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On the SAT 9 math test for the spring of fourth grade (or the fall of fifth), you’ll find items talking about “number sentences,” a remnant of New Math that has been resurrected in the New New Math. Go visit the mathematics department of a good university and ask a senior mathematician if “math is a language,” with “sentences” and the like. Chances are fair that someone will start screaming at you, or even might just beat you up.
Your State Test Was Not Divinely Inspired
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I don't know if New Math came up with "number sentence", but it certainly was not at all a "math is a language" sort of trend. If anything it is exactly the opposite: math for its own sake. "Math is a language" refers to the idea that math is the language of the sciences and is often associated with a heavy on the applications approach to learning it (which, again, is almost the opposite of New Math).
My kids have been doing "number sentences" for the last few years.
The term annoys me to no end, but once I figured out it was nothing but a problem with the answer, I have learned to let it go.
I am also guessing, some readers might not know what a number sentence problem looks like.
Typical number sentence homework.
If John has three apples. He gives two apples to Jane. Write a number sentence for this problem.
Answer: 3 - 2 = 1
This is one of the few articles I've ever seen taking the state tests to task. I'm amazed that it isn't brought up more. With all of the finger pointing that goes on, why are the states left uncriticized? Invariably, the federal gov't. gets blamed, but not the tests themselves.
In IL they are clearly tied to the new curriculums (or vice versa, no one can say for sure). The "teaching to the test" phenomenom has to do with teaching kids how to answer critical thinking and open ended questions with exactly the "proper" language that only the new curriculums teach. In a rote way, no less.
The pressure on school districts to jump through these curriculum hoops becomes enormous.
The newspapers and parents aren't the only ones to miss this. Teachers still (as we see constantly) put the blame squarely on NCLB itself as the culprit even though the states are left to actually set their own standards.
I went through school during the Space Race era, and we did New Math. I have never heard of a number sentence. What is it?
We did set theory, bases, operations and functions, that sort of thing. Our parents hated New Math because they had no clue what we were doing. It had its problems (the curriculum was overly-ambitious and possibly too rigorous, for one thing, and there was an overemphasis on process, for another), but whatever its faults, we did learn math.
A number sentence is an equation or an inequality that involves numbers but not variables.
Mathematics may not be a language, but there are strong similarities. An equation is like a sentence. The left side is the subject, the right side is the direct object, and the equals sign is the verb. Good writing uses complete sentences, while good mathematical writing uses complete equations. Solving a problem with many steps requires careful organization, which is similar to constructing a paragraph or an essay.
but whatever its faults, we did learn math.
Well, you did, professor,
Some of us regular math types completely shut down.
I was in 6th grade when the new math came and it was exactly as you described. I remember my father throwing the book after numerous times of trying to help me. I was confused, but it also took me away from higher grade level arithmetic, which I needed much more practice. My budding math phobia can be traced to that year.
OTOH, my brother (an engineer/inventor type) thought it was fine. But then, he doesn't understand why people need to memorize their times tables. It's all right in there, right?
I'd say that the right side of the equals sign is more like a predicate nominative, rather than a direct object.
prof,
In the texts that I have "number sentence" is used in lieu of assertion.
I can only imagine that some people familiar with successful programs that have used this terminology have gutted the contents (in their own program) and simply sprinkled the phrase in without any advancement of the topic. This then gives the appearance of teaching math but really turns math into a vocabulary class.
It's interesting how new terminology is introduced to make math easier for the kids, some kids don't pick up on the terminology and then math class turns into some sort of review for that particular gimmick. The math that underlies the vocabulary is then completely forgotten. The real victims of this are the children who can do arithmetic but fail to pick up on the invented terminology. I have noticed that my own son understands a more formal definition but will fail to understand the poorly defined educational gimmicks which were intended to help him.
Another example, my state has some sort of fettish for the phrase "fact family." A "fact family" is:
2 + 3 = 5
3 + 2 = 5
5 - 2 = 3
5 - 3 = 2
The theory is that if kids know "fact families" it will be easier for them to remember "number facts." But rather than directly ask the kids " 2 + 3 = _____" on the state-wide test; they test knowledge of the term "fact family." How do we know if this fact family gimmick is working? How do we know if the number sentence gimmick is working?
