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No one would deny that establishing the validity of ideas is critical to mathematics, both for professional mathematicians and for students. But how do people establish "truth"; how can they prove things? According to Martin and Harel (1989), in everyday life, people consider "proof" to be "what convinces me." Most mathematics instruction and textbooks, however, lead us to believe that mathematicians make use only of formal proof -- logical, deductive reasoning based on axioms.
But mathematicians most often "find" truth by methods that are intuitive or empirical in nature (Eves 1972). In fact, the process by which new mathematics is created is belied by the deductive format in which it is recorded (Lakatos 1976). In creating mathematics, problems are posed, examples analyzed, conjectures made, counterexamples offered, and conjectures revised; a theorem results when this refinement and validation of ideas answers a significant question. Hanna (1989) argues that because mathematical results are presented formally by mathematicians in the form of theorems and proofs, this rigorous practice is mistakenly seen by many as the core of mathematical practice. It is then assumed that "learning mathematics must involve training in the ability to create this form" (pp.22-23). The presentation obscures the mental activity that produced the results.
In fact, according to Bell (1976), personal conviction grows out of internal testing and forming a judgment about whether to accept or reject a conjecture. Later, one subjects this judgment to criticism by others, presenting not only the generalization formed but evidence for its validity in the form of a proof. For a mathematician, often this internal testing can take the form of proof as one attempts to perform the socially accepted criticism of one's argument.
In sum, formally presenting the results of mathematical thought in terms of proofs is meaningful to mathematicians as a method for establishing the validity of ideas. However, does proof convince students? Do they see it as a way to establish the validity of their ideas or, as Hanna (1989) suggests, as a set of formal rules unconnected to their personal mathematical activity?
Let me guess.
No?No, students do not see proof as a way to establish the validity of their ideas?
Is that it?
ConclusionI had a feeling.
Ironically, the most effective path to engendering meaningful use of proof in secondary school geometry is to avoid formal proof for much of students' work.
By focusing instead on justifying ideas while helping students build the visual and empirical foundations for higher levels of geometric thought, we can lead students to appreciate the need for formal proof. Only then will they be able to use it meaningfully as a mechanism for justifying ideas.Only then, after sophomore year has come to an end and so has geometry.
Geometry and Truth
by Michael T. Battista and Douglas Clements
Here's a question.
How many sophomores in high school have mathematical ideas?
15 comments:
I think that you are falling into the trap of thinking that just because TERC is a bad curriculum (I don't really know what it is, by the way) that everything the TERCers say is risible.
To me this all rings basically true, albeit expressed in a bloated manner. You often need to have an intuitive understanding of something in mathematics before a formal proof can make sense. Sometimes a formal proof helps you understand why a mathematical truth is so, and sometimes you need the intuition to actually understand the proof.
"Only then, after sophomore year has come to an end and so has geometry."
Yup. Them students aren't quite ready for real work yet. They need more time for "appreciation". Maybe they can figure out some fun in-class games for a real indirect route. It couldn't be that algebra gave them enough appreciation by requiring them to master careful, step-by-step solution of equations.
In fact, let's just turn all of high school into in-class group projects that focus on concepts and appreciation. That will get all students so primed up that college will be a breeze.
Actually, the Glencoe Geometry book that I will use with my son next year has only one chapter on proofs. This is now typical, for better or worse. The authors raise an issue of formal proofs, but they don't define it in terms of the full typpical course in geometry. They are just using formal proofs as a way to lobby for some other kind of geometry, such as using the Geometer's Sketchpad.
No thanks! This is another case of misdirection. They talk about better, but what they really mean is different.
For me, I'll take the transformation matrices.
You often need to have an intuitive understanding of something in mathematics before a formal proof can make sense.
I don't know enough about it to have an opinion -- and I wouldn't be surprised to learn that intuitive understanding of mathematics is part and parcel of writing a proof!
What I object to is the argument that child novices should do what adult experts do.
That will get all students so primed up that college will be a breeze.
13 years of priming.
Oh really .... I don't know where to begin!!!!
