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Re: Ratsense
Posted:
Aug 17, 2013 2:51 PM



Back when I reviewed the Shute book on algebra, it did a much better job than Dolciani using intuitive understanding (my version) to lead into formal understanding. It had a better balance of the two and played them against each other more nicely. The Dolciani book was clumsy in this regard, introducing formality too soon and often with little or no air time to the accompanying intuitive notions.
I would say that the majority of the trouble with students not making it far in mathematics lies with their lack of intuitive understanding (my version). Intuitive understanding is not innate and it is certainly not the same thing as concrete understanding (which is innate). So, how do we make students more intuitive? I think we have to make them think more. But that requires that we challenge them more and make the quiz and test problems harder which then will cause the grades to be lower. I have thought of eliminating grades in elementary school but human nature being what it is, I think that would just backfire. I think the answer is to schedule elementary school in 3 sections, reading/writing, math and all the other stuff (science, history, etc). And students would remain in these classes for multiple years and move on to the next level when they are ready. This can buy us a year or two without the stigma of normal retention (where the kid is held back entirely for a year). Because kids are in these classes for 2 or 3 years, it isn't that noticeable when a kid moves on to the next class a year early or a year late. And each section is different. A kid may advance a year ahead in math but remain where they are in reading.
By middle school we are then in the normal block pattern and can focus on the students' strengths and by high school there should be many paths available, including non academic paths.
Bob Hansen
On Aug 17, 2013, at 1:58 PM, Robert Hansen <bob@rsccore.com> wrote:
> By the way, I thought of these three "understandings" as... > > Concrete Understanding (1, 2, and 3) > Intuitive Understanding (4) > Formal Understanding (5) > > Can I vote that this is how we label these things going forward? > > Bob Hansen > > > On Aug 17, 2013, at 1:48 PM, Robert Hansen <bob@rsccore.com> wrote: > >> >> On Aug 17, 2013, at 1:21 PM, "Louis Talman" <talmanl@gmail.com> wrote: >> >>> You are scamming yourself here. "An intuitive understanding" (the words Dehaene used) isn't the same thing as a "formal understanding". It is the step from intuitive to formal that kids have trouble making. >> >> Fine, in the interest of time, I will use your terms. >> >> 1. When a 5 year old says that order doesn't matter it is because it doesn't. What else are they going to say? >> >> 2. A 5 year old picks the larger number first because it is easier. That it works is simple. Order doesn't matter. >> >> 3. The "intuitive" understanding represented in (1) and (2) above isn't understanding at all. It is simply the way things are and thus we are conditioned to these truths. >> >> 4. Understanding goes beyond recognizing that order doesn't matter. It begins when the child feels that order CANNOT matter in addition. In other words, the child has some justification in their head, beyond concrete examples that commutativity falls out from what addition is. I vote we call this just "understanding". >> >> 5. There is not a big step at all from (4) to "For every a and for every b, a + b = b + a". (formal understanding). >> >> 6. There is an infinite step from (3) to (5). >> >> Is this better? >> >> Bob Hansen >> >>> >>> Most kids can tell you that it doesn't matter which order you add a pair of numbers in. There's the intuitive understanding. And you haven't dealt at all with Dehaene's observation that kids, by the age of five, already intuitively select the larger of two numbers to begin counting from when they add by counting. If that doesn't demonstrate a grasp of commutativity, I don't know what would. >>> >>> But it's a big step from that simple understanding of the commutativity of addition to an understanding of the statement "For every a and for every b, a + b = b + a" \it{and its connection with the simple, intuitive statement.} One of the conceptual difficulties here is the formal notion of a fixed, but *unspecified,* number. That notion appears twice in the "For every" statementpossibly even three times, if we think about the sum itself. That notiona fixed but unspecified numberis one of the biggest hurdles kids have to pass in going from arithmetic to algebra. >> >



