In designing a new corner cabinet for our family room, my family and I had to figure out how deep to make it so that the TV we currently have would fit. We want the new cabinet to be the same length on each side (along the two walls). Here is an overhead view:
How long should each side of the cabinet be? Show all of your calculations and explain how you approached and solved this problem.
source:
Exemplars K-12
And you thought kids hated math now.
Catherine, I rather like this problem! I wouldn't be upset to see this on an assessment. To me it's just a run-of-the-mill "story problem."
ReplyDeleteI would object, though, if the teacher wheeled in a couple of AV carts with a TVs on them, and told the kids to get into two groups and push the things into a corner, get out the tape measure...you get the idea.
I didn't read the whole thing - it was endless - but the idea seemed to be to use Geometer's Sketchpad to solve it.
ReplyDeleteThe math people around here will have to tell me this: the concept behind performance based assessment seems to be that all assessment should be done in the form of word problems.
ReplyDeleteIsn't that "wrong" - wrong in the sense that you've turned math ed into one big Math Olympiads?
Also, performance based assessment, from what I can tell, leans heavily on low-level statistics, probability, and find-a-pattern type questions.
If they really turned it into Math Olympiads that wouldn't be too bad at all.
ReplyDelete"P and Q represent numbers, and P * Q means (P + Q)/2. What is the value of 3 * (6 * 8)?"
And the kids could use calculators, tape measures, or Geometer Sketchpad, Chismbop, Russian arithmetic, Egyptian fractions or whatever other "math" they are learning these days to come up with the solution ;-)
Problem from "Math Olympiad Contest Problems for Elementary and Middle Schools"
I think I remember a Ralph Raimi rant about Geometer's Sketchpad.
I think I've seen several rants about it actually from different people.
Like Vicky, I don't dislike this problem, but I hadn't studied the site.
ReplyDelete"Math Exemplars gives both teachers and administrators a way of assessing students' problem-solving and communication skills and provides classroom-tested, real-world problems for instruction. It can be integrated into your existing curriculum to facilitate both assessment and instruction."
Well, regular math classes have word problems that are graded, and unfortunately, push communication skills. So, what's the big deal?
They are pushing a product. I guess I never looked at the education market as one that's perfect for making big bucks with supplemental products and training services. Everyone has an angle and a product to sell. We all know how products can be pumped up with fluff, but I still have a hard time with this in the education world. So, when teachers talk to me about Everyday Math, am I getting what they believe as educators, or am I getting just the marketing hype. Both, I suppose. Everyone is looking to buy a solution. The big fix would be something like Everyday Math, and the little fix would be something like Exemplars. It's something with a name that they can show parents. It seems like schools would much rather talk with sales reps who tell them what they want to hear than to talk with parents who will tell them something else.
So, Exemplars can be integrated with any curriculum? Right. Just add more time to the day. If you thought that missing 35% of EM was bad enough, just wait until they add in Exemplars.
My view now is to equate things like "performance based assessment" with snake oil.
It's pretty bad.
ReplyDeleteIn fact, it's horrific.
We just got our preliminary Tri State Consortium report (Steve - you may want to take a look because I'm sure your district will be having one, too.)
I've only skimmed so far but it looks to me as if the report takes aim at the high school & middle school:
"The Tri-State team’s concern relates to the challenge to maintain the critical thinking skills that students have acquired in the K-5 program as they move into the district’s more traditional math environment. With a clearly articulated balance of integrated critical thinking skills and traditional math, the student results will be a deeper and more enduring understanding of mathematical procedures and concepts."
It sounds to me as if Tri State will say that we need to bring 6-12 math into alignment with Trailblazers, instead of bringing Trailblazers into alignment with 6-12.
The idea of performance-based assessment isn't just word problems; often it's word problems you can't do:
"Performance-based assessment involves students, working individually or in groups, solving a problem that may take 15-30 minutes up to a few days."
I'll get another post up later on.
For the math experts here: what relationship do word problems have to a coherent curriculum?
What I read over and over and over again is that the pinnacle of mathematics achievement is the "ability to solve problems."
