Friday, March 18, 2011

Multiplying by 10

My latest epistle to the Head of School at Natural Friends:

Teacher Cranium -- I recently had the following conversation with a "New New Math" educated kid:

Me: You know the shortcut for multiplying by 10?

Kid: Yeah, you just add a zero.

Me: OK, what's 53.5 x 10?

Kid: Um... 53.50?

Me (horrified): That's the same value!

Kid: Oh, right. Is it 530.5?

Kids need to understand that it's not just about "adding a zero", it's about shifting the place value. In an ideal world, you should be able to ask the kid, "why does 10 have this special property, but not, for instance, 8?" and eventually get the answer, "... because we're in base 10."

Sincerely, FedUpMom.

43 comments:

  1. I'm astounded at the math INabilities of cashiers and even nurses when discussing refunds and dosages, etc. When one considers that nurses are part of our safety system, that they are the ones in hospitals who are to know when a prescription seems "not right" and alert someone, it's a scary thing indeed.

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  2. PsychMom has a story...

    When I was 12 I got my first summer job in a country store....made $7.50 a day! I remember a customer being totally freaked out at me one day because he was buying 10 lbs of butter (he owned a resort and was in a pinch) and he had this dumb cluck kid (me) at the cash register putting in the price 10 times. "Don't you know the total cost of 10 lbs of butter?"
    I did, after he said that. But it didn't occur to me before that. In fact those understandings didn't become "matter of fact" to me until later...maybe into university and statistics courses.

    Maybe there's a better way to teach it...I certainly didn't get it in the 70's. And my 9 year old has no grasp of this base 10 concept either.

    Even with registers that calculate the change, I find that people (old and young) can't make change. They get totally confused if you round up to a more convenient number, like giving them 10.35 for an order that's 9.31 for example. For some people these things seem to be as difficult as if it was foreign currency.

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  3. PsychMom, whatever enlightenment you arrived at in the country store could have been done a few weeks earlier at school, right?

    If you didn't know it already, it means you were taught badly. As I recall from your earlier comments, you were a good student, so if you didn't get something, probably most of the class didn't get it.

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  4. The "New New Math" folks claim that they teach deep conceptual understanding, but conversations like the one I had above show that's not true.

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  5. PsychMom says...I wonder about that enlightment happening in school, FedUpMom. I wonder, because there was the emotional embarrassment to go along with the "Ahha" moment (it was more like an "ohhhhh...yeah" moment) which would not have happened to me at school. This man had me in a pressured moment because there were other customers waiting and they're all watching me ring in the 10 lbs of butter one by one. I happened to be secure enough to be able to take the pressure and actually be able to figure it out, but another type of kid could have been shut down emotionally and not been able to even deal with it. You know, like the pressure at the blackboard to solve math problems?

    I guess what I'm trying to say is that humans learn in different ways and depending on the personality type as well, things may be easy or difficult to learn. I'm thinking that context does play a role too. I'm reminded of that essay you reprinted near the beginning of your blogging days about the famous authors (who escapes me at the moment) describing his boarding school days. Some things were learned under extreme duress...but they were learned well.

    So if I regroup and go back to your original post above....if you had put the problem in terms of 10 times $53.50, would the youngster have faired better? My guess is the error would not have been made. And if he or she were measuring a piece of cloth or a length of board and had to cut a piece ten times larger than 53.5 cm, they'd immediately know it wasn't the same amount.

    There is a concept of 10 in there..in their heads, I mean, it just didn't come out in that particular example.

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  6. Wow, PsychMom, it's not often I get to get to feel like a gruff old traditionalist, but I do now!

    53.5 x 10 is not a difficult problem. If a bright 6th-grader like the one I talked to can't do it, I hold the school accountable.

    The essay we looked at earlier was "Such, Such Were the Joys" by George Orwell.

    Such, Such were the Joys

    Maybe I'll take another look at it!

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  7. PsychMom, Let's try a thought experiment -- if you met a bright 6th-grader who was about to graduate from an expensive private school, and then you discovered the kid couldn't read "Dick and Jane went to the sea shore", you'd be concerned, right?

    That's how I feel about this level of math. It isn't hard, and there's no good reason why a 6th-grader shouldn't be able to do it. It's educational malpractice.

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  8. "Kid: Oh, right. Is it 530.5?"

    That's a good one!

    This is a classic learn by rote problem that the new, reform, discovery, understanding math curricula supposedly don't do. What happened here? In spite of an "understanding" approach, a method was learned by rote. The justification for replacing traditional approaches to math with discovery and understanding approaches was to avoid this problem. What happened?

    Perhaps something else is going on. Perhaps it's a simple matter of competence. Perhaps it's a matter of schools not taking responsibility for ensuring that what they teach is learned. They specifically tell Everyday Math teachers to just keep moving through the material and "trust the spiral". There is no need for mastery at any one time. Apparently, that includes the mastery of understanding.

    Low expectations = rote learning.

    In the private school my son went to, kids were getting to fifth grade without learning their times table (Everyday Math). Some had difficulty adding 7+8 quickly. These kids were "pre-selected" and quite capaable of learning. This is also not about the stress of learning. Most schools spend an average of one hour per day teaching math. There is no rush or pressure here.

