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I'm wondering, now, whether GATE people have internal bar models, too.
Remember the 6th grade boy who was given the problem about the whale?
A whale swims 40 miles in 1 1/4 hours. How far does he swim in 1 hour?
This boy, a GATE type kid, instantly knew the answer, but didn't know how he got it -- and didn't know how to set up proportions. (I'm pretty sure his dad said he didn't know how to set up and solve a proportion, at least.)
A couple of weeks later I talked to a GATE mom, and told her about the boy. I rattled off the problem quickly, and she, too, instantly knew the answer. She said, "Well you talk kind of fast [true], but I think that's the answer, right?"
This was over the phone.
I asked her how she knew the answer, and she said, "Well it's 1 1/4 hours, so that's five parts."
Now that is not the way I would have thought of this problem until I did a lot of bar models. (I didn't think of the bar models when I first heard the problem. Had to set it up as a proportion. But if I'd been "in practice," in practice meaning I'd been doing some bar models recently, I a bar model would have popped into my mind's eye.)
This mom also told me that it would take her a lot longer to set the problem up as a proportion and solve than it did just to "know" the answer, which she was pretty sure she knew because 1 1/4 is 5 parts.
I find this fascinating.
Normal human beings, and that category excludes all you math brains, do not think 1 1/4 is 5 parts.
Normal human beings think 1 1/4 is two parts: 1 and 1/4.
It's a well-known fact that normal human beings fall off the math cliff when they hit fractions in 5th grade. Maybe the presence of an internal number line (not proved, btw) and the absence of an internal bar model, tell us something about why normal human beings fall off the math cliff when they hit fractions and not, say, when they hit multiplication.
I'm inclined to believe this, simply because my guess is that Dehaene will turn out to be right; numbers are represented spatially in the brain in some way. Makes sense that fractions would be, too, in people who "know" the answer to the whale problem without having to set it up.
Assuming this is true - which is what I'm going to assume, since I have nothing to lose if I do assume it to be true - I draw three conclusions:
1. Hung Hsi Wu's recommendation that one teach fractions first and foremost as a point on a number line (pdf file) is a fine idea. Hence: lots of practice with rulers!
2. The bar models in Primary Mathematics are indeed one of the core reasons for the effectiveness mathematics instruction in Singapore. (pdf file)
3. I am going to carry on insisting C. do Primary Mathematics 3rd grade bar model problems.
trip down memory lane
Back when I was working my way through Challenging Word Problems Book 3, I could feel my brain changing as I did those bar models.
I loved the bar models so much I started a solution manual for Challenging Word Problems Book 4. Didn't get too far; not sure where it is today.
Writing this post I'm feeling a lure that I must resist back to that project.....
The Number Sense by Stanislas Dehaene
Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4-6 Texts Used in Singapore by Sybilla Beckmann
Sybilla Beckmann articles
Mathematics for Elementary Teachers by Sybilla Beckmann (1st edition)
Chapter 2: Fractions by Hung Hsi Wu (pdf file)
Cassy T on bar models
Drat These Greeks on bar models
Challenging Word Problems, Book 3
Spatial attention and the internal number line
66 comments:
Could digital clocks be the problem?
When I read or hear an hour and a quarter, I see the minute hand pointing to twelve and then chunking around five times. I don't see bars. (Maybe I see wedges .... )
When I see a two-digit number, I am usually aware of its factors; I'm not sure why. So my brain realizes that the five quarter hours in the time and the factor five in the number of miles might make this an easy "mental math" problem.
If the problem had read, "A whale swims 37 miles in 1 and 3/13 hours, how far does he swim in 1 hour?", I would have to use a standard algorithm.
I've just added an addendum to the post.
I think digital clocks are a huge problem.
I don't like credit cards, either -- I think Google Master first made me aware of how little "real world" math C. was doing, since we always use a credit card.
I'm pretty sure a huge amount of incidental arithmetic learning has disappeared from everyday life.
When I see a two-digit number, I am usually aware of its factors; I'm not sure why.
That supports the idea of something like a mental bar model...doesn't it?
I'm not aware of factors; I'm not aware of "divisions" of any kind.
A number, for me, has a "oneness."
hmmm....
No, I don't think that's quite true.
"Fifty two" probably, to me, is a fifty and a two.
But that may be because it's two words linguistically.
My daughter's mind works this way. I'm usually a step behind because I have to figure it out while she sees it. Her challenge is showing her work because it so clear to her as to be obvious. She can't understand what there is to show.
My math kid said that he doesn't see pictures. He didn't know what I was talking about.
When he was in the 5th grade, I had him do the Singapore 6 word problems. There was always an answer and no work or drawings. The vast majority of the time he was right. He couldn't really expain how he knew.
