When I was in school, I’d frequently get marked down for “not showing my work.” Same thing.
You’d rather we start as early as possible making children think of mathematics as a meaningless call-and-response ritual?
… yes. Only for a small set of problems, though (addition, subtraction, and multiplication with numbers up to 10 or 12).
And it’s not like finger counting and modeling with objects needs to be totally off the table. But that shouldn’t be the focus … just a teaching aid. In the end, the child should be tested on the ability to fill out an addition/subtraction/multiplication table, not on the ability to model the problems on that table. If they use modeling of some kind as they learn the tables, great. If they rely on models too much and never learn the small stuff by rote, their efficiency in solving more advanced math problems suffers greatly.
No, no, I could totally get on board with the bordelond method of arithmetic. What’s 10 - 6? Easy, 4. What’s 23 - 17? 4. What’s Sum(n=-1…inf,2^(-n))? No problem, 4.
You have to admit, it would make math a lot easier to learn.
First and third check out. Second seems fishy…
Why is efficiency in solving more advanced math problems a concern?
Maybe I’ve missed something due to a personal eccentricity: do other people typically not learn small-number arithmetic by rote? Sure, we were taught about the number line, and using an abacus, and some other modeling methods. But, at test time, what got me through was having memorized that, say, 9 - 5 =4. Or later, that 9 x 3 = 27.
Chronos, your post put words in my mouth.
Example: my middle-school daughter has but one class period to complete her math tests. If she couldn’t instantly recall that, say, 9 x 8 = 72, she’d have to model it out every time she needed to invoke multiplication (e.g. during long division).
Drawing out arrays, dots, whatever just takes too long once the numbers get large. Others obviously have difference experiences, and that’s fine.
Among the issues in contention is whether we should be testing for such skills as “Can answer ‘What is 9 times 8 in decimal notation?’ and ‘What is 756/9 in decimal notation?’ in mere seconds, without a calculator” in the first place. What is the purpose of doing so? Unless that is clear, that one can test to such standards does not provide strong justification for choosing to both test and teach to such standards.
Well if a kid works out that 9x5=5x9 because they can see that when you turn an array of X’s on it’s side the number of Xs don’t change, then they shouldn’t be marked down if they drew it with 9 rows, or 5. And better yet, if they work out that it is also the same as 10 x 5 - 5, which is also something they apparently are eventually taught, then that should get extra points, not marked down, because it is showing a very solid understanding of how this all works.
Honestly, I am behind the whole concept of kids really understanding the how and the why of it, but it seems to me that there are lots of adults that are trying to teach them (parents and teachers) that don’t have enough grasp of it to tell if the kid is not there yet, or has moved beyond the exact concept they are trying to teach.
And I don’t think it is common core that is the problem. I think the root cause is that people that can really grasp this stuff can make a lot more money as a programmer or engineer than they can as a teacher.
Around here, the purpose of doing so is so that math tests in future school years can be reliably completed in the time allotted. At least that’s the way it’s turning out for my daughter.
Now then. If math were regarded in such a way that the use of calculators wore be acceptable … then sure, no need to know multiplication tables.
BTW, that 756/9 example you posted is one I wouldn’t expect any student to know by rote.
Concerning the 5x3 and 3x5 thing linked to by Senegoid above, my general line on this (and a lot of similar stories) is that there is not enough information here to pass judgment on the teacher, because there’s not enough information here to even know exactly what the teacher is doing. Lots of different equally plausible stories could be told about the context of this photo which end up ranging from making this teacher an unthinking monster to making this teacher a pedagogical genius.
I want kids to learn commutativity as a cool fact about rotating rectangles (for example) rather than as a boring meaningless fact about how you can make your teacher happy by switching some numbers around an x symbol.
But in order for kids to learn about commutativity as a cool fact about rotating rectangles, it’s going to help if first they’ve had their attention drawn to the differences between rectangles before and after rotation first.
The approach exemplified in the photo may be part of something like that.
Why do I want commutativity to be introduced as (something like) a cool fact about rotating rectangles? Because I want kids to understand commutativity for what it is–a result of the fact that things that are different can nevertheless evaluate to the same value, and it’s important to know when to think about how things are different and when to think about how things are the same. This will lead to better problem solving later on, as it allows for flexibility and creativity even in the use of strict standards.
The main mistake I think a lot of people make when looking at this photo is to assume that what we’re seeing here represents what it is intended for the students to believe the rules of mathematics deliver unto them always and for all time. Instead I think it’s more likely that something like this is done as one pedagogical step in a larger process of bringing students to see how to use more and more tools. It’s not, in other words, part of an authoritative statement about how math is and always must be, but is instead, a partial statement of what we’re going to be doing in math today, which we may learn more about tomorrow.
The latter can easily be part of an excellent pedagogy. The best pedagogy often (usually?) does not follow the most efficiently logical laying out of a subject.
From the photo we have no way of knowing whether the kid wrote what he did as a result of an insight or as a result of a mistake, though.
Also I’d say that if the kid is as insightful as all that, it won’t necessarily hurt the kid (or even hurt her feelings) to do things on particular worksheets without the benefit of that insight for a while. Of course if the teacher simply had the attitude of “no, that’s wrong wrong wrong and you just don’t get this!” that would be a problem. But there’s not much of a reason to assume that’s what’s happening here.
Third grade teacher here, and I agree almost entirely. There’s an ideal mix of rote memorization and conceptual development, and it probably varies from child to child. The NCTM, in my opinion, goes a bit far toward conceptual development, and I’ve had district math coaches who elide memorization from our curriculum entirely. “Skip lesson 2.1 in Unit 2,” they’ll tell us, referencing a lesson in which students practice single-digit addition facts.
