I looked at my fifth grade son's EM homework for the night on adding and subtracting fractions. (They are way behind on their second book, so other fifth grade parents might have seen this before.)
This is a sample.
1/3 + 1/2 = ?
1/4 + 1/5 = ?
The answer to the first is 5/6
The answer to the second is 9/20
Explain what pattern you see.
"Pattern?", you say? Well, don't feel bad. I had to figure out what on earth they were talking about. I finally figured it out. To get the numerator of the answer, you add the two denominators. To get the denominator of the answer, you multiply the denominators. What we have here is rote pattern work that masquerades as understanding.
Raise your hand if you see half of the kids applying this pattern to:
1/4 + 2/5 = ?
OK. How about a different pattern for subtracting fractions, like
1/4 - 1/5 = ?
1/2 - 1/3 = ?
The numerator of the answer is always 1 and the denominator of the answer you get by multiplying the denominators.
They conveniently ignore problems like 1/5 - 1/4 = ?, and, of course, any other problem where the numerator is not 1.
What other patterns does EM need to cover all of the bases of adding and subtracting fractions?
I had to spend some time making sure that my son knows the formal rules for adding, subtracting, multiplying, and dividing fractions. I want his understanding to be based on general, mathematical rules, not stupid, error-causing patterns. I wonder what my son thinks when even his teacher tells the students that she is teaching them a method from EM because she has to.
Whenever anyone talks about mastery versus understanding, just think of these patterns. They are classic and show exactly what the fuzzies mean by understanding. When they talk of understanding, they are not talking about mathematical understanding. They are not talking about any sort of understanding that will help students when they get to algebra.
Talk of balance or understanding gives them credibility that is completely undeserved. We need to talk about competence, not philosophy.
This is shocking.
ReplyDeleteI've never seen anything like it.
Plus....how much time would it take your basic kid (and parent) to figure this out.
No wonder parents are rioting.
I don't think TRAILBLAZERS has anything like this.
ReplyDeleteTRAILBLAZERS just seems to be silly and slow (although they seem to have great little word problems).
I've been wondering whether TRAILBLAZERS is the best of the bunch, or whether it's as bad as the rest but not so torturous.
I'm starting to think it's the former.
What a bizarre idea.
ReplyDeleteI always assumed that "patterns" meant "rules" -- that constructivist texts were having kids discover rules by looking for patterns.
But no!
They're finding exceptions to the rule!
Or, umm, rules with many exceptions.
In keeping with the recognition that there is no one right answer, I guess.
Math is so diverse.
ReplyDeleteMath has little to do with finding patterns. You might think that pattern work will lead somewhere, but it doesn't. It's a constructivist end in itself. Pattern recognition in place of abstract mathematical understanding.
ReplyDeleteHey, at least you're into the second book! It's mid-April, and my 5th grade son is still putzing around in the first one! I'm not sure if that's a bad thing or a good thing...
ReplyDeleteDid you catch the "Short History of Mathematics Instruction" in the first book? It's a hoot.
"Throughout our nation's history, students have learned mathematics in different ways and have spent their time working on different kinds of problems. This is because people's views of what students can and should learn are constantly changing." Um hmmm.
They proceed to give examples of math instruction over time.
1790s: Apparently, children under 12 didn't learn math back then.
1840s: Mental arithmetic--did you know there was a child in Connecticut who could mentally multiply 314,521,325 by 231.452,153 in 5 1/2 minutes? As if that has anything to do with the kind of math instruction he received!
1870s: "Many textbooks were step-by-step guides on how to solve various word problems. Students were given problems and answers. They had to show how the rules in the textbook could be used to produce the given answers." I'm thinking 1870 looks pretty good...
1920s: "Elementary mathematics emphasized skill with paper-and-pencil algorithms." All those shopkeepers, you know. "Clerks had to add up sales, but there were no cheap, easy-to-use calculators." Duh.
And, or course, the pinnacle, the 1990s! "Today the emphasis is on solving problems and applying mathematics in the everyday world." As long as they can figure out how much it costs to rent a motorcycle, they'll be just fine. Yup, that's sure what I want for my kids--why would they want to bother themselves with math in college, anyway?
