoli, she interrupted them to remind them that eating broccoli was compulsory? Who would do such a thing!
I firmly believe that we can't make anyone do math (or eat) and that we can cause great harm by trying to force these things. And I also believe we have a great responsibility to do what we can to set kids up for an ability to enjoy these things, while knowing that we have only influence and not control and that a good life can look many ways, with or without prime factorization and broccoli.
It's a pleasure discussing education with people who care about it so much. Thank you!
As I follow this series of posts, I'm starting to rethink history as more of a conduit for learning rather than a subject in and of itself. History as I have experienced it might more aptly be identified as government. Do you plan to fold math, science, government/sociology, geography, art, world religions, language, etc into the History 100? Ooh... as I'm typing this I'm thinking perhaps it is the reverse? Maybe everything we learn in every subject is history, which would explain why whittling down to 100 stories feels impossible! "This is [how/who/what/where] [people/things] [thought/tried/wrote/experienced/looked]; Find things that inspire/challenge/bore you and build on it! Hmmm...
Back to math. In some ways this feels like a backdoor to math and numbers that fills in a gap I didn't realize was missing! I'm excited to get started on our link list. Maybe as we go we move a boss problem to the link list? When the answer is a whole number between 0 and 100 at least.
>> " Do you plan to fold math, science, government/sociology, geography, art, world religions, language, etc into the History 100?"
Y'know, I suddenly realize that "the History 100" sounds like "History 101" — that is, the introductory course of history at a university. Which is actually sorta like what this is!
Okay, sorry about that. Now to answer your question: some of those yes, and others, no.
The plan (right now) is to do government (including political science) and sociology as part of history.
Math, science, geography, art, world religions, and language (both foreign and domestic!) will be their own subjects. (See the recent geography posts, by the way. The others are coming soon.) They will, however, interact with the history curriculum. As an easy example, when we do an origin story (of a math theorem, scientific discovery, artwork, religious movement, or language) we'll be open to putting that into the Memory Palace. (If you're worried about your walls becoming too messy... yeah, that's a potential problem! Folk might want to those "in your mind", like a traditional memory palace, or to refresh the palace frequently. This is a practical hurdle that we'll need experience to jump over...)
The making friends with numbers idea sounds like something the 8 year old will absolutely love. Considering numbering the history stories thusly maybe (which might be a terrible idea for reasons I haven't thought of yet).
Just placing the cards with the number associations alongside the 100 history stories so that we are counting up from the first history story we pick to the one furthest in the future. But because the history stories are in order but not ordered items on any list (besides the one we're making) and because we're for very good reason not using a linear time scale I'm not sure whether or not it makes sense to strongly associate the 46th of the history stories we put on our wall with the number 46.
My *hunch* on this is no, only because the history events already have their own number — the year they happened. These, too, can be helpful for memory (whenever I'm playing ping-pong and the score is 12-15, I think "aha — the formal break between Roman Catholicism and Eastern Orthodoxy!"). (TMI?)
I have Opinions. 😜😂 I think in some areas, I am more of a traditional/classical homeschooler, like with history. However, when it comes to math, I am full Poppy. Phrases like "rigorous daily practice" and a "need for continuous practice to ensure fluent mastery" make me itchy. I'm curious, does anyone know if Egan enjoyed math? Was he, personally, enchanted with numbers? I feel like if he were the sort of person who did math for fun, he wouldn't have said those words.
Regarding the practices, I LOVE the concept of ladders. The math that I made for my own kids uses this concept, except rather than going through a ladder in a day, for example, the ladders are tackled simultaneously and spread out over the year. Essentially, there's one ladder for each area of math (decimals, fractions, geometry, probability, time, graphs, rounding, etc.) each divided into 36 steps. The kids just do one step on each ladder each week, going back to the previous week or two if necessary. It's like built-in spaced-repetition and, in my opinion, significantly reduces the need for "continous, rigorous practice" which frees up times for math games and other ways of making developing math fluency enjoyable. However they are used, ladders are so effective, because they build confidence in a nonstressful way while letting the student do all the actual mental effort.
I think the "making friends with numbers" and the "origin stories of math" practices are also brilliant. I haven't ever seen those suggestions before, but I love them and wish my kids were younger so it would make more sense for me to geek out on them. I don't know where it fits, but I feel like connected to these practices somehow should also be math metaphors and stories. I'm not sure, but I think they are pretty popular in Waldorf. I don't have a good reference for them unfortunately, but whenever I'm tutoring, I always try to come up with *something* because it makes whatever concept we're working on more friendly and memorable. Here are some examples I can think of off the top of my head:
- Long division - A certain number of people (the divisor) go to the house of Count Divide and hand him their objects that need to be divided among them (the dividend). He takes the objects inside his house, does his calculation, then writes the answer on his roof for how many each of them should get (the quotient). -- This was the first math story I ever saw. I believe it was in Oak Meadow's first grade curriculum that I used over a decade ago. I read it to my then first grader and was like, "this is genius, we'll just do this for everything in math."
