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Maybe I can make a convincing argument that a student who only thinks of multiplication as iterative addition and can't multiply 24X86 has neither procedural nor conceptual knowledge.
There is more to the "concept" of multiplication than iterative addition. (Try applying iterative addition to 1/8 x 2/5.) Perhaps iterative addition is appropriate for 2nd and 3rd graders learning their multiplication tables (or is it 3 and 4th graders these days?) But "the" concept of multiplication includes the fact that it distributes over addition (and that it's associative as well). The multiplication algorithm invisibly makes use of the distributive "concept," and does not employ an iterative "concept." Perhaps I'm overdoing the disdain quotes but I've been lied to too many times by people telling me that something is the "concept" of a procedure or rule and it turns out not to be.
A child with a conceptual knowledge of multiplication, and a lot of time on his hands, could successfully multiply two digits numbers without the multiplication algorithm:
24 X 86 means that
(20 + 4)(80 + 6) which means/implies that...
Etc. You see where I am going with this. One of the benefits of Singapore is that the kid does end up with a conceptual understanding of multiplication, and can apply his knowledge of concepts to come up with correct answers.
Notwithstanding operations on super hairy numbers, he is capable of doing the algorithm on paper when he needs to and can resort to "concepts" when he needs to do mental calculations.
the multiplication algorithm invisibly makes use of the distributive "concept"
I love that!
I love the whole Comment, in fact. People like me -- people who value liberal arts education in general and mathematics education in particular but who aren't expert in mathematics and probably never will be, have no way to get at these things.
I intuitively grasp the notion that there is some kind of "starter understanding" a person can have without being fluent in procedures. Seeing that 6x4 is the repeated addition of 6 4s or 4 6s as the case may be (I've spent quite a bit of time muddled over that one!) strikes me as superior to not seeing it. (I had no idea multiplication could be called repeated addition until I started reteaching myself math, and then I noticed it on my own.)
But at the same time I am gripped -- and gripped is the correct word -- by the conviction that a starter understanding is not a real understanding.
And yet because I lack a real understanding I have no way to express this and thus no means of combating the forces of reform math when they threaten to overrun my son's education.
I'm logging this post under Greatest Hits so I'll know where it is when I need it.
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I need to hear posts like these so I don't throw our Singapore books out the window. I'm in Investigations -----> Singapore transition hell. I used to think it was kind of funny that traditionalists vengefully call constructivist math "guess and check" but after this past two weeks of watching my 1/2 through 3rd grader try to literally guess his way through these 2B level Singapore math lessons. I just want to cry. And he wants to cry. There's a lot of stomping and slamming things around these days. I'm not sure I'm doing more good than harm...
Glad to hear I'm not the only person who thinks Guess and Check is used to excess. I was trounced on once for saying that in an article I wrote about Singapore Math.
Katy - if you're interested in forming an online "positive behavior management" support club I'm up for it.
Seriously.
As I write my chapters on animals I read over and over again how CRITICALLY important it is to manage animals (and kids) positively....and it is unbelievably difficult to do.
My major self-improvement project for the new year is going to be figuring out behaviorism for my kids and for me.
I am all for a positive behavior management support club.
And, Catherine, when you figure out a good plan for behaviorism, could you let me know?
My house is meltdown central these days all because of a "write to learn" science project.
Thank goodness tomorrow is Friday.
RE: positive behavior management support club - YES LET'S. Please.
I know in my rational head what I am supposed to say and do with my kids. In the applied world, if I am tired or overstressed, it goes out the WINDOW (along with the math books). This is more often than I'd like.
I've actually started taking boxing lessons - mostly to get in shape. But it's very satisfying to get to pummel the bag or my sparring partner's mitt. It calms me down a little - gets me in a better frame of mind except on days I have to get up at 5:30 am for the boxing class - then it's kind of a wash by homework time. The tiredness cancels out the calm from my theraputic bashing.
Positive behavior management support club. pbmsc!