We still have major problems with him not showing his work, especially when it's a tiny step. This is an ongoing problem and getting lower grades doesn't seem to convince him.
His ability to think abstractly and merge steps at a younger age seems to have made a difference in how he handles problems involving arithmetic. He also seems to be able to view problems in different ways without getting lost.
The thing that drives me nuts is that he often misses the easy ones (the ones I can do) over and over again. I mean, really easy ones.
I'm usually a step behind because I have to figure it out while she sees it. Her challenge is showing her work because it so clear to her as to be obvious. She can't understand what there is to show.
That's exactly what this dad told me (though I'm not 100% sure he meant that his son had never even seen a proportion before - he might have seen it in class for the first time; something like that...)
But for this kid, setting it all up as a proportion was a HUGE ordeal.
He knew the answer.
Period.
That's what the math-brain mom said, too. She said that setting the problem up would be much more work than just knowing the answer.
(Well, yeah....!)
The thing that drives me nuts is that he often misses the easy ones (the ones I can do) over and over again. I mean, really easy ones.
Interesting.
I've heard (don't know if it's true) that mathematicians aren't necessarily good at arithmetic...
btw, I don't think the math mom actually had an image in her head, but that doesn't make a difference to what I'm thinking about mental bar models.
Regular people probably do have number lines in their heads, but those aren't especially visual, either. (I don't see my mental number line...I don't have an image of it.)
Math mom said to me, "I think I had an image," but her tone was doubtful.
What she definitely had was an instant perception of 1 1/4 as 5 equal parts.
I'm pretty sure a huge amount of incidental arithmetic learning has disappeared from everyday life.
Have you seen the new electronic monopoly game? No more cash.
That was a favorite way to get the kids really competent in addition and subtraction.
Gone.
The big switch, of course, is the electronic banking. Each player starts with $15 million in their bank card account, and the banker adds amounts (such as when the player passes go) or subtracts (such as when the player pays to get out of jail) by inserting the player's card in the plus or minus slot of the Banker Unit. When one player has to pay another player, the banker debits or credits each account as appropriate.
The electronic banking is neat and easy to handle, and as an added educational bonus, the player who takes on the banker role will quickly become familiar with basic bookkeeping concepts. The bank cards operate more like debit than credit, so players will learn responsible money management, not how to run up credit card debt.
Well, it's a little off topic, but I saw it advertised a couple days ago. Kinda depressing.
I'm thinking "see" was proabably not the best choice of words. I'm not sure my daughter would describe it that way. When she sees the equation or word problem, she just knows the answer. The awful thing is that it takes me a bit to figure out if her response is correct, because I need to work it out. She then gets to say "I told you so" which she enjoys thoroughly. I then get back at her by saying, "Yeah, well, show your work."
What is apparent to someone with good number sense is that 1 and 1/4 is 5/4, that is five of something called "one fourth." There's no work to show, though you could explain that if five somethings equal 50, then one something is 10 and so four somethings are 40.
Since the numbers are easy, you can do it in your head; no need for visual models or bar models or algorithms. But then, I was an algebra person, not a geometry person.
It works just as well when the whale swims 37 miles in 16/13 hours except you probably can't compute (13)(37/16) without writing it down or asking the calculator. Knowing what you need to compute, however, is exactly the same.
I've been enjoying your blog for a few weeks now. My son is in 3rd grade and attends a charter school with a constructivist curriculum. For the most part this actually works - their reading and language scores are high and the kids are bright and good readers. The math however is abysmal - their scores are OK but the pace at which they work (both through the curriculum and through each problem) is torturous for me. They are using Investigations which I am fast coming to despise though I liked it - or at least understood the premise of it in the early grades. We do lots of extra study at home especially in math. It’s a hard curriculum to teach over, though the teachers do not discourage the students from using traditional problem solving methods. I’m trying to build an arsenal of persuasive information to initiate a change in math curriculum – it is a parent run school. There is SO much information out in the ether to sort through and of course I have to bring whatever I bring in a positive way. I’m glad that sites like this exist to help our efforts at home – thank you.
I see factors (like SusanJ), not bars or pictures. I don't know if that's in spite of or because of my education. My son seems to see the factors too, but as SusanJ says, it doesn't help much for things like 1 and 3/13 hours.
"I'm not aware of factors; I'm not aware of "divisions" of any kind."
How about a number like 12? I immediately think of 6, 4, 3, and 2. If I had to guess, this factoring has to do with all of the practice I've had with mental arithmetic.
So, am I good in math because I practiced, or did I practice because I was good in math (good with numbers)? Success breeds success. The question is how much success (innate or otherwise) does a child need to trigger this spiral? I don't think it's much. I can see it happen with my son. Small successes can make huge differences. As a parent, my job is to keep the school from screwing that up.
"They are using Investigations ..."
