That’s very easy to say when you know how to do it.
Explain it to a nine-year-old. Go ahead. Trust me, it’s not as easy as you think. Everyone’s a real genius when they don’t have to try to get this across to two dozen fifth graders, many of whom aren’t the math whizzes you guys are.
Fifth-graders are 10-11. Not nine. And yes, you can continue teaching kids to count on their fingers all the way to high school. That’s how we get to be the last among developed countries in math education.
Challenge:
Prove that technique works.
Next challenge:
Prove that technique works to a 4th grader when they ask, “Why?”
Next Challenge:
Justify why students do not need to understand how math works in order to learn to be problem-solvers in an authentic setting.
This seems obvious to you, but I promise it is not obvious to me.
When you tell me “dividing something by a quarter is the same as multiplying it by four,” nothing in my brain goes “ah, yes, that makes sense.” Statements like that get mashed up in my extremely visually oriented brain, and the thought process ends up “Ok, so wait…apparently multiplying is like dividing…but this is with fractions…so wait if I have three quarters of something that is like…uh…of multiply by something. What was that rule again? Ok, invert. So I’m picturing three quarters of an apple and I have four of them, that’s four three quarters of an apple. What do I invert? Screw it, let’s just guess. This is so pointless.”
Or, I can draw a picture and get it immediately. THEN I can learn the rule, because I understand what it looks like and can visualize it if I get stuck. It stops being an arbitrary rule made up by an evil wizard, and starts being a sensible rule based on reality.
Now, it could be that I’m stupid, but I don’t think that’s it. It’s just different ways of making sense of things.
The technique is a stepping stone to “invert the fraction and multiply”. They don’t go to counting the boxes and then move onto something else. If they are, then your problem is with the teacher.
It’s not. I never said don’t teach it both ways, I said don’t throw out the reciprocal method in favour of the visual method.
You’re right, people learn in different ways. At school I was forever getting frustrated with long-winded “visual” explanations rather than simple, easy-to-remember rules.
It’s like those supposed “memory aid” methods that teach you to remember a long and convoluted story in order to remember a phone number or something. How is memorising a whole story with a cast of dozens of characters more efficient than memorising, say, eight single-digit numbers? That has always baffled me!
What makes you think they aren’t teaching the reciprocal method anymore? I imagine it’s taught also.
Because not everyone is as good at memorizing things as you are. Those mnemonics do help some people.
Because attaching a layer of meaning to something that’s purely syntactic provides another layer of significance that can help trigger a memory.
For example:
Memorize the following list of letters:
djklfdgjiernkdfjkkermkljldfsjljldfjklasfddftadfgaglfdsljfiuewrbvc
Now memorize the following list of letters:
I think that assigning a meaning to a string of letters can make that string of letters easier to remember.
(Of course, in this example, English speakers don’t need to construct the layer of meaning; it’s already been provided by the language).
For another (personal) example:
I’ve never been one to set about memorizing digits of pi, but several years ago (on this board, I think), someone mentioned putting the digits of pi to the tune of Frère Jacques:
3 point 1 4 (3 point 1 4)
1 5 9 (1 5 9)…
Ever since I saw that, I can regurgitate this immediately:
3.14159265358979
because (in my mind) there’s a tight mesh between the digits and the melody.
I teach math and get frustrated that the vast majority of students merely memorize lots of different formulas to do a variety things that, conceptually, can be covered with a handful of concepts. If you have those concepts down (a “meaning” attached to them), the formulas (the “syntax”) come quite naturally. If you put a little effort into understanding, it makes everything more natural and easier to follow. But I think that for a lot of my students (college students), the “memorize everything” strategy has become so ingrained that it’s very difficult to change.
(I have had college students that need calculators to do things like 16+1, or 20. They don’t even recognize something’s wrong if they hit a wrong key and get 16+1=18, or 20=2. From my experience, this is not terribly uncommon (and in COLLEGE, mind you)).
I’ve seen similar questions on my kids’ homework. Teaching the concepts is fine. A lot of what is being taught is not intuitive to me, but if they are going over the lessons in class it may make more sense.
I find it surprising that in math they aren’t teaching tables, addition, subtraction, multiplication, hardly at all. Maybe that’s later but I’ve found that while it’s great to get concepts for higher level questions, some things are really about memorization.
Definitely. From my experience in teaching college-level physics, the students in algebra-based physics could be counted on to have mastery of arithmetic, and the students in calculus-based physics could be counted on to have mastery of algebra.
No, we get to be the last among developed countries in math education because we mostly don’t even teach math at all until tenth grade, instead just teaching meaningless algorithms like “multiply by the reciprocal” before then. If you’re not explaining why the algorithms work, then you’re not teaching math. Counting on one’s fingers might be a very trivial explanation, but at least it’s an explanation.
Several points, from an elementary math teacher and math writer.
The United States is not “last among developed countries” in math in the world. It is not close to being last among developed countries. (Unless, of course, you believe that countries such as Australia, Germany, and the UK are not developed.) This is not a statement that anyone who chooses to talk about education should be making. Especially on a board dedicated to fighting ignorance. So please stop. Luckily, it’s easy to educate yourself on the subject.