What's guess and check?
"...a starter understanding is not a real understanding."
Reform math thinks that repeated addition or pictures of fractions is all of the understanding you need. All of it is a concrete "starter" understanding, not an abstract understanding. I saw this in Everyday Math. They use pie pictures for fractions, but when it comes to multiplying fractions, they just give you the rule.
As I've said before, we give them way too much credit. Up through sixth grade my son's school goes on and on about all sorts of fuzzy constructivist understanding talk, and then everything changes in seventh grade pre-algebra without a bat of the eye. My son now gets (thankfully) homework sets that have about 50 problems every other day. He is now working on exponents. The textbook introduces the rules, provides explanations, and then gives them lots of practice. There are a few silly "writing" questions, but not too many.
They finally get serious about math in seventh grade (in our town), but it's too late for many kids.
"What's guess and check?"
Anti-culture.
It's the lack of "transmission of our knowledge to the next generation so when we die the stuff we know doesn't die with us."
Thank you Catherine.
Culture isn't necessary. All you need is "basic literacy and numeracy" and constructivism.
My son's school is having the kids (sixth grade) write in an online (safe?) blog so that they can develop their "voice". Voice without knowledge or culture. Opinion and voice trumps knowledge.
Without knowledge, everything is opinion, and we all know that all opinions are valid. Therefore, no knowledge is required.
I really have to stop before I get on a roll and go out and shovel. I don't like leaving it for the morning. It will be as solid as a rock.
For years proponents of constructivism denounced traditional math by calling it "drill and kill", claiming that the way to get kids to learn math is to get them to love it - with math "games" I suppose - instead of driving the concepts into the ground with repeated drills.
So in the backlash, constructivism got dubbed "guess and check" in retaliation. Quite accurately it seems. Both terms are accurate. There's got to be a best of both worlds scenario. We should be getting better at teaching our kids math, not worse. It feels like we're devolving a little bit as a society - in some areas at least. Just like the gap between rich and poor widens, I think the gap between education levels is widening as well.
"The textbook introduces the rules, provides explanations, and then gives them lots of practice."
Just curious. What textbook is he using?
One of my high-SES tutees is in 7th grade and uses McDougal Littell's Pre-Algebra. It looks like a good math book. Things are explained. Then there is lots of practice and review. The way I expect a good textbook to be. Oddly, Lee Stiff is one of the four authors listed. I remember Stiff for having made some incredibly fatuous comments about learning math. So this is a bit surprinsing. Maybe his influence was kept in check. I guess. Must check.
"Without knowledge, everything is opinion, and we all know that all opinions are valid. Therefore, no knowledge is required."
This pretty much sums up postmodernism.
I addressed some of it here:
History is knowable
Maybe his influence was kept in check. I guess. Must check.
Love it. Thanks for the chuckle.
"Just curious. What textbook is he using?"
Glencoe Pre-Algebra (2008)
I could probably pick out a better textbook, but compared to Everyday Math, it's a dream. I complained about a few issues in the past (dealing with signs and writing), but otherwise, I'm happy.
This textbook is being used for the first time, having replaced CMP! The biggest driving force for that change (as far as I can tell) was the backlash from parents and kids entering high school. They weren't prepared for honors courses. They tried to blame it on the kids and the lower school's desire to give "do-overs", but they couldn't gloss over a clear gap in content and skills.
Now, the 8th grade algebra text being used is exactly the same text that is used for honors algebra in high school. If the high school requires a grade of 80+ (I think) for this course to get into honors Geometry, then CMP in middle school just doesn't cut it. Parents asked why this algebra wasn't taught in 8th grade. (Duh!) Flowery ed school talk covers up a lack of common sense.
It would be interesting to hear from others what their middle schools are doing. Are they sticking with fuzzy programs like CMP, have they gone to real math like pre-algebra and algebra textbooks, or do they provide a mix, presumably with the top students getting the real math. If the top students get the real math, then what do the schools say about the other programs?