My condolences!
"I’m trying to build an arsenal of persuasive information to initiate a change in math curriculum – it is a parent run school."
KTM has tons of information.
"There is SO much information out in the ether to sort through and of course I have to bring whatever I bring in a positive way."
Yes, well, that's a big problem. You could get stuck talking about basic math algorithms or discovery learning. Most discussions, however, tend to gravitate towards a vague idea of "balance", and who could be against balance? But what does it mean? What are the details.
It's perhaps easiest to avoid philosophical issues and keep it as simple as comparing curriculum A versus curriculum B. You could buy all of the Singapore Math books for K-6 (it's not that expensive) and compare them side-by-side with the TERC workbooks.
At a proper forum, you could put TERC and Singapore Math books for each grade side-by-side and give parents a chance to compare the two. Perhaps that's too much. To show the cumulative effect of each curriculum, you could select sample problems from grade six. The differences are most visible and dramatic by sixth grade.
I guess my advice is to tell them what you want (or propose), and keep the message simple. Singapore Math is a better curriculum any way you want to look at it. The goal is not to supplement TERC. The goal is to replace it.
If you want to stay positive, don't trash TERC (although that's pretty easy to do), but offer a better curriculum that prepares kids for a proper course in algebra in 8th grade.
I'm fascinated by the kids who can get the right answer faster than the parents but not explain why.
If it were me, I think I'd be tempted to work out the problem when the kid's not around and come up with a "plausible" explanation of how to get a "plausible" wrong answer. Then when we did the problem together, I'd argue that my answer was correct until they proved me wrong at which point I'd let them feel very superior.
And BTW, don't kids play with Legos anymore? I think that's where my boys likely got their number sense. They had names for all the different types of blocks and would say things like, "I need a two-ee."
On the topic of how people represent numbers mentally and how they solve problems, if you haven't read the late Dr. Richard Feynman's humorous essay, "It's as Simple as One, Two, Three" be sure to check it out. I believe it's included in an anthology entitled "The Pleasure of Finding Things Out" (don't know how to include a link here). Very amusing but it makes some good points especially about empirical verification of what is actually going on when people perform such an apparently simple mental exercise as counting.
BTW on this problem I "saw" the five parts before I even finished reading the question. I always "see" numbers as multiple personalities, according to their factors...no bar model needed. There's a (possibly apocryphal) story about the mathematician John von Neumann, asked at a party for the answer to a problem about a bumblebee flying back and forth between the handlebars of two cyclists approaching each other from opposite ends of a course. He gave the answer instantly, surprising his inquisitor, who confessed that most mathematicians didn't see the easy way to calculate the answer, they tried to work it out using a complex formula for determining something based on an infite series of decreasing sums. Von Neumann shrugged and said, "But that's how I solved it."
Vive les differences.
"And BTW, don't kids play with Legos anymore?"
Legos are huge, especially the smaller size. There are all sorts of specialized kits, from Harry Potter to Transformers. They are a favorite present on the birthday circuit.
There is also the serious side for robotics. The central processing unit (the "brick") is a computer with input and output ports and it's covered with lego bumps. The full kit (Mindstorms $200+) contains the brick and all sorts of lego parts, wheels, axles, gears, motors, input sensors and output devices. It also includes a program that provides a graphical programming language. Kids build a working model out of Legos, write a program, and then download the program to the brick. It's a fully autonomous robot.
Then, there is the FIRST Robotic League (national contest) where teams of kids have to build a LEGO robot that performs specific tasks. They also have to write and present a report on a technical topic. Last year, it was on nanotechnology. I was a coach for my son's team last year and there was another KTM contributor (I'm sorry I forgot who it was.) who was also deeply involved in the contest.
We had some interesting threads about it last year. I love the kit, but there are some issues with the child-centered "discovery" nature of the FIRST LEGO League. In talking with coaches from other teams, the ones who did the best (and learned the most) were the ones who controlled the discovery process very carefully. The kids are supposed to do everything themselves, but that just doesn't work. As a coach, you're always trying to find the right balance. If you leave it all up to the kids, nothing will get done and they will hate the whole process.
I'm not involved this year. My son is at a new school and they only allow 7th and 8th graders to be involved. I'm not unhappy. My son's not unhappy. Actually, he would prefer just an after-school robotics club, not a formal league or contest with very contrived rules that meet the ideals of child-centered discovery learning.
Many of us are big fans of Dr. Richard Feynman. I don't remember that essay. I'll have to look it up.
Some might think that Feynman liked the idea of discovery as it's practiced in K-12 schools. That just isn't the case. What Feynman liked was applying fundamental concepts, rules, or skills to what you see everyday. This is very hard to do. K-12 discovery is used to try and discover basic rules. This is backwards.