Maybe, after explaining fraction addition, and doing some examples, the teacher could comment that the sum of two fractions whose numerators are both equal to 1 takes a particular form. And perhaps ask why? Or maybe the textbook could comment to that effect. It is an interesting aside, and a nice lead-in to algebra.
ReplyDeleteAs it stands, the addition example is open to a number of misunderstandings, such as that the pattern occurs only when the denominators differ by 1. There's also the issue of "lowest terms." It's true that 1/2 + 1/4 = 6/8, but that would usually be written as 3/4. Presumably the teacher would set such potential misunderstandings straight?
I must disagree with SteveH that "Math has little to do with finding patterns." It does play a role. For example, Gauss first formulated the quadratic reciprocity theorem by spotting patterns. Of course, he then went on to rigorously prove his hypothesis. This approach is not uncommon in number theory.
VickyS, I agree that the 1870s math instruction looks pretty good. I wonder whether it was also explained to the students why the rules in the textbook worked? That would have been even better!
Math being about patterns is something the fuzzies appropriated from mathematicians without understanding what it was they were appropriating. When mathematicians talk about patterns, they mean the big picture view that links things like our ordinary arithmetic operations to an axiomatic approach (groups, rings, fields) that then generalize to far beyond ordinary arithmetic. In K-12, math is about the basics that students need to know to eventually get to that point if they choose to pursue mathematics as a major.
ReplyDeleteThe "pattern" that EM talks about in adding fractions would be harder to show with a problem like 1/2 + 2/3 + 4/7 wouldn't it?
I just saw an "explanation" on the Internet of the rule for adding electric resistance for resistors in parallel, given by the equation:
1/R = 1/R1 +1/R2 + ...1/Rn
The explanation said that for two resisters, the total resistance can be found by R = R1 x R2/(R1+R2) but that this relationship only holds for two resisters. Beyond that, it is more complex. Well, the formula they present comes out of the above formula; it's not that you need separate formulas. Looks like the EM mode of explanation is catching on.
"...mentally multiply 314,521,325 by 231.452,153 in 5 1/2 minutes?"
ReplyDeleteI did it in less than 5 1/2 minutes, but I had to use paper and pencil. The hard part was keeping the columns lined up.
"Math being about patterns is something the fuzzies appropriated from mathematicians without understanding what it was they were appropriating."
Thanks Barry, you beat me to it. The fuzzies use patterns in place of real mathematical understanding. That is my point.
In this case, there is absolutely no justification for having kids see this pattern. It is rote pattern recognition that will end up confusing many kids in the long run. The pattern does not lead to a development of the mathematical governing rules.
It reminds me of the way many students do math problems. They look at a problem and try to remember another problem they did before that looks the same (pattern matching). Then they just try to plug the numbers into that pattern.
It's one thing to sneak up on a good understanding of a complicated mathematical algorithm, but adding fractions is not one of them. EM sneaks (spirals) around all over the place, but never jumps in and gets to work - a sure sign of fuzzy math. The direct and general explanation ends up being so much easier and better than sneaking around.
Another example of this is converting a mixed fraction to a regular fraction.
For 2 3/4, you take the denominator (4) and multiply it by the whole number (2), and then you add in the numerator (3). This is put into the numerator and the old denominator stays in the denominator. What good does recognzing this pattern do? Do they use it as a lead-in to a proper mathematical understanding? No. It should be the other way around. Learn the general rule and then (perhaps) play around with interesting special cases.
Constructivism is all about pattern recognition, but they go nowhere with it. It's an end in itself. It's like combination and permutation problems. Discovery seems to work well with simple problems, but the discovered patterns do not help one bit as the problems vary or get more difficult.
As with calculators, the fuzzies use pattern recognition to avoid real work and understanding. They dance all around learning and never jump in and get to work.
rote pattern recognition
ReplyDeleteabsolutely
Hi Steve, you know that I share your definition of competence.
ReplyDeleteBut,
We need to talk about competence, not philosophy.