I've also noticed that some publishers offer two levels of 7th and 8th grade algebra courses. Rather than call it something different, like CMP, they call both series "algebra", although one might have some extra real-world words. Prentice-Hall uses "Tools for a Changing World". That's a clear giveaway. They also have algebra courses that are labeled: "California Mathematics" (that's good!) as opposed to mathematics for the rest of the states, I guess. But then, Prentice-Hall is part of Pearson, and Pearson provides CMP (oops, New! Improved! CMP2!). Schools don't know enough math to evaluate textbooks and publishers tell them whatever they want to hear. They don't care. They have a product for you.
Now, the big hurdle is to get from Everyday Math in sixth grade to pre-algebra in seventh grade. Apparently there are some questions (complaints) about Everyday Math because the school sent home a notice that there will be an informational meeting in January. Parents need some way to organize and compare notes.
I found an example of guess and check online. The first sample problem I came across was this, "Amy and Judy sold 12 show tickets altogether. Amy sold 2 more tickets than Judy. How many tickets did each girl sell?"
This kind of problem would be a typical problem in the Singapore curriculum and would be handled by a bar diagram. There would be no guessing. "Guessing the right answer" is anti-math at worst and acts of desperation of a multiple choice test at best. Why bother learning math at all if we are going to guess, "How much money is in the bank account? I dunno let's guess and then ask the bank. Does the square root of two expressed as a decimal expansion ever end? I dunno, let's guess and check."
In solving the above word problem the child would be expected to use a chain of reasoning that leads to the correct answer and there would be several similar word problems that they could practice with.
Ideally they'd look at their bar diagram and think, "Say Amy did NOT have two tickets more, then the two girls would have an equal amount. But if Amy had two tickets less then the total would be two less than 12. Therefore, their two equal number of tickets would represent a total of 10 tickets. 10 divided into two equal groups means there is five in each group. So, each girl has 5 tickets However, Amy does have 2 more. That means Amy has two more than five which is seven and Judy only has five.
The reasoning looks more sophisticated in prose than it does when it's happening verbally.
I don't know how this is taught in Singapore but I have always done all this thinking out loud for a few sample problems and then turned the kid loose to work a few on his own. No guessing. If the kid comes up with the wrong answer it is because his reasoning went wrong and this can be directly addressed, not because he got unlucky and didn't guess the right answer.
It would be interesting to hear from others what their middle schools are doing.
Our middle school is pushing CMP2 big-time. I've been told they are supplementing heavily with some computerized scaffolding program of which the name escapes me. It seems that if any math learning is taking place, that's where it's happening.
For the first time, there was no math placement testing before at the middle school or the upper elementary (5-6th) because it seems they are working to phase out the tracking completely. I heard that about 3% of the 7th graders are taking algebra, while everyone else is using CMP2.
The pool of students able to take algebra in 7th grade is shrinking because the current 6th grade class has had Everyday Math all the way through. You are what you eat.
*That should be about 3% taking pre-algebra in 7th and algebra in 8th.
"That should be about 3% taking pre-algebra in 7th and algebra in 8th."
3%? Wow! That's low even with fuzzy math.
I think our schools dropped CMP because it's a small town and they can't support multiple curricula.
The interesting thing is that if you open the sixth grade Everyday Math book at random, you might see problems that my son is getting in pre-algebra. It's just that Everyday Math jumps all over the place, doesn't require mastery, and doesn't explain new topics in detail. It's just constant introduction.
Way too low. Unless parents do something to change the direction of where the district is heading, the group of children who enter high school with a solid understanding of algebra may disappear altogether.
A child fed a steady diet of Everyday Math K-5 and CMP2 6-8 is doomed to mathematical mediocrity.
At this point, that represents all children K-6 in our district (unless they moved from somewhere else, participate in tutoring, or are extraordinary enough to overcome the hurdles placed before them.) Sadly, at some point, 3% may actually be high.
