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  1. #21
    Senior Member Gabe's Avatar
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    Quote Originally Posted by Orangey View Post
    I agree with this. Perhaps there should be a supplementary history of math (or math principles, whatever you want to call it) to go along with regular math. This way you could understand exactly why and how the concepts that are so familiar were thought up and proven.

    At my university they had a special set of math classes for elementary education majors where the instructor would make them go back through the principles of the basic elements of math (+,-,x, etc...) so that they could understand them thoroughly. It was amazing to see people who had taken advanced calculus courses struggle to understand why the basic division algorithm works (not the procedure, but the concept). Of course, they may have just been a dumb class.
    That doesn't suprise me at all. I bet barely anyone knows how the division and multiplication algorithms work. Personally I think they should teach those in 8th grade.

  2. #22
    Senior Member edcoaching's Avatar
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    I actually train math teachers to differentiate lessons for S and N learners and am involved in a research project filming students doing fractions tasks to document the very different approaches S and N learners take.

    • The algorithm vs. constructivism approach (traditional curriculum vs. Chicago Math and Investigations) is very much an S-N battle
    • A majority of elementary teachers prefer Feeling and are math-anxious (Many Feeling types love math but math anxiety tends to be way more prominent with F's) and spend as little time as possible wiht math instruction
    • To use the new curriculum formats, you have to recognize when varying approaches still will lead students to understanding and when they are off track. This is really difficult for math-anxious people who memorized to pass classes rather than understanding
    • The "spiraling" was meant to mean, "We do fractions every year" etc. But it's become an excuse, as in, "If the child doesn't get it this year, there's always next year..." which is disastrous for Sensing students since a consequence for many is, without a main idea to pin what they've learned to, they forget it all.


    Type REALLY helps teachers see their blind spots and why students aren't learning the way they've been teaching. I'd gladly share an article written for lay people from the NZAPT Bulletin on what we found on the first round of filming students if you want to know more...
    edcoaching

  3. #23
    Strongly Ambivalent Ivy's Avatar
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    edcoaching, I'm glad that someone is familiar with Investigations. It has definitely struck me as an S/N debate, and as usual, I'm straddling the fence. Why not learn a multitude of ways to go about approaching math? (Actually that's kind of what the Investigations curriculum purports to do, at least from my parental perspective-- I've been mostly very impressed, and I don't mind supplementing a bit myself.)

    Edit: PS, ygolo, that's pretty brilliant stuff IMO. Wish I had had more math teachers like you.
    The one who buggers a fire burns his penis
    -anonymous graffiti in the basilica at Pompeii

  4. #24
    Senior Member edcoaching's Avatar
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    Quote Originally Posted by Ivy View Post
    edcoaching, I'm glad that someone is familiar with Investigations. It has definitely struck me as an S/N debate, and as usual, I'm straddling the fence. Why not learn a multitude of ways to go about approaching math? (Actually that's kind of what the Investigations curriculum purports to do, at least from my parental perspective-- I've been mostly very impressed, and I don't mind supplementing a bit myself.)

    Edit: PS, ygolo, that's pretty brilliant stuff IMO. Wish I had had more math teachers like you.
    Like all subjects, math requires S and N for proficiency--you're spot on. Kids who lack math confidence, though, need to get grounded through techniques that use their own styles. If teachers really understand the concepts (the countries with the best math scores concentrate on this in teacher training, as was suggested somewhere in this thread, something we really work on in our trainings) then they recognize what the students need. S students need more time with manipulatives that make sense to them; a lot are poorly chosen because the teacher doesn't get what they're teaching. N's need some room to roam.

    But...teachers teach from their strengths. Those strengths come from their own learning style, which is tied to their type preferences for gathering information and making decisions. Without a neutral framework like type to help them understand what diverse students need, teachers tend to pick and choose from curriculum, leaving their opposites high and dry.

    And if the teacher is math phobic...well, one example from the Everyday Math (Chicago) is, the drill is embedded in some of the games in the early grades. And the math phobic teachers say, "I can skip the games, they're just fluff" and rob Sensing students of the chance for mastery (and many INtuitives as well...)

    It's really fun to watch the proverbial scales fall from teachers' eyes when we do type alike math exercises--or when they watch the films of students that demonstrate what they need and where their misunderstandings come from...
    edcoaching

  5. #25
    Strongly Ambivalent Ivy's Avatar
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    You wouldn't believe how many school administrators and teachers I've encountered who seem to think differentiated instruction is like gnomes and fairies and unicorns--nice to think about but completely unrealistic. I don't accept that.

    Wish we could bring you to our school for a seminar!
    The one who buggers a fire burns his penis
    -anonymous graffiti in the basilica at Pompeii

  6. #26
    Senior Member edcoaching's Avatar
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    Quote Originally Posted by Ivy View Post
    You wouldn't believe how many school administrators and teachers I've encountered who seem to think differentiated instruction is like gnomes and fairies and unicorns--nice to think about but completely unrealistic. I don't accept that.

    Wish we could bring you to our school for a seminar!
    That's because most differentiation texts either having you differentiate for all levels of Bloom's Taxonomy, 3 ability levels (even though "all students can learn" , and multiple intelligences, resulting in a matrix of about 100 boxes to fill in during planning. Or, it's all little tricks that don't get at student needs.