    There has to be some absolute scale used to judge school effectiveness. Unfortunately, the level states define for NCLB is very low, and this minimum has become the maximum. Our state defines a proficiency index that is the percent of kids who get over the minimum proficiency cut-off. This is a very low cutoff. The questions are simple and the raw percent correct score needed for proficiency is very low. Therefore, the proficiency index percent is high for many schools. It's kind of like proclaiming that 95% of the kids get over a raw score of 60% correct on a simple test. People are shown the 95% number, not the 60% number, and our schools claim that they provide a good education. I tell people to look at the questions on the test and look at the raw percent correct score needed for proficiency.

    I can accept the argument that there is only so much a school can be expected to do, but there has to be a better way to define this on an individual basis using an absolute scale. At some point, the results don't make sense. Schools don't want those expectations to be set very high, but they fight charter schools that set high standards. The argument our schools give is that since our proficiency index is so high, kids should not be allowed to go to charter schools. They won't do more for the willing and able kids, but they don't want them to go elsewhere because those are the kids who make their numbers look good. And it's the parents at home who make the best students look good. I have gotten a few notes from schools telling us parents to work on math facts with our kids. I have met too many people who think that their poor math skills are because "they are just not good in math". If you keep pushing (spiraling) kids along, then eventually you can blame all of the problems on them, and they will believe it.

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  9. What I see is that "conceptual understanding" translates to kids as something along the lines of "just try a few things and go with what feels right or doesn't make the teacher wince." Witness your daughter's foray into answers.

    I had the privilege (I now know) my first year of teaching to teach middle school math under a principal who wanted to the children to get it and the teachers to get it done. It was a tough school and had lots of kids labeled as sp. ed.

    I did some hands on conceptual stuff (or drew pictures or...), but I also drilled. We did a warm-up everyday that was about 2 or 3 questions of already covered topics and 1 or 2 questions of the material from the day before. I honestly believe that that repetition day after day was one of the biggest factors.

    While I had often felt overwhelmed and as though the kids weren't getting it, they did well on the tests (compared to past performance) and did seem to have at least internalized a FEW of the basic concepts. Before many problems I would ask them what a sensible answer would look like -- for your problem above: "Will the answer be bigger or smaller than 53?" "Will it be closer to 60 or 500?" They did get better at not just guessing and thinking first as the year went on.

    The next year we had a new principal and were told to follow the new scripted program to the letter. That new program would work fine for kids that already knew their math -- and pretty much only for those kids. Warm-ups were almost randomly chosen on one single topic, and we were told NOT to correct answers then (keep to the time, keep to the time!) but to "take notes" about who did and didn't get it. Then the kids were to do some sort of hands-on/group think about a problem for the day -- a problem that used the skills you hadn't taught yet. And on and on. The actual "lesson" part of the hour plus time was often almost invisible amongst the activities.

    While I had often felt overwhelmed and as though the kids weren't getting it the first year, they did well on the tests (compared to past performance) and did seem to have at least internalized a FEW of the basic concepts. Before many problems I would ask them what a sensible answer would look like -- for your problem above: "Will the answer be bigger or smaller than 53?" "Will it be closer to 60 or 500?"

    --The second year? Well, right now, during a school day I'm home typing on the internet. Yup, I am one of those young (if old in age) teachers who left early on. I couldn't do something that wasn't working. Being held accountable when I had no say in what I was doing just made no sense to me. Hoping to be employed teaching again, but it may be tutoring for a while.

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  10. Whoops, even with preview, I managed to move something AND leave it where it had been! Forgive the repeat!

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  11. PsychMom says...I think we may have found something we don't see quite the same FedUpMom.. Bound to happen.
    Three years ago, I would have agreed with you about these "simple" concepts.

    And I'm running out of time...I'll have to pick this up later...it may not be til tomorrow sorry...

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  12. No problem, PsychMom! Write back when you get a chance.

    I don't ask anyone to agree with me 100%.

    Here's another way to think about this: if the school was truly far-out and progressive, and the kids spent their time devising their own curriculum, or herding sheep, or whatever, I might have more sympathy.

    But the school is basically traditional. They think they've taught the kids all the standard math that they'll need to go on to 7th grade in the surrounding traditional schools.

    This is something I see all over the place -- the goals of progressive ed haven't been met, because the kids don't have much autonomy or responsibility for their own education. The goals of traditional ed haven't been met either, because the kids don't have the content knowledge that they should. It's neither fish nor fowl.

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  13. Oh, and @Anonymous, a couple of points --

    The "Kid" in the dialogue isn't my daughter, but a kid at her school who I'm tutoring.


    Anonymous says:

    ***
    Being held accountable when I had no say in what I was doing just made no sense to me.
    ***

    Do you see how this is what school feels like for the kids, too?

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  14. Also @Anonymous -- I've been trying to both "warm up" and "cool down" with review. I read somewhere that "intermittent repetition" is the strongest way to study.

    In other words, it's better to do 2 reps every other day than 10 reps in one day.

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  15. Hasn't that always been true, re how much say kids have in school, especially in how the lessons are taught?

    If anything, I'd say that currently kids are thought to be given a lot of say in what they're doing -- they don't need to follow one algorithm, they feel that many answers are right, that any problem can be done anyway.

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  16. Anonymous, yes, it's always been true that kids have very little say over what they do in school.

    The reason I brought it up is that so often the complaints I hear from teachers apply just as well to the students. "I can't stand any more of this pointless paperwork!" -- is that a teacher or student speaking?