To the extent that the US lags behind some of the rest of the world, which it does, the issue seems not to be so much a difficulty with following rules, formulas, and algorithms, but a difficulty understanding the larger picture: under what circumstances do you apply those rules, where do those rules come from, etc. In other words, if a question is phrased as “Solve 8 divided by 3/4,” the US tends to do better (relative to the rest of the world, anyway) than if the problem requires students to take a real-life situation, determine that it can be solved by dividing by a fraction, and then carrying out the division. In OTHER other words, the issues we have in the US spring from teaching people to solve problems in the formulaic way Terr was taught, not in the more focus-on-understanding way Terr’s kid is being taught.
People really do learn in different ways. I see it quite literally every day in my classes. People like even sven are not stupid at all (and it pains me to see her suggesting such a thing); she simply is not wired to see things that are readily apparent to people like Colophon. Guess what: she’s not alone. The Colophons of this world, who see the connection between reciprocals and inverse operations from the numbers alone, are rare next to the even svens, especially in fifth grade but among adults as well; if you doubt it, ask the next not-especially-mathy grownup you see to explain to you why 8 divided by 3/4 is the same as 8 multiplied by 4/3. Or ask said adult to give you a real-life situation in which 8 would be divided by 3/4. Odds are reeeeallllly good that the answer will be a shrug, and vanishingly small that you will get an answer one-eighth as good as Colophon’s.
As for the CC math standards, what is taught and how it is to be taught do not represent much of a change for most states. Including NY. Really. Truly. I promise.
And as for the OP, everything in the 5th grade math standards (which I have worked with quite a bit) should be easy to solve for an NY state high school graduate. Which I suppose doesn’t mean that graduates actually CAN solve them.
i think protests like Colophon’s stem from the idea that these are taught as the only way to solve a problem, and are expected to be used in an examination in lieu of the faster way. it might perhaps be coloured by examples of homework which requires said methods because they’re homework - meant to gauge if you understood a perticular lesson.
otherwise i don’t see why anyone would protest teaching maths so that everyone could understand it and/or learn to grasp it in a different way, as opposed to simply “just do as I say”.
Perhaps. And yes, I accept that I am not a visual person, I like easy-to-remember rules rather than hand-wavy “pretty picture” illustrations.
My major problem with the fraction example is that it appears that this is the method that is supposed to be used in a test in order to “show your working”, hence the boxes being drawn on the paper. I may be wrong about that, but if not then the idea that you would lose marks by not working out the answer in the approved, time-consuming way seems rather ridiculous and would penalise people who prefer to follow quick and simple rules.
i was also referring to Terr’s post. i certainly hope that is not the case. Shodan’s link is reasonable, but those are multiple choice questions.
http://www.oecd.org/unitedstates/PISA-2012-results-US.pdf
Page 7.
Australia, Germany and the UK rank higher than United States. Consider yourself educated on the subject.
Context is crucial. I think that a student who is taught to divide the boxes and count and then taught that the short cut for that is invert and multiply will have a much better understanding of what is going on that someone just taught invert and multiply.
Most of the examples I’ve seen in the media of the horrors of common core math (excluding the obvious typos) seem to be providing a basis for undertanding what is going on versus “just follow these steps”.
Indeed. I would also hazard to guess that understanding the underlying problem and what it’s actually asking when you’re dividing 5 by 1/8 helps in retention vs simply remembering the reciprocal method. I always preferred the teachers who taught both concept and algorithm vs just algorithm because it increased my understanding of what was going on and made that knowledge more flexible. To this day, I could come up with how to find the volume of an area under a curve rotated about the x-axis or y-axis because I can visualize what is being asked and how to get there via integration as opposed to having memorized a formula which would be long forgotten by now.
That said, as mentioned above, Common Core is not a teaching method. It’s a set of standards. A lot of the teaching methods I see people bitching about online I actually approve of and, in many cases, are variations for how I learned myself. That said, I don’t think there’s a “one size fits all” solution for teaching math. Some kids are going to be better learning it formulaically and learning the underlying concept afterwards, and some, like me, like to know what is going on and what question is really being asked when I see a problem like 5 divided by 1/8. When you have that conceptual understanding, it also gives you a sanity check for the answer. You should be able to look at the answer and see if it seems like it’s in the right ballpark or not. If you’re just robotically working through algorithms, with little understanding of what is going on and why, you don’t have that kind of error check.
Before someone criticizes these Common Core worksheets, a few things to keep in mind:
Is it a conspiracy to destroy our children’s sanity? Or did someone just fat finger the keyboard? Never attribute to malice what can be explained by stupidity.
Just because you understand abstract math doesn’t mean everyone does.
If you weren’t there for the class, why are you getting pissed off by the worksheet? If it doesn’t make sense to you, did you ask the teacher to explain?
America’s grades suck compared to other developed nations. Why are you getting mad at teachers for trying something new? The old way of doing things obviously doesn’t work so well.
Common Core is a set of standards. They have little to no control over who writes the worksheets and how the teachers teach the class. I’ve had plenty of instructors who are inept retards, even at the graduate level. Refer to rule #1.