"The first sample problem I came across was this, 'Amy and Judy sold 12 show tickets altogether. Amy sold 2 more tickets than Judy. How many tickets did each girl sell?'"
We've talked about these pre-pre-algebra problems. With algebra, they are trivial. The big question is whether or not these problems are useful (in any Zen kind of way) before algebra. My opinion is no. I might make some exceptions, but in general, I don't like them. Save the time and get to algebra sooner.
If you give these problems, there are variations on what tools you provide the student. Singapore Math provides bar modeling, so it's really not guess and check. This is fine, since math is about providing tools and skills to make problem solving easier.
Fuzzy math, however, loves to give these problems with only the vaguest kind of solution advice. They require you to discover (guess and check) some sort of solution method. I call it anti-math.
Our middle school is in flux, also. There was a re-alignment of sorts to better work with the grade school's reform math program.
That's always a big red flag.
My son's pre-algebra text was also the Glencoe and we were pretty happy with that. His first algebra 1 text was pretty mediocre, but the second one for accelerated math is a Dolciani text. It is supposed to match the high school (9th grade honors algebra)in speed and rigor.
At the end of the year the 8th grade algebra teacher recommends which tier of honors the kid will enter the next year.
This is now. I don't know how long it will last.
Our tough-as-nails Boomer math teachers are retiring right and left. I have no idea who will replace them. My son is on his third retiree. I'm just glad he had them when he did. It will make a difference when he goes to the high school.
constructivism got dubbed "guess and check" in retaliation
I didn't know that!
That's the first catchy anti-constructivist slogan I've ever heard.
We need more.
The middle school principal here spent his entire first year trying to get rid of the 6th grade accelerated math course.
He failed.
But I hear the kids enrolled in it are having a hard time. They are the first group of kids to have had Trailblazers all the way through K-5.
At the focus group a teacher told Ed, "The (K-5) teachers like TRAILBLAZERS."
That was it.
The teachers like it, so we've got it.
Our district routinely justifies academic and spending decisions on grounds that "the teachers like it." A blanket "the teachers like it" statement is EOC.
end of conversation
I found an example of guess and check online. The first sample problem I came across was this, "Amy and Judy sold 12 show tickets altogether. Amy sold 2 more tickets than Judy. How many tickets did each girl sell?"
The kids were directed to solve this via guess and check???
Catherine, it was the first example I found. Don't know if it's representative of how guess and check works, since I'm not sure of what guess and check is:
http://www.mathstories.com/strategies_guesscheck.htm
"Guess and Check" is presented in some textbooks as a technique to be used to solve problems such as the Amy/Judy ticket problem. In many U.S. texts (for example, Houghton Mifflin’s "Math" and "Math Expressions"), students are exposed to a variety of techniques to solve problems. Among the techniques are “draw a picture” (which theoretically could include bar modeling, but generally texts do not provide guidance for how to represent problems pictorially, relying instead on students discovering their own way), work backwards, solve a simpler problem, or use the “Guess and Check” method of trying combinations of numbers until the right numbers are found that satisfy the conditions of the problem. So kids are usually not directed to solve an Amy/Judy type problem in any particular way, since they are expected to choose from any number of inefficient methods that they have been shown how to do. As Myrtle points out quite well, Singapore's bar model approach is a pictorial approach that is efficient and accurate.
Text to be displayed
Math Expressions, by Houghton Mifflin. Work sheet from 5th grade. These are Guess and check problems.
I was searching for the incredibly fatuous comments made by Lee Stiff, past president of NCTM, but can't find them. Instead I ran into this description of how math used to be taught according to Stiff, past president of NCTM:
STIFF: Parents are upset because, when they visit classrooms, they see activities that they're not used to. When they were students in school, they probably sat in rows neatly lined up, and the teacher just talked and talked and they used paper and pencil, and that's how they learned their mathematics.
When they see students engaged and talking with one another, when they see teachers allowing students to question and think thoroughly about the mathematics and the relationships, they wonder if the basics are going to be achieved. But the test results show that they are, their students are learning the basics.