- Fractions - obviously, you're going to a pizza party
- Borrowing/Carrying - the different place values are actually apartments for different types of creatures. There are doors between the apartments and they can visit their neighbors, but they have to first transform into the other creature type according to rules, like 10 goobers (single units) makes a doodle (a ten).
- Negatives/Positives - with younger kids I use a squirrel holding an acorn in a tree (positive) or burying it (negative). with older kids, I use money. earn money (positive), borrow money (negative).
- Perimeter - you're a farmer walking around your sheep pen checking for holes in the fence.
- Functions - are factories. You dump in your raw material, X, and some magic happens on the inside and it spits out Y on the other end.
- Solving for the Unknown in Algebra - A number commited a crime and doesn't want you to know who he is so he's wearing an X costume. You have to be a detective and use the clues in the rest of the problem to figure out his real identity.
- Mutivariable equations - You're throwing a party and your best friends get awesome gift bag X while everyone else gets mediocre gift bag Y.
Anyway, Egan was all about stories. I feel like it would be a shame for an Egan math curriculum to not incorporate them somehow when they can be so helpful!
Warmly,
Michelle
>> "Was he, personally, enchanted with numbers? I feel like if he were the sort of person who did math for fun, he wouldn't have said those words."
That's a good question — and I don't know the answer. I'll be talking in the near future to someone who knew him well; I'll ask her.
"The kids just do one step on each ladder each week, going back to the previous week or two if necessary. It's like built-in spaced-repetition and, in my opinion, significantly reduces the need for "continous, rigorous practice" which frees up times for math games and other ways of making developing math fluency enjoyable."
This sounds like continuous rigorous practice though! Continuous rigorous practice doesn't have to be miserable, it just has to be frequent and thoughtfully chosen.
Fair! I completely agree, Becky. 😊 I just think traditional math educators and I have different ideas about what constitutes "frequent." I'm pretty sure if I told my school's administrators that my kids worked on math once a week for about 30 minutes they would have something to say about the rigor and frequency of my kids' math education. 😂
I think Egan (and Brandon) are exactly right when they say that math can and should be meaningful, magical, and enchanting. What I'm pushing back against is they both seem to then add a caveat to ensure they are taken seriously acknowledging that OF COURSE not ALL math can be enjoyable and there will need to be some time spent on tedious, regular practice in a rigorous math education. Maybe I'm wrong and this isn't what they are saying. I admit I'm reading between the lines. Or maybe it is what they're saying, and I respect that, but since I don't have the goal of trying to get people to take me seriously, I have the freedom to politely, but adamantly, disagree. 😂 I do not think that any part of a quality math education, particularly in elementary school, needs to feel tedious to the student. I think Egan had it right with the first sentence of his quote and should have just stopped talking. 😜
Warmly,
Michelle
>> "they both seem to then add a caveat to ensure they are taken seriously acknowledging that OF COURSE not ALL math can be enjoyable and there will need to be some time spent on tedious, regular practice in a rigorous math education. Maybe I'm wrong and this isn't what they are saying."
Ooh, this is good! I'm not sure if this is what I'm saying, or not. It's certainly close — and insofar as it is what I'm saying, I'm happy to defend that. But I feel like there's something we're missing, in this. Wanna record a conversation, and post it for paid subscribers?
Sure! I have a feeling that finding a time that works for both of our crazy schedules could prove tricky, but I'm game!
Warmly,
Michelle
Interesting! I think I'm interpreting the words tedious, frequent, and rigorous slightly differently. I think math inherently has tedious parts, but that people enjoy lots of things that objectively contain tedium. Kids often like coloring and repetitive craft projects after all! Both mine went through a phase of delighting in prime factorization. I have fond memories of row reduction. All of these can be described as hard work, but that hard work doesn't necessarily equate to distress. My own belief is that if kids see the kids and adults around them delighting in the joy mixed with the work and the tedium they will more easily be able to do the same. And of course I think the Eganization helps too. As for frequent, I interpret that as frequent enough that kids aren't becoming demotivated by having to relearn what they've forgotten rather than any particular frequency. And as for rigorous, that we've charted a coherent course through the world of math rather than declaring that baking counts as math and we're done here. But I absolutely agree that mathematical misery is counterproductive!
I think we are in near perfect agreement, Becky. I am fully on board with your points regarding rigor, frequency, and the value of an inspired teacher. There is definitely joy to be found by many in a challenge and satisfaction that comes from hard work. Also, I too can appreciate the beauty of a clean Gaussian elimination. I'm right there with you. I think the word tedious is where I diverge with you just a tiny bit, because tedium is in the eyes of the beholder. I completely agree that tasks many find tedious and monotonous can be sources of pleasure and satisfaction for some. When I was in grad school, one of my favorite ways to decompress after a long day was to clean my skateboard bearings. I've been a knitter since I was a teenager. I'm very familiar with finding bliss in mindless repetition where others find tedium.