Isn't that wrong?
How does that square with "the ability to write proofs"?
I'm asking because I don't know.
It does seem to me, though, that performance based assessment turns math into applied math.
I'm fairly hostile to Math Olympiads because that's what my district uses for the gifted kids. They get a bunch of Math Olympiad problems, and that's it. No acceleration allowed.
ReplyDeleteNow they're using material from TIMSS, I gather:
The integration of TIMSS rubrics and assignments is evident in the elementary level enrichment program. The use of the TIMSS program, an international benchmark, encourages students to engage in authentic activities that are directly connected to acquisition of higher level mathematical concepts.
It does seem to me, though, that performance based assessment turns math into applied math.
ReplyDeleteAnd success hinges on foundational fluency which in most cases isn't there. Unless the child has been heavily tutored/supplemented or is a "math brain", a problem that takes up to a few days is only a lesson in frustration.
yeah:
ReplyDeleteThe Tri-State team’s concern relates to the challenge to maintain the critical thinking skills that students have acquired in the K-5 program as they move into the district’s more traditional math environment.
The Trailblazers kids are emerging from 5th grade with critical thinking skills.
Then they hit the middle school and the concepts are gone.
What the Tri State people don't know, or don't care to know, is that the K-5 grades have outsourced "computational fluency" to parents. The 4-5 school sends home newsletters telling parents to teach their kids the math facts & giving them a deadline by which to complete this task.
From time to time (I don't know how often) they send home "challenge" problems the kids don't know how to do. The parents and their kids work on them together.
I wish I could remember the one my friend told me about (I posted it in a Comment, but don't remember which one..)
I was something about 1/2 of 1/3 - it was a fraction problem.
The kids hadn't been taught how to do it and I believe the entire grade was given the same problem.
My friend, who is good at math (and likes math), taught the concept of equivalent denominators to her kid.
She is also giving him daily worksheets entirely because I've been yacking about distributed practice and fluency for lo these many years. She has her own worksheets; the school didn't provide them.
So, yeah.
Some or many of the K-5 kids are going to show up at the middle school with decent computational skills and pretty good comprehension.
That's not going to be thanks to Trailblazers.
That's going to be thanks to the fact that Trailblazers was adopted in a highly-educated community where parents could teach the concepts their kids were supposed to discover.
Very few parents can teach pre-algebra and algebra no matter how well educated they are.
This is one of those ironic situations where the school, in taking credit for work parents have been heavily involved in, will paradoxically end up removing credit from middle & high school teachers who are dealing with the consequences & who have to get the kids in shape to pass Regents Math A & B.
Proofs are about solving problems in math. "Word problems" are solving problems in science.
ReplyDeleteHe he. Classify this word problem:
"Seven plus eight equals a number. What is that number?"
I'm looking for a philosophical discussion on how to catergorize word problems. Anyone have any links? Seems like some things that are problems expressed in words count as what we think of as word problems and others don't.
What is the point of age problems in algebra that go like this, "Albert was half the age of his brother two years ago...blah"
when in real life we never encounter age problems like that. We don't encounter coin problems either and I've never had to figure out when two planes flying to Chicago will meet or how long it takes three pipes to fill up a pool. For that matter, my tv is in the corner and there was never an issue about figuring out dimensions (well there was, we wanted to know if a bookshelf would fit and my husband who has his MS in math pushed the bookcase over to the space to see! It did not and it got pushed around to other areas to see if it would fit there.)I've never used Geometer's Sketchpad for figuring out if furniture fits, I use a measuring tape!
What is the point of presenting this problem as a "real life" problem when it's not solved in a real life way? If we aren't going go "real life all the way" how about presenting it as a geometry problem and solve it using the methodology used in geometry?
And then the other problem with the "real world" problems that they come up witI don't encounter any of them in my real world. Any word problem in a book is not in the real world, but in a book.
The only time I ever use fractions is in cooking and sometimes at Home Depot. Estimating the cost of recarpeting the house is the trickiest "real" problem I've come across outside of a paid job. Recently, a friend of mine who is an attorney uses so little arithmetic in real life that she asked me how to add unlike fractions.