    The deal where students "can use any algorithm" doesn't really pan out in practice. I've heard plenty of stories of kids being penalized for using standard algorithms taught to them by their parents, for instance.

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  17. Anonymous again, in hopes of working againMarch 22, 2011 at 1:48 PM

    Oh, see, the curriculum we had didn't involve "pointless paperwork" really, so I didn't think of it as coming off that way.

    More so, it was a curriculum written by teachers and administrators who like math to look good -- to be asking "the big questions" all along during the lessons, so that "deep understanding" was happening. Unfortunately, if you haven't taught some content to a bit of mastery first, it's hard to get to the understanding part.

    We were given rather contradictory advice on the using different algorithms -- though I will say that most teachers wouldn't directly penalize correct, but not in the curriculum, algorithms unless they had to due to what the question asked.

    See, the midway quiz and then end of unit test were written by the curriculum writers too and handed to us, along with very specific grading rules. I could tweak it a (very) little bit in the grading, but I could do nothing about the questions.

    One question showed the same multiplication problem done using two different algorithms, each with a mistake.

    The kids were to pick one incorrectly done version, show it done correctly (using that algorithm) and tell what the mistake was.

    This problem had a lot of explanation spread over the page, took up the entire side of a piece of paper and was at the end of a fairly long test. You can just imagine what grading that was like. And did it tell me as the teacher anything about their real ability to multiply 2-digit numbers that I didn't already know? Nope.

    But it sure did leave us all peeved!

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  18. PsychMom asks

    Would it irk you, FedUpMom, if you asked that same 6th grader..."I need my lawn cut 10 times this summer and I'm willing to pay you $15.50 each time. How much is that in total?" and let's say 6thGrader knew the answer without flinching.

    What would you think then? Is it the same problem (with education) or is it a different problem.

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  19. If 6th Grader could answer the lawn question, I'd be pleased. I wouldn't bet on it, though.

    If she could do the "real life" problem, it would help her understand the theory.

    She's a bright kid, and both the real life and theoretical versions of this problem should be straightforward for her, with a decent education.

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  20. Now that you mention it, when I was looking at this with Kid, I walked her through this version of the problem:

    53.5 x 10 = 10 x (53 + 1/2) = 530 + 5 = 535

    to show that this is the answer you should expect to get.

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  21. PsychMom says:

    But if she gets the real life problem (and I'm not so sure she would either) but doesn't understand the theoretical, is there still something wrong with her schooling?

    And what if this is just a maturation problem? What if 6thGrader never forgets the encounter with you and has forever changed the way she thinks about base 10 because of you, just like what happened to me?

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  22. PsychMom says..

    The way you explained the answer is algebraic isn't it? (see how slow I am still, to this day)

    How does this help her understand a base 10 concept?

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  23. PsychMom, this version:

    53.5 x 10 = 10 x (53 + 1/2) = 530 + 5 = 535

    doesn't teach base 10 at all. My point was that you get the same answer using the above approach or using the base 10 approach.

    Ideally, the 6th grader should be able to get the right answer either way.

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  24. "Me: OK, what's 53.5 x 10? Kid: Um... 53.50?"

    Errors like this tell you that there is something very wrong going on. Errors can do that. However, if "Kid" gets this one question right, that might not tell you much. Since this error happened with a 6th grader, that is not a good sign.

    There are many levels of understanding. What is 53 X 10 for base 8? What is the algebraic representation for this? That's not a reasonable expectation of understandoing for a 6th grader. However, everyone (!) can expect that a 6th grader should be able to do the decimal problem.

    Understanding doesn't mean much if you can't do the problem and doing one problem doesn't mean that you understand much. However, current discovery math curricula give a lot of lip service to understanding, but never ensure that the skills are mastered. Mastery of skills can evolve into better understanding, but understanding without skills is meaningless.


    53.5 x 10 = 53.50 shows that there is no skill and no understanding.

    If I gave a child a whole series of problems to do and he/she did them correctly, then I would know something about his/her understanding. If I talked with him/her about understanding, that would not tell me as much.


    "And what if this is just a maturation problem?"

    You have to base this on some absolute scale. What if a 6th grader doesn't know what 6*7 is? This comment sounds too much like the education mantra: "they will learn when they are ready". It's the basis of the Everyday Math philosophy of "trust the spiral". It puts the entire onus of education on the child.

    Kids should not be in 6th grade math if they can't solve 53.5 X 10 easily. It doesn't do them any good and it doesn't do the other kids any good. Maybe if schools actually try to ensure mastery on a grade-by-grade basis, they might find that kids are "ready" much sooner.

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  25. PsychMom, this version:

    53.5 x 10 = 10 x (53 + 1/2) = 530 + 5 = 535

    doesn't teach base 10 at all.


    Oh, I'm not sure I completely agree with that. There is a crucial step in the middle, where 10 x 53 becomes 530. Why does that happen? Is it because (as Kid said in the original post) when you multiply by 10 you "add a zero"? Or is it because the "5 tens" in 53 become "5 hundreds", and the "3 ones" in 53 become "3 tens"?

    Here's another way of handling the original convesation:

    Me: You know the shortcut for multiplying by 10?

    Kid: Yeah, you just add a zero.

    Me: Are you sure? I thought when you add a zero to a number it doesn't change.