Thanks for your comments steveh. I do love Feynman, have since high school, really got into him in college (my degree is in Physics and English). The funny thing is when I was studying physics in college our professor (who studied under Feynman) used some of the discovery teaching methods - in a very controlled manner with HIGH expectations - and I think that is one of the reasons I loved it. It really made things click with me.
BUT I think it was important that I had a strong math base to back it up - I'd learned the traditional way so it was an easy transition to start intuiting with a high degree of accuracy.
I just don't think you can or should divorce solid (rote, if you will) math knowledge from explorations.
It's interesting that you all brought up legos in this thread. They're huge in our house, my son has been adept at both following the directions to create the indended models as well as building his own for years. We went to the FIRST Lego League World Competition last year to observe just because it was closeby. There's some interest in starting a league but it didn't get off the ground this year. My concern was actually wondering if the kids could do as much by themselves as was expected.
Obviously I have bigger fish to fry with the curriculum before I can concentrate on FLL.
"My concern was actually wondering if the kids could do as much by themselves as was expected."
It depends. Without proper guidance and help, the experience could be a disaster. I think that schools should focus on robotics rather than the contest. This could be as simple as an after-school robotics program with no specific goals or deadlines.
For our team last year, the specific goals and deadlines got in the way of learning and fun. All of the kids had to specialize early into builders, programmers, or technical presentation workers. You couldn't stop to explore side issues or really understand how the robot worked. There was absolutely no time to allow all kids to learn how to program the robot. If you take away the programming part, then it might as well have nothing to do with robotics.
One fundamental issue we ran into was accuracy and repeatability. Sometimes the robot wouldn't do the same thing twice in a row. Is it a friction problem, are the tires slightly different sizes and did someone flip them around, is the pivot wheel causing an error, or are the batteries getting weak?
No Time! Sure, it's good for students to experience deadlines, but not if they're doomed to failure. That's when the coaches start helping more.
Having an on-going program will make it easier for students following on in the next year, but they won't get the chance to discover or understand these problems if there is a big rush to get the project done.
My feeling is that the contest is not time for discovery or for kids who are new to LEGO robotics. If a school wants to get involved, they should start an after-school robotics club with no set goals or agenda other than learning and having fun.
As the kids get into 6th, 7th, or 8th grades, some might want to commit to the FLL competition. They have to be willing to put in the time, they have to have some experience, and they have to be willing to get started by the beginning of September. The team that won in our state came from a school-independent robotics club that operates year-round.
Have you seen the new electronic monopoly game? No more cash.
That was a favorite way to get the kids really competent in addition and subtraction.
C. pointed the add out to me.
I was horrified.
Last summer's discovery that C. couldn't figure 10% off a price was a huge wake-up call for me.
Schools like mine used to be able to depend on this kind of incidental learning of arithmetic, and rightly so.
I bet it hasn't occurred to schools or to math teachers any more than it has to me that kids today aren't getting it.
I wonder if disadvantaged kids still are experiencing this kind of real-world learning?
I have no idea how much their parents rely on bank cards.
Hi Katy--
Wow!
A parent-run school!
Can you tell us more about that??
Is there a possibility of simply offering choice inside a parent-run school???
Could you have a Saxon or a Singapore Math "track"?
BTW on this problem I "saw" the five parts before I even finished reading the question. I always "see" numbers as multiple personalities, according to their factors...no bar model needed.
Interesting.
I have to get all this stuff up front.
The more I read here, the more I think bar models are a brilliant invention that allows those of us who don't perceive 5 parts to learn to do so.
How about a number like 12? I immediately think of 6, 4, 3, and 2. If I had to guess, this factoring has to do with all of the practice I've had with mental arithmetic.
Now that is interesting, because I believe Primary Mathematics does lots of mental math, too.
I did essentially no mental math as a child or as an adult.
THANKS
I'm going to start pushing some mental math around here, too.
I've noticed C. is completely calculator/paper and pencil dependent.
When it comes to perceiving factors, it amounts to the same thing.
Katy S --
You can find all the Singapore Math threads easily....you should be able to pull everything up quite easily.
There's a huge amount of material here, so you'll probably be able to find things that would work to help persuade other parents.
You'll also find a fair amount of anecdotal material about specific children learning math (which, thanks to Steve, we now know to call "case studies.")
I just don't think you can or should divorce solid (rote, if you will) math knowledge from explorations.
Call it procedural knowledge!
That's typically what it is, and the word "procedural" will stand you in better stead.
"Rote" knowledge is....without any meaning at all. Purely memorized, "parroted" back. (Or "regurgitated" as constructivists like to say!)
(And, btw, parrots aren't so parrot-y.)
The classic example of procedural knowledge is knowing how to ride a bike.
Knowing how to add, subtract, multiply, and divide and when to do it can't be rote. It's not deeply conceptual, but it's definitely not rote.