Your philosophy, or specifically, your epistemology dictates the competence you seek in children.
For the educator who wants children to be competent to perform a planned role in a new social order, children must learn reading and writing and arithmetic unencumbered by the old social order, which means the old algorithms. Algorithms that embody somebody else's idea of efficiency, reliability, and generality. Value judgments the educator doesn't want you to make on behalf of children, not even your own.
The educator's desire for this new competence operates beyond the reach of your rational appeals to old competence.
A long time ago, Catherine asked if there was any connection between constructivism and socialism, and standing back I think that the deep connection is that constructivism is a useful tool for socialists. Or anybody trying to build a new social order.
Constructivism is the epistemology that cuts the tie that binds children to their parents and traditions.
It's no use to hope that you and a constructivist share the meaning of the word "competence."
To the constructivist, a child who has been directly instructed in the use of somebody else's efficient, reliable, and general mathematical algorithms is seen as a victim of that instruction, and the child remains incompetent to boot. From first principles.
Does ANYONE ever finish both EM "math journals" in a year?Through 1st-4th grade, I don't think Meg ever got through Ch. 8. My first grader just brought home her mostly-completed Vol. 1 the other week. She saw me about to throw it promptly in the trash and stopped me. "No Mommy! Ms. S says we HAVE TO FINISH the other pages at home for practice in our spare time." (This is the age where all teacher comments are divinely inspired and MUST be followed) I asked if they had to hand it in. No. So into the trash it went and I think we'll be doing lots of facts practice and Singapore "Challenging Word Problems" this summer. Ms. S is just 2 years out of school and looks as young as my teenage nieces and spouts all the latest ed school stuff about "taking ownership" etc.
ReplyDeleteGuys, I share your view that kids should be shown how to (in this case) add fractions, from first principles. Once they have mastered the technique, and maybe understand why it works (*), a nice supplement is to spot any interesting patterns that arise. The smart kids should spot the "1/a + 1/b" pattern, anyway. For the teacher (or textbook) to highlight it would be a bonus.
ReplyDelete(*) Understanding why a technique works sometimes comes later than actually mastering the technique. That's another reason why repetition is important.
I should mention that I'm a Brit, and I'm not familiar with the techniques currently used in US (or even UK) schools. If you're saying that spotting the "1/a + 1/b" pattern comes before the algorithmic technique, then I agree; this is likely to confuse all but the brightest (or best prepared) students. And waste a lot of time, especially in the hands of a teacher who has only a hazy understanding of the subject.
Barry, I believe that spotting and using patterns is an important part of mathematics. Adult mathematicians make use of "big" patterns, as you say; kids learning math can use patterns on a smaller scale.
Once you pass a certain point in mathematics, you begin to come across problems that are unlike any you've seen before. That's what math is about; it's not a closed system! To solve such problems you need both (a) a mental library of tried and trusted techniques (I assume you agree with me here), and (b) the ability to relate the new problem to a more familiar problem, to restate or translate it in some way, to break it down into sub-problems, and sometimes just to sit there with a blank sheet of paper hoping for inspiration! Both elements are important. For (b), it helps to have some experience of spotting patterns.
The "1/a + 1/b" pattern would indeed be tricky to apply to a problem like 1/2 + 2/3 + 4/7! That's why the teacher should emphasise that the pattern applies only when adding two fractions whose numerators are both equal to 1, and why this should only be attempted once the basic technique is mastered.
For two resistors, R = R1 x R2/(R1+R2) is a nice shortcut, and equivalent to 1/R = 1/R1 + 1/R2. For more than two resistors there is a generalization in terms of elementary symmetric polynomials, but I think the reciprocal form is easier to remember, and follows the derivation.
SteveH, re your example of converting a mixed fraction to a regular fraction, do you mean that kids should first be shown that 2 3/4 = 2 + 3/4, and then 2 = 8/4, so that 2 3/4 = (8+3)/4 = 11/4? If so, I concur. Once this is mastered, though, the approach you describe is a nice shortcut.
So, here are a couple of questions for you guys. Who believes that:
1) There is no such thing as cross-multiplication?