    Kid: No, I mean, not add, but... put a zero on the end.

    Me: Why do you do that?

    Kid: I don't know.

    Me: What if I told you that you're actually supposed to put a 4 on the end, not a 0. How would you know whether I'm lying to you?

    Kid: Um, I don't know.

    Me: What if I told you 3 x 10 is 34? How would you know whether I'm lying?

    Kid: Because it's 30!

    Me: How do you know it's 30?

    Kid: Because 30 is three tens!

    Me: Ah, now we're getting somewhere...


    Unrealistic? Maybe, but I have had conversations like this with middle schoolers before. Conversations like this ought to take place in 2nd grade. By the time kids reach 6th grade, this stuff should be so basic that it doesn't even need to get mentioned.

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  26. FedUpMom, the "neither fish nor fowl" quote really resonated with me. I'm pretty much curriculum agnostic, my job is to help students and teachers learn problem-solving skills and strategy, and figure out how to learn through problem solving. Way too often I've observed "rote" teaching where kids are discouraged from making sense of problems ("ours is not to question why, just invert and multiply"). And on the others hand "discovery" teaching where kids were either allowed to spend forever solving a single problem, and when they had an answer that sounded ok, they moved on to another, seemingly unconnected problem, with no sense of mastery; or, they went through the motions of discovery before being told an algorithm. In all three cases, kids just sort of stuck a new idea on top of a shaky pile of ideas and misconceptions.

    I feel like observant parents, good teachers, thoughtful curriculum writers, etc. actually do have an idea of what real learning looks like -- kids' misconceptions are exposed and they have an opportunity to make sense of the material and then practice with it until it's solidified. We know what it sounds like... "Oooooh! I see! It's like... this, right? Wait, let me say that again. Can I reproduce it? Yeah, yeah, this is easy. Awesome, I get it!"

    To me, it doesn't matter all that much whether kids are shown an algorithm, practice it, and then have opportunities to make sense of it; or whether kids do contextual problems that they can use problem-solving strategies to figure out and then abstract an algorithm from there. What matters is that kids feel like, "math makes sense, I can understand it, and when presented with a mathematical thing, I have routines and strategies for making sense of it."

    The most important curriculum is teaching kids to be math learners. We should be teaching strategies like estimation (as Anonymous did), checking their work, solving a simpler version (53.5 * 10 is hard, but 53*10 is 530 so the answer is near that), changing the representation (as you did when you wrote 10 * (53 + 1/2). We should also be teaching kids to be aware of their understanding: I know "add a zero" -- is that a conceptual understanding, an invented method, or a procedure? If that's a procedure, do I understand the concept behind it? If it's an invented method, can I get good at it so it becomes and automatic procedure? And how can I use my conceptual understanding to check my work at all times.

    Phew... sorry for going on... it's just that thoughtful conversations about "discovery" math vs. "rote" math always make me want to add my 2 cents.

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  27. "...it's just that thoughtful conversations about "discovery" math vs. "rote" math always make me want to add my 2 cents."

    It's really all about low expectations versus high expectations; about real math versus math appreciation.


    "The most important curriculum is teaching kids to be math learners."

    I don't know what that means. How do you define that in grade level expectations of mastery? How do you ensure that all kids have the opportunity to reach a real course in algebra in 8th or 9th grade at the latest?

    Curricula define content, pacing, and expectations, or at least they should. Unfortunately, most math curricula found in schools only go through the motions and don't ensure much of anything. You could drop Singapore Math in our schools and they would still let kids get to 5th grade not knowing the times table, but at least it would define a proper grade-level pacing. Part of the problem would be fixed.

    I would say that that is the main point of the thread. There are specific skills and understandings we should be able to expect from kids at each grade level. If they are not close to that level, then they shouldn't be passed along to create new problems on top of old ones. Schools and teachers can't just go through the motions; they have to take responsibiity for results.

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  28. "There are specific skills and understandings we should be able to expect from kids at each grade level."

    I don't understand that. First of all, all kids are different, so it's not going to be realistic to expect them all to be able to do the same things at the same ages. But moreover, these expectations seem entirely arbitrary. When someone says that all kids at age x should be able to, say, divide by fractions, where is that coming from? It's not based on any need that kids of that age have. It seems to be pure assertion, based entirely on the desire of (some) adults that kids be able to divide by fractions by age x. The justification always seems to be along the lines of, "because otherwise they won't be able to meet the expectations we have for kids at age x+1."

    What would be so terrible about not getting to algebra until the 10th or 11th grade? For that matter, how would kids be any worse off as adults if they didn't have any compulsory math instruction until after elementary school? How would they be harmed -- how would their adult lives be affected? Wouldn't a seventh-grader starting from scratch reach our "7th grade level expectations" in way, way less than seven years, and maybe even less than one? Isn't that alone a justification for waiting?

    These "grade-level" expectations in elementary school all seem to grow out of the idea that doing something to kids is always better than doing nothing to them. We have the kids in the building for six hours; therefore we have to make them learn things -- regardless of whether there's any reason to think it benefits them in the long term. Letting the kids use that time the way they want to use it -- well, that would be irresponsible! (And how could we justify our salaries?)

    I realize this is the unschooler in me coming out. But I'm not against compelling kids to do things. I just think the default, when it comes to compulsion, should be doing nothing, unless there's a strong reason to think that doing something is clearly better for the kids. Where's the strong reason to hold kids to these arbitrary grade-level expectations?