One more comment about Legos. My kids were born in the late 1960's, back before all the fancy Lego kits and most of their Lego time was when they were pre-schoolers. So that was why I was thinking it helped their number sense long before they studied arithmetic.
"I'm going to start pushing some mental math around here, too."
The only thing I remember from school was having races to see who could interpolate the fastest from the tables in the back of the math book. With calculators and computers, nobody learns mental interpolation at school. Even though I don't have to interpolate logarithms any more, I still use mental interpolation for all sorts of things.
For other types of estimating, I do the calculation left to right. I can stop when I get the accuracy I need. If I need an exact answer, I use the traditional algorithm which works right to left. (on paper) If the problem is not too difficult, I can continue the left-to-right estimation until I get an exact answer.
I don't even know what interpolation is.
Although I have recently taught myself logarithms.
Never saw a logarithm in 2 years of algebra in h.s.
"I don't even know what interpolation is."
B.C. (before calculators), some calculations were too difficult to do exactly by hand, so math books had tables in the back. It might be a table of sines by degrees. If you wanted the sine of 42.3 degrees, you needed to go to the table and find the sines of 42 and 43 degrees and figure out the sine value for 30% between the two numbers. (Calculate the difference of the sine of 42 and 43 degrees, take 30% of that number, and add it to the sine of the first number.)
We had races.
"I don't even know what interpolation is."
Suppose I have a table of data
such as
x, y
.
.
.
10, 275
14, 345
18, 415
.
.
.
and I want to know what the value y is at say 12.5 I would interpolate to get the data. In this case it is very easy since there is a linear relation.
There is also extrapolation where let's say you have the same data
x, y
.
.
.
10, 275
14, 345
18, 415
and 18, 415 is your last piece of data and you need to know the value of y at say x = 20. You would extrapolate the known data into the unknown region.
Again this one is straight forward since it is linear. Non-linear data can be difficult to extrapolate. It is not so bad when you interpolate since if the known numbers are close to each other you can assume linear data and get close. For example
x, y
.
.
.
10, 275
12, 350
14, 440
.
.
.
Would not be hard to extrapolate to get the y value of 11, you would assume it is linear and find the approximate value at 11. It won't be exact but should be close.
The data
x, y
.
.
.
10, 275
20, 500
30, 1000
.
.
.
would cause you to have large errors trying to extrapolate to say find y when x = 18 due to the non-linear nature of the data.
This is where a program like excel is nice since you can plot the data and put a trend line in that will give you the equation of the line. You do have to know what the trend is though. Quadratic, exponential, etc.
Hope this helps
Good example too Steve. I was typing mine as you were putting yours in.
Hi SteveH. I'm a computational scientist and I remember having trouble understanding the need for nonlinear interpolation. Anyway, the point I wanted to make is that people who develop algorithms still need to understand how to do this so its not an out-of-date skill.
Qualitatively speaking, interpolation is just a way of formalizing the guesstimating of values that are missing from a well-behaved table. It's also what you do when you are measuring something and there's not a line on your ruler that exactly matches up.
"There is also extrapolation .."
I forgot about that!
SusanJ
As I said in my post, even to use Excel people need to understand what the curves mean. I have seen students try to do linear trend lines in excel on data that are definitely not linear.
Oops in my last example of my first post I meant to say "interpolate" not "extrapolate" so it should read as ...large errors trying to INTERPOLATE to say find y when x = 18 due...
Sorry about that
"... interpolation is just a way of formalizing the guesstimating of values that are missing from a well-behaved table."
That's a good way to put it.
I've written a lot of routines to do numerical curvefitting and interpolation over the years.
I like to tell people that chess programs aren't a good indication of the power of the human brain. If you take a group of unequally-speced points on a piece of paper, a human will always fit a better curve through the points than any computer program.
B.C. (before calculators), some calculations were too difficult to do exactly by hand, so math books had tables in the back. It might be a table of sines by degrees. If you wanted the sine of 42.3 degrees, you needed to go to the table and find the sines of 42 and 43 degrees and figure out the sine value for 30% between the two numbers. (Calculate the difference of the sine of 42 and 43 degrees, take 30% of that number, and add it to the sine of the first number.)
That's so cool!
I always wondered how students did that before calculators!
THANKS EVERYONE FOR EXPLAINING INTERPOLATION.
(must get these up front, too....and write a book....)
Jeff, good point about proper use of Excel. I think that's a case where you can begin to understand what numerical analysis is even in high school. (I, unfortunately, was not exposed to the concept until grad school.)
BTW, we were all posting at the same time so didn't see all the posts.
"I always wondered how students did that before calculators!"
I don't know, maybe taught students how to use their minds instead of a calculator? Just guessing.