2) In proving an identity, you must transform one side into the other; it is never permissible to "meet in the middle?"
Nick - great comments; I hope you stick around and continue to contribute your thoughts. We are a friendly people, and I promise that our hazing rituals only rarely involve trips to the emergency room.
ReplyDeleteAnyway, I'm not sure I'm interpreting your questions correctly, but here's my attempt:
1. Cross multiplication is indeed a useful shortcut for working with fractions, but, like all shortcuts, it's subject to misapplication if the fundamentals are not there. A student might survive a good long time knowing only how to cross-multiply; eventually, though, they're likely to be completely lost when they get to functions like the trig identities, or logarithms.
2. It's perfectly fine to 'meet in the middle', as long as you understand what is actually happening in each step of the transformation, and why you're allowed to do it. Again, if you teach the shortcut without the underlying reasoning, the student is very, very likely to not only misapply the lesson, but also not understand where he went wrong.
Patterns can be useful, but are worse than worthless if all you learn is the pattern but not the underlying math. I say worse, because the student not only doesn't know the material, he in fact knows something which is wrong, and will continually try to apply that wrong thing in the future.
"A long time ago, Catherine asked if there was any connection between constructivism and socialism, and standing back I think that the deep connection is that constructivism is a useful tool for socialists."
ReplyDeleteI've come to the conclusion that I have never met a fully-developed practitioner of that faith. Even when they spout off discovery math ideas they don't get it right. At my son's school, they are now arguing about practice versus understanding. They can't even get that right. I offered to give the curriculum committee my views on the subject, but have been studiously ignored. They are the experts, but nobody gets review copies of Singapore Math even though some think it's interesting. It seems to me like it has to do with academic turf. All they have from ed school is child-centered group discovery. If you take that away, they have nothing.
It's almost as if the ed school professors are the high priests and the teachers are the poor students. The teachers go to the schools, don their coconut headphones, and wait for a landing. Becky, weren't you the one who created that hilarious picture of Lucy Caulkins? It's a classic.
I understand what you are saying, but I'm always surprised by how bad they are at doing even what they think is best. If they can spout off a lot of philosophy, then many won't realize that there is nothing there. They try to corner the market on understanding because they have nothing else. Some might have a well-developed social justice plan, but most of the teachers I know just do not want anyone to call their bluff.
" ...spotting the "1/a + 1/b" pattern comes before the algorithmic technique,..."
ReplyDeleteYes.
"... kids learning math can use patterns on a smaller scale."
Not the way it is currently being done in K-8 math. Patterns and "real world" problems are used to AVOID true mathematical understanding. In all of my 30+ years of mathematical programming, I have never done what they do.
What do they do? They don't teach basic skills first. They assign "real world" problems, they break the kids into very mixed ability groups, and then they expect them to look for patterns to solve a problem. This is somehow better than learning basic skills first.
I never, ever do that. I research the literature, I learn new skills, I recreate other people's work so that I bring myself up to their level. I don't reinvent the wheel. Scientists don't look kindly on technical papers that don't cite current and historical research. Ignorance does not improve creativity or results.
" ...do you mean that kids should first be shown that 2 3/4 = 2 + 3/4, and then 2 = 8/4, so that 2 3/4 = (8+3)/4 = 11/4?"
Yes, but that's not what they do. Fuzzy math does not like rigor so they substitute patterns and explanations that fail when the problems become more abstract, as in algebra.
"1) There is no such thing as cross-multiplication?"
ReplyDeleteNo such thing. I suppose I could make one up and justify it, but why create problems? If I had a nickel for every time a student applied that technique (in its various forms)... well, you know. Shortcuts and simple cases may be very useful, but they are also very dangerous.
x/2 = 3/5 + 5
I've seen too many students solve this with "Cross Multiplication" to get:
5x = 2*3 + 5
Like adding fractions with ones in the numerators, why bother with special cases when the basic rule is simple and much more powerful?
"2) In proving an identity, you must transform one side into the other; it is never permissible to 'meet in the middle?' "
I'm a pragmatist, but I still expect rigor.