    I agree with FedUpMom's "neither fish nor fowl" comment. I think the people who make these progressive math curricula believe, deep down, that all elementary schoolers need is a little exposure to some math concepts, and that they don't need any real mastery of anything at that age. But they could never sell that, so the thing ends up a collection of half-baked, confusing, unclear expectations that drives the kids and their parents nuts, with the result that by the time the kids do get to seventh grade, they break out into hives at the mention of the word "math." So, yeah, I think the far-out approach and the traditional approach are both preferable to an incoherent hash.

    Here's a question: What percentage of the adult American population do you think meaningfully understands the concept of base 10? Or could even begin to think about what base 8 is? I'm guessing it's a very, very small number. Do you really think that's because the schools are doing something wrong, and that if they just did some things differently, that number would be *much* higher?

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  29. "First of all, all kids are different, so it's not going to be realistic to expect them all to be able to do the same things at the same ages."

    You have to have some level of learning and mastery expectation for each grade. I said nothing about age. However, it's the educational fashion nowadays to track kids by age.


    "When someone says that all kids at age x should be able to, say, divide by fractions, where is that coming from?"

    It has nothing to do with age. It has to do with grade level. Grade-level content is not arbitrary. In fact, most state proficiency cut-offs are too low for most kids.



    "What would be so terrible about not getting to algebra until the 10th or 11th grade?"

    The problem is that if they didn't get to algebra by 9th grade, then something has gone wrong along the way. If kids really do have difficulty with math, then they should be identified early and not age-tracked along with all of the other kids. The problem is that schools make absolutely no attempt to determine if a child is having problems because of the child's own issues or the school's issues. They just push the kids along by age and assume that kids "will learn when they are ready".


    "For that matter, how would kids be any worse off as adults if they didn't have any compulsory math instruction until after elementary school? How would they be harmed -- how would their adult lives be affected?"

    Why educate at all? Why have any school or state-imposed expectations? Go for it with your own kids, but stay away from my son. Unfortunately, I don't have any choice. My son (and I) had to put up with years of Everyday Math.



    "Wouldn't a seventh-grader starting from scratch reach our "7th grade level expectations" in way, way less than seven years, and maybe even less than one? Isn't that alone a justification for waiting?"

    Absolutely not!. Show me a school where they do this. You go ahead and start a charter school that does this and see how many kids enroll. Prove it.

    Waiting really isn't the issue. The issue is whether schools know if what they are doing works. If kids get to 6th grade and don't know that 53.5 X 10 is 535, then something is wrong by normal expectations. My son was in a school where many of the kids did not know the times table in fifth grade. These were perfectly capable kids. It was clearly the school's fault.


    "Letting the kids use that time the way they want to use it -- well, that would be irresponsible! (And how could we justify our salaries?)"

    Once again, go start your own school. Give parents full choice and see if they accept your ideas.

    "I realize this is the unschooler in me coming out."

    Why do you expect others to buy into this idea?


    Much of this discussion would not exist if parents had full school choice. I'm more than happy to have others pick un-schools.


    "I think the people who make these progressive math curricula believe, deep down, that all elementary schoolers need is a little exposure to some math concepts, and that they don't need any real mastery of anything at that age."

    And astoundingly, they think that all kids should be forced to follow these ideas. It's extraordinarily arrogant.



    "Do you really think that's because the schools are doing something wrong, and that if they just did some things differently, that number would be *much* higher?"

    Absolutely! How math is taught in K-6 borders on incompetence. After suffering through Everyday Math with my son for those years, I can be very specific. More importantly, you have to consider more than the math needs of the average adult. What about all of the individual STEM career opportunities that were never realized because K-6 schools screwed up math so badly. Kids end up thinking that they just aren't good in math. The blame themselves. Few ever recover from the damage of K-6 math.

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  30. "Go and start your own school" isn't an argument. Don't tell me that's it's unrealistic to convince people of those ideas. What do I care? Tell me why you're not convinced by them, or why I shouldn't be -- that would be engaging the argument.

    "You have to have some level of learning and mastery expectation for each grade." That's just an unsupported assertion. In fact, you don't have to have any expectations at all for what someone should know about math by the end of first grade, any more than you need to have an expectation for what they'll know about molecular biochemistry by the end of first grade. The only reason to have such an expectation is if you have reason to think it will make a difference in the kid's later life.

    "Why educate at all? Why have any school or state-imposed expectations?" Translation: "I dislike your conclusions; therefore your reasoning must be wrong." That's not a real argument, either.

    "Astoundingly, they think that all kids should be forced to follow these ideas. It's extraordinarily arrogant." I'm no great defender of progressive math people (I dislike Everyday Math, too), but I think that statement is unfair. Someone in a district with a traditional curriculum could just as easily make the same accusation of arrogance. But the people who make the curricula don't play any role in deciding how much we have school choice; you don't know what their opinions about offering options are. As for me, I don't want to force you to put your kids in my kind of curriculum. I just want to reason it out with someone who disagrees, and see where the discussion goes.

    "Absolutely not!. Show me a school where they do this." The experiment described here is the best I can do. It's not my fault that no one else (as far as I know) has tested the idea. (There are also lots of anecdotal reports of homeschoolers testifying to how much more quickly math can be learned if you wait until the kids are older and more interested in the subject, for whatever that's worth.) But seriously, you think it would take a twelve-year-old *seven years* to get to "grade level"? Wouldn't you agree that it would least be valuable to know the empirical answer to that question?

