Hmmmm… you weren’t in the Mccarthy-Town (Acton MA) school by any chance? My daughter ran into exactly the same thing. She told the teacher that she learned about negative numbers at home; the teacher said, “Well in THIS classroom the answer to " 3-5=” is “not possible”
I told my daughter to lie low; she now has a PhD in Engineering.
Exactly.
Asserting that order matters completely undercuts learning about the commutative property of multiplication. It’s a core point.
That order of terms is relevant for some potential future when a student is learning about a different mathematical construct, matrices, is completely irrelevant for a 2nd to 4th grader learning multiplication.
Maybe it would be important to talk about frames of reference and relativity at this point? Perhaps the kid was using a frame of reference for his array with axes orthogonal to that of the worksheet…amirite?
In grade 1 I was in some kind of enriched math program and we had an assignment where the teacher asked us to make up our own subtraction question with an answer of 7. Mine was something like 3-100+104. She said, “But you can’t take a bigger number away from a smaller one” and I said, “But don’t you just get a negative number?” She said something like, “You sure do,” but the look on her face said something more like, “Oh crap, what am I going to do with this kid?”
Look, do these kids even know the Peano Axioms? I think I’d give them at least -1 for lacking rigor. I mean, until they define 3 as S(S(S(0))) and 5 as S(S(S(S(S(0))))) where S is the successor function, I don’t even know how they begin.
The worksheet is right there. If the questions aren’t clear even to grownups reading them, then perhaps saying that a child’s answer is “wrong” is, itself, wrong. Math is a language designed to facilitate precision. That’s the whole point.
Structuring a question to require an answer in the form of a typical equation (as is done here) sets a specific expectation. If they want a different answer, they might ask a different question. Even asking it as “what is five times three?” rather than “5x3=___” might lead in a productive direction if they want the student to be thinking of 5 sets of 3 instead of 3 sets of 5.
But most importantly, implying that order matters in multiplication, as the teacher does with their demerit here, is seriously at odds with a core principle of multiplication, the Communitative property. Kids at this stage are drilled on it.
Kids at this stage also tend to be drilled on finding convenient shortcuts, like simplifying calculations using 5s and 10s if you see them. Structuring something as 3+3+3+3+3 instead of 5+5+5, would normally be inviting trouble for no upside. The real question is why would you do it this way?
Indubitably!!
My life partner is a teacher, so most of my exposure to criticism of common core has been from teachers.
I don’t know much about the specifics of CC, but from what I can gather, many teachers believe CC is yet another ed-reform-flavor-of-the-month racket – a way to suck education funds out of schools and into private education consulting companies.
Folks, I’d like to you look at the instructions. It says to use the repeated addition strategy. A quick search shows that when a problem is set up as a*b, repeated addition involves repeating b. 5x3 is 3+3+3+3+3+3, not 5+5+5, using this method. Now fast forward to college algebra. You learn 3 methods to solve quadratic equations (factoring, completing the square, and the quadratic equation), yet the teacher will often tell you to use a particular method to solve an equation. If you solve with the quadratic equation and you were told to use completing the square you lose points. This is no different. Part of learning is knowing how to use multiple different methods to solve a problem. Sometimes you will be able to solve with the method of your choice. Sometimes you must demonstrate that you understand a particular method.
Now that I’ve addressed that, those of you griping ought to realize that you are exactly why this country does so poorly in education. You do not support the teachers who are just doing their job, and then you wonder why your kid gets to college and has no clue what they are doing. You are the reason my mom would get 7th grade English students who never learned to write an essay, because their parents pitched a bitch fit every time their grammar, spelling, or structure got corrected. All you are doing is hurting your kids by teaching them that if you whine enough the world will reorganize itself to your whims. The truth is that any real job is going to involve a process far more complex than this one that has to be followed to the letter, and you don’t get to freestyle it if you don’t like the method.
Without overflow errors, 1+1+1+1+1 = 101. /nerd
And in all those cases the teacher is wrong.
It’s one thing to teach a few methods (e.g. of quadratic solving), it’s quite another to insist that the kids use a specific method on a quiz or test – as opposed to homework.
Besides which, as any number of posts have already tried to explain to you, “repeated addition” , if it truly requires repeating “b” and not “a” is dead wrong, too. That’s not how addition, or multiplication, works.
Sounds like a great way to prep for the sort of job that is rapidly being replaced by automation. I’ve actually been paid pretty well to “freestyle” it, or to change he method if I came up with a better one.
Wow, sounds like a great way to incentivize kids to do exactly what teachers tell them to do and most of all not to try to think through problems on their own!
We learned this freshman year of high school. Apparently, my freshman year high school teacher had a much higher opinion of his students than your college algebra instructor had for his or hers. We proceeded the way this is usually taught: start with easy examples that can be easily factored in one’s head to demonstrate the factoring method, then introducing more difficult examples where factoring isn’t effective. At this point, the teacher introduced completing the square. Finally, the teacher used completing the square on an arbitrary equation: 0=a^2x+bx+c to derive the quadratic formula so that we would understand why the quadratic formula is exactly equivalent to completing the square and understand why we could always just use the formula to get the answer.
