Math Fact Practice ≠ Fluency

Discussion in 'Debate & Marathon Threads Archive' started by TeacherShelly, Mar 9, 2014.

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  1. TeacherShelly

    TeacherShelly Aficionado

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    I had the wonderful experience of meeting Jo Boaler of the Stanford "How to Learn Math" course and mathematics professor at Stanford. She made a comment, math fact fluency does not come from practice, it comes from understanding.

    I asked her what she'd say in a parent conference when a parent says practice is important for at least two things: music and math. She said you don't need to understand how a piano works to play it.

    She said memorizing facts is not as useful as having number sense.

    Now I'm {almost} excited to have conferences.

    :)
     
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  3. TamiJ

    TamiJ Virtuoso

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    I am so jealous that you met her! I loved that course. It changed how I look at math and a lot of what I am doing in math. Lucky you!
     
  4. EdEd

    EdEd Aficionado

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    That's interesting and thought-provoking, but I think misleading. I do think "understanding" is important to fluency, but practice leads to understanding.

    I'd also say that the idea that a child with an understanding of number sense and addition, but with no memorized facts, will not achieve high levels of computation fluency in the area of addition.

    Did she offer any specific responses to these kinds of thoughts? Certainly open to being challenged and corrected :)
     
  5. ready2learn

    ready2learn Comrade

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    This is what I was thinking. I might be misunderstanding this, but it sounds like one of those things that is nice to say but in practice, you need fact practice to have fluency. Of course, that does not mean forget about understanding.

    Thanks for bringing this up. Made me think.
     
  6. TeacherShelly

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    She said that memorization (done by practicing) does not lead to understanding; but if the practice uses mental math, taking apart and rebuilding numbers, that will build fluency because you will be able to do it faster.
     
  7. Missy

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    Did she suggest any resources on how to teach this?
     
  8. ajr

    ajr Rookie

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    I don't know my times tables and frequently forget addition. You are completely correct - I have very low computation fluency.

    There is an interesting cart/horse problem here, because fluency in computation only matters insofar as the curriculum demands it; fluency matters because teachers grade on it. Any curriculum beyond expects this set of knowledge that isn't in-of-itself useful as a shortcut.

    In terms of Actually Doing Math (tm), it doesn't seem to affect anything. I don't have problems doing data analysis at work, nor do I have problems in my classes (I'm a math major).

    The question I've had for a long time is how much does memorizing facts assist understanding in a practical sense; you have to start somewhere. If you don't start with ANY knowledge, what bits do you have to get someone to understand to progress to higher tiers of understanding? How big of a role does age play - i.e., do we teach facts because the people we're teaching not have the mental development to support the abstractions we want them to have?

    The type of "practice" matters when you're discussing nearly any subject, and it seems to be no different in math. In training for a marathon there's the concept of "empty miles" - you run, but you don't improve because your strategy for running is poor. You run some negative combination of too fast, too slow, too far, too near, too long, too briefly.

    In math, you can memorize the times tables up to 20*20, know a thousand formulas, and memorize a hundred theorems and still not understand anything. Doing hours of this kind of work blindly - which is what people do since that's how it's graded - doesn't help understanding, whereas one carefully crafted example that a person reflects on for ten or twenty minutes can catapult that person well past understanding. I can tell a lot more about what a person knows in math by asking them questions verbally, rather than having them solve equations. This problem is brutal when trying to tutor people who are going into college. They have years of the wrong type of practice.

    Asking someone to solve a problem allows them to lie to me - they produce the right answer, but I don't care about the answer. I want to know how they think. They can get the wrong answer and still know what they're doing, and can get the right answer consistently and have no idea what's going on.

    I think it's easier to assess some of these questions teaching adult learners, because some have a more fully formed ability to deal with abstraction and you can carry on with the understanding and ignore rote practice until the subject has been sufficiently motivated.
     
  9. TeacherShelly

    TeacherShelly Aficionado

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    Missy, yes she has a book called "What's Math Got to Do With It?" which tells how to teach number sense. She is a strong advocate of Number Talks, too and recommends a book called Number Talks by Sherry Parrish. I really want to get the book, but it's, of course, expensive (~$50).