    As for raising the number of adults who understand base 10 (or base 8!), this is an empirical question that neither one of us can actually know the answer to, but my guess is very different from yours. I think the main reason most adults have limited math fluency is that they have concluded (often quite reasonably) that they can get along just fine without it. (It's the same reason so few American adults speak a foreign language.) Nothing you do in K-12 education will change that reality.

    (Continued in my next comment)

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  31. The point of my question was that, unless you at least ask about whether a different kind of math instruction is likely to have any real effects on the kids' adult lives, you're basically saying that kids' time and freedom and autonomy are entirely without value. I'm not accusing you of saying that, but I do think that's the unspoken premise behind a lot of discussions of education. People act like it's enough just to show that making the kids do something will marginally increase the chance that a small percentage of kids will retain something into adulthood -- or even just to suggest that it's somehow "good for them" to be made to do those things, regardless of whether it will have any long-term effect -- and that it automatically follows then that we should do that thing, no matter how much it burdens the kids. No one suggests that adults should be treated that way -- that the government should make people do things any time there's even the slightest rationale that it would be "good for them." But with kids, it's as if there's nothing at all on the other side of the scale.

    I think kids' time and freedom has inherent value that should be taken into account before you decide to pile hours of compulsory instruction on them. But even if I didn't, I'd at least want to ask how sure I am that the thing I am compelling is more educationally valuable than the thing they might choose to do on their own. To try to make people learn something against their will is to start off with two strikes against you. Shouldn't we at least consider whether *that's* why adults' math skills are what they are?

    In the end, I guess you just have more faith than I do that changing curricular policies will transform America's math-literacy (numeracy?) profile. Has there ever been a time when people weren't complaining that Americans don't know enough math, and that therefore math education must be to blame? The suggested remedies always seem to involve piling more instruction on the kids, to the point where my first-grader has hour-long math classes five days a week. I think it's delusional, and just not justified by any realistic prospect that it will make any difference in her life.

    "What about all of the individual STEM career opportunities that were never realized because K-6 schools screwed up math so badly." Here, I partly disagree with you and partly agree. Doctors and scientists may need to know cell biology, but it doesn't follow that, if we don't force elementary schoolers to learn cell biology, we'll have no doctors and scientists. People are perfectly capable of self-selecting and then learning the things they need to know to do the things they want to do.

    But I'd agree that "few ever recover from the damage of K-6 math." Not creating math-phobes should be the first priority of elementary school math teachers. I'd agree that a traditional math curriculum is probably less likely to create math-phobes than a neither-fish-nor-fowl approach like some of these progressive curricula. But a genuinely student-driven approach -- where math isn't forced on kids when they're not interested in it, and kids are subjected to constant tests and grades that are bound to generate performance anxiety -- would be even less likely to create math-phobes. The question is whether it would render them less likely to develop math skills that would be useful in their adult lives. If you could convince me that it would be, and that the gain would be significant enough to justify the cost to their free time as kids, you'd win me over. But I guess I'd throw the same question back at you: show me the school, with any curriculum, anywhere ever, that has generated kids who, thirty years later, know how base 8 works.

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  32. Chris, you might convince me to leave math instruction till somewhat later (late elementary school?) and then compress it into a few years. In a sense, that's what I'm doing with the kid I'm tutoring.

    As for the amount of time kids spend doing math, I think a good curriculum would mean LESS time in the classroom. And I'm sure you know how I feel about homework, including math homework, for young kids.

    As for the effect I'm looking for, I want kids who like math, or at least don't hate it. I want them to have, at minimum, the math skills to handle their finances intelligently as adults.

    I also think that math is useful as a discipline. I think educated people should know what a proof looks like, and how logic works.

    I care a lot about kids' time and freedom, but I want them to learn math too. Bad curricula eat huge amounts of time and energy while failing to impart learning.

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  33. FedUpMom -- I'm pretty much with you on those points. I get the sense that SteveH, too, is not suggesting that we need more math instruction, just better math instruction. Part of this discussion is just about what approach will maximize the likelihood of the outcomes you describe. That's an empirical question that I can't claim to know the answer to, but I do think people underestimate (or ignore) the possibility that making people learn things that they're not interested in, not ready for, or bored by might actually make those outcomes less likely, rather than more likely.

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  34. Chris, my feeling is that a good teacher can impart her own interest to the kids. That is, maybe the kids didn't start out interested in math, but a good teacher can kindle their interest. Similarly, if the kids aren't ready, a good teacher will figure out what they're missing and fill in the gaps.

    There's no good reason why school should be such a dreary experience for so many kids so much of the time. Authentic learning is a pleasure.

    Oh, I meant to respond to your earlier comments about the expectations for each grade. I think Steve H's point is that the curriculum needs to be structured in such a way that the kids have the skills they need to undertake each successive step.

    In math, that means it's no fair to put a kid in algebra class, at whatever age, if she can't do basic computations with fractions. It's setting the kid up for failure.

    Similarly, it's no fair putting a kid who's reading at a third-grade level into a high school lit class.

    And I've said before that at Natural Friends, the lack of clear curricular goals results in the kids not learning anything in particular. I wish I could say that the kids were deeply engaged and just learning different stuff from the standard curriculum, but that's not what I see.