The part that does not quite sit right with me is that there are a lot of people who WILL find repetitive mathematical exercises tedious. No matter how inspired of a job you do at presenting prime factorization, there are those who would still rather go outside and play than *practice* these skills. (Not that I'm one of them. I'm just saying they COULD theoretically exist. 😜) Where I'm pushing back is on the idea that repetitive mathematical exercises are necessary even when a child finds them tedious. I think you hit the nail on the head when you said that "mathematical misery is counterproductive." As homeschool parents, we can put the worksheets away when our kids hit a wall and I have a feeling you and I are probably more alike as homeschool parents than we are different. We know to stop long before misery sets in. I am just putting my stake in the ground (I *think* alongside you) saying that if the child is NOT a willing participant in planned mathematical activities, enforcing a plan of "rigorous daily practice" to ensure "fluent mastery" has the potential to do more harm than good. Actually, I think that language is not strong enough. I believe it WILL cause actual harm AND is unnecessary in achieving your goals. (You know, provided you have a more coherent plan than declaring your math curriculum to be solely baking. 🤪)
Yes, I think we do agree. I once had a conversation with someone in which I mentioned my younger one's amusing habit of asking me every morning if he had to do Beast Academy and then delightedly doing it as soon as I said that of course he didn't have to. "My son is older, so I just don't think it's enough for him to do math every day because he chooses to. I think he needs to know that it's required," she answered. I wanted to ask her if, as her kids ate broccoli, she interrupted them to remind them that eating broccoli was compulsory? Who would do such a thing!
I firmly believe that we can't make anyone do math (or eat) and that we can cause great harm by trying to force these things. And I also believe we have a great responsibility to do what we can to set kids up for an ability to enjoy these things, while knowing that we have only influence and not control and that a good life can look many ways, with or without prime factorization and broccoli.
It's a pleasure discussing education with people who care about it so much. Thank you!
As I follow this series of posts, I'm starting to rethink history as more of a conduit for learning rather than a subject in and of itself. History as I have experienced it might more aptly be identified as government. Do you plan to fold math, science, government/sociology, geography, art, world religions, language, etc into the History 100? Ooh... as I'm typing this I'm thinking perhaps it is the reverse? Maybe everything we learn in every subject is history, which would explain why whittling down to 100 stories feels impossible! "This is [how/who/what/where] [people/things] [thought/tried/wrote/experienced/looked]; Find things that inspire/challenge/bore you and build on it! Hmmm...
Back to math. In some ways this feels like a backdoor to math and numbers that fills in a gap I didn't realize was missing! I'm excited to get started on our link list. Maybe as we go we move a boss problem to the link list? When the answer is a whole number between 0 and 100 at least.
>> " Do you plan to fold math, science, government/sociology, geography, art, world religions, language, etc into the History 100?"
Y'know, I suddenly realize that "the History 100" sounds like "History 101" — that is, the introductory course of history at a university. Which is actually sorta like what this is!
Okay, sorry about that. Now to answer your question: some of those yes, and others, no.
The plan (right now) is to do government (including political science) and sociology as part of history.
Math, science, geography, art, world religions, and language (both foreign and domestic!) will be their own subjects. (See the recent geography posts, by the way. The others are coming soon.) They will, however, interact with the history curriculum. As an easy example, when we do an origin story (of a math theorem, scientific discovery, artwork, religious movement, or language) we'll be open to putting that into the Memory Palace. (If you're worried about your walls becoming too messy... yeah, that's a potential problem! Folk might want to those "in your mind", like a traditional memory palace, or to refresh the palace frequently. This is a practical hurdle that we'll need experience to jump over...)
A math curriculum I like that lets kids teach themselves arithmetic through algebra in a year is youteachyou.org
The making friends with numbers idea sounds like something the 8 year old will absolutely love. Considering numbering the history stories thusly maybe (which might be a terrible idea for reasons I haven't thought of yet).
Ooh, that sounds interesting — can you give an example for "numbering the history stories thusly"? (But also, props for saying "thusly")
Just placing the cards with the number associations alongside the 100 history stories so that we are counting up from the first history story we pick to the one furthest in the future. But because the history stories are in order but not ordered items on any list (besides the one we're making) and because we're for very good reason not using a linear time scale I'm not sure whether or not it makes sense to strongly associate the 46th of the history stories we put on our wall with the number 46.
My *hunch* on this is no, only because the history events already have their own number — the year they happened. These, too, can be helpful for memory (whenever I'm playing ping-pong and the score is 12-15, I think "aha — the formal break between Roman Catholicism and Eastern Orthodoxy!"). (TMI?)