It seems to me that keeping this caricature of traditional math teaching alive plays a vital role in perpetuating fuzzy math. By setting up a false dichotomy, the caricature provides the rationale without which the fuzzy project would collapse. Barry's latest articles confront this caricature head on.
Barry, I have to say I was a little bit mortified by what I saw when I followed that link. I'd always heard the term used disdainfully about constructivism. I didn't realize it was an honest to life "strategy".
It's times like these that make me feel curmudgeonly old because I just want to say "what has the world come to?"!! Honestly why even try to provide a public education if this is what we're offering?
Barnacles!
The use of that sort of caricature is known in logic as a strawman argument. It's one of the most basic of logical fallacies.
The only question is whether the strawman is the result of ignorance or malice -- I'm not sure which I find more disturbing.
The use of that sort of caricature is known in logic as a strawman argument. It's one of the most basic of logical fallacies.
Exactly right. And Lee Stiff might have served as the role model for Sherry Fraser (of IMP fame) who delivered testimony to the National Math Panel, opening with words remarkably similar to Mr. Stiff's:
“How many of you remember your high school algebra? Close your eyes and imagine your algebra class. Do you see students sitting in rows, listening to a teacher at the front of the room, writing on the chalkboard and demonstrating how to solve problems? Do you remember how boring and mindless it was? Research has shown this type of instruction to be largely ineffective.”
In this case she doesn't set up the dichotomy, though. She just outright states that there is research to show that "this type of instruction" (a rather loose term) is "largely ineffective". As I wrote about in the article that Instructivist referenced above, I wrote Sherry a note asking her for the research only to be told essentially to do my homework. Which I did.
Friends of mine in the math wars who know these techniques and arguments all too well asked me why I would write an article that in their view just explored the obvious. Many things are obvious to those of us who deal with this kind of propaganda on almost a daily basis. But I wrote it because there are many people for whom it is not obvious and who actually believe these lies. Yes, lies.
There is a ploy that I'm sure Doug Sundseth knows well, which is a speaker or writer assuming tacit agreement of the audience. At school board hearings an official may start the discussion about a controversial textbook by saying "Well, we all know that the traditional method of teaching math doesn't work." This presumes that the speaker and the audience agree on some obvious points
(e.g. "traditional math doesn't work", "desks in a row is typical of bad teaching") and then has no trouble finding the conspiracy
(usually ivory tower mathematicians) to explain it. What is happening is that those in the audience who are new to the game may be thinking "I didn't know this, but everyone seems to agree with him (or her), so if I ask about this I'm going to look like a fool. I better keep quiet."
I sought to provide to inform the debate and am happy to hear from some readers that they have notified school board members of the article.
Concerned says:
"Our middle school is pushing CMP2 big-time. I've been told they are supplementing heavily with some computerized scaffolding program of which the name escapes me. It seems that if any math learning is taking place, that's where it's happening."
There must be many computerized programs that offer practice of real math. I found a free program that offers practice problems for specific grade levels. The topics are comprehensive.
http://www.aaamath.com/
Some kids are magically drawn to computers. They might get excited doing the same things on the computer that they might otherwise find less thrilling.
"There is a ploy that I'm sure Doug Sundseth knows well,..."
What is, "Begging the Question", Alex?
Can I have "Basic Logical Fallacies" for $600, please?
8-)
"Do you see students sitting in rows, listening to a teacher at the front of the room, writing on the chalkboard and demonstrating how to solve problems? Do you remember how boring and mindless it was?"
This scenario sounds more like an idee fixe, a hallucination or just a plain lie.
What teacher would teach math without encouraging student participation through questions and having students work problems or come to the board?
Just yesterday I achieved amazing success with a small group of usually refractory and definitely lagging 7th and 8th graders through a combination of direct instruction and the Socratic method.