    Now, if you wanted to set up some kind of creative unschooling, where kids could discover and follow their passions, with someone willing to step in and make sure they've got a reasonable set of basic skills (like reading and math), I could be interested in that.

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  35. FedUpMom -- Yes, and I agree that the schools are responsible for not asking kids to do things that they haven't prepared the kids to do. I just think that at some point you also have to step back and ask what the point is of all those expectations, whether they actually serve their own goals as much as a different approach would, and whether they justify the cost to the kids.

    I don't entirely agree that a good teacher can make the natural disadvantages of compulsion disappear. If someone suddenly told me that I was required, as a 46-year-old adult, to attend compulsory math classes because it's good for me, I doubt the best teacher in the world could overcome my resistance to that enterprise. There's no reason to think that kids don't, at least at some level, experience compulsory learning in that same way.

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  36. There are two separate issues here: philosophy of education and effectiveness. The thread was originally about effectiveness. It's clear that much of K-6 education is not effective for the amount of time spent. Kids in homeschools can do much more in less time. The solution to a problem in effectiveness may be just to wait, but that assumes that what schools are doing is working. This isn't true. The solution for many kids is not to wait. The solution is to provide better instruction. Waiting might work for some kids, but how do you decide that on an individual child basis? Also, what kind of education do you provide or ensure whenever you do decide to start? How will you know it's effective? Unschooling approaches have lower external expectations and measures of effectiveness. They are based on what what the student wants to do.

    Many parents have different ideas about what constitutes a proper education. They have to accept what public schools give them or else pay huge amounts for private schools where it's likely they will get the same ed school approach to education.

    I'm more than happy to let others choose unschooling options, but that's not a reasonable solution to those who have different philosophical ideas. Those with different views of education will have different opinions of what constitutes success, so any discussion of effectiveness with unschoolers will be meaningless.

    My son is and always was a sponge for knowledge, but I would never just leave it up to him to decide what to learn or not. I want him to know specific content and to master specific skills. I want to keep all career doors open as long as possible. I want him to value hard work and understand that all learning is not going to come naturally. Waiting or a natural approach to education has its own risks, but my goal is not to change anyone's mind. My goal is to make sure parents have proper options.


    "I just think that at some point you also have to step back and ask what the point is of all those expectations, whether they actually serve their own goals as much as a different approach would, and whether they justify the cost to the kids."

    This is fine on an individual parent/child basis, but not as a systemic solution. You can't just assume that a problem of 53.5 X 10 = 53.50 is a waiting issue. If you do wait however, how will you determine whether the student is reaching his/her potential and whether all individual career doors are kept open? Competition and supply and demand are not natural. Life requires hard work. My son's piano teacher once held his hand low and told him that he was trying to have too much fun "down here". Then he raised his hand high and said that if he worked really hard, he would have much more fun "up here". The "working hard" part might not be effective, but our solution would not be to wait.

    There are three things my wife and I want for our son. The first is to care about other people. The second is to know the value of hard work. The third is to be happy. My mother said that if he does the first two, then the third will take care of itself.

    Learning to work hard is not natural.

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  37. SteveH -- I agree with a lot of what you've said. You and I have different ideas about how readily we should intervene in kids' lives. I enjoy the idea of hashing that disagreement out online, but I don't want to force that activity on anyone. And I agree that we should try to create a world where parents had more choices about what kind of schools their kids would go to.

    "Learning to work hard is not natural" seems like an unsupported assertion to me, and one that doesn't coincide with my intuitions about the world, and with my own experience. My parents were largely hands-off -- maybe because I was a pretty good student anyway, and maybe because I was the youngest of six -- but to the extent I ever learned to work hard, it came from the satisfaction I got (and still get) from doing something well.

    I could probably make an even longer list of things I would like for my kids, but I have much less faith that I can "make" them have certain values or learn certain lessons, or that my attempts to do so wouldn't have unintended consequences. To me, the less coercive techniques -- modeling good values as much as I'm able, providing an intellectually stimulating home environment, responding genuinely to their interests, showing faith in their general good sense, and giving them time to achieve things on their own before intervening -- seem the most likely to be effective and the least likely to backfire. But in the end, we're all just making the best guesses we can.

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  38. I do think, though, that whether you're an unschooler or a traditionalist or something in between, you can't just assume that the things you want are things that you can make happen. It seems to me that, in many debates about what our educational policies should be, no attention at all is given to the limits of what can realistically be achieved, to the burdens on the kids' time and autonomy, and to the possibility of unintended consequences, including the possibility that kids will resist and rebel against educational coercion. No matter what your goals or educational philosophy, don't you have to at least take those possibilities into account?

    In other words, I will admit that there is a risk to waiting before intervening to make the kids learn something you want them to learn. But isn't there also a risk to not waiting?

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  39. PsychMom says:

    "Learning to work hard is not natural"

    If you had seen the sweat running down my 17 month old kid's face as she learned to walk, you might have a different perspective. Even today, as she learns a skating move, the drive to get it right is all her...same with the multiplication tables and the spelling words.....there's no one who can "make" her learn anything. No one instilled the drive in her..she came with it.