    SHe also uses dot cards which cluster a number of dots in different patterns. The students discuss how they saw the number. They are not supposed to count, but visualize the number. The idea is to make numbers more visual and to teach flexible thinking - that there are multiple ways to see the same thing.

    HTH!
     
  10. gr3teacher

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    Fact fluency is incredibly important. It may not be more important than number sense, but fourth grade math will be a miserable experience to any kiddo that isn't comfortable with their facts.
     
  11. Pashtun

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    I tell the parents in a letter before the year begins. "if you want to set your child up for a successful 4th grade year, have them memorize their mult. facts"
     
  12. gr3teacher

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    I do something similar... it's tough because all my kids skip 3rd grade math and start right with 4th grade math... and parents don't really seem to believe that the kids need to know their multiplication facts... and then the second unit of the year is fractions (with finding equivalent fractions, reducing fractions, LCM, GCF, adding/subtracting with unlike denominators...) and the kids that don't know their facts are miserable and I'm ripping my hair out.
     
  13. mathmagic

    mathmagic Enthusiast

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    Think of it connected to reading: to be a good reader, you have to be able to read the words, but then also understand them. That takes those foundational skills, but also strategies to comprehend. Naturally, the ability to comprehend is vital, but those foundational skills are needed as well.

    In math, sure, I could go through and have great number sense, but if I'm trying to do a / learn 3-digit times 2-digit multiplication problem, I'll be spending far too much mental energy on multiplying single digits that I won't be able to attend as much to the learning of that new concept.

    Needs to be a true mix. If I had my own classroom, kids would be using Xtramath once a day for the fluency, but we'd also be doing number talks several times a week. Both are vital.
     
  14. DrivingPigeon

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    I've always heard that once students have an understand of number sense and operations, they are ready for memorization of facts.
     
  15. a2z

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    While I haven't heard this said, I have observed in students that lag with memorization of facts addition or multiplication that once the number sense comes to that of a typical 2nd or 3rd grader the memorization of facts and the understanding of how they are related seem to come together. This makes fact recall easier. For those that lag with fact fluency, I find that consistent practice with a chart is the way to solidify accuracy of recall. But even then, those with weak number sense or poor memories, tend to not develop the same fluency as those with number sense and fluency.

    You can really see the lack of number sense when you start to work with fractions or factoring. It is painful and near impossible for kids even when they have a chart to work with.
     
  16. a2z

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    You can pound out notes on a piano without knowing how it works, but all of the best pianists know how the instrument works because it allows them to understand the subtly of how to play the instrument to its ultimate capacity.

    You can memorize math facts until you can spit them out in a spit second and struggle with fractions and factoring and multiplication of large and small numbers because you lack number sense.

    Most need both to be truly proficient. Of course, there is always the exception to every rule.
     
  17. DrivingPigeon

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    I'm really struggling with my students and 2-digit subtraction. They understand the process, but when the ones spot has 16-9, they freeze up, have to draw a picture or use their fingers, and often get the answer wrong.

    They understand what subtraction is, and they understand the process of ungrouping. However, not knowing their basic math facts is keeping them from being able to get the correct answer.
     
  18. Missy

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    Thanks! I will check with our curriculum dept to see if they have either of these titles. Also, sometimes they are able to buy requested books and make them available for teachers to borrow.
     
  19. TeacherShelly

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    DrivingPigeon, I tried to like your post :)

    Today we had a PhD who was Jo Boaler's mentee come to our staff meeting. Her thesis was on place value and subtraction, and how children develop skills with them. She said these are the things 2nd graders are just understanding, or misunderstanding.

    A2Z, the point about the piano made sense to me. A pre-algebra kid can fake math proficiency by memorizing (like a piano player can play just by muscle memory and some music reading); but a mathematician and a pianist share an understanding far deeper and more flexible than those who memorized.

    Also, it is so much more boring to see math as a bunch of numbers with formulas; than to see how it works in many different scenarios. I was taught in such a boring way.
     