Nick, it sounds like you are NOT a parent of a child learning math with a program like Everyday Math. Some of us parents at KTM get quite cranky after seeing what comes home from school and working untold hours teaching our kids what they really need to know. All I can say is that you have to see it to believe it. As is commonly repeated here: "It's worse than you think."
As kathyiggy says:
"Ms. S is just 2 years out of school and looks as young as my teenage nieces and spouts all the latest ed school stuff about 'taking ownership' etc."
I can't even imagine my neice as my son's teacher.
Nick: I appreciate your comments and questions. I think many have been answered above, but let me just say that I agree that observing patterns is important. They make sense when they are put in context and guide the student to the next step. For example, in teaching division of fractions, many texts (including Singapore's)establish the pattern of invert and multiply long before division of fractions is presented as a topic. Students first learn that fractions can express division; i.e., 1/3 of a dozen eggs is 1/3 x 12 = 12/3. Or, working backwards, 12 divided by 3 = 12 x 1/3. They next learn that 1/4 divided by 2 is the same as 1/2 of 1/4 or 1/2 x 1/4. Still later: 5 oranges cut into quarters = 5 x 4 = 20. So how many quarters in 5 oranges? 5 divided by 1/4 = 5 x 4. By the time they reach the unit on division by fractions, they have seen the pattern of invert and multiply many times. It is thus a small leap to the general invert and multiply rule. Some books attempt to teach why it works; some students will get it, some not. But make the effort and move on. It's been presented in context,with understanding of what it means to divide a fraction by another fraction, rather than just teaching a rule in isolation.
ReplyDeleteThis is Steve H's point: teach with understanding. EM for all its blather about teaching kids understanding presents rules without sufficient development. Any sequential development is entirely coincidental given how much the program spirals.
Regarding whether one must work one side or the other of an identity rather than working to a happy medium, I am for rigor as well: so if you work to a happy medium you must show that all steps taken to reach the equivalent expression must be reversible.
Kathy -- I think my daughter's class will finish both journals in Everyday Math this year. At this point (April in 5th grade) they are mid-way through Unit 10 in jouranl 2, that leaves only 2 units-- volume and probability. We've covered probability ad nauseum this year, so it should be a snap.
ReplyDeleteHer class devotes 1 1/2 hours everyday to math.
I looked back in her journal at the patterns Steve highlights (lesson 8.3). My kid never saw the pattern. She just carried on solving all of the fractions and wrote out a long explanation on how to add and subtract fractions.
When you fly through the journal at lightspeed, apparently the teacher doesn't have time to notice when the kids are not seeing a "pattern." Of course, maybe it's indifference, if she doesn't see the pattern now, don't worry, she'll spiral through next year.
Actually, I don't think I asked about the connection between constructivism & socialism -- I think it was someone else.
ReplyDeleteI probably asked about the connection between Russian constructivism and ed school constructivism (that is, I'd like to know whether there is a connection).
The Russian constructivists produced my favorite definition of art: the familiar made strange.
That thought changed my life.
Seriously!
Funny the topic is fractions. My son's third grade teacher emailed me to say he is having difficulty with fractions and would benefit from more practice at home.
ReplyDeleteThe difficulty is with mixed fractions. I do not recall him bringing home any homework regarding mixed fractions.
It must be the osmosis learning requirement again.
Lord, give me strength.
This comment has been removed by a blog administrator.
ReplyDelete***************
ReplyDeleteSo, here are a couple of questions for you guys. Who believes that:
1) There is no such thing as cross-multiplication?
2) In proving an identity, you must transform one side into the other; it is never permissible to "meet in the middle?"
