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Topic: Re: It Works for Me: An Exploration of 'Traditional Math'
Replies: 11   Last Post: Jan 21, 2008 10:55 AM

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Fred Siegeltuch

Posts: 19
Registered: 10/7/05
Re: It Works for Me: An Exploration of 'Traditional Math'
Posted: Jan 18, 2008 2:15 PM
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But what if you think of the square root function as a process? Then becoming familiar and comfortable with it is more than language. The issue then becomes is it being taught in such a way as to provide a richly connected context that it will be integrated with other knowledge.

Fred Siegeltuch

Alain Schremmer <> wrote: I am not sure I understand what's with "the meaning of […] square
root" and, in fact, I think that this raises an important distinction.

There is the real world and there is the language we use to represent
the real world.

Given the sentence

Jack is the brother of Jill

it alludes to a real-world situation and may or may not be true
depending on what the actual real-world relationship between the
individuals whose names are Jack and Jill is. The point here is that
we can allude to exactly the same situation with the dual sentence:

Jill is the sister of Jack

and both will be true or both will be false. Similarly,

the cat chases the mouse


the mouse is chased by the cat

allude to exactly the same real-world situation.

Again similarly,

9 is the square of 3 and 3 is the root of 9


12 is double of 6 and 6 is half of 12

allude to exactly the same real-world situations.


1) The issue is a syntactic one, much related to the "passive voice"
and I think that the "meaning of square root" is no more to be
discovered than the meaning of "table" or "tree". While the
linguistic aspect of mathematics is crucial, this is absolutely not
specific to mathematics and common to any trade in which a
specialized language is required. And, indeed, this aspect is
completely ignored by mathematics teachers as well as English teachers.

2) This type of focus, such as on "the meaning of […] square root",
is detrimental to an organized understanding of the language.
Moreover, it reinforces in the students the belief that mathematics
in particular and sciences in general are just collections of facts
and topics.


On Jan 18, 2008, at 9:54 AM, Edward Laughbaum wrote:

> All,
> I think Laura has captured the essence of what many of us do.
> However, several items in her list I would describe as
> investigations, explorations, or projects, which to me are not
> necessarily meant to lead a student to a mathematical generalization.
> A few weeks ago, someone on this list (sorry I can't remember who)
> said that there is a place for memorizing the meaning of, for
> example, square root. Below is a guided discovery activity that I
> would use as homework before I would discuss square roots in class
> and diminish the need for memorizing the definition, but rather,
> focus on meaning. DISCLAIMER: I am on vacation, so I did not want
> to spend lots much time on the activity. So it may be incomplete.
> But I think the idea can be discerned.
> (sqr will imply the square root sign)
> The goal of this activity is to discover the meaning of square
> root. It takes 4-5 minutes, and can be an individual assignment or
> group.
> Directions: Use your graphing calculator to find the missing
> values. (I suggest the graphing calculator so students can see the
> table being created on the home screen.) It is important that you
> not think of the calculator as an "answer getter," but instead look
> for a pattern in the answers that may lead you to a conclusion
> about square roots. (Note, this comment is not used with students
> who are used to looking for patterns because the teacher has always
> emphasized pattern recognition as well as correct answers.) Sorry
> if the formatting is destroyed through email.
> # | sqr(#) (if your students
> know the meaning of a variable, replace # with x)
> ________________________
> 4 | ____ How is your answer
> related to the number 4?
> ____________________________________________________
> 9 | ____ How is your answer
> related to the number 9?
> ____________________________________________________
> 16 | ____ How is your answer
> related to the number 16?
> ____________________________________________________
> 25 | ____ (at this point the
> average brain has made a conjecture on the meaning - right or wrong)
> 49 | ____
> Describe, as best you can, the meaning of sqr(#), where # is a
> positive
> number._______________________________________________________________
> _____
> Try the following:
> 20 | ____ How is your answer related to
> the number 20?____________________________________________________
> 54 | ____ How is your answer related to
> the number 54?____________________________________________________
> 6^2 | ____
> Describe, as best you can, the meaning of sqr(#), where # is a
> positive
> number._______________________________________________________________
> _____
> Without using a graphing calculator, make a conjecture about sqr
> (-4).________________________________________________________
> ========================
> This question is left open, as it will be analyzed in class the
> next day, as will related material. Those students who made
> incorrect conjectures will have a chance to discuss their thinking
> in class. This activity is also used as a device to prime the
> unconscious learning module and will decrease conscious learning time.
> The table format is used because we will connect this to the square
> root function at a later date.
> Regards,
> Ed
> =======================================================
> At 06:02 PM 1/17/2008, Laura Bracken wrote:

>> I have much better performance from students, both in initial
>> understanding and in retention, when I frequently use "guided
>> discovery activities."
>> I agree with Ed that working from pattern recognition to
>> generalization is an effective strategy. And this is one reason
>> that guided discovery works. But I think that there are other
>> things in play here, too. Like most ed research, it is darn hard
>> to figure out what is working for which students.
>> 1. Guided discovery learning requires students to be active rather
>> than passive. Students tend to learn better when they are paying
>> attention and not daydreaming.
>> 2. In collaborative group settings, students discuss their
>> mathematical insights and understandings with each other in common
>> language -- that helps them build mathematical knowledge,
>> vocabulary, and notation onto the understandings that they already
>> have. When I write an activity, I often ask students to explain a
>> concept or outline a procedure in writing. There is such a
>> dramatic difference between being able to look at something and
>> say "yeah, that makes sense, I get it" and explaining it.
>> Sometimes as I am looking at their work, I'll ask a student to
>> read their response out loud -- this helps them think through what
>> they've said and figure out what is missing.
>> 3. If students feel that the group situation is "safe," they will
>> often ask questions that they would not ask in front of the whole
>> class. They are far more willing, in my experience, to guess and
>> make mistakes.
>> 4. In my role as coach, I get to know students faster and at a
>> more personal level than when I am lecturing. Students ask for
>> help more often. Students also get to know each other in the
>> class. I think that these personal connections help students to
>> decide to go to class (improves attendance and persistence). They
>> also realize that they are not the only one who is struggling.
>> Developmental students can have pretty good poker faces, even when
>> they are clueless.
>> 5. As students get to know each other, they often form study
>> groups. They talk about things such as "did the math lab help
>> you" and "who is your favorite tutor." They help each other with
>> homework questions before class. More opportunities to learn math.
>> 6. As we all get to know each other, the classroom is more
>> relaxed and friendly. There is a lot more audience participation
>> in whole-class lecture situations as well. Many of my students
>> have had negative experiences in front of a class, whether at the
>> board or being asked a question by the teacher. Especially at the
>> beginning of a term, I don't call on students by name. I ask
>> questions and let the bolder ones reply. In time, the chorus gets
>> much louder.
>> Of course, these activities are not easy to write. I've written
>> some real duds over the years. I can't always predict how
>> students will perceive a situation or interpret a question. But,
>> I would never go back to just lecturing and asking questions. I
>> just enjoy watching them work and think too much.
>> Laura
>> -----Original Message-----
>> From: on behalf of Alain Schremmer
>> Sent: Thu 1/17/2008 3:10 PM
>> To: Edward Laughbaum
>> Cc:
>> Subject: Re: It Works for Me: An Exploration of 'Traditional Math'
>> On Jan 17, 2008, at 10:18 AM, Edward Laughbaum wrote:

>> > Mike, Alain, All,
>> >
>> > IMHO:
>> > I often work with two-year college professors and high school
>> > teachers, and find similar approaches as described by Mike.
>> > However, one needs a very definite mathematical outcome for guided
>> > discovery to work well.

>> Yes and no and this depends on what the desired "very definite
>> mathematical outcome" is.

>> > This has a tendency to lead to a correct generalization. As for
>> > Alains' concern about the concept of guided,

>> Not exactly, I just wrote that " I am not sure "guiding" is quite the
>> right word but then it might be that, for me, the term has bad
>> connotations." What did you have in mind?

>> > the way I look at this is that guided discovery is the good choice.
>> > It may not always work, but we cut the odds of it not working by
>> > knowing our goals, having a mindful teaching experience, and an
>> > understanding of how the brain functions relative to learning. The
>> > facts are that the brain has a pattern recognition functionality
>> > ("The cortical areas corresponding to these buffers (short-term
>> > memory) project to the striatum, part of the basal ganglia, which
>> > we hypothesize performs a pattern-recognition function." Anderson,
>> > J. R. et. al. (2004). Psychological Review. Vol. 111, No. 4,
>> > 1036-1060). Further, the brain generalizes constantly. (. the
>> > brain's capacity to generalize is astonishing. I have previously
>> > suggested that there are two main modes of thought-logic and
>> > pattern recognition. . the brain can function by pattern
>> > recognition even prior to language... " Edelman, G. M. (2004).
>> > Wider than the sky: The phenomenal gift of consciousness, 38-39,
>> > 47. Yale University Press. New Haven, CT.)
>> >
>> > So the guided discovery activities are short, and use pattern
>> > building to lead to a specific concept generalization. On the other
>> > hand, I had a very prominent mathematics teacher educator suggest I
>> > was wrong in my use of guided discovery activities because every
>> > brain is different, and so I was misleading some students. In my
>> > defense, I would argue that brains are mostly different in content.
>> > But every brain does pattern recognition. Every brain generalizes.
>> > (*unless maybe on drugs, has a mental disease, or has been injured)
>> > We simply do our best at knowing where our students are, and where
>> > we want to take them. And use guided discovery to develop a pattern
>> > our students can use to generalize.
>> >
>> > So I continue to ignore his advice that I not use neuroscience
>> > research results to enhance teaching and learning.

>> As skeptical as I usually am about "educology", I would be interested
>> in pursuing this a bit. In particular, I would like to know what
>> "guided discovery activities" are. (I really don't know.)
>> By the way, the fact that a "very prominent mathematics teacher
>> educator suggest[ed] [you were] wrong in [your] use of guided
>> discovery activities because every brain is different, and so [you
>> were] misleading some students." is quite typical of the lack of
>> logic in Educologists, prominent or not. By that "suggest[ion]",
>> there is nothing that I can do since nothing I can do is going to
>> work for all my students.
>> This "suggest[ion]" reminds me of the time when I had written a text
>> for a course based on Model Theory (See AMATYC Review, Fall 2000) as
>> an alternative to the Mathematics for Liberal Students course usual
>> in the 1970s. The question eventually arose as to which of the two
>> course we ought to recommend to the students. One of my colleagues,
>> while admitting that by any factual standards the Model Theoretic
>> Introduction to Mathematics was a much better course, insisted that
>> since "it wasn't perfect" it could not be recommended in place of the
>> Mathematics for Liberal Students course even though the latter
>> "wasn't really working". I wasn't the only one to be somewhat stunned
>> but the colleague went on to become one of the only two full
>> professors in the department (Disclosure: I am not the other one.)
>> Clearly, whatever "mathematical outcome" we want our students to
>> achieve, we cannot just lecture and give the final exam (as used to
>> be the case, though, once upon a time). So, one way or the other, we
>> must affect the students' activity while learning.
>> The question then, and I would be very interested in a discussion on
>> this question, is how to do that with a long term effectiveness-but
>> in a non-intrusive manner.
>> Regards
>> --schremmer
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