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NCTM News Bulletin, March 2008
"Is 4 × 12 closer to 40 or 50? How many paper clips can you hold in your hand? If the restaurant bill is $119.23, how much should you leave for a tip? How long will it take to make the 50-mile drive to Washington? If a 10-year-old is 5 feet tall, how tall will the child be at age 20?"OK. I've generally ignored it when people talk about "number sense", but here it is from the president of NCTM. He says: "Number sense is important and needed—right now." What is it, exactly? Is it estimation? No, it seems to be more than that. Does Mr. Fennell define it? No. He just gives the examples above. Let's look at each one.
"Is 4 × 12 closer to 40 or 50"Do it exactly in your head. It's part of the times table.
"How many paper clips can you hold in your hand?"How accurately do you have to make this estimate to show number sense? He doesn't say, but I don't think he is talking about plus or minus 25% accuracy. I don't think I could guess that closely.
"If the restaurant bill is $119.23, how much should you leave for a tip?"Does he think that those who have mastered the traditional method of multiplication are stuck doing this calculation right-to-left on paper? This is a straight estimation problem, and my traditionally-taught wife takes pride in calculating these things down to the penny in her head.
Is number sense more or less than estimation? It seems to be both more and less. Number sense means more than just estimation, but it doesn't require you to provide accurate estimations.
"How long will it take to make the 50-mile drive to Washington?"About an hour? What are the assumptions? Is number sense equal to estimation with common sense added in? Apparently the common sense level is not very high. How about the number sense to determine how long it will take to drive 400 miles on the highway, accounting for stops for gas, eating, and traffic? What if I gave you the exact times for stops and the lower speed in the traffic? Mastery of the basics leads to number sense, not the other way around.
He seems to be making the case that there is no linkage between mastery of the basics and number sense. But then he really isn't talking about mastery of estimation. Schools could hand out "Arithmetricks" and practice, practice, practice. No, he seems to be talking about some sort educational number sense osmosis. Low expectations.
"If a 10-year-old is 5 feet tall, how tall will the child be at age 20?"He goes on to say:
"Students who have a good sense of number are able to provide a reasonable response to the examples above, including the driving example. And they know that there is no proportion-driven response for the final example."Sure there is. If a 10 year old child is 5 feet tall, then proportion tells you that there is some other effect going on when they get to 20. If you were talking about some unusual species of tree, then proportion or number sense is not going to help. You need content knowledge, and we all know what schools think of content. I think I'll coin a new term: Content Sense. That's what we need-right now! Just like with fractions on the real number line, kids need to be able to place major historic events on a timeline.
After all of this, I still don't know what he means by number sense or what performance level is required. Whatever it is, it seems pretty low. Mr Fennell can't define it, but he wants it fixed "right now".
There is another example:
"A sense of number emerges that is built on the foundations discussed above, which"the pieces were bigger in fourths"?
yield responses such as, “I knew 3/4 was more than 3/5 because the pieces were bigger in fourths.” This is what all math teachers want. Such “aha!” classroom moments remind us about the importance of understanding."
So number sense is something other than math; something other than mathematically knowing why 3/4 is greater than 3/5. And "understanding", according to him, is something other than mathematical understanding. What if the student said that 3/4 = .75 and 3/5 = .6? Is that number sense? Does that show understanding or is that just rote knowledge?
Math is all about tools and methods that you can rely on to give you correct results in spite of the fact that what you might be doing defies common sense. That's the power of math. As I've said before, let the math provide you with the understanding. If you're worried about estimation, teach it directly. Number sense, whatever it is, will take care of itself.
44 comments:
Oops. That's what I get when I cut and paste from Notepad. Usually I catch this.
don't sweat the formatting,
SteveH ... you're on a roll here.
my "analysis" (of your recent posts):
the thing to remember is that
politics turns words into their
own opposites almost inevitably.
"sense" means "mind-numbing bilge",
for example; "equity" means
"whoever has the gold,
makes the rules".
and "for all" means "there exists".
(one thing motivational speakers
[like our esteemed mr. fennell]
tend to do that annoys me (even more
than most of the *rest* of what
they do): they keep saying that
for such-and-such project to work,
*all* the members of such-and-such
community will have to do so-and-so.
when *obviously* this will never happen.
it's like they're admitting right there
that they know their BS idea
is doomed to fail ...
which i suppose (hey! a true thing!)
should annoy me less.
the annoying part i suppose is that
they do it in the serene confidence
that if you point out that what
they've just said is moronic
and insulting, they'll be able
to make *you* the badguy effortlessly.
try it and see sometime.)
v.
I think you are correct that phrases like "Number Sense" can be used without enough precision to really understand what is being spoken about.