The problem I put on the board was a circle inscribed in a square, one of my favorite mini think problems (an alternative is two circles in a rectangle). The task was to calculate the area not covered by the circle. Only the measure for a side of the square was given. The creative jump was to see that subtracting the circle area from the square area would get to the answer and that the known length of the side of the square would reveal the radius of the circle.
It was amazing to see that with a little bit of prodding and filling knowledge gaps the students actually got the answer with FULL UNDERSTANDING.
There was no need for constructivist texts to achieve this breakthrough.
My 9th grader's Geometry teacher wants the class to master the properties of quadrilaterals. Almost every day this week, she has given them a quick quiz at the beginning of class. Then she tells them that it doesn't count for a grade, and to be prepared for the quiz to count the next day. Today, M. commented that she was getting to the point where she could finish the quiz both quickly and accurately. (She said it with a faint air of disgust.)
"Ah, that's the point," I said. "Your teacher seems to want you to have both mastery and automaticity."
"The only question is whether the strawman is the result of ignorance or malice -- I'm not sure which I find more disturbing."
I've always used the choice of ignorance versus arrogance.
"But I wrote it because there are many people for whom it is not obvious and who actually believe these lies. Yes, lies."
Yes, lies.
I've come to the conclusion that it's more than arrogance. I don't know if I would call it malice, at least not towards students, but they are lies, nonetheless. They are lies in that they KNOW that there are good arguments against their position, but they still spout out the same rhetoric to those who don't know better. They are lying to their audience. Then they they ignore (as best they can) those people and arguments they cannot handle.
The latest tactic can be seen with the math panel which is constrained to talk only in terms of valid research. As I've said before, schools get to select curricula based on any whim they might have, but you need valid research to make any changes. This is all wrong. With a monopoly on education, the onus is on public schools to prove that they should continue the status quo.
"This scenario sounds more like an idee fixe, a hallucination or just a plain lie."
Even after my last post, I'm still struggling with this. Perhaps it's none of the above. It could be that (because of ed schools), it's all they have. If you take that away from them, they have absolutely nothing. Besides, it's a whole lot more self-inflating to talk about constructivism than flash cards at cocktail parties.
This is neat.
You don't often see the divisibility rule for seven. 2, 3, 5...no sweat. But seven?
Numbers Divisible by 7
To determine if a number is divisible by 7, take the last digit off the number, double it and subtract the doubled number from the remaining number. If the result is evenly divisible by 7 (e.g. 14, 7, 0, -7, etc.), then the number is divisible by seven. This may need to be repeated several times.
Example: Is 3101 evenly divisible by 7?
310 - take off the last digit of the number which was 1
-2 - double the removed digit and subtract it
308 - repeat the process by taking off the 8
-16 - and doubling it to get 16 which is subtracted
14 - the result is 14 which is a multiple of 7
Erm, wouldn't it be easier just to divide by 7 and see if it comes out even?
That's neat about 7. But why does it work? Wait, wait. Don't tell me!
My son and I were doing something like this because he had to determine if a number was prime or composite.
For numbers divisible by 3, we could add up the digits of the number and see if that was divisible by 3.
723 ==> 7 + 2 + 3 = 12
Which is divisible by 3, so 712 is divisible by 3.
We were struggling with rules for 7 and 9. Instructivist's rule for 7 looks interesting, but I think that dividing in my head would be easier. As I'm dividing, all I have to remember is the remainder, add the next number, and divide again.
For the prime/composite problem, we started looking at the last digit in the number.
If a number ends in 1 then the two factors could end in 1 and 1, 3 and 7, or 9 and 9.
If a number ends in 3, the two factors could end in 1 and 3, or 7 and 9.
If a number ends in 7, the two factors could end in 1 and 7, or 3 and 9.
If a number ends in 9, the two factors could end in 1 and 9, 3 and 3, or 7 and 7.
This reduced the number of combinations we looked for, but that's as far as we got.
OK, so much for constructivism. Time to look up prime number solvers.