"establishing the validity of ideas is critical to mathematics"
I know straight away that my blood pressure will need to be checked by the time I get to the end of this!!!
But wait, there's more!!!
"Most mathematics instruction and textbooks, however, lead us to believe that mathematicians make use only of formal proof -- logical, deductive reasoning based on axioms."
Of course, that should read "Most mathematics instruction and textbooks AND ALL MATHEMATICIANS ..."
"evidence for its validity in the form of a proof"
By this stage, it's pretty obvious to me that the author isn't a mathematician. "Validity in the form of a proof" ..... what other type of validity is there??
Of course, the trained mathematician should have their 'proof by obfuscation' alarm bells ringing by now.
"For a mathematician, often this internal testing can take the form of proof as one attempts to perform the socially accepted criticism of one's argument."
Is this even English?
"However, does proof convince students? Do they see it as a way to establish the validity of their ideas or, as Hanna (1989) suggests, as a set of formal rules unconnected to their personal mathematical activity?"
They'd better see it as (ahem) "a way to establish the validity of their ideas" or their teacher hasn't really communicated the difference between Science and Mathematics too clearly.
Worthy of mention is the desire to "convince students" .... you may accuse me of semantic nitpicking here, but it's important!!
"Ironically, the most effective path to engendering meaningful use of proof in secondary school geometry is to avoid formal proof for much of students' work."
Which roughly translates as "In order to save the village we had to destroy it!"
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I think Melanie Philips (British author) summed it up best in her book 'All Must Have Prizes':
"A fundamental shift in emphasis from knowledge transmitted by the teacher to skills and process 'discovered' by the child has undermined the fundamental premises of mathematics itself. The absolutes of exactness and proof on which the subject is based have been replaced by approximation, guesswork and context."
Melanie Phillips, All Must Have Prizes
Of course the 'mathematician' tinkers with their ideas before writing a formal proof! Nobody ever sat down and wrote a proof, straight up; not Einstein, not Pythagoras, not Pascal, not anybody. But, you can be damn well sure they knew their tinkering had to lead to one and this knowledge is what drives the tinkering itself. Tinkering in math is, by definition, the act of building a set of provably correct truths that are the outline for the proof.
I know Terc. There primary domain is elementary school. Proofs are not taught in elementary school anyway so this whole argument is a straw man set up to validate their approach to geometry which is lacking in rigor. It's this rigorless (rudderless) geometry that they hope to foist and quelling the strawman throws a veil over their approach.
Here's an example. Starting in first grade they introduce shapes; triangle, square, hexagon, circle, etc. It's not until much later (sixth grade for most kids) that a serious discussion of angles takes place and in all of Investigations (their premier offering) I don't know that they ever define a line or point.
If you're serious about learning geometry, whether you call it a proof or not, you should at least learn things in an order which provides for building a meaningful progression of concept and the vocabulary to accompany it. Much of what Terc does is upside down like this, especially in geometry. This of course allows states, school districts, and administrators to tout their new and improved math which miraculously has kids doing 'geometry' in the first grade.
Unfortunately, teachers who get Terc kids in middle school have to unteach much of what they have inappropriately conceptualized by the time they arrive on their shores.
To conflate the thinking of mathematicians with the thinking processes in elementary school is disingenuous at best and misleading at it's worst.
Before you can say to a student that a polygon is comprised of line segments you had better be sure you've had a conversation about what a line segment is. Otherwise you're just standing in the room going blah, blah, blah, like Charlie Brown's teacher. If you don't explicitly make such connections in your teaching, instead relying on a child's discovery processes, you'll get kids with no sense that their even is a hierarchy to what they are about. People who disparage this as a proof, have little understanding about math and how to teach it.
You can see why it becomes important to use a fuzzy math middle school program so that these omissions of learning remain largely undiscovered.
Is there any other country in the world that has to develop terms like "Authentic Algebra" and "Pretend Geometry" to explain the nonsense that is being pushed as math in the classroom.