Let's think about the comparison between 3/4 and 3/5 that you quoted. I think that number sense has something to do with being able to make that comparison, in one's head preferably, with the aid of a conceptual model, such as a fraction circle or a number line and that that model will aid in the understanding of other elements related to the kinds of numbers and operations that are being studied.
I don't think that I agree that "Math is all about tools and methods that you can rely on to give you correct results...". To me Math is all about ideas and conceptions that provide beauty and sometimes practical applications. I do agree that sometimes the best way to understand the math is to "just do it" for a while, at least.
In my teaching practice, the direct teaching of estimation would be one of the prime areas where I felt I had absolutely no success. It seemed students didn't know why they were doing it and that they didn't have the flexibility or number sense to make it intellectually satisfying. Maybe you have some specific suggestions about making it more palatable. One of the most interesting ideas I have seen is about using 'strings' of questions as proposed by Cathy Fosnot.
I have been involved with a project that is trying to inform the topic of fractions with a conceptual framework. It is called CLIPS. Clip 3 of the fractions cluster addresses comparison directly. The dropball game in Clip 5 might be an example of an experience that would build number sense in a (do I dare suggest it) entertaining way.
I have also commented on the Math Advisory panel at my blog.
"the annoying part i suppose is that they do it in the serene confidence that if you point out that what they've just said is moronic and insulting, they'll be able to make *you* the badguy effortlessly."
Would it work better if I nicely ask for a better explanation and a suggested solution? I thought all of the NCTM-supported math curricula were centered on the vague idea of number sense. I didn't go to the extreme of calling it moronic and insulting. I guess the answer is that it doesn't matter what I do.
Perhaps I'm naive about what is going on in their heads. These people are making a huge number of assumptions that have nothing to do with research. It has to do with control and their own opinion. Do they feel justified because their intentions are sincere? Is this kind of like the idea that all opinions are equal, but the sincere ones are better?
I've said before that it seems like an academic turf battle. I get the feeling that they don't care what they teach as long as they are in charge. Push them about mastery, and they talk about balance, but they want to control the details.
They have definite opinions about education, they know that others disagree, but they push on regardless. This has nothing to do with research. I've mentioned before that I told two school committee members that they should hand out Hirsch's Core Knowledge Series books to parents and tell them that this is NOT the education their children will receive.
I get the distinct impression that the education field has long ago circled the wagons. They are protecting their turf. If they don't have their opinions, they have nothing.
"Let's think about the comparison between 3/4 and 3/5 that you quoted. I think that number sense has something to do with being able to make that comparison, in one's head preferably, with the aid of a conceptual model, such as a fraction circle or a number line and that that model will aid in the understanding of other elements related to the kinds of numbers and operations that are being studied."
But knowing the reasons for finding a common denominator is not a proper conceptual model? What about the relative positions on the number line for 4/7 and 25/33? Is there no linkage between number sense and mastery of the basics? Mastery does not imply rote.
"To me Math is all about ideas and conceptions that provide beauty and sometimes practical applications."
"sometimes"?
Do you know what I find beautiful? Vectors, matrix transformations, dot products, cross products, linear spaces, and the fact that I can write a routine to find the intersection of two triangles in space. These are not ideas or concepts. They are mathematical tools I can use to solve a vast array of problems. Math is not a survey course to study its conceptual beauty or ideas. At some point students have to exchange conceptual (picture) models with mathematical (abstract) models.
"In my teaching practice, the direct teaching of estimation would be one of the prime areas where I felt I had absolutely no success. It seemed students didn't know why they were doing it and that they didn't have the flexibility or number sense to make it intellectually satisfying. Maybe you have some specific suggestions about making it more palatable. "
Students don't know why it's good to figure tips out in their head, or how to figure out if they have enough money to buy what they need in the store? I can come up with all sorts of real-life examples for practicing estimation. Arithmetricks is a great book, but it's not necessarily entertaining or fun. As for palatable, this is a separate issue. You don't change content or expectations just to make something palatable.
"I have been involved with a project that is trying to inform the topic of fractions with a conceptual framework. It is called CLIPS."
"inform the topic"?
I looked at it and it says that it's for grades 7 - 12, but my son has even seen this same material in Everyday Math in 4th and 5th grades. CLIPS seems remedial. I saw a video showing 8th or 9th graders wondering why fractions are important. Others should look at it and decide for themselves.
You need to understand what the parents at KTM want. We want curricula like Saxon and Singapore Math. My question now is what, specifically, is missing in these two curricula that would improve number sense? What is number sense and how is it different from estimation?