I actually like "fact families". I made up flash cards for all the addition/subtraction fact families 1-10 and taught them to my 1st grader.
It helped her see the relationship between addition and subtraction.
All and all, the terms "fact family" and "number sentence" aren't in of themselves damaging. They refer to basic math concepts... they are nothing more than new terms... not new math.
I will say that I thought the fact family idea was goofy until I had to use it alot with my special ed son. Saxon is big on it and brings it back constantly.
Moving back and forth quickly with the inverse of operations seems like a no-brainer when you're an adult, but it actually can get lost with some kids, especially strugglers.
By hammering it in for years I had a much easier time tutoring him in more complicated critical thinking problems (for him, 2-3 steppers are very complicated) because he was solid on thinking of inverses. Other techniques I might use would confuse him because "new" language confused him.
I often would say, "What is the inverse of that?" and if he paused, I would then say, "think of your fact families," and it would fly out. If those stupid things hadn't been pounded in he never would have been able to quickly access them. In special ed, speed and quickness are rare, but beautiful things. I never take them for granted.
They really can help down the road when you tutor your child about how to think about a problem.
I believe that math is a language. However, if we're going to have math sentences, we should be introducing units as soon as possible.
Also, the equals sign is so important. It actually means something. It should be in bold and a large font size until students understand equals means equals.
Well, I'm so glad we've gotten rid of that New Math! Nowadays in this enlightened age, everyone can get a PhD -- you know, so no one's feelings get hurt or anything.
Seriously, that's it -- that's the subject -- take it or leave it, like it or not. New Math is as close to math handed down to society by its mathematicians as it gets. Reality is not going to change. Math is not going to change. And, the fact that rigor is required is not going to change. These things are facts of reality -- they aren't going to go away just because we wish it so. We can stamp epsilon-delta proofs out of our calculus programs as much as we like. That will never change the fact that empirically plugging in numbers to a function and graphing it or some such thing is not a legitimate way to establish whether or not it is continuous. And, as long as that's true, there will always be some people out there willing to do what it takes to do it right. In other words, rigor is never going to go away.
We are only hurting *ourselves* when we decide: "Well, they can do that if they actually choose to major in math." Don't blame New Math for over-abitiousness -- that blame lies squarely in the society that received it. What we should be doing is trying to figure out how to teach as much of it as we can -- not trying to figure out ways to defend watering it down so that we can teach yet another stupid human calculus trick.
It is an immutable truth of reality that is as true today as it was centuries ago: "There is no royal road to geometry." (Not even if we throw a bunch of psycho-babble at it.)
"The term annoys me to no end, but once I figured out it was nothing but a problem with the answer, I have learned to let it go."
I've learned to let it go too - most of the time. Fact families and triangles don't bother me much. It's the rest of the low expectations, little mastery, and lack of mathematical understanding that's the problem. What strikes me most about modern reform math are the low expectations and slow pace.
Math as a language is, at best, superficial. I think of it more as a tool box. You apply different tools to create a solution. You apply techniques like a Taylor series expansion to linearize an equation, or you apply a Newton search technique to find the roots of a nonlinear equation. I like the idea of tools because it implies a need for mastery. Nobody gets worked up about "drill and kill" and a need to understand your hammer. As a language, teachers will begin to look for whether the kids are developing their mathematical "voice". Creative math will be like creative writing - no one right answer.
Math "as a language" is a tiny bit better than "math is about patterns". The people who propagate the latter have no idea what they're spouting.
As far the "New Math" of the 60's, yes it was conceived by mathematicians, and some of it was very good. Its drawbacks were that there was too much emphasis on theoretical (an axiomatic approach to the basic operations via mappings and binary operations) that were not appropriate for the early grades. When I took such courses in college as a math major, my eyes were opened as to how our ordinary operations were part of a much larger picture and saw value in teaching kids this view. Yes, a 20-year-old tends to be idealistic, but the mathematicians working in SMSG really should have known better. Still, as I say, there were very good things to ocme out of it. Another thing that happened though was that since the SMSG texts were in public domain, publishers would then appropriate portions of the text and put together their own math books using authors who knew very little about math and not knowing what the material was that they had "appropriated". The result was a bunch of very bad math books which sold well because they were labeled "new math", which was the brand du jour at that time.