Steve H -
Can you get a copy of Jurgensen's Geometry? It's part of McD Littell's Structure and Method series and it is excellent. It has as much proof work as you could desire for your son, wonderful, interesting word problems (one of the authors is with NASA), and incorporates Algebra review into Geometry at regular intervals. I think you and anyone else supplementing Geometry will be pleased with this resource.
I'm working with my daughter in geometry, using Moise-Downs "Geometry" textbook. Math is not her favorite subject, but she is enjoying geometry so far. She likes the formality of things taken for granted. M-D Geometry is very proof-oriented, and does it in a way that it introduces students to what proofs are about. In the early chapters it shows three addition problems, like 9+5 = 14, 5+7 = 12, 3+7 = 10 and notes that the addends are odd and the sums are even. It then asks if the student can think of any two odd numbers for which the sum is not even? My daughter thought for a bit and said "no". It then asked if that proved that the sum of any two odd numbers is always even? She said "No, because you haven't added all the possible odd numbers." This is correct. And a good introduction to some of the aspects of proof. The book will keep upping the ante with questions about proofs, and by the time they get to the chapter on simple proofs of congruence, the students have a fairly good idea of what proofs are for. I don't see anything wrong with this.
"Jurgensen's Geometry"
"Moise-Downs 'Geometry'
Thanks! I know these were mentioned before, but I never wrote them down. I've been meaning to do my homework on what to use for supplementation. I remember doing two-column proofs in school, but I always liked the application end of geometry more than the proof end.
My whole career centers on geometric modeling and computer graphics. You might think that I would like a Sketchpad approach to geometry, but I don't. It might be good for a homework assignment or two, but not much else. I will, however, make sure my son knows about vectors, transformation matrices, and parametric equations.
boy - lots of great stuff here - I want it ALL posted up front!
Quick question:
Barry: can I use Moise and Downs for me, do you think?
Everyone: what do you think of the ALEKS geometry course? C. and I are doing it now - and C. will be using the Jurgenson text at Hogwarts next year.
I'll see if I have the IBSNs for Moise & Downs & for Jurgenson handy.
I also have possibly the only remaining copy of the Moise & Downs solution manual in captivity. I don't know what to do with it; for several years now I've had it here on my desk with my most-treasured books, where I can see it.
I can see it now.
Somebody has to tell me how to get it scanned.
Google Books, maybe?
But I hate to let it out of my possession.
"establishing the validity of ideas is critical to mathematics"
I know straight away that my blood pressure will need to be checked by the time I get to the end of this!!!
RIGHT!!!!!
bky & Paul are right (I presume) - mathematicians, like everyone else, tinker with ideas, use intuition & "implicit memory" (which may amount to the same thing), etc.
But the entire thrust of this passage approaches the "math is everywhere, math is dance!" school of thought.
"No one would deny that establishing the validity of ideas is critical to mathematics"??
Why would you say that?
You would say that because in fact many from the TERC corner of the world would deny this, which we can infer from the fact that these authors, in the very next sentence, enclose the word "truth" in quotation marks.
"math is everywhere, math is dance!"
Even dance is a discipline that requires training and rigorous practice. Students of ballet don't intuitively "find" Vaganova training all on their own. It takes hours of training under the guidance of an expert intructor who follows a meticulously crafted and tested syllabus to develop a ballerina who makes all that work appear so effortless.
To be good at most anything in life requires hard work. Period.
Barry: can I use Moise and Downs for me, do you think?
Yes; it's very readable and the explanations are top notch. They take their time getting to proofs; congruence proofs first appear on p. 100 for students to do. But the lead up is worth it; they've actually given you quite a lot of info about proofs without you realizing it. Once they get to the congruence proofs, then the rest of the course is very much proof-based. Like watching a Buster Keaton movie; the first part is all set-up, and then when the action begins the build up is all very much worth it.
I did a video on the distinction of discovering new math and the form it's presented/taught/published.
http://www.youtube.com/watch?v=re_Xv1SbTcY
I think it's important to let the students know that exploratory math/looking for new results is a different process from how one presents it.
Perhaps I need to do a followup to say that no one wants to see the slapdash exploration, especially as that can lead you into error. Hmmm.
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