"If you take a group of unequally-speced points on a piece of paper, a human will always fit a better curve through the points than any computer program."
I just had a thought about curve fitting. At work, I use a draw program (Adobe Illustrator) all the time that draws curved lines with Beziers. I suspect that using the Bezier-curve tool would be a very good way to get a fairly good fit to non-linear data.
Once you have a curve, interpolation and extrapolation both become pretty trivial.
Now that tool doesn't give an equation for the curve, but it might be useful to write a tool that used a similar UI and that would give you an equation for the curve that resulted.
Use the tool for what it does well and the brain for what it does well.
I don't know, maybe taught students how to use their minds instead of a calculator? Just guessing.
Well, I have a question about this.
Saxon teaches sine and cosine using a scientific calculator --- what do you think of that?
It seems OK to me, since you're doing a lot of long division to come up with the various ratios, but what do you think?
Sine and cosine: two other concepts I didn't learn in h.s.
I took 3 years of math, and most of Saxons's second high school book is new to me.
I didn't even learn the classic word problems. No number problems, no consecutive integer problems, no age problems, no coin problems, no trains leaving the station at the same time problems.
I also didn't know that a linear equation actually described a line.
Now I wonder what the heck we were doing all that time.
Of course, I'm sure this explains why I learned what little math I learned very, very well.
Catherine,
I probably should have been a little more clear. I was speaking more about interpolation/extrapolation in general not specifically Steve's example, sorry.
But to answer your question, if they teach what the cosine, sine, etc really mean I have no problem with a calculator. What I have experienced is students who do not conceptually understand trig functions because all they did was hit buttons on a calculator. When I got them in physics I had to teach them what the trig functions meant because in physics you need to know more than how to punch the buttons.
Does that help?
Catherine, as for the parent run school it is mostly wonderful with some really frustrating parts. It started as a group of parents in a recently gentrified inner city neighborhood who did not want to move when they had kids and so started up a charter school - the schools had not caught up with the real estate.
The school is a public charter school with a constructivist curriculum. The other defining element of the school is that parental involvement is required - you have to sign in blood, sweat and tears that you will be involved and your "volunteerism" is tracked punitively.
The board of directors is made up of parents (2 reps from each grade) with faculty, community and committee (also parents) representatives. The principal (non parent) serves as the executive director. We have certified teachers, low student/teacher ratios, and incredible diversity. Anyone can attend as long as 1. there’s room – school and class size is limited, 2. they live in the district and 3. their parents are willing to be involved. There is no tuition. It is run like a regular public school only we do not have to answer to the public schools system in terms of curriculum or other administrative issues. On the other hand there is a lot of reinventing the wheel going on.
My frustrations are mostly due to standards. I fear there is an elitist undercurrent emerging socially (hence the volunteer tracking) yet the same dominant parents that are the driving force behind those policies are the same ones who seem to resist efforts to raise academic expectations. It’s like they want the school to be this warm fuzzy extension of preschool without acknowledging that at some point you have to get serious about learning and not every little moment can be fun fun fun all the time. I know that sounds mean. I do love the school.
There have been some wonderful people who’ve done wonderful things…but it’s also hard to put in hundreds of hours of time into the school and still feel like standards aren’t high enough. I could go on and on – I’m sure that’s more than you wanted. It’s been a long row to hoe but it beats moving to the suburbs and facing 3 hours in the car every day. I think.
if they teach what the cosine, sine, etc really mean I have no problem with a calculator
Great.
That's the way I felt about it, but I'm not confident of my opinion --- also, I don't necessarily know what "trigonometric functions" are!
What do you think about graphing calculators?
I do have an opinion there (not really based in experience yet) which is that you have to gain some experience seeing how various equations translate into a particular curve before playing around on a graphing calculator...
What do you think?
Saxon definitely teaches what sine and cosine are (imo)
Actually, I would love to hear you go on and on!
I've been thinking about parent-run schools quite a bit....
The dominant parent issue is probably unavoidable...which makes me feel we need a "new paradigm" - that is to say, we need a "school creed" concerning choice that would serve as a constraint on dominant parent types.
Dominant parents get a lot done; they're the movers and shakers of schools.
But the type of personality that makes one a dominant personality isn't necessarily easy-going or oriented to choice.
Plus we, all of us, are constantly dealing with the universal assumption that schools are One Thing and One Thing Only.
Everyone, parents, teachers, administrators, & board members alike, assumes that a school has to have One Approach for everyone.
It's inconceivable for a school to offer a "Singapore Math" option.
A 3-hour commute would not be good.
"I suspect that using the Bezier-curve tool would be a very good way to get a fairly good fit to non-linear data."