  20. iteachbx

    iteachbx Enthusiast

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    We've implemented number talks this year in our school. It's great! I'm amazed at how my students can manipulate numbers and how deep their understanding of the four operations is. In turn I think it's made them more successful problem solvers. I can't say enough great things about how great math is going in my classroom this year- number talks is a big part of it.
     
  21. TamiJ

    TamiJ Virtuoso

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    I am doing math talks too in my room and it seems to reay be helping the kids develop a greater understanding. I even have math journals for them now where they illustrate math problems. We have recently just started doing number bonds and I noticed that this year they quickly seemed to really get number bonds, something that is usually challenging to them inititally. I do believe it is all the math talks that have prepared them for number bonds.
     
  22. John Lee

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    In many ways, it reminds me of Writing. Students practice everyday, doing DLR (Daily Language Review) questions. They also have Spelling assignments. And most can do well in these assignments. But when it comes to actually spelling and punctuating, and capitalizing (i.e. all writing mechanics), they are awful at it.
     
  23. Pashtun

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    So the two should go hand in hand? You need quick recall of facts and understanding of operations and how they can be applied and manipulated?
     
  24. ajr

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    Foreward: This is not a condemnation of teachers; this is a broad condemnation of what we, as a society, value.

    There's a bit of is/ought fallacy going on in this thread.

    Math fact proficiency is, in blunt fact, a requirement for success in math. However, this requirement exists because it's taught that way, not because of an innate requirement in the subject. It exists because we adults say it does, and for no other reason.

    That success in a grade is predicated on the successful memorization of "math facts" in a prior grade is not surprising and does not require mention, because that's how the curriculum is written. More condemning, it's how the people teaching it understand the subject to be. Thinking that math is the ability as an adult to calculate tips, distribute pies evenly to friends, or any similar task is so wrong as to be nearly criminal.

    If you catch yourself saying something like "People are coming to me without being able to do <task>," that is not an indication of poor understanding of math. That's an indication of nothing more and nothing less than a misalignment of expectations. The expectation that someone "ought to be able to do this" isn't automatically valid just because someone thinks it. Nor is "That's how it was when I was in school!" an argument, it's just a historical statement, as valid in the discussion as "Prussia was a country in Europe."

    You can teach operations without numbers at all, so the insistence that these "math facts" are inherent to the subject is patently absurd. This view seems to exists only because we do not expose teachers of primary math to more complex ideas. It's a failure of the imagination and our education, not an inherent requirement.

    Am I arguing that we teach Lie groups to kindergartners and forgo numbers entirely? No, I'm not. What I am suggesting is that this narrow pedagogical hyper-focus during the initial teaching of the subject is harmful, and is part of the reason why most people stop all mathematical exposure as soon as they're able. Young children are allowed to express themselves and their experiences in their writing as they learn vocabulary, and yet the depth of self-expression that's possible in mathematics is completely forgone in favor of "How many pies does Missy have?" - the direct analogue of doing nothing but vocabulary exercises for year after year.

    Lockhart's criticism of primary math education is exactly what I am making reference to: you have a group of people who have never done any math whatsoever responsible for the initial exposure of math to students. It would be absurd if you hired teachers who had never written an essay to teach English, and yet we feel collectively, as a society, that this exact thing is what ought to be done for mathematics.

    The questions that started this thread are extremely heartening, because the ideas about math talks, journals, discussions, etc., are closer to what mathematicians do than anything else I've seen mentioned in any other thread in this forum.

    Afterword: The addition, subtraction, multiplication, and division of numbers is an ugly, brutish subject, and should be understood by students of math as such. Everything we learn about it is intended to speed the process, allowing us time to think of other things. These other things should be our true focus: they are interesting. So we should seek to dwell on the first, vulgar things only as long as is necessary, to allow us the most time for what is interesting and beautiful.

    The only subject uglier than numeric manipulation of the Reals is differential equations, which requires both a strong stomach and true grit to consider for long.
     
  25. Missy

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    I take exception to your statement that those who teach primary math have never "done any math whatsoever."

    Where on earth does this gross generalization come from?
     
  26. gr3teacher

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    You only take exception with that part? I take exception with basically that entire post.
     