"But knowing the reasons for finding a common denominator is not a proper conceptual model? What about the relative positions on the number line for 4/7 and 25/33? Is there no linkage between number sense and mastery of the basics? Mastery does not imply rote."
I take it that you are interested in Mastery learning and not in rote learning. That appeals to me. Finding a common denominator is one of the best examples, in my opinion, of something that is taught by rote in schools without any concern for "reasons". One of my pet peeves is that we don't use a common denominator for division, which is when it really makes sense.
I agree that students need to master some basics. I have watched, in frustration, students reaching for a calculator to help them find out what 2-4 is. I think that number sense might allow me to figure out 6 times 8 in a variety of ways that made sense to me, if I forgot that 6 times 8 is 48 - the way that I might forget that the capital of New York is Albany.
Do you know what I find beautiful? Vectors, matrix transformations, dot products, cross products, linear spaces, and the fact that I can write a routine to find the intersection of two triangles in space. These are not ideas or concepts. They are mathematical tools I can use to solve a vast array of problems.
Are you sure? The platonic vector is a pure idea with no visual representation. It is certainly not a tool - though a graphing calculator or computer could be used as a tool to model and operate on vectors. These things are beautiful to me because they touch the imagination.
Math is not a survey course to study its conceptual beauty or ideas.
I think this might be a crying shame. I love music but have no great aptitude for creating it. I am glad that there are avenues for enjoying what others do and have done without being left in frustration holding my squeaking recorder. The vast majority of students will not create much mathematics, but imagine if we could help them appreciate what others are creating!
At some point students have to exchange conceptual (picture) models with mathematical (abstract) models.
Sure, but I think Math teachers are way too quick to assume that students are ready for more abstract conceptions.
You don't change content or expectations just to make something palatable.
I think we are in a world where teachers are competing with all kinds of compelling content. I think that we have some responsibility to make our content engaging.
Fascinating discussion.
I looked at it and it says that it's for grades 7 - 12, but my son has even seen this same material in Everyday Math in 4th and 5th grades. CLIPS seems remedial.
The CLIPS fraction cluster is meant for students in Grades 7 to 10 who are still struggling with basic fractions. The Grade 11 example is for students operating at grade level. You didn't say what level you got to on Dropball. Twenty or Twenty-One is all that I can manage.
Ross
Here is my uniformed opinion about what math is:
More precisely, math is about the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. It may be more accurate to say that teaching is the art of explanation, philosophy is the art of truth, and aesthetics is the study sensori-emotional values, such as charm and beauty.
The activity which students are denied most frequently in math class seems to be (and I have an uninformed opinion to be sure) is philosophy, more specifically epistemology, e.g. How do we know that a particular mathematical result is true? Why are a particular set of assumptions needed? Can the assumptions be derived? Are these properties independent of each other? Memorizing facts and formulas denies the student a philosophical approach to the study of quantity, structure, space, and change. However, intuitive speculation about limits, triangles, and irrational numbers, just because it's fun, does not guarantee math either. I don't see any other way around learning the details of what makes a mathematical argument sound other than to memorize them. Well, another way is to have your proofs picked apart by a mathematician who is lawyering your argument to death for no good reason other than to be a math meany. You'll learn to anticipate all that pedantic nitpicking and insert the right sorts of phrases such as (where x does not equal zero) in your assertions. But that is learning the hard way.
History is full of false conclusions due to lack of mathematical rigor and it is precisely because mathematicians are interested in truth, they want to be right—and not gayfully roll about in beauty and charm--that they pull out the much maligned “mindless formalism.” And so here is where I think “constructivists” have it wrong. They seem to think that any assertion made about a mathematical concept (see definition in first sentence), regardless of the methodology which produced it, or its truth value, counts as math. So while math without philosophy isn't math, speculation and intuition without rigor isn't math either.
Here is a nice quote saying this with better rhetoric than I ever could and when you come across the word “computer” if you mentally replace it with graphing calculator the assertion is still true:
If one were to remove “proof” from mathematics then all that would remain is a descriptive language. We cold examine right triangles, and congruences, and parallel lines and attempt to learn something. We cold look at pictures of fractals and make descriptive remarks. We could generate computer printouts and offer witty bosevations. We could let the computer crank out reams of numerical data and attempt to evaluate those data. We could post beautiful computer graphics and endeavor to asses them. But we would not be doing mathematics. Mathematics is (i) coming up with new ideas and (ii) validating those idea by way of proof. The timelessness and intrinsic value of the subject come from the methodology, and that methodology is proof.” --Stephen Krantz
I have made a mental note that nary a mention is made that a proof must be charming in order to be a proof. Perhaps it must be charming in order to hold one's attention, which makes it a good teaching gimmick, one I immensely enjoy and hope to use, and perhaps it serves as a good promotional gimmick to attract new students to the field, but “charm” and beauty in and of themselves do not determine what counts as math... or so it seems to me.