Bezier curves are a polynomial-based curvefit. Actually, the curve doesn't pass through the defining points (except at the ends) so the curve is guaranteed to be smoother than the defining points. Interpolation means that the curve has to pass through the defining points and it's very sensitive to slight changes in position. You can get wiggles in the curve between the data points. The higher the degree of the polynomial, the more wiggles (inflection points) you can get.
My specialty is NURB (Non-Uniform Rational B-splines) curves and surfaces, which also use a polynomial-based fit of data points, but the degree of the polynomial isn't related to the number of data points, as it is for the Bezier curve. NURB curves and surfaces also don't normally interpolate (pass through) the data points. You can force the curves through the data points, but you will probably get wiggles.
I wrote a program once that read data sets and allowed the user to try out different interpolations or regressed curvefits through the data. When a good one was found, the program would give the coefficients of the equation so that the user could interpolate or extrapolate just by plugging the numbers into the equation.
Ah you nailed it on the head. Dominant parent types DO get a lot done and they are responsible for the major accomplishments. And also 90% of the conflict. (I say “they” but I am not excluding myself as dominant, hardworking, and possibly obnoxious).
I’d never considered the option of having 2 tracks for math. I think you get into money there, it would be more expensive to run two programs and there would be resentment between the two groups perhaps? Or not? I plan to start a committee to review the math program and will definitely bring that idea to the discussion. Part of me sees the logical conclusion to my efforts and wonders why bother, and the other part of me sees this as a battle that needs fighting. In case you’re wondering I predict this conclusion: 1. people will resist because they’ll see it as a domino effect problem so they’ll tell me to 2. start an afterschool math “club”. Foregone conclusions are so mentally exhausting.
They key to my success will be demonstrating that Singapore math can complement a constructivist (and integrated) curriculum. It's logical in my mind but I can't articulate it yet.
Email me at uralva at gmail dot com if you have more questions about the school.
"It’s like they want the school to be this warm fuzzy extension of preschool without acknowledging that at some point you have to get serious about learning and not every little moment can be fun fun fun all the time."
Elitists with low expectations? Wow. Don't they have any older kids? Won't they be surprised when their kids hit middle school.
Actually, I've come across parents who have warm and fuzzy ideas about K-6. They will probably bend over backwards to help their kids make the transition to high school without a thought about how many other parents can't possibly do that.
First, I should have been more clear that I was thinking of two different sorts of data sets. In the first, the data is (or could be) algorithmically determined. In that case and given two endpoints to play with, I'm know I can get a Bezier curve to hit four arbitrary data points exactly. If the underlying curve doesn't have lots of inflection points, I suspect I could hit rather more. This could be fun, but probably isn't all that useful. You would be better off figuring out the algorithm in most cases.
The more interesting case, and the one that I was thinking about when considering the use of such curves with a graphical interface, is that of noisy data. The usual sort of first approximation for such data is linear least squares, which doesn't work well when the data isn't linear. That doesn't stop people from using a linear model, of course, which perhaps occasioned your "a human will always fit a better curve" comment. In that case, using a Bezier to approximate the data might be useful; it's certainly easy to do on the fly when you have the right interface.
It looks to me as though you have a much deeper understanding of curve-fitting than I do, though, so perhaps I've missed something.
Same basic idea, in math they need to really understand what the curve looks like from the equation and vice versa. Graphing needs to be mastered by hand first so that the student understands the physical meaning of the curve.
I will say that in physics I would let them use any type of calculator they wanted. Why? I am teaching them physics not math. I would use the calculator to speed up the math so we could play "what if", like what if I change the initial velocity, what if we do this on the moon, and so forth. Same with Excel, I would have them do a lot of modelling in Excel so that they could play "what if" on HW assignments not in the book. In fact a lot of college text books for engineering are doing this in the problems at the end of the chapter. You solve the problem, then using Excel you change conditions and plot trends. These are things I could not do when I was in college becasue Excel did not exist. Calculators and Computers are tools to be used when you understand the basic concepts, they are not a replacement for really understanding the math at a deep level.
Just an aside, I speak about teaching and engineering. I started out as an engineer for 15 years, went into teaching high school physics and engineering for about 10 years and now I am back in engineering. In case anyone is confused at what I do when I leave a post.
I left teaching due to administration related silliness not due to the type of students I had. I can tell you I had some very intelligent students, I taught at an all girl's college prep school and still I saw some things in their math ability that made me want to cry. At some point I will put together a synopsis about what I saw, I just want to make sure I do nothing to let on what school it was.
Dominant parent types DO get a lot done and they are responsible for the major accomplishments. And also 90% of the conflict. (I say “they” but I am not excluding myself as dominant, hardworking, and possibly obnoxious).
Exactly, and me, too ---
One thing about folks around here, which I've come to respect enormously, is that "dominant parents" are very able to....deal with conflict, bounce back, refrain from being defensive.