While I appreciate that math is used to solve problems in science and home ec, if math should be about "the real world" why don't we see these being done in math and home ec classes? Math has turned into science class. Science has turned into reading class. Reading has turned into multiculturalism.
---Myrtle (I couldn't log in for some reason)
As for this particular problem, I don't like how their explanation changes this from a simple problem into a large project. I don't dislike Geometer's SketchPad, but students must be able to solve this problem without using graph paper, computer-based or otherwise.
ReplyDelete"A journal entry was done with the following diagram included:"
They are creating a journal(!) for this problem. A student with a proper math background should be able to solve this in a minute or two. If they are new to the subject or skills, then they should first work on the basic skills. This is typical top-down, rather than bottom-up learning.
"What This Task Accomplishes
This tasks puts the student in the role of designer, using specifications from a diagram. They must employ (a variety of) techniques and develop appropriate strategies for solving the problem. A meta-cognitive aspect is built into the task by requiring an explanation of the approach and consequent solution."
Aaaaaaarrrrrrggggghhh!
"meta-cognitive"?
"consequent"?
How about having the kids spell that and write a definition.
"Time Required For The Task
The students had thirty minutes to complete the task, and it seemed sufficient."
Low expectation alert!!!
"Interdisciplinary Links
As with most of my performance tasks, this one comes from my real life experience. I believe my students enjoy that connection and work hard on the problems I have had to wrestle with myself."
The author had to "wrestle" with this task?
The problem I have is that this is done near the beginning of the learning process. Rather than first mastering how to find any angle or side of a right triangle, they dive right into a problem like this.
Constructivism = Frustrationism.
Students need to be taught that if they have any two pieces of information about a right triangle (angle or side), they can find any other dimension. And if they don't have a right triangle, they need to learn how to create one with the dimensions they have.
"The Tri-State team’s concern relates to the challenge to maintain the critical thinking skills that students have acquired in the K-5 program as they move into the district’s more traditional math environment...."
ReplyDeleteI laughed when I read this. "...maintain the critical thinking skills...". They are in their own little world.
I'm fairly hostile to Math Olympiads because that's what my district uses for the gifted kids. They get a bunch of Math Olympiad problems, and that's it. No acceleration allowed.
ReplyDeleteThis could be done well or this could be done poorly.
The Olympiad problems represent classes of problems the solutions to which can be taught in a systematic matter. And if taught in that way the student would be accelerated. Number Theory and Combinatorics are not really found in the standard engineering/physics math curriculum that high schools are moving the kids towards. A kid with knowledge and skill in combinatorics is accelerated in that speciality area of math compared to the kid that is in the standard scope and sequence.
A poor approach to Olympiad problems is to just throw them at the kids without giving them instruction or insight into them and use them as "busy work."
"For the math experts here: what relationship do word problems have to a coherent curriculum?"
ReplyDeleteWord problems are critical. They develop the ability to create equations out of words and are usually the only place you get to learn about units. This doesn't mean that I like what many curricula are doing. You don't use word problems (top-down) to learn and master the basics.
"I laughed when I read this. "...maintain the critical thinking skills...". They are in their own little world."
ReplyDeleteI did, too. That whole memo is unbelievable. I hope you post some of it. The idea of K-6 dictating to the high school is unbelievable arrogance. But that is what is going on. First, they start with the middle schools, though.
"I'm fairly hostile to Math Olympiads because that's what my district uses for the gifted kids."
ReplyDeleteI've always hated the idea of math as a competition, but that's me. I would hate to be forced into that environment. I think it should only be an optional after-school program, not a main part of the curriculum.
A meta-cognitive aspect is built into the task by requiring an explanation of the approach and consequent solution.
ReplyDeleteIn other words, if the kid comes up with the correct answer then absolutely any explanation (approach) he uses is considered meta-cognitive-critical thinking and appropriate.
There is no role whatsoever for mathematical rigor in this curriculum, even for applied problems.
