Posts
"Randolph Heights has been teaching the Core Knowledge curriculum since 1999, and was recognized as an official Core Knowledge school in December 2006. "
Well, that's nice. What does it actually mean? The CK site is clear that it does not dictate methods or materials use; Core Knowledge is about what students should learn, not how they should learn it.
What's the typical use of CK at Randolph Heights look like? Here's the 5th grade web page excerpt:
Everyday Math will help your child learn many new concepts such as Number Theory, Estimation, Geometry, Division, Fractions, Decimals, Percents, Exponents, Negative Numbers, Coordinates, Area, Algebra, and Probability.
We will using the Reader’s and Writer’s Workshop models for our Literacy Block. Your child will be involved with Guided Reading Groups, Book Groups, and reading independently. We will be learning reading strategies such as making connections, visualizing, making predictions, questioning, inferring, and summarizing through mini-lessons, authentic literature, and personal responses to their own reading.
Your child will be gathering seed ideas and developing ideas in a writer’s notebook. From these ideas, we will be writing and publishing Personal Narratives, Procedural Writing, Informational Writing, Writing Conventions, and Response to Literature. We will use Mentor texts to study author’s craft. Vocabulary will be developed with Word Wall and Spelling activities, and students will be practicing their handwriting skills with the Handwriting Without Tears curriculum."
Where does CK fit into this? The sixth grade web site makes it more clear:
OUR DAY
8:55 am -9:30am Homeroom & Morning Meeting
9:30 am- 11:10 am Readers’ & Writers’ Workshop
11:10 am- 12:00 pm Art, Phy-Ed or Science
12:05 pm - 12:35 pm Lunch
12:35 pm-1:45 pm Everyday Math
1:45 pm-2:05 pm Recess
2:05 pm-3:05 pm Core Knowledge Curriculum
3:05 pm-3:35 pm Social Studies
41 comments:
Allison, we have to connect!
Randolph Heights is merely, according to the Core Knowledge website, a "Friend of Core Knowledge" school. According the CK website:
"Friends of Core Knowledge are schools that are implementing Core Knowledge at any level