"I think that number sense might allow me to figure out 6 times 8 in a variety of ways that made sense to me, if I forgot that 6 times 8 is 48 - the way that I might forget that the capital of New York is Albany."
I expect my son to always know 6 times 8 and the capital of New York. No excuses.
When you get to something like algebra, you need to know your identities and know the many ways they are applied. If you get confused solving an equation one way, you need to know other approaches that can clear it up. I wouldn't call this number sense. I would call it mastery of the basics.
Number sense seems to be applied at a very low level on things that should be automatic. Automatic does not mean rote.
"The vast majority of students will not create much mathematics, but imagine if we could help them appreciate what others are creating!"
Schools should never select K-8 math curricula based on this idea. Only math brains or kids with help at home get a chance to do more?
"Sure, but I think Math teachers are way too quick to assume that students are ready for more abstract conceptions."
I think it's the other way around. The expectations are way too low.
politics turns words into their
own opposites almost inevitably
It really is the opposite, isn't it. Or just about.
It's time for me to re-read Orwell.
Content Sense
I love it!
Beyond basic literacy and numeracy, it has become next to impossible to predict what kinds of knowledge people will need to thrive in the mid-21st century.
[snip]
A resolution has been in the direction of undertaking to teach not simply knowledge itself but the skills of knowledge acquisition—skills that will equip a new generation to learn what they need to know to adapt flexibly to continually changing and unpredictable circumstances….
After examining possible alternatives, I make the case that the only defensible answer to the question of what we want schools to accomplish is that they should teach students to use their minds well, in school and beyond (Kuhn, 2005). The two broad sets of skills I identify as best serving this purpose are the skills of inquiry and the skills of argument. These skills are education for life, not simply for more school (Anderson et al., 2000). They are essential preparation to equip a new generation to address the problems of the day.
EDUCATIONAL PSYCHOLOGIST, 42(2), 109–113
Is Direct Instruction an Answer
to the Right Question?
Deanna Kuhn
"the skills of inquiry and the skills of argument"
...but you have to have something to inquire and argue about!
All this number sense stuff reads like a page out of the Everyday Math play-book. My second grader brings home EM assignments with questions like the paper clip one all the time. Even in the fourth grade EM homelinks my daughter brought home last year, I often found the word problems vague, poorly written, and almost always pointless.
The activity which students are denied most frequently in math class seems to be (and I have an uninformed opinion to be sure) is philosophy, more specifically epistemology, e.g. How do we know that a particular mathematical result is true? Why are a particular set of assumptions needed? Can the assumptions be derived? Are these properties independent of each other?
This is absolutely my experience.
I stand foursquare behind committing math facts broadly defined to longterm memory, which takes time & lots of it.
But in my own efforts to learn math, and in C's courses thus far, there is too much strictly procedural teaching for my taste.
I don't know what to think about this issue on a practical level -- there's so much sheer stuff to learn in math that I see why most textbook authors and teachers conclude that we don't have time for philosophy.
The one idea that has stuck with me is what a graduating math student told Ed. His father, who worked in a math-related field, had addressed the issue of overly-procedural teaching by having his son derive every formula he used.
I don't know whether that's a good idea or bad, but that's what I've been trying to do (seeing as how I haven't gotten to proofs yet -- or to proofs strictly defined).
C. has a test coming up on graphing quadratic equations that is quite challenging; I was having a lot of trouble with the material myself.
A lot of my problems were resolved last night when I printed out the purplemath pages on quadratics and studied them. One I had derived the formula for the axis of symmetry I was in much better shape to teach this material to C. (who missed the entire week of school during which the material was taught).
I meant: "once I had derived"
..but you have to have something to inquire and argue about
That's what Ed always says.
But we would not be doing mathematics. Mathematics is (i) coming up with new ideas and (ii) validating those idea by way of proof. The timelessness and intrinsic value of the subject come from the methodology, and that methodology is proof.”
This is the kind of thing I was referring to when I asked the question of whether "performance based assessment" and "problem solving" are math as opposed to applied math...
Here's a very recent of Everyday Math second grade "number sense" question:
Circle the best estimate for the weight of each object.
1. newborn baby
8 pounds
20 pounds
70 pounds
2. Thanskgiving turkey
1/2 pound
20 pounds
70 pounds
3. bag of apples
5 pounds
35 pounds
65 pounds
4. adult Golden Retriever
6 pounds
20 pounds
65 pounds
I just don't think this kind of "number sense" they are trying to develop artificially plays out like this. I don't believe it's something you explicitly teach. I consider number sense (if this is what they're concerned with developing) as something that develops as a result of having coherent foundational knowledge and a context into which to put things. Steve's reference to history in a timeline is an excelent parallel. Children need to learn things in context whether it's history of mathematics.