"The 4-5 school sends home newsletters telling parents to teach their kids the math facts & giving them a deadline by which to complete this task."
ReplyDeleteWow! A deadline? Do you have this in writing? That's a smoking gun type of message. It's one thing to ask parents to help out, but it's quite another to set a deadline.
It used to be that the school dealt with teaching the basics and the parents were in charge of enrichment. Now, it's the other way around.
but the idea seemed to be to use Geometer's Sketchpad to solve it.
ReplyDeleteI'm not sure what that is, but the solution should consist of a single vertical line drawn down the middle of the triangle.
The base of the small triangle on top is 27 inches; drawing a line down the middle splits it into two smaller triangles with a base of 13.5 inches. On an isoceles triangle two sides are equivalent length; on an isoceles right triangle, those two sides are always the one surrounding the right angle. Therefore, base = height = 13.5 inches.
The height of the entire triangle is therefore the height of the square plus the height of the triangle. 24 inches + 13.5 inches = 37.5 inches.
The diagonal line is the hypotenuse of the triangle formed by drawing the line down the middle. Since the base = height = 37.5 inches, and the square of the hypotenuse is the sum of the square of the other two sides, we get:
37.5^2 + 37.5^2 = h^2
h = SQRT(37.5^2 + 37.5^2)
Alternately, the student could calculate the sum of the two smaller hypotenuses:
13.5^2 + 13.5^2 = h1^2
24^2 + 24^2 = h2^2
h1 + h2 = SQRT (13.5^2 + 13.5^2) + SQRT (24^2 + 24^2)
Personally, I'd stop the question right there; I don't think calculating the problem out to get the exact number is all that useful, and the logarithms which might help reduce the question are generally (and rightly) taught after geometry. Ideally, the question would have used 'friendlier' numbers for the lengths of the bases.
I think the idea behind the question is ok - it pretty effectively tests a student's ability to identify the different triangles, whether he understands their properties, and how to integrate them. I just don't know why they chose the lengths they did. If it were a calculator-based question, the calculation of the lengths is irrelevant; so why ask it in the first place? If it's not calculator-based, why have the student spend twenty minutes doing a needlessly complex calculation?
"This is one of those ironic situations where the school, in taking credit for work parents have been heavily involved in, will paradoxically end up removing credit from middle & high school teachers who are dealing with the consequences & who have to get the kids in shape to pass Regents Math A & B."
ReplyDeleteThis is an easy thing to check, but our state's parent questionnaire has not a single question about how much time or money is spent on teaching at home. Not one. What we do get are questions about whether we parents feel able to help with math homework. Perhaps they will decide to have parent classes so we are better able to do their job.
You think that high schools would scream about how poorly some kids are prepared, or wonder why they see a gaping chasm between the haves and have-nots. When our K-8 school was using CMP in 7th and 8th grades, there was a clear content and skills gap getting to 9th grade algebra and a huge gap getting to geometry. At best, the high school told the lower schools that they should eliminat "do-overs". In other words, it's the kids problem. It's almost like they don't want to criticize fellow teachers.
So, what does the high school do? They provide remedial math classes that try to bring the kids up to speed. And they get awards for this even though the problems started years ago.
George, you are right about the solution, but as a kid I was dense and made these problems unnecessarily difficult to solve.
ReplyDeleteFor example, I would not have had the key insight that the triangle was isosceles, nor would I have assumed that. What knowledge is required in order have this insight? The base angles of the triangles are not marked as congruent.
I wasn't able to solve any sort of problem like this until tenth grade geometry when I was explicitly taught about the properties of triangles. In my obviously deficient geometry class we didn't use any technology at all except a straight edge and compass. We solved the same sorts of problems only without units or real world contexts.
"What knowledge is required in order have this insight?"
ReplyDeleteThat's a proper way of thinking. Most students would look at the picture and decide without proof.
But this is the key sentence.
"We want the new cabinet to be the same length on each side (along the two walls)."