    I think we work very hard, naturally, at things that stimulate our minds and bodies. What we find stimulating changes with age, so what Chris says makes sense to me. What stimulates a child at 6 is not the same as what interests them at 8 or 9. And introducing age 9 topics at age 6 is probably a waste of time.

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  40. "But isn't there also a risk to not waiting?"

    This should be an individual decision made by parents. At most, they get to decide on whether to wait a year for Kindergarten or whether they can find (or pay for) a school that takes a more unschooling approach. Our public schools use full-inclusion which is all about natural learning. They will introduce the material, but if you don't learn it, that's OK, you will see it again the following year. No pressures; no angst. That's why there were capable kids who got to 5th grade without knowing their times table. I don't know of any sort of natural process that will fix that. How can you tell if a school is doing a good job or not if they follow an unschooling approach?


    "'Learning to work hard is not natural' seems like an unsupported assertion to me,..."

    It depends on your definition. It's clear that my wife and I have much higher expectations than many other parents. But no, I don't think that learning what you need to keep all doors open is natural at all. But it's also not necessarily unnatural. We've gotten very good at knowing where a proper push boundary is for our son.

    ".. you can't just assume that the things you want are things that you can make happen."

    We don't, but that doesn't mean we don't set standards for how we expect him to act and how hard we expect him to work. He has lots of opportunity for natural learning (he reads math and physics books on his own), but there is also a timeline involved here. It starts with the requirements for college and works backwards. To leave this competitive process to a natural process risks closing many doors. However, a "natural" advocate might argue that this will produce the best result ... by definition. This difference in philosophy offers no avenue for discussion.


    "... in many debates about what our educational policies should be, no attention at all is given to the limits of what can realistically be achieved, to the burdens on the kids' time and autonomy..."

    Look at the actual online math test questions for NAEP for 4th grade and the results. Are kids really that stupid or not ready yet? How would you define these "limits", especially in a "natural" setting?

    Your presumption is that there is something here that can inform the debate over school standards and expectations, when in reality, these decisions should be strictly the domain of parents.

    We can both advocate for our own philosophical ideas, but the real solution is full choice. The problem comes when ed school opinion is crammed down parents' throats in the guise of "best practices".

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  41. PsychMom, great point. I've seen my Younger Daughter work very hard at learning to swing on the swingset, and learning to ride a bike with no training wheels.

    I'm no fan of work for the sake of work -- I think we've got too much of that in our culture already.

    For me, the bottom line is that we could achieve more real learning of real content with a lot less fuss, bother, and, yes, work, if we started with a smart curriculum.

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  42. SteveH -- I'm with you in wishing that the school system provided a lot more choice to accommodate a lot more different educational and parenting philosophies.

    "However, a 'natural' advocate might argue that this will produce the best result ... by definition. This difference in philosophy offers no avenue for discussion." You keep making that point, and maybe there is somebody who might argue that, but that's clearly not what I'm arguing. If I knew in advance that using a light hand would turn my kids into heroin addicts, I surely would take a different approach, and "naturalness" could be damned.

    What I've been saying is that I don't have nearly as much confidence as you have that I can "make" good outcomes occur by intervening with a heavier hand. That's mainly an empirical question, not a philosophical one -- though one that could probably never be conclusively pinned down. How can you be so sure that your efforts to make your kids learn won't make matters worse instead of better?

    The normative part, for me, is that I think I should err on the side of not butting in unless I have a relatively strong reason to believe that the butting in is justified by the benefit to the kid. That's certainly how I want to be treated by people who have authority over me.

    I can't tell whether you agree with me on that, and are just quicker to conclude that you're justified, or whether you really think the burden should be flipped -- that you should err on the side of intervening unless you feel very confident that staying out will do no harm. Or maybe it's just one long spectrum of parenting styles.

    "It starts with the requirements for college and works backwards." This just begs the question. Where is your evidence that taking a lighter hand with math at younger ages will result in poorer skill levels at grade 12? You have no more evidence on that question than I have. Again, in the end, most of us are making these decisions more on intuition than on evidence, and are just hoping that we guess right.

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  43. Interesting side-note: John Holt, who a lot of unschoolers see as the founding father of the unschooling approach, started his career as a math teacher in a conventional elementary school. This thread made me go back and look again at his first book, How Children Fail, which he wrote early in his teaching career. FedUpMom, have you ever read it? I think you would get a kick out of it if you ever see it on sale at the used book store. I certainly don't agree with everything he says, but his description of the kids' efforts at pleasing, placating, and evading their math teacher (as opposed to learning their math) is a great read. I'm sure it would inspire some posts!

    In the book, he makes the to-be-expected points about how mere memorization of formulas is not enough. (I don't take anyone on this site to be suggesting that it is.) And overall, he seems pretty enamored of "progressive" approaches to math teaching -- it was the Sixties, after all. But on the issue of whether the algorithm should come before or after the conceptual understanding, sometimes some ambivalence sneaks in. Looking over the book in light of this thread, this passage leapt out:

    "Nobody 'explained' 10, or the function of base and place in our numeral systems, when I was little. I went to a very old-fashioned school where they just showed you how to do problems without ever trying to explain why they did them that way, or to convince you that this made any sense. This was probably hard on the children who weren't very good at parroting. But I was great at parroting, and the advantage for me of this system is that I was left alone to make sense of 10, and a lot of other things, in my own time and my own way."

    He concludes: "Bad explanations are a great deal worse than no explanations at all."

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