Teaching "number sense" in such an artificial and incoherent manner is extremely counter-intuitive. As adults, these leaps in undertanding seem so simple, but when you're 7 or 8 years old, you have little experience when it comes to how much newborn babies or adult Golden Retrievers weigh or, for that matter, Thanksgiving turkeys or bags of apples.
"The vast majority of students will not create much mathematics, but imagine if we could help them appreciate what others are creating!"
This is something I've thought about a lot....based in my own experience there's pretty much no way to appreciate the math others are creating without actually studying and learning and practicing math for years.
I don't even bother buying "math appreciation"-type books any more (I'm referring to good books about math written for people who don't know math).
They're not the same thing as actually being able to do the math.
... and I meant
Children need to learn things in context whether it's history OR mathematics.
--The activity which students are denied most frequently in math class seems to be (and I have an uninformed opinion to be sure) is philosophy, more specifically epistemology, e.g. How do we know that a particular mathematical result is true?
This is because the current teachers HAVE NO IDEA. They don't know WHICH IDEAS are the foundations, which ideas are derived. Prof. Wu has a great paper where he asks his "students" at a how-to-teach-math seminar, who are all high school math teachers, the difference between a theorem and a definition, an axiom and a theorem. NONE of them knew the answer.
--Why are a particular set of assumptions needed?
This is lacking because we start with the "exploratory" math stuff, but nowadays, never graduate to anything else. It makes sense that in K-3 grades, kids are learning about numbers, and how to manipulate them by procedures, and by memorization. At some point, you need to abstract, and at that point, the issue of "what do I know when I can no longer count it out on my fingers" is the issue.
Prof. Wu argues quite clearly that fractions are the FIRST place where such abstraction matters, and where we're going to learn how to know the difference between knowing something procedurally and knowing something is TRUE generally. From there, you can learn what kinds of things are TRUE, generally, and what kinds of things can be SHOWN TO BE TRUE.
And as you move farther and farther, you make your assumptions more explicit. But high school math teachers NEVER talk about what they are assuming. they simply dont' know.
Honestly, I think the reason classroom teachers (as opposed to ed school theorists or textbook writers) gave up teaching long division and embraced NOT teaching it was because they Simply Didn't Know WHY IT WORKED. and then it's quite natural that it was too hard to explain it to someone else struggling. Might as well skip it--seems like hocus pocus anyway.
[...but you have to have something to inquire and argue about!]
What these anti-knowledge gurus like this Kuhn person are advocating is connecting the dots without the dots, as I like to say.
[Beyond basic literacy and numeracy, it has become next to impossible to predict what kinds of knowledge people will need to thrive in the mid-21st century.]
Knowledge has value beyond the purely utilitarian. It can be enriching and exciting. Make life more interesting. Help us understand the world around us. Even from a purely utilitarian point of view, I doubt that the fabled, mysterious, impossible-to-predict 21th century skills will look kindly on the tabula rasa these gurus want.
[A resolution has been in the direction of undertaking to teach not simply knowledge itself but the skills of knowledge acquisition...]
What these anti-knowledge gurus are missing is that knowledge acquisition requires effort and time. No one can become an expert in, say, math or chemistry on an ad hoc basis.
Why waste precious 12 years of schooling spent handwringing about what the 21th centiry will bring? Why not spend that time building a foundation?
Prof. Wu has a great paper where he asks his "students" at a how-to-teach-math seminar, who are all high school math teachers, the difference between a theorem and a definition, an axiom and a theorem. NONE of them knew the answer.
Is that right???
gosh
I wouldn't have guessed that.
I'm heading towards knowing what these things are now. If I had to bet I'd bet that my informal definitions would pass muster as informal definitions.
all high school math teachers, the difference between a theorem and a definition, an axiom and a theorem. NONE of them knew the answer.
Help me to understand this aspect of math education.
These people have taken a math class in the math department of an accredited university, have they not? How is it that university mathematicians are letting us out of their classes (with an A nonetheless!) without knowing why the product of two negatives equals a positive? That is not a rhetorical question.
My classes were in the math department, not in the college of education.
But did my teachers way back when know how long division worked?
I bet they didn't - yet they taught it well enough that I never had a moment's trouble with it (without understanding it myself, either).
My guess as to why long division got taken out of the curriculum would be that at some point addition, subtraction, & multiplication were taught so poorly that it became impossible to teach long division.