OK, but how do you translate this into something you can use? It's kind of like a trick question. You have to then say that the length of C->J and J->H (I think those are the letters. It's annoying that in the solution they also use P, Q, and R too.) are equal. If they are equal and the top of the triangle is 90 degrees, then the two side angles must be 45 degrees.
So, one student could make this assumption without proof and figure out the problem and another could get stuck because he/she couldn't prove the result.
I've seen problem pictures that show angles that look like 45, 60, 90, or equal degrees, but aren't. They catch lots of students because they can't prove it.
Again, I don't like top-down learning.
Myrtle - you're absolutely right... On reading it, I'd assumed that this was a 10th-grade geometry problem, which would have been apropriate; it honestly never occurred to me that it would be anything else. On following the link, it appears that it's an 8th grade problem, at best. There's just no way you can expect a normal 8th grader to solve this in a reasonable amount of time; they wouldn't have the tools with which to solve it.
ReplyDeleteFurthermore, you're also right about my other assumptions. It's not obvious that it's an isoceles triangle. There are only two ways to know this: prove it using geometry rules that wouldn't have yet been taught to an 8th grader, or infer it from the text of the word problem (a corner tv stand is going to be symmetrical, therefore isoceles). Both of those require pre-requisite knowledge that an 8th grader wouldn't have (and I'd argue that the latter is unreasonable for any student).
This is a good problem for a 10th grade geometry student. It's outright torture for an 8th grader who hasn't been taught the pre-requisite knowledge.
You gotta love the rubric. This is what is expected to achieve expert ranking:
ReplyDelete"Uses multiple approaches to the task to find correct solutions. Written explanations show the student's thoughts clearly. Diagram usage is solid and correct."
So, in the logic of performance assessment, if the student utilizes only ONE efficient and effective approach, finds the correct solution, and shows their work, they are not an expert per the rubric. They have to use multiple approaches in order to achieve the ranking of "expert".
This is the "expert" ranking:
ReplyDeletehttp://www.exemplars.com/materials/samples/math_9-12_exp.html
compared to the "practitioner" ranking:
http://www.exemplars.com/materials/samples/math_9-12_pra.html
Practioner: Uses accurate formulas to find correct solutions to the problem. His/her written explanations are clear and straightforward. Diagram work is accurate.
Steve,
ReplyDeleteI didn't mean to imply that a kid would have needed to have been taught synthetic geometry in order to find the solution to this particular problem, so much as recollecting that it wasn't until I took synthetic geometry that I had the skills it took to solve such problems.
"If they are equal and the top of the triangle is 90 degrees, then the two side angles must be 45 degrees."
Prior knowledge needed for this problem would be that the sum of the angles of a triangle is 180, that drawing a line through the vertex and the midpoint of the opposite side bisects the angle, the pythagorean theorem. Another thing is that you have to really infer a lot from the diagram. You have to realize that the tv can't go in skewed based on the diagram, you have to really appreciate the importance of that 90 at the top of the triangle. Now the way that I did the problem required me to recognize that dropping the perpendicular from the top of the triangle to its base was also perpendicular to the top of the square and to the bottom of the square which was coincident with the base.
I would have never known where to start as a kid, unless I had been explicity taught the prior knowledge and I would have needed practice applying it in multistep problems before even having a chance of getting this problem correct (Geometer's Sketchpad notwithstanding)
By the time I was in high school I could do this. I couldn't do it in junior high. They were still teaching us long division in junior high.
In high school I didn't even see the connection between these implicit p's and q's and all the the chains of reasoning that are used to solve multistep problems---the kind of thing that is routine in Singapore word problems.
The good math students are good natural reasoners too, they pick up on this sort of thing quickly, they just "know" it. I didn't. To this day I have no idea how to negate, "For every epsilon greater than zero, there exists a delta greater than zero such that whenever |x-a| is less than delta, |f(x )-f(a )| is less than epsilon."
Smart folks just pick up that skill along the way, I guess. Completely lacking in natural ability I'm finding direct instruction in logic useful and math problems of all kinds are becoming obvious.