  27. TeacherShelly

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    AJR, I applaud and agree with almost everything you wrote. Your presentation, however, will probably not win friends (not that you were trying to). Here's how Jo Boaler put it in What's Math Got to do with It,:
     
  28. TamiJ

    TamiJ Virtuoso

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    Ditto. I think often times as teachers we put an emphasis where there shouldn´t be. Having true number sense and understanding how to manipulate numbers will make one superior to his or her mathmetician counterparts who have only mastered math facts but lack number sense. Jo Boaler´s class was extremely interesting because she interviewed real professionals who reflected on their own math use and it was their ability to manipulate numbers (as opposed to having memorized math facts) that helped them be successful in their careers.
     
  29. DrivingPigeon

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    This is an interesting discussion. I have been trained to teach math in the way that many of you are talking about (manipulating numbers, developing a strong number sense, exploration, real-world application, etc.). My district uses the program Math Expressions as a resource for teaching math. I like following the scope and sequence, but I often supplement with activities from this resource book, which my additional training was focused on: Teaching Student-Centered Mathematics. I would highly recommend it, no matter which curriculum you use.

    Anyway, I often find myself just trying to get through the next Math Expressions lesson, rather than working more slowly and meeting each child at his or her developmental level. This thread is a good reminder to step back and slow down.

    Math facts are still a tough one, though. I understand what everyone is saying about building number sense and the fact fluency will come, but when there is pressure for them to be fluent in addition and subtraction by the end of 2nd grade, the reaction is to work on quick memorization of facts.
     
  30. gr3teacher

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    The parallel I see is with reading. You want kids to think deeply about texts. You want them to comprehend what they read. You want them to make connections, make predictions, infer, visualize, etc.

    But they ain't gunna do any of that if they can't decode the words on the page, and they ain't gunna do any of it WELL if they can't decode fluently.
     
  31. ajr

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    Since there's a lot of different themes going on, I'm going to reply to each theme in a different section, not sure how better to do it. Apologies!

    This is also a short novel. I apologize again. I'm not Oscar Wilde and lack the skill to make my points with more brevity.

    On English and math facts

    To clear some of this up, I'm not arguing against math facts, full stop. I memorize things as required, then forget it later. I still NEED those facts for a period of time. However, my sophistication as a math person increased as I try to remember less; it means I use more sophisticated mental models and am more particular about the information I'm holding. This applies even to basic things like multiplication tables, which I do not know at any given time.

    I'm arguing against math facts over everything else, because that's how math education seems to go. Rather than teaching tools to pick up facts as required, we demand students memorize an expanding list, and woe unto he who forgets any one of these holy relics. This will be on the test.

    To continue the analogy with English:

    When first learning language, we have an intense desire to communicate our experience to others, and understand their experiences in kind. Children do this constantly, and respond well to being read to and writing short stories about themselves. This motivates vocabulary; it may suck to learn and not everyone likes it, but there is an understandable need. Someone who is crappy at vocabulary is usually gets a much, much better grade than someone who is equally isn't 'getting' math. I argue the reason for this is the intrinsic motivation that communication gives to vocabulary. In the end, we stop asking people to memorize vocabulary words. We move on to essays, writing, analyzing thoughts.

    In English, we try to learn to understand, and learn to be understood.

    In math, we have nothing at all like this; no creative expression being taught. There is nothing at all being taught that would prompt deep introspection. Every weekday for twelve years, we're doing vocabulary and nothing but vocabulary.

    Finding out how many pies Missy owes John has is not a creative expression. All the word problems in the world won't make rote calculation creative; the pig remains a pig, no matter what fantastic robes we dress it in.

    Being allowed to think incorrectly is part of this. There is no room in math education for being unequivocally wrong. Yet this is how mathematicians come to understanding. We use our intuition, realize errors, and investigate it systematically. Was the error in our intuition, in our model, somewhere else? I may have a friend who thinks algebraically, and I may prefer a geometric or topological view. These types of discussions are where understanding math - the understanding of patterns - are built.