Then, as is customary in the world of education, the cause of the problem was attributed to students (division is intrinsically hard to learn) rather than to teaching.
I constantly see the damage done by an institution in which by definition all negative events must be explained as a function of the "client" of that institution, not of its actors.
The Galen Alessi paper may be the only paper anyone needs to read to grasp the core of what is wrong. When you rule out of bounds 3 of 5 universally-agreed upon explanatory factors (curriculum, pedagogy, administration) how often are you going to analyze problems correctly?
In a complex system the answer may be "never."
Myrtle
I know your question isn't rhetorical & I have no answer; I don't even have a guess.
I assumed that any math teacher focused enough on his work to be attending a seminar with Wu would be able to define these terms and to write proofs.
Ed said, this morning, when I read him the Kuhn quote that for her to make her case she would have to show that the content-rich traditional education people received in the 19th and 20th centuries didn't contribute to the phenomenal advances & successes of those two centuries.
When Saxon works on number sense or estimation in an explicit way, it ties it into the concepts being worked on at that moment. This is why it works where others fail.
SusanS
I just checked the definitions and, yes, I do know what these terms mean.
At least, I know their definitions well enough that I'd be able to raise my hand and volunteer a definition if I were sitting in Professor Wu's class.
I should add that I was taught the meanings of these concepts in high school.
-- assumed that any math teacher focused enough on his work to be attending a seminar with Wu would be able to define these terms and to write proofs.
hm, I'm not sure I understand this assumption. Prof. Wu was part of some extensive California professional development for teachers. It was a funded element of the CA budget at that time, and those funds were turned into summer institutes which he and other teachers taught at. In some cases, the attendees received credit for these courses just as college courses.
They would set it up so he'd lecture in the morning, and then they'd do exercises and manipulatives and discussion in the afternoon. Does that mean he's going to get the cream of the crop? Or going to get teachers who feel fearful, overwhelmed, unable, etc.? I don't know. It's an interesting question.
http://math.berkeley.edu/~wu/pspd4c.pdf
is a cowritten paper about some of the California's summer institutes for teachers in grades 4-6. In the Geometry institute, where grade school teachers spent 15 days doing geometry--lectures in the morning, discussion and exercises in the afternoon. Here's one bit:
"The main difficulty with the Geometry Institute, and the relative lackof success thereof, was the teachers’ unfamiliarity with anything geometric. With but mild exaggeration, some teachers literally trembled at the sight of
ruler and compass or when they were handed a geometric solid. As mentioned in the preceding paragraph, we were prepared for teachers’ being ill-at-ease with geometric reasoning and lack of geometric intuition, but not for the degree to which both were true. School education in geometry is in deep trouble. It goes without saying that, given such a low starting point, every step of the instruction in the institute was met with considerable resistance.
It was especially true of the teachers’ encounter with proofs about parallelgrams, rectangles, and circles, although this difficulty is less surprising when
one considers that our high school teachers also have the same difficulty."
I don’t have a good definition of number sense and have not researched any references to attempt to find one. Where I would be tempted to apply the term is in the context of error checking or “sanity checks.” In other words, there are errors that students make that you would think would not slip by them if they had more “number sense.”
For example, if you multiply a number by an improper fraction, your product should have a magnitude greater than the original number. So, if I make an error in multiplying 419 by 5/2, I might get a number like 168. Now I make errors all the time, and I don’t catch them all. This one, however, I would catch because I have enough “number sense” to know that I should get an answer greater than 419. You could argue that this is knowledge of some rule about proper and improper fractions rather than some “sense,” but then I would counter by saying that it was my number sense that led me to reach for the rule to apply.
Similarly, a story problem might tell me that a company with revenue of 1.84 billion dollars invested 4% of revenue in research and ask how much was invested in research. I might make an error and multiply the revenue number by 4 instead of .04. Seeing that my result was greater than the total revenue number would trigger my number sense to tell me something didn’t look right. In geometry, when computing the length of a hypotenuse, one should get a number larger than either leg of the triangle. In trigonometry, number sense would recognize that something was wrong if you applied the tangent function when you should have used the inverse tangent.
I guess the antithesis of number sense, to me, is when a student computes the third side of a triangle with sides of length 20 and 30—by the Pythagorean Theorem or the Law of Sines or however—and his calculator tells him that the third side is something nutty like 0.561 long. If he had any number sense, he’d know to not trust that answer and re-check his work. I guess a more specific case might be computing a hypotenuse with the Pythagorean Theorem, but failing to take the square root as the last step. A right triangle with sides 3, 4, and 25 (instead of the correct 5) makes no sense.
That’s my two cents’ (sense?) worth,
Dan K.