By this point, most students have figured out that for problems that can be factored, doing so is quicker than either of the other two strategies; otherwise, the formula is probably the best (honestly, there are very few situations where completing the square is going to beat either of the other two). Requiring anyone to use any particular method seems counterproductive – it seems much better to let students try the different methods for themselves to see what works best.
The foregoing is a much better way of learning to use multiple different methods to solve a problem than your example, IMO. The most important reason is that different methods are used for different reasons, and in my example, the students learn that as part of the process of learning the methods in the first place. This way, you justify the existence of multiple methods instead of making it one more arbitrary detail that must be memorized.
As a student, I hated arbitrariness and became interested in mathematics exactly because everything can be formally justified in math. Your method of teaching would have turned me right off of mathematics and I’d probably have a philosophy degree or something at this point.
I only taught high school math for one year, but I can attest that your method contributes to poor long-term understanding of mathematical concepts leading to repeated elementary errors. My best example is “cross multiplication”. Kids are taught these methods or algorithms for solving equations such as cross multiplication, but they never seem to retain why these methods work, and as a result continually try to apply them in situations where they do not work. I spent months saying to my students: "No, you are not “moving 5 to the other side and making it negative. You are subtracting 5 from both sides of the equation” and “No, you are not cross multiplying – you are multiplying both sides of the equation by the products of the divisors.” After a few months of sticking to that, they could actually do algebra – no thanks to “multiple methods” applied for arbitrary reasons.
Are you a teacher? Do you know many teachers? Would it make any difference to you if I found a bunch of teachers who disagreed with you about this?
Different teachers have different strategies and opinions – they are individual people. Some of them are stupid, and some of them are wrong. I support teaching and education in general, but I do not support terrible approaches to instruction like the one in the OP.
Simply not true. In fact, most “real jobs” above the median wage involve creating processes and formal methodologies where none exist – which requires experimenting, not going back to the “Big Book of Approved Algorithms”.
And of course, any kid who was really interested in math would pick up the concept of imaginary numbers at some point. At this point, a good teacher would offer a salient lesson in function domains and ranges. This particular problem is undefined with respect to the positive integers (and the natural numbers, but I assume that first graders are familiar with zero)
If you had 3 dollars in your bank account (with no minimum balance requirements) and attempted to withdraw 100 dollars,the bank wouldn’t let you. If you first deposited $104, and then withdrew $100, you would be left with $7.
so many high fives.
#So Many High Fives.
I think the commutative property technically states that multiplicative
order doesn’t change the result. So five sets of three and three sets of
five both constitute fifteen objects but that doesn’t mean they are the
same thing. The notational expression is a useful simplification of the
rule. Perhaps there is value in being aware of a notational difference that
is of use in ways we are not thinking. Perhaps the student was
well-instructed on this point prior to the assessment.
If this is a bad teacher, it is because of the assumption that he/she is
correct about something without considering the possibility of being wrong.
How is that different than the author of this blog post and the commenters
jumping on board?
aren’t imaginary numbers the ones written on my paycheck?
Commutativity of multiplication means that a × b = b × a. Saying 3 × 5 = 5 × 3 = 15 mean that all three are precisely the same thing. In the real world you can’t say that 3 boxes of 5 cupcakes is the same thing as 5 boxes of 3 cupcakes - boxes are real objects that matter.
But it wasn’t a word problem about cupcakes (and boxes), it was a question using standard mathematical notation. If the students are being taught that × is not the multiplication symbol, but rather a specialized symbol that for the purposes of this class in non-commutative, then I don’t see how that is doing them a service. If the student had merely written down 15 that would be a correct application of the method, since they would have applied the “repeated addition” method to 1× 15 which is precisely the same thing as 5 × 3.
Here is the rule that all math teachers should always obey: full marks for the right answer.
Eastern suburbs of Pittsburgh, PA.
The sad thing is that some people actually take the “you can’t do 3-5” thing to heart. While in the math tutoring lab at Michigan State (where I got my PhD) I once ran into an older man who was taking some math requirement for his horse breeding certificate. He just did not believe me that 3-5 had an answer.
Or differential forms. Then one of those two answers should be -24, but it could be fixed by turning the paper upside down.
I don’t want to say that this is a bad teacher. That’s probably an over the top title for the post. Who knows what else they do outside of this example?
But this is an example of a bad question and poor grading. The takeaway for the child is most likely going to be that order matters. As has been said elsewhere, it’s prioritizing the algorithm over the outcome.
Presumably, if the student had written 5x3=3+3+3+3+3=3x5= 5+5+5 = 15, that would have been ok. Of course, the concept of equality enables the skip in the repeated 3s and still shows repeated addition as in the instructions.
But somehow, repeating a long string of unwieldy numbers that get you no calculational benefit is a good idea.
If that is the appropriate answer, the teacher should find a different question to ask.