Then again, there really is no reason why you HAVE TO do this kind of factoring:
391x + 34 = 17(23x + 2)
I wouldn't be too pleased to see that on a test.
"My son and I were doing something like this because he had to determine if a number was prime or composite."
I could be wrong, but I think you can make the prime/composite test very, very simple. You only need to test for divisibility with the primes 2, 3, 5 and 7, all of which are a piece of cake except for seven.
If a number is divisible by 2 (all even numbers), then it automatically eliminates all numbers divisible by 4 and 8. So no need to bother with 4 and 8. This takes care of half the population.
Then tackle 3 (sum of digits). This eliminates all numbers divisible by 6 and 9. So no need to bother with 6 and 9.
5 is superfriendly (all numbers ending in 0 and 5).
This only leaves the nettlesome 7.
All numbers that yield to 2, 3, 5 and 7 are composite.
The rest is PRIME, since what's not composite is prime (except for one, of course).
I think this is a foolproof and extremely simple way to tackle the intimidating prime/composite challenge.
I developed (discovered?) this insight that the divisibility test can be limited to only four primes (2, 3, 5 and 7) while rehearsing the Sieve of Eratosthenes with my students.
An interesting fellow, that Eratosthenes. He is the guy who found a brilliant way to calculate the circumference of the Earth to a high degree of accuracy when flat-earthers were still roaming the Earth.
Now, how many stadia is it from Alexandria to Syrene?
"All numbers that yield to 2, 3, 5 and 7 are composite."
I should have added that this simplified test is good up to the square of the next prime number, which in this case would be the square of 11 and so on.
What about numbers that only have large prime factors?
391 = 17 * 23
My son's pre-algebra book seemed to get carried away with having students find the factors of any large number.
Make a list of squares of primes to set the parameter. Then test the number that falls below a given parameter by using the respective primes.
Here are squares of primes 11-29:
121
169
289
361
529
841
Give me some examples to test for primality.
As an alternative, you may also want to consider the Rabin-Miller Strong Pseudoprime Test.
[What about numbers that only have large prime factors?
391 = 17 * 23]
Same difference.
Identify the upper boundary. In this case formed by the square of 23 or 529 (391 falls below 529). Then test for divisibility with the 8 primes preceding 23 (2 3 5 7 11 13 17 19). You can rule out the first three primes on the spot. Testing is now reduced to only five primes. In this scenario, if a number below 529 does not respond to any of these primes, you are dealing with a prime number.
The approach reduces the necessary test operations considerably.
Instructivist and all,
re http://www.aaastudy.com/
Some kids are magically drawn to computers. They might get excited doing the same things on the computer that they might otherwise find less thrilling.
BOY! I wish that'd been around when the Ditzlings were in k-5. All three of them struggled with handwriting/fine motor issues, which made any hand-written homework just torture. (The older two are now 29 & 27).
All eventually got to fluency, but having a practice method that did not involve actually forming the numbers would have helped the process along a lot.
All learned to keyboard early. Oldest Ditzling still has barely legible handwriting and does not like to write, but is fine with keyboarding.
Glad to hear I'm not the only person who thinks Guess and Check is used to excess. I was trounced on once for saying that in an article I wrote about Singapore Math.
Cute, Barry. Except that you didn't write that it was being "used to excess." You insinuated that it was the norm.
Nice try. I'll highlight the "give away" word for you:
"TYPICALLY, in U.S. texts, students are taught to use a method called “Guess and Check”—trying combinations of numbers until the right numbers are found that satisfy the conditions of the problem—a method that many professional mathematicians consider inefficient."
Typically? To Excess?
I'm not sure what the issue is.
It IS typical. My son got it with Everyday Math and with Mathland. Most math curricula use it. It's been a big topic on KTM for ages.
The Amy and Judy type of problem for pre-algebra guess and check is a classic. My son got one that used marbles. They worked in groups to figure it out without any prior guidance or tools like bar modeling.