Myrtle said:
--These people have taken a math class in the math department of an accredited university, have they not?
They certainly didn't in ed school. They didn't have to in order to become a credentialed teacher in the state of California.
Here's the Grad School of Ed at UC Berkeley's web site for some special math science ed + credential program, which gives you a master's in ed and a credential, supposedly preparing you specifically for math or science teaching in secondary schools:
http://www-gse.berkeley.edu/program/macsme/macsme.html
Not ONE math course in ther math department. Now, these folks are supposed to come from "math or science undergrad" degrees, but that's a HUGE swath, and it certainly doesn't mean that the undergrad degree ever required a math course past calculus and maybe a stat course.
The path for an elementary school teacher to a master's in ed and a credential requires NO math courses outside of the Ed department, and states this, explicitly:
"The program culminates with the completion of a master's project that is concerned with the application of developmental principles to classroom practices. Although the DTE program draws heavily on psychological theory and research, a background in the social sciences is not required for admission. Experience working with children, preferably in a public elementary school setting, is expected of all applicants."
http://www-gse.berkeley.edu/program/DTE/dte.gse.html
The clearest explanation of number sense I've seen is here: http://www.didax.com/newsletter/pdfs/gains.pdf which is really a cut from an educational supplement sold at http://www.didax.com/shop/productdetails.cfm/ItemNo/2-149.cfm
Number sense is detailed vaguely in NY State's Math Standards,which seem to be heavily influenced by the State of Virginia's standards which actually spell out details on what the child is expected to master http://www.greatschools.net/content/stateStandards.page?state=VA
>>"In my teaching practice, the direct teaching of estimation would be one of the prime areas where I felt I had absolutely no success. It seemed students didn't know why they were doing it and that they didn't have the flexibility or number sense to make it intellectually satisfying. Maybe you have some specific suggestions about making it more palatable. "
Not having seen your teaching, it would be hard to comment. However, the way my district's text and teachers mess up estimation is by not including the rules for selecting compatible numbers and by not including the rules for handling estimations when the numbers involved cannot be rounded to the same place value (ex. 1245 X 23, the answer key will only give one possible answer, say that for 1200X20, when the unguided student will estimate 1000X23, which is even easier to crunch with mental math).
"Not ONE math course in ther math department."
Allison is right. I've checked a number of college web sites and it's the same thing.
They don't know how to teach kids because they don't have any experience with what classes the kids will have to take in college. They've redefined K-12 math and it's become low expectation math appreciation.
I find the trend to have all requirements met by ed school courses quite abhorrent. They are in their own little low expectation world.
When you talk of the importance of content, I am reminded of one of the less-valuable academic experiences in my life: a unit study of the circus in 4th grade, because the whole elementary school was going to go and clearly it had to be tied into the curriculum somehow (which now seems silly to me -- why can't that just have been a fun field trip?). Now that I know the phrase I am certain it was a "learning to learn" project -- really, were the vocabulary words on circus jargon really the most critical thing for us to learn? And then having to write a story incorporating them (I believe mine was a mystery hinging on toxic clown makeup)?
I think the only one I still remember is "roustabout."
-m
this is a great comment thread!
i believe steveh is much too kind
in imputing sincerity to "reform" leaders.
it seems to me unlikely that
anyone in the propaganda business
could survive for long -- never mind
achieve such wild successes at it
as NCTM has done -- while believing
their own lies.
meanwhile, at the classroom level,
one has teachers, who, yes,
sincerely believe that, for example,
high-tech tools can eliminate
the need for calculational skill,
but are every bit as guilty
of the same kind of bad faith.
their outward message is
"math is interesting and important
and everyone should learn to love it"
while inwardly they believe
"math is boring and useless
and i hate it". indeed, i've come
to believe that there's a widespread
attitude to the effect that
"math is something over there
and actual life (where things
have *meanings* and can sometimes
be *explained*) is this thing
over here ... and they have
nothing at all to do with each other.".
on my model, this accounts for
the kind of phenomenon Dan K
was talking about upthread
(not doing "sanity checks").
of course this doesn't make 'em
master propagandists.
just lousy math teachers.
& it's sort of fascinating, really ...
but it sure can be frustrating.
v.
Prof. Wu was part of some extensive California professional development for teachers. It was a funded element of the CA budget at that time, and those funds were turned into summer institutes which he and other teachers taught at.
These institutes may have been part of the California subject matter projects. Ed headed the first one devoted to History/Social Science.
I'll ask him to confirm, but my impression from listening to him talk about these seminars over the years was that the teachers who attended them were above average in terms of their subject matter knowledge.