    In English, we allow students to make tremendous errors of grammar and vocabulary, and they still get a B. Make the same types of mistake in math, and you are told just how irredeemably wrong the answer is. The same amount of technical wrongness is punished disproportionately in math.

    For explicit emphasis:
    A student who gets every single problem wrong adding fractions has done zero math, and yet is being implicitly but unequivocally told they're bad at math.

    Noam Chomsky pointed out a problem with questions and discussions of this sort. You can question things like "should we have standardized testing of mathematics or not?", but if you question a base assumption of the debate - whether or not certain things should be taught at all - the discussion degenerates to name calling and questioning the asker's sincerity.

    Where the idea that K-12 teachers don't know math comes from

    First, it is imperative that the comment be understood as a social commentary, and is not an insult to teachers. Any person in any field is going to learn and focus on the material required to do their job, and probably not focus on underwater basket weaving. I have not focused on late 13th century fashion, because it is not relevant to what I want to do.

    When a mathematician says "Teachers don't know math," they are commenting that we are teaching people things like calculus, differential equations, probability, and lower-level linear algebra. To a mathematician, these subjects hardly register as math at all.

    This is troubling, because when we look to medicine, we see what it is that doctors do - that is medicine. When we look to a politician, what they do is politics.

    A mathematician does mathematics, and yet society sees no problem when there is an overwhelming consensus that "ya'll ain't teaching math." Imagine if doctors came out to med school and said, "Hey guys, what are you doing, because this sure isn't medicine."

    Look at the curriculum of most (not all) secondary ed and primary ed programs. Some secondary ed programs go a little farther and dip their toes into proofwriting, and maybe a rigorous class on linear algebra or analysis. Many stop at the same place undergrad engineers and physicists stop; rote calculation, but rote calculation in fancy dress and coattails.

    Most of the time, the emphasis for our young teachers is on the solved fields of mathematics - the things that haven't changed for awhile, in some cases in centuries. Someone who studies these fields, to the exclusion of all else, isn't being exposed to what a working mathematician usually considers "math." Even analysis is a crappy hodge-podge of subjects considered by most to be an introductory course, giving young math majors some basic background and terminology required to approach more interesting problems.

    My personal failures in knowledge

    I'm not a primary or secondary ed teacher.

    This is critical, because while I can sit back and criticize the system, I lack any and all knowledge required to make specific suggestions to actually improve the situation.

    I do not understand what it is like to teach ten year olds. I don't know what they're capable of understanding, what personal difficulties they have, how they relate to each other, what their concerns are.

    This means someone who isn't me has to sit down and think about these problems a great deal - what SHOULD we be teaching in math, how should it be approached, and when should we approach it? I just don't have the appropriate knowledge required to make an improvement.

    Some offhand ideas about timing in math education

    This is all extremely hypothetical and is only meant to serve as a point of discussion, not a concrete suggestion for moving forward.

    What if we didn't teach numbers beyond their literary value until middle school? Sure, teach numbers and how they are written, but save algorithms until later.

    Take classical education - reasoning was key, not computation. What if we focused on logic and geometry in the early years, instead of computation?

    Teach computation to middle schoolers, and teach it faster than it's taught in elementary. By this point, the need for computation should be obvious, similar to the need for vocabulary in English. Introductory financial and business math could be focused on here. Practical stuff.

    In high school, completely forget algebra initially and let students take math classes that align with their interests. Many high-level math classes are pretty easy, and have major implications for diverse fields. There are fields in math that apply to English, art, music, and a host of other areas. Bring calculation in again towards the the end of highschool, having motivated again the need for algebraic calculation to advance.
    Why I write like a jerk

    I have noticed that on this forum, if you want replies that are serious and measured, you have to be blunt and willing to agitate.

    If I post something that isn't inflammatory in the slightest, it will be glossed over. The only replies will be short "I agree!" or "I disagree!" statements, without much discussion.

    I like discussion, and I think both teachers and what we teach are important. Toward that goal, I'm willing to be the bad guy.
     
  32. Pashtun

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    Very very interesting.
     
  33. gr3teacher

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    So basically, you're trying to be a troll. Duly noted. I appreciate the honesty.
     