******************************
(thus far nick.)
thanks for asking.
i tend not to *say* "cross-multiply"
without some squarequote-equivalent
like "so-called" or what have you.
obviously, if a/b = c/d,
one has ad = bc; it might've even
been useful at some point to have
given this fact a name.
somebody else has already given
an example of a student using
the word "cross-multiply"
without knowing the context
in which it should apply;
suchlike mistakes are common
in and out of mathematics of course.
the peculiar annoyingness
of "cross-multiply", as it seems
(gropingly) to me now, is that
it seems to be used by students
as a tool to *avoid* thinking about
whatever they need to do
to understand what's really going on
(in this case: one can multiply
both sides of an equation by
any nonzero constant & obtain
an equivalent equation).
it's really weird. a lot of people
would rather (so they believe) learn
just about any number of ad hoc tricks
if only they never have to come to terms
with any general principles.
or maybe not so weird.
general principles in ordinary life
might easily be seen by certain temperaments
(including my own, i guess)
as hard to find and typically useless.
the nature of mathematics makes for
a stark difference with ordinary life:
definitions, for example, are allowed
(required!) to mean *exactly*
what they say (this is of course
usually impossible in extramathematical contexts).
the enemies of clarity are of two kinds:
(1) the merely ignorant, who simply
have never imagined that the kind
of corrrectness attainable in maths
exists at all; &
(2) those who *do* understand
that perfect clarity is sometimes
attainable, and don't like it.
promulgators of "find the pattern"?
just as likely to be type (2) as not ...
as for proving identities ...
again, somebody else already nailed it:
presenting a collection of
equivalent equations, the last
being some "obvious" identity",
is (obviously) a perfectly valid
way to establish the validity
of a proported identity.
trouble is, the *words* ought to be there.
and we who are fortunate enough
to be teaching maths very often
will rightly run screaming from
assignments that require us to
help students with *writing*.
the reason so many of us insist
that a proof that "a = z"
use the classic
"a = b
...= c
.....
...= z"
style is that this style doesn't require
any verbal embellishments to be
a "correct" proof.
students that stubbornly resist
rewriting their "both ends against
the middle" proofs aren't willing
to accept that part of the project is
"writing a good proof" rather than
"showing understanding".
oh, that was me, vlorbik. hi, nick.
ReplyDeletehttp://vlorbik.com
um ... markov?
ReplyDeletewhat good would 1/a + 1/b = (a+b)/(ab)
have done egyptians if they *only*
used unit fractions?
1/a + 1/b would then simply be
*the* right way to talk about.
everybody else:
i should know better than to respond
to trolls; sorry.
try searching for "clueless vitriol"
in google sometime ....
This comment has been removed by a blog administrator.
ReplyDeleteHi Markov--
ReplyDeleteWe have a rule that commenters & members not call each other names in posts or threads, so I'm deleting your 2 comments.
Independent George, I agree that cross multiplication is a useful shorthand. Of course, as SteveH notes, the student must understand that the shortcut applies only to equations of the form a/b = c/d. But once mastered it minimises effort and saves time.
ReplyDeleteVlorbik also mentions that "cross multiply" can be used as a tool to avoid thinking. This can indeed be the case, and in many other areas too. I recall one teacher who explained that an equation such as x - 5 = 9 can be solved by "taking the 5 to the other side and flipping the sign." This would have seemed utterly mystifying had I not previously been taught the underlying concept, which in this case calls for adding 5 to both sides of the equation.
SteveH, you're right -- I'm not the parent of a child currently using Everyday Math! I've been passionate about mathematics since I was a young child. I stumbled across this blog and thought I would add a few comments.
Barry and Independent George, re proving identities, I concur that the steps must be reversible when working towards the middle. But there seems to be a widespread misunderstanding about this.
A few months ago, as a result of a throwaway remark, I got into an extended online debate with several people, all of whom maintained that working with both sides of an identity is not rigorous. I was shocked that they were able to quote three textbooks (Lial, Trigonometry 8th Ed., Larson, Trigonometry 7th Ed., Sullivan, Trigonometry 7th Ed.) and an online reference: Proving Identities Rigorously, in their favor. Apart from being incorrect, it is misleading to argue as these authors do, and possibly damaging to the student who suspects something is amiss but lacks the confidence to question the textbook. (And the teacher?) Rather than state that manipulating both sides of an identity is always wrong, the textbook should maybe *advise* students to work from one side to the other, for clarity.