I've never seen a professional development workshop in math -- math per se -- that teachers were forced to attend.
roustabout!
an excellent word
oh gosh - just read the rest of Allison's comment, including the quotation on geometry
I would bet money those teachers were the best in their schools.
You're not going to find too many elementary school teachers who are afraid of geometry and willing to spend a good portion of their summer trying to learn it.
Those are dedicated souls.
"I think that number sense might allow me to figure out 6 times 8 in a variety of ways that made sense to me, if I forgot that 6 times 8 is 48 - the way that I might forget that the capital of New York is Albany."
I expect my son to always know 6 times 8 and the capital of New York. No excuses.
Did you expect your son to do this out of the womb? or can you imagine a time when he knew that 10 sixes were 60 and 2 sixes were 12 and could reason that 8 sixes must be 60 subtract 12? That is an example of what the notion of number sense is to me - using the properties of number in a reasoned way to expand or verify. I think that you can use the properties to estimate as well, but that estimation would be but one of many subskills of number sense. Yet, I offer this without looking yet at the links to articles about number sense above or looking at any curriculum documents.
I really appreciate Myrtle's first comment above, though I do think that formalism belongs later in mathematics education just like it occurred later in the history of mathematics. In Ontario, the idea of proof has been broadened (I can here SteveH say 'watered-down') to the mathematical process of reasoning. My example above might be an example of the kind of reasoning that an eight year old might accomplish - taking accepted mathematical results and producing new results from them.
I was a lot firmer about every student just needing to remember all kinds of things (being very good at such memorization myself) before I had a son with short-term memory difficulties. There are all kinds of times when knowing that the capital of New York is Albany (which is a pretty irrelevant thing to Ontarians anyway) will be a lot less important than knowing how to find out what it is (say with a google search from his mobile phone) and the understanding of what goes on in capitals that is different from other cities. Ah, but I dilute the train of inquiry again...
6 times 8 is too simple of an example to prove your point. Albany is borderline at best. Facts are never mere.
You can't Google everything, and foundational knowledge does inlcude facts. Number sense requires content knowledge and "sense" means that you don't have to look it up. The more foundational knowledge, facts, and skills you don't have to Google, the better off you are. Why should kids ever need mastery of the times table if they have a calculator? No one ever thinks this is an either/or question. I expect my son to know facts AND meaning.
So the question is where should this borderline of automatic facts be located? A major complaint of K-8 education is that this line is set very, very low. As I have mentioned in this blog before, I made the mistake of telling my son's first grade teacher that he loved geography and could name all of the state capitals and find any country in the world. She said: "Yes, he has a lot of superficial knowledge." Ouch! The ironic part of the story is that my son had to show the student teacher later in the year where Kuwait was during their thematic unit on "Sands From Around The World". He also had a Kindergarten teacher who rambled on and on about how some kids can read encyclopedias but they don't know what they're reading. But did she ever test my son on comprehension? No.
Facts aren't random bits of knowledge. They form a framework in your brain that makes it easier to place and remember new knowledge. As with the teacher's preemptive parental strike, many educators feel that this fact and skill line is very, very low.
"(I can here SteveH say 'watered-down')"
No. I'm an applied geometry guy, remember? Vectors, transformations, cross products?
Actually, most high schools have two variations of geometry class. The honors classes (properly) set higher expectations for proofs. I do remember spending hours trying to trisect an angle with a straight edge and compass.
Problems arise when this sort of trade-off is done in the earlier grades. In the US, the Math Panel has properly defined what they call "school" math, as opposed to "real-world" math. They are fighting the trend to make these low expectation (door closing)trade-offs before kids get through with algebra.
You can't Google everything, and foundational knowledge does inlcude facts.
If you don't know something about a field you can't Google anything because you don't know the search terms.
Wish I could think of an example off the top of my head - can't at the moment - but there have been periods when I've gone a year or two without being able to Google a topic because I didn't know what people in the field call that topic.
OH!
An example comes to me.
Last spring I needed information FAST about what IDEA (special ed law) has to say about a district's legal duty to identify kids needing services instead of just waiting until the parent brings up the subject.
I was pretty sure there was such a provision in the law, but I didn't know. I feel their had to be such a provision, and I needed to know instantly. (Won't go into details - didn't involve my kids but someone else's.)
I Googled for hours and found nothing.
Finally an attorney told me that the term I was looking for was "Child Find."
IDEA says that schools have a legal duty to identify kids in need of services. That provision is called "Child Find," and if you want information about it you need to Google Child Find.
If I hadn't known a school board attorney -- an attorney living in another state, no less -- I wouldn't have been able to find the information I needed in time.
I've had this experience over and over and over again.
People need foundational knowledge, which means facts & vocabulary & a schema, to use Google.
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