  34. ajr

    ajr Rookie

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    Uh, I'm not trolling in the slightest? I'm being very honest about my perception of how discussions go on this forum. That is, it's extremely partisan and unless you pick your wording carefully, exactly zero productive debate happens because nobody bothers to reply with any depth. Which is exactly what is happening now, since I pointed it out at the end of the post, so it's foremost on people's minds.

    I am confused how you both got "he's a troll" from someone who clearly put a huge amount of effort into trying to be understood.

    I just apparently made the mistake of being too honest about why I choose to use the tone I did, and now I'm going to be insulted and called names for it.

    edit:

    I even made DIRECT reference to this phenomenon in my post, but I guess that everyone is so hasty to dismiss opinions different from their own, it's not relevant.

     
  35. gr3teacher

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    http://en.wikipedia.org/wiki/Troll_(Internet)

    In Internet slang, a troll (/ˈtroʊl/, /ˈtrɒl/) is a person who sows discord on the Internet by starting arguments or upsetting people,[1] by posting inflammatory,[2] extraneous, or off-topic messages in an online community (such as a forum, chat room, or blog), either accidentally[3][4] or with the deliberate intent of provoking readers into an emotional response[5] or of otherwise disrupting normal on-topic discussion.[6]
     
  36. ajr

    ajr Rookie

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    And I'm doing none of that.

    Are you interested in discussing the original topic - which is what I'm attempting to do - or are you just going to continue to derail the topic and continue to persecute me for an imagined wrong?
     
  37. gr3teacher

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    Mar 16, 2014

    No, I'm not interested in discussing the original topic with you, unless you'd care to do it in a way that is not deliberating insulting and (by your own words) "like a jerk" and "willing to agitate."
     
  38. TamiJ

    TamiJ Virtuoso

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    Mar 16, 2014

    Hence balanced literacy. This is why I still have my students practice math facts occasionally, and I even assign xtramath.org (math facts) as homework. However, that´s not my focus. In class I am worried more about the manipulation of numbers and developing number sense.
     
  39. ajr

    ajr Rookie

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    Mar 16, 2014

    Actually, I'm in the private sector. I'm not going to say what I do or give a ton of details, because my current sector is a small community. That said, I get to deal with average adults doing math in their jobs on a day to day basis at a bunch of different companies.

    I care about this because of what I see in my job, day to day. I see what kind of education we're cranking people out with and what the average adult is able to do. I'm in a better position that either you OR a college professor to know what our educational system is turning out, because I'm the private sector, dealing with vast ignorance of mathematical principles.

    When someone in finance comes to me unable to deal with a simple subtraction problem because all they know is formulas, there's a problem. When this kind of thing happens constantly, across different companies, with college educated adults, there's a BIG problem.

    And the only reason I was suggesting I can't fix the problem is because the solution requires a multidisciplinary approach.

    It's stupid to have someone who doesn't know early childhood development try to come up with curriculum, on his own, for early childhood. Just like it's stupid to have someone who DOES know early childhood development come up with a curriculum for a subject they don't understand.
     
  40. gr3teacher

    gr3teacher Phenom

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    Mar 16, 2014

    That I can agree with. Other than a two day "multiplication boot camp" I run every year because my district thinks it's a good idea to introduce GCF, LCM, and fractions with unlike denominators before my students have ever officially done multiplication, I don't spend time in class working on facts... similarly to how most third grade teachers aren't spending much time working on phonics. But if students don't come in KNOWING their facts reasonably fluently... third grade will be similar in difficulty to them coming in not knowing how to decode.
     
  41. TamiJ

    TamiJ Virtuoso

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    Mar 16, 2014

    AJR, I think you have a lot to offer here on A to Z given your experience. I do believe there is a better way to teach math, though I don´t pretend to have all the answers. I will say that making math a little more open ended through math talks and math journals where students can illustrate their thinking has seemed to help. I also agree that formulas have no real value if one does not understand what they should do with it, or why they are using it. Making mistakes and looking at why those errors don´t work have helped my students as well to see where they went wrong (in my class we embrace mistakes because we know that´s when learning occurs) in their thinking.
     
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