    With type, if teachers start with their own style and teach to their opposites, they can actually reach all students to a far greater degree. That makes sense to them and is worth trying. Give them a 2-day seminar and they become true believers...
    edcoaching

  7. #27
    Senior Member edcoaching's Avatar
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    Quote Originally Posted by ygolo View Post
    G
    1) Give students a fairly involved problem that they will struggle to solve without knowledge of a particular concept (like integration, for instance), and have them attempt to solve it. Some may even succeed, but either way they will have a deep insight into the concept about to be presented.

    2) Then place them in groups, and have them try to solve the problem as groups. Encourage them to be creative and to transform the problem in to equivelent ones, or approximate ones, or smaller ones, etc. Have them go at it as if they were trying to solve a puzzle. Many more will likely solve the problem, without the aid of the concept being taught. Keep track of the approaches the students take that are most promosing, or interesting. Have them discuss/present the most interesting approaches.

    3) Develop the concept from its founding principles. This is the usual exposition. Then try and relate the theoretical foundation to the approches students took on the problem. The teacher will likely trip up in several spots trying to do this. That's OK, and actually prefered; it shows that even proffesionals have to struggle with math sometimes, and shows how they struggle with it. Point out that there "maybe some connection here or there." But then go back and summarize the core theoretical concept concisely. Don't show them the solution to the orignal problem, yet.

    4) Place them back into groups, and have them have one last go at the orignal problem, now armed with the concept just presented. For those who did just understand the concept may marvel at the "magic" of using the new concept in making the problem easier.

    5) Present theoretical solution to the problem. Take questions for clarification, etc.

    6) Quiz/Test the individuals on many similar problems in different contexts to the one solved. If a review of the concept is needed, then repeat steps 1-6 and a different appropriately chosen problem.

    7) Quiz/Test individual on extremely diverse sets of problems requiring the same concept. If a review of the concept is needed, then repeat steps 1-7 and a different appropriately chosen problem.

    To have time for this approach (which I estimate to be 3x-4x normal exposition), give students a short time to prepare for a pretest of review information, and only allow those who "pass" to take the class.

    Sorry, but without the foundations, going further is often futile.
    This is really right on the track of what the "reform" curriculum try to do. It can take quite awhile to get the kids to believe they can figure things out though, since they're so used to the "Watch me and do what I do" approach to math. What's really cool is when the teacher gets the students involved through a "Math Congress" where the different groups can report what they've learned.

    If you pick a great problem, kids can even zoom at their knowedge levels. Like with fractions, I've seen kids divide up subs. THe kids who still are trying to get what a fraction is divide each one by the # of kids--3 subs, 7 kids, give each kid 1/7 of each. The kids who get it go right to 3/7 but they all can work on it (actually the real problem involves fair/unfair...if 3 kids get 4 sandwiches, 4 get 5 and so on is it fair? This is a huge concept if you're in 5th grade).

    And yeah, the time is so huge. any ideas for making room for it?
    edcoaching

  8. #28

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    Quote Originally Posted by edcoaching View Post
    And yeah, the time is so huge. any ideas for making room for it?
    Wow! I guess I have been ignoring most of the forum during the Gauntlet.

    I think, the main way to make time for this is to be very strict about mastery of lower level concepts before moving on to more advanced things. This way, we don't waste too much time on review, and the students are more likely to succeed.

    This a has a few drawbacks in that what is "more advanced" may be subjective. That's why I believe math curriculum has to be made more "modular." Where each of the basic concepts can be built up on their own in rather expedient ways.

    Also, I would cut out some of the "drilling" that is done (though not all of it). We live in an age of computers (that are usually not human). We ought to teach estimation, "number-sense," and good ways to perform "check-sums" instead. These things take a lot less time than actually memorizing things, and making the math automatic. At lower levels, we still need drilling, but I am not sure how much is enough.

    Also, I can't stress how important it is to show people that it is OK to screw-up in math (as it is in anything), and to show how professionals go about checking/correcting mistakes.

    In some ways, a bumbling teacher/professor that the students often correct is a good thing. But the trick is finding a bumbling teacher who is actually competent, and can maintain the trust of the students.

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  9. #29
    Senior Member celesul's Avatar
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    Students still need to have some sense of basic math, and be able to do a lot of it in their heads or on paper, but math facts were torture for me. If I hadn't had a few amazing math teachers later, I'd hate math. Also, I didn't even learn my math facts then. I learned my math facts when I was in Algebra I and Geometry because I never remembered to bring my calculator to class. To this day, I'm extremely thankful that I did, because my math teacher has a no calculator section of every test. For me, it would have been best to spend more time on the concepts, perhaps even starting Pre-algebra very early, but spending a long time on pre-algebra through algebra, and teaching arithmetic at the same time.

    That said, I adore my math class right now. I was in Pre-calc last year, and we started calculus (we were learning differential calculus) and I was enjoying myself so much that I figured out some of the concepts before they were taught, because I was playing with the numbers after I finished my classwork. Teaching math as a set of patterns and rules, like a game, instead of equations to memorize makes a huge difference though. Thus, the variability of the enjoyability of math class based on the teacher. (math and history are the most variable that way, in my experience)
    "'You scoundrel, you have wronged me,' hissed the philosopher. 'May you live forever!'" - Ambrose Bierce

  10. #30
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    I don't dislike the way math is taught more than any other subject... I actually had a fun calculus class that left all of the learning to us and classes were basically just work time for people or a chance to ask questions about things they didn't understand. And unlike other classes, we went into the history of the mathematical theories and how people came up with them... that was the best part for me, because it made more sense then, and rather than just knowing how to do a problem I got why it had to be that way.

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