"To excess" implies a judgment perhaps, but it requires more explanation. I would say that it passes that test too. It's a fundamental philosophical idea of reform math. It's the idea that there is no one correct solution, or that there is no one (rote) way to solve a problem. This is why they don't like algorithms or procedures. They consider them rote techniques. Guess and check fits very nicely with their philosophy and they can talk about all sorts of vague brain processes that happen when students find the answer. It makes them sound important. As I said before, this is anti-math. You would think they would LOVE the idea of bar modeling, but they don't.
Typical doesn't bother me. The excess part does.
I have also mentioned ages ago that engineers use guess and check a lot, but it's called searching. I have many books on my shelf on just this topic. For example, if you have a nonlinear equation, like
x^2 - 1 = 0
you can solve this algebraically to get x = +/- 1. But this is also a good example for searching.
You start by guessing a value for x, say x = 0 and plug it into the equation. The left hand side comes out to:
(0)^2 - 1 = -1
So x=0 is not a solution. Since you want to find where the left-hand side is 0, you have to keep making guesses. Since -1 is less than zero, you might want to try a guess for x that is greater than zero. So you try (pick a number out of the blue) x = 10 and plug it in.
(10)^2 - 1 = 99
X=10 gives you a number above zero, so you know that the(?) answer is between the two. I used the '?' because the student might not have a clue that there are two answers. This process is called bracketing the solution. Once you've bracketed the answer, you can find the answer. You do this be splitting the difference between the two guesses that bracket the answer and run the test again. With the new result, you reduce the bracket by one-half. Keep doing this and you will converge to the answer. (There are many, many other ways to search for a solution.)
The key problem is that you will find only one solution or root, so it's very important to select an initial guess that is "near" the desired solution. This can often be more difficult than actually solving the problem.
I wrote one computer application that searches for a solution of a problem that can have tens of thousands of equations. A slight change in initial guess parameters can cause it to not converge.
If you know that the nonlinear shape is smooth in the area of the solution, you can use the derivatives (slope) of the equation to speed up the searching.
So, is this sort of smart guess and check what the math reformists want the students to construct? In some respects, it is. But what if they don't construct any sort of smart search technique? What if they just stumble on the answer by brute force? What if they don't learn any specific technique? Do they tell the students afterwards? No. Do they work on mastering the skill? No.
Is constructivism a means to an end, or is it an end in itself? It fails on both counts.
As a means to an end, it is neither necessary or sufficient, and there is no guarantee that all students will achieve the light bulb effect. Kids will have to be directly taught.
As the end in itself, it is anti-math. Math is all about skills and procedures that make your life easier. If you don't codify the results of constructivism into specific procedures or algorithms and then master them, then you aren't doing what math is all about.
We've all had teachers who lead us down the garden path to discovery, but that was a means to an end, not an end in itself, and for many students, the teacher has to come out and explicitly teach them.
Guess and check is more than "typical" or "to excess", it's a guiding principle of reform math.
TYPICALLY, in U.S. texts, students are taught to use a method called “Guess and Check”—trying combinations of numbers until the right numbers are found that satisfy the conditions of the problem—a method that many professional mathematicians consider inefficient.
a) Without reading the original article, the logic of that sentence is that it is typically taught not that it is the typical method used. But, even if that were what he said, it is still pretty much true. Math in American secondary schools is generally taught a lot more like an empirical science than it is the formal a priori subject that it actually is.
b) It isn't just inefficient. Such an empirical approach to mathematics is normally fallacious. In some cases, where you have, say, an equation with a finite number of solutions, then it is at least possible to use such a tactic in a non-fallacious manner. But, relying on such methods does, indeed, train students to approach mathematics fallaciously which is one reason students that do go on to do math as their specialty are in for a real culture shock and have a lot of trouble understanding what it takes to actually prove a theorem. (And, proving theorems is 90% of what they will be doing and almost entirely how they will be evaluated if they do try to go on.)
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