You're in a particularly difficult situation, because I presume you're dealing with founders -- is that correct?
If so, you're dealing with "founder syndrome." Which is a whole other kettle of fish.
I've been in two founder syndrome situations. The one really useful concept I took away was "spaced repetition," i.e. saying my own idea so often that it became "naturalized," and seemed to belong to everyone.
I think you could go pretty far just pushing the idea that Singapore Math conforms to the requirements of a constructivist curriculum.
There's tons of stuff around here and on the other site....and I'll think about it, too.
There are lots of good pieces of persuasive writing about the curriculum, too.
Here's one: it's a "problem-solving" curriculum (true).
Also, be sure to get the AIR Singapore Math study.
That will be a huge help.
It's written by one of the major NCTM constructivists, Steven Leinwand.
I can get you the link later if you don't find it.
Google "Singapore Math" "Steven Leinwand" "American Institutes of Research"
Check out the latest progress reports for the National Math Panel as well.
http://www.ed.gov/about/bdscomm/list/mathpanel/8th-meeting/presentations/progressreports.html
The Conceptual Knowledge and Learning Processes are a must read!
Favorite quotes:
"International studies show that high achieving nations teach for mastery in a few topics, in comparison with our mile-wide-inch-deep curriculum. A coherent progression, with an emphasis on mastery of key topics, should become the norm in elementary and middle school curricula. There should be a de-emphasis on a spiral approach that continually revisits topics year after year without closure."
"The learning of algorithms to solve complex arithmetic problems is influenced by working memory, conceptual knowledge, degree of mastery of basic facts, and practice. Learning is most effective when practice using algorithms is combined with instruction on related concepts."
"Curricula Should provide sufficient time on task to ensure acquisition and long term retention of both conceptual and procedural knowledge."
Findings and Recommendations
• The Task Group affirms that algebra is the gateway to more advanced mathematics and to most postsecondary education.
• All schools and teachers must concentrate on providing a sound and strong mathematics education to all elementary and middle school students so that all of them can enroll and succeed in algebra.
• It is much more important for our students to be soundly prepared for algebra and then well taught in algebra than to study algebra at any particular grade level.
BTW on this problem I "saw" the five parts before I even finished reading the question. I always "see" numbers as multiple personalities, according to their factors...no bar model needed.
Catherine - I had the exact same experience as palisadesk here. I just...knew, immediately...that the whole thing revolved around 5, and from there, the arithmetic was easy.
I wouldn't say it had anything to do with bar models, since I don't really know what bar models are. Growing up, my math curriculum was anything but consistent, being as I'm an Army brat and all, and I moved around alot, especially in the early grades. That said, I do know that I did a lot of the timed worksheets, and that my dad made me practice my math facts. He would quiz me on them every morning. I remember sitting on the hamper in the bathroom while he shaved, and we would go through all the number families. Every time I paused on one he would turn away from the sink to look at me, and if I said, "I'm thinking!" (all whiny-like, you know how kids are), he would respond "You can't stop to think about it - you just have to know it."
This was said with the same degree of seriousness as, say, "Don't take drugs," and to this day, I know my "number families" by heart. (Also, I've never done drugs...)
I have lots of good stories about my dad contributing mightily to my math education, but this is probably the lesson that I value most.
Not sure where all this fits into the current discussion, but it seemed relevant when I started...
"You can't stop to think about it - you just have to know it."
Your dad was exactly correct, and you are reaping the benefits of that today. You can't be using working memory for basic arithmetical operations if you want to do anything complex. Your brain starts to do the wetware equivalent of disk thrashing and your net processing speed drops to about 10%* of normal.
It's bad when 2 hours of homework takes 20 hours to finish.
Drill keeps you alive in shipwrecks and advanced math classes.
* "10%" chosen because it sounds good.
Thanks so much for all the help and suggestions. I'm on it. Probably be back tomorrow with a ton of questions.
Catherine again you nailed it. Founders syndrome is precisely what I'm dealing with. I like "spaced repetition" - it's what I've been doing with a limited audience since last January and now I have the encouragement to keep it up and expand it!
Founders syndrome is precisely what I'm dealing with.
OK, yeah.
That is huge.
Having dealt with founder's syndrome twice, I probably know exactly what you have to do, and that is:
* spaced repetition
* always framed in the founder's terms
Always, always, always.
Singapore Math is a constructivist curriculum.
I would play up the fact that Leinwand is the author of the Singapore Math study.
He's the guy who said long division was damaging to kids.
If you take a group of unequally-speced points on a piece of paper, a human will always fit a better curve through the points than any computer program.
I wonder whether that's the same phenomenon Jeff Hawkins is talking about in On Intelligence (e.g. humans being able to recognize the curvy letters in Comments moderation boxes).
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