It’s terrible, and it’s akin to not getting kids to develop a sight vocabulary for reading. If my students, on trying to read a sentence, are having to decode every word on the page, they’ll have no cognitive power left over for understanding the flow of the sentence, the meaning of the paragraph, the thrust of the story. Similarly, if a student attempting to solve a three-digit subtraction problem is having to hold up their fingers and count backwards in order to solve 374-9, they’re going to have no cognitive power left over to solve 374-189, much less determine whether the story problem I’ve given them is best solved by finding the difference between those two numbers, much less go back and check their work.
First grade is a beautiful time for students to develop a strong conceptual understanding of what addition and subtraction entail. By the end of first grade, every student ought to be able to figure out the sum or difference between any single-digit numbers.
But then they need to learn their freakin facts. By the time they hit third grade, they should know what 8+5 is without using their fingers; they should know not only that 16-7=9, but also that that means 86-7=79 and 586-9=577.
So yeah. I spend a lot of time teaching concepts, and I think overall common core lays out concepts very well (I have some issues with a few pieces at the edge, but overall it’s great). But I also spent September giving my kids 60 addition facts to do every morning, and October giving them 60 subtraction facts. They need to know those puppies without having to waste any brain power on them, so they can save their cognition for more complex stuff.
Thanks for weighing in, LHoD. What you say sounds totally reasonable.
Extremely basic common sense.
To understand something you need to know it exists first. First you learn that 2+3=5, before you learn why 2+3=5. This is not exclusive to math, all knowledge works this way. You learn that a certain side won a war before you learn why they won. You learn that mixing two chemicals provides a precipitate before learning how the reaction occurs.
Memorization comes before understanding the way that you need eggs before you can make an omelet. It’s how human beings learn. You don’t need any “research” beyond being a person.
I understand wanting to avoid only teaching children to memorize facts, I’ve tutored many people and helped write training materials for computer professionals and my goal has always been to guide people to understand how computer systems work rather than just memorizing “stuff”. But memorization is essential as a basis for that understanding. Hence, I teach mnemonics like “California cows won’t dance the Fandango” to memorize the steps a laser printer goes through (cleaning, conditioning, writing, developing, transferring, fusing) before explaining what each step is and how it leads to a successful print job. If I only did that without giving someone an easy way to remember what the steps are and in what order, it can make it so much more difficult to understand.
Just a note: I don’t see anything in the photo we’re talking about that suggests this teacher does not also have his or her students memorize the math facts.
Extremely basic common sense isn’t research, and is wrong often enough that it own’t do as an answer to a request for a research foundation for someone’s empirical claim.
I think rote memorization of basic math facts is fine, and probably necessary for most kids. But somehow being more “foundational” than the conceptual understanding? No, I would need to see more than “extremely basic common sense” to take a claim like that on board.
This is not true. You can learn that 2+3=5 by generalizing from a lot of instances illustrating why 2+3=5.
Again, I am not making a claim that one way is better or more effective in this particular post. I am taking issue with the claim that one is necessarily more foundational.
Yes, if students, from a very young age, don’t have all those arithmetic “facts” down pat, then they will forever get bogged down in the petty details when they should be focusing on the more advanced work. Not knowing the basic facts by heart will be a boat anchor around their necks. Similarly, as students get into more advanced work, they are required to memorize various formulas, algebraic identities, all those trig identities, derivatives, and pages after page after page of integration formulas.
What bordelond said a few posts up:
I had a math teacher who said substantially that, referring to formulas that we should memorize, but could work out from scratch instead every time if we preferred. And that was in my Differential Equations class!
If these kids grow up in the habit of pulling out a calculator every time they had to work out a trivial grade-school arithmetic problem, they will be bogged down. I tutored college math students (in Introductory Statistics) who were like that.
You end of with this sort of thing happening. (Famous New Yorker cartoon.)
To be sure, this is possible – but if so, it might just raise other arguments about what and how the subject should be taught.
There’s an issue here that hasn’t been mentioned in this thread nor in the article I linked nor any other source I’ve seen in this thread, and it has to do with the interpretation of 5 x 3 versus 3 x 5. By one point of view, those aren’t synonymous, and it is relevant to describe the first (5 x 3) as a shorthand notation of 3 + 3 + 3 + 3 + 3 while the second (3 x 5) is shorthand for 5 + 5 + 5, and to not get those cases mixed up.
There’s a common convention, seen in math more advanced grade school, that in the product ax the second factor is the number being multiplied and the first number, or coefficient is a scale factor indicating how much bigger the second factor is being scaled up.
In the Old Way of teaching, multiplication is first taught as simply repeated addition, with those rectangles being useful as teaching props as well as showing why it’s commutative. The idea of multiplication as repeated addition doesn’t literally work well when you try to multiply fractions (how do you add 5 up 3½ times?) so when you start to teach that, you need to teach a new extended concept for multiplication that works for fractions too.
In the New Way of teaching, you teach multiplication as a process of scaling up one number. The typical visual aid is a slide projector which can be moved forward or backward to make the image on the screen bigger or smaller. Multiplication, in concept, does that. This also introduces a fundamental asymmetry in the factors in a multiplication problem: One number is distinctly the number being scaled (and by convention, this is the second factor) and one number (conventionally the first factor) is conceptually the scale factor. The “New Way” argues that this concept of multiplication should be taught right from the very start.
That’s where all the fussing is coming from that 5 x 3 is not the same problem as 3 x 5, and that it matters which way you draw the rectangle to correspond with the given problem.