Vlorbik, re working both sides, usually the only "embellishment" required is a "<=>" symbol (or "iff") instead of an implicit "=>" !
the nature of mathematics makes for
ReplyDeletea stark difference with ordinary life:
definitions, for example, are allowed
(required!) to mean *exactly*
what they say (this is of course
usually impossible in extramathematical contexts).
the enemies of clarity are of two kinds:
(1) the merely ignorant, who simply
have never imagined that the kind
of corrrectness attainable in maths
exists at all; &
(2) those who *do* understand
that perfect clarity is sometimes
attainable, and don't like it.
Well said, vlorbik.
nick:
ReplyDeleteyou're quite right of course
as to <=> (or "iff").
and students at the trig level
(where "proving identities"
is likely to have arisen)
will typically have enough
"mathematical maturity" to be
well-prepared to deal with, e.g.,
the distinction between
equations and logical equivalences.
still, once we decide to present
such material, time will have to
be found ... other material omitted.
if i designed my own courses,
i'd very likely want to discuss
things like proof techniques
a good deal more than i can.
of course, just to say,
"this is the only right way"
when it's *not* is contemptable.
authoritarians shouldn't teach math.
Good point, Vlorbik. Teachers often do not have much leeway in what or how they teach.
ReplyDeleteProving identities by working toward a "middle" is what is done in combinatoric proofs. They are very sophisticated--not to mention hard to do. So there are times when working toward the middle can be extremely rigorous.
ReplyDeleteNick: A student of mine (who I tutor) has a trig book by Lial et al which has "working toward the middle" as a strategy for proving identities, and specifies that in doing so, the steps must be reversible. I guess I mis-spoke when I said I liked rigor so preferred working one side or t'other. Sorta like the supposedly inflexible rules English teachers preach such as "never use the passive voice", "never split an infinitive" and "never end a sentence with a preposition." I break those rules on a daily basis. (On purpose, might I add.)
Nick, quoting Vlorbik: "Vlorbik also mentions that "cross multiply" can be used as a tool to avoid thinking."
Lastly, on the subject of "cross multiplication" and its automatic use leading to wrong thinking, I quote from Wu from his monograph on "Whole Numbers" in which he states:
"As to the non-thinking aspect of these algorithms, there is at present a perception that if anything can be done without thinking, then it does not belong in a mathematics classroom. This is wrong. If mathematicians are forced to do mathematics by having to think every step of the way,
then little mathematics of value would ever get done and all research mathematics departments would have to close shop. What is closer to the truth is that deep understanding of a topic tends to reduce many of its sophisticated
processes to simple mechanical procedures. The ease of executing
these mechanical procedures then frees up mental energy to make possible the conquest of new topics through imagination and mathematical reasoning. In turn, much of these new topics will (eventually) be once again reduced
to routine or nearly routine procedures, and the process then repeats itself. There is nothing to fear about the ability to execute a correct mathematical
procedure with ease, i.e., without thinking. More to the point, having such an ability in the most common mathematical situations is not only a virtue but an absolute necessity. What one must fear is limiting one’s mastery of such procedures to only the mechanical aspect and ignoring the athematical
understanding of why the procedures are correct. A teacher’s charge in the classroom is to promote both facility with procedures and the ability to
reason. In the teaching of these algorithms, we should emphasize both their routine nature as well as the logical reasoning that lies behind the procedures."
I think that's what we're all trying to say here, but I want to point out that when I work with my daughter (who has a learning disability), I work to teach her the understanding behind the procedure. Sometimes she gets the understanding, sometimes not, but she does get the procedure. If she misuses the procedure, I look at that as a learning opportunity to show why this is a mis-use and it helps underscore the understanding. But in general, if she understands a procedure well enough to carry it out correctly, I do not stand in her way. I told Wu this in an email exchange on the topic of "cross multiplication" yesterday. He responds:
"You are a good father, if I may say so. "if she knows a procedure and uses it correctly, I try not to stand in her way." How many educators would say the same?"
Wu has helped me a great deal through the years, so I put a lot of trust in what he writes and advises. Wu rules!