Ironically, NCTM sought to distance itself from the New Math, yet some of the NSF-sponsored texts teach some of the very same "alternative" algorithms (like Everyday Math and Investigations) used in the new math SMSG books of the 60's. Now that it has the NSF/NCTM brand, it's legitimized again. (The difference, however, are the careful explanations of why such algorithms work--something the SMSG books at least attempted).
Speaking from a different place than many of you, I think my complaint about the old new math had to do with coherence and sequence, timing, and yes, overkill.
11-year olds who aren't firm in base 10 probably shouldn't spend a lot of time in other bases.
Like the new-new math, the idea wasn't bad so much as the implementation.
And while gifted math heads can survive that kind of thing, the rest of us are not so lucky.
And while gifted math heads can survive that kind of thing, the rest of us are not so lucky.
I agree Susan. My apologies if my message seemed to imply otherwise. While there were good things in it, it was not implemented well. And like the new new math, some of the problems were an emphasis on underlying theory rather than basic facts and skills. Unlike the new new math, however, was that the math (at least in the SMSG books) was first rate.
math is not a language
math and language, again
I LOVE the fact families!
I had no idea addition & subtraction were inverse operations until I did the fact families with Christopher.
(You haven't seen "fragmented knowledge" until you've seen the arithmetic I learned back in central IL.)
I also like Saxon's approach of having kids write them down.
"Use the numbers 1, 2, & 3 to write 4 fact families."
I had no idea that's what a number sentence is.
I thought it was an equation.
Recently encountered this -- the whole "number sentence" thing is actually WORSE than simply a change in terminology. What elementary schools are being presented with is something like the following:
10 + 4 + 45 - 27 = ______
And then they are being asked to add the numbers. All fine and well you say, this is just an "equation".
But it is NOT so straight-forward, because the teachers are then PREVENTING and specifically PROHIBITING the students from "stacking" the numbers to do the actual math. So, if a child actually understands math and does this:
10
04 +
45 -
27
-----
32
They will be told not only that they are doing it "wrong" but they are likely to be reprimanded for doing so. (One child I know of was brought to tears by a VERY public reprimand. Fortunately her mother realized what was going on and IMMEDIATELY removed her daughter from the public non-school and is teaching her at home now).
Whatever the purported "intent" what this IS accomplishing in fact is a wholesale and complete undermining and destruction of any root-level understanding of the DECIMAL system of numbering.
The end result will be children (and later adults) completely illiterate of even basic addition or subtraction (and you can forget about any "higher" understanding developing from this.)
Mathematics will become some "magic" that is done by computers and calculators (which even the first graders are being encouraged to use).
Sheer Insanity.
"They will be told not only that they are doing it "wrong" but they are likely to be reprimanded for doing so."
What happened to their idea that there is no one way to do a problem. If a child stacked the numbers but wrote a nice explanation, I think they would love that.
It just shows the fallacy of their arguments. They are implementing fuzzy math in a rote fashion. K-6 teachers don't really want to know math. They want a process that tells them what to do. They take service classes. They are told what to do. They do it. If kids do something else, they have no way of evaluating its effectiveness.
I don't think that they want kids to think that math is magic. They are trying to get kids to follow their pet curriculum that claims to develop mathematical understanding. Since they really don't know what that means, they are stuck with rote fuzzy math; no mastery and no real understanding.
I think that's what I'll call it from now on; Rote Fuzzy Math.
Saying that math is a language because it has sentences would be like saying literature is a language because it has sentences. Literature is not language, it's something else. It uses language. So does math. I like the terminology of number sentences because, echoing what someone else said, 1 + 2 = 3 doesn't look like an equation to me, whereas 1 + n = 3 does. Fact families are a great idea too, likewise Singapore Math's number bonds. These are all good tools to help kids learn math. If the tests end up testing knowledge of this unessential terminology, that, clearly, is a bad thing, but the tools are still useful.
About the issue of writing addition problems horizontally instead of stacking the numbers, that is usually used in situation when the curriculum is asking students to practice mental math rather than using pencil and paper. Could there be some missed communication on that point?
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