I can see why kids hate math if they are expected to "just see" the reasoning and "just remember" the assumptions without having practiced it because it was mentioned once by the teacher in passing sometime at the beginning of the year.
I don't really see why you even need to bother bisecting the whole thing. To me, it's a problem that simply tests a^2 + b^2 = c^2 and the ability to use the equation in both directions.
ReplyDeleteYou find a hypotenuse for one triangle; you do the reverse for the other triangle.
Obviously there are some proof steps missing, but I'd be happy with that solution.
In all seriousness, at the lower levels I think it is perfectly realistic for kids to assume that the drawing is somehow to scale: i.e. that if it looks like everything is equal (which it does) then it's equal. the concept of drawing what looks like a right triangle while actually assigning value that make it something else is too advance for 10th graders.
(not incidentally: 10th grade? Yikes. This would have been an early fall problem in my 9th grade geometry class, if we didn't learn it for the hell of it in 8th grade.)
Alright, I'm gonna break down for all you guys right here! The Answer! The Truth! ;oP
ReplyDeleteAt any rate, I think the key lies in what abstraction basically is and how it is achieved. It is nothing more than a statement that applies to two (or usually more) particular things or instances of a thing at the same time. In order to pull off something like that, you have to be able to consider if it applies to the first thing and then figure out if it applies to the next thing. Real simple. But, the issue is that we just willy nilly jump out to infinitely many things, speaking of classes naturally and intuitively almost ab initio. So, if you are dealing with a difficult, non-obvious generalization, say, about widgets, what do you do? You think of a particular instance of a widget and ask yourself if the statement is true about that widget. Then, you think to yourself, "But, then there are these other completely different kinds of widgets, too. Is this true about them, as well?" And, you keep going on like this until you have covered enough cases of different kinds of widgets to persuade yourself that the statement is probably true about all widgets.
Of course, if you are trying to really be rigorous, here, you will make sure you have a valid deductive argument that really does account for every conceivable widget and not just try it out on what appear to you to be some paradigm examples (depending on whether or not such a thing is possible). This distinction lies at the heart of the problem, in my opinion, with all of math education, including most college level education. Essentially, we are not rigorous like that and, more particularly, don't limit ourselves to topics in which we can achieve that kind of rigor. Instead we just do the other method which is essentially an empirical approach to math. And, that is fallacious in the case of mathematics. So, we are literally teaching not just our children, but most college students, even most math majors throughout most of their undergraduate degree, to reason fallaciously. That is not an education. That is an anti-education.
At any rate, I submit that we all do this and do it this way. Sometimes we'll shoot for a general deductive argument, but as soon as the problem gets terribly difficult and time in the day and resolve in the student being limited as it is, we resort to the more expedient empirical approach. What the smart students can do that the not as smart students cannot do is simply just go through a lot more cases in the same amount of time, asking themselves "Yes, but is it true about this case?" Or, they just finish it and move on to the next problem sooner. The bottom line is they just get through more problems, getting the correct answer more often, than the less bright students in this manner because they can work faster. It all comes down to something as simple as being able to carry out trial and error faster.
Every time a student doesn't "get it", it is because they cannot imagine just the right paradigm examples or cannot go through them fast enough. In fact, even imagining the right paradigm examples is mostly a matter of starting with some random one and going through it so fast that you can infer a better one that you zoom through to infer a yet better one and so on. Even Einstein's wonderful imagination just comes down to that, in the end. Maybe this kind of thing allows us to get more answers in a shorter amount of time, and maybe if we make sure the problems are sufficiently hard, we can even promote a certain amount of reliability in our students as they impliment such a technique. (We can ensure that they consider enough examples.) But, in the end, it is all one big fallacy. That isn't knowledge. No matter how true your beliefs may be or how persuasive your rationale for believing in them may be, that kind of fallacious reasoning can never ever lead to actual knowledge about an a priori subject like math.
And, in any case, that is what works to make this problem a difficult one. Simply observing that the shortest path to the answer isn't that long, doesn't make on bit of difference because it is nevertheless quite difficult to find that path. You may ask yourself, "Well can I squash the triangle down?" And, then you think "Oh no, it says it has to be a 90 degree angle up there." And, then you ask yourself, "Can the TV be in the corner at an angle, here?" And then you realize, "Oh no, it says that the face of the TV is flush with the bottom of the triangle and that the edges are both perpendicular." You keep going on like this until you eventually realize just what the simple chain of resoning is to solving the problem. But, it takes time -- a lot more time and effort for the less bright students. And, ultimately, if it take enough time that it surpasses the student's motivation to think about it that long and capacity to remember the cases they have gone through mentally in their head, the student will exhibit this phenomenon of "just not getting it".
In all seriousness, at the lower levels I think it is perfectly realistic for kids to assume that the drawing is somehow to scale: i.e. that if it looks like everything is equal (which it does) then it's equal. the concept of drawing what looks like a right triangle while actually assigning value that make it something else is too advance for 10th graders.
ReplyDeleteI disagree; I think that it's important to emphasize, from the start, that you can't assume a picture is to scale, and that you have to prove the properties before working on the problem. I suspect the reason it becomes difficult for 10th graders is because they're used to assuming the picture is to scale, and getting them to do otherwise goes against both common sense and habits which have been reinforced for ten years. It's like teaching pilots to fly on instrumentation - you have to ignore what you're seeing (or not seeing) through the windows, and trust what the dials are telling you.
I am wandering off-topic, but Myrtle Hocklemeier mentioned that he didn't know how to negate a statement with nested quantifiers. This is actually quite easy if you know the rule. You just move the "not" past each quantifier, changing each "for all" to "exists" and vice-versa. That is, not(for all x, P) is equivalent to (exists x, not P), and not(exists x, P) is equivalent to (for all x, not P).
ReplyDeleteIn Myrtle's example we have three quantifiers: for all e>0, exists d>0, for all x, if |x-a| < d then |f(x)-f(a)| < e. The negation is: exists e>0, for all d>0, exists x, |x-a| < d and |f(x)-f(a)| >= e.
Well, if it is so simple, then why don't we teach calculus this way -- that's what I want to know....
ReplyDeleteI will hear no negatives of Geometer's Sketchpad! It is a MAGNIFICENT PROGRAM, and the idea of creating a program that allows you to specify PROPERTIES of geometric objects that remain true under various transformations is BEAUTIFUL AND BRILLIANT.
ReplyDeleteGSP is a terrific tool for those of us who have graduated from college in math, science, engineering, etc. I managed to reproduce a fair number of newton's Principia proofs using it, and I could never have shown them as prettily or convincingly without GSP.
Its wonderfulness has nothing to do with the misuse of GSP.
GSP in grammar school is just like every other "technology" in grammar school: it's useless at helping you form truths in your mind until you have achieved enough other mastery to really learn what it is you known vs. what you don't know.
Since grammar school students don't have that yet, and certainly won't have it in any curriculum that substitutes using GSP for doing the arithmetic yourself, its usage is a disaster. But don't blame GSP!
Just a word in defense of word problems -- they can be really useful in helping students get ready for science and engineering work. Students who can't initially understand a limiting reagent problem with chemicals and scientific measures can understand a completely parallel recipe problem (e.g. How many gallons of lemonade can you make given these ingredients and this recipe). The problem is that at the K-12 level, math classes are preparing students to do higher level math but are also preparing them for later science course work.
ReplyDeleteOf course, all this is of limited use when teachers don't understand WHY it is useful for students to master these problems, or what skills students need at the higher levels!
SteveH sound bites alert!
ReplyDeleteFrom the preceding comments, I found a couple of nuggets.
The problem I have is that this is done near the beginning of the learning process. Rather than first mastering how to find any angle or side of a right triangle, they dive right into a problem like this.
Constructivism = Frustrationism.
My kid would be spared much grief if they taught her instead of wasting so much time letting her struggle to discover the lesson herself.
It used to be that the school dealt with teaching the basics and the parents were in charge of enrichment. Now, it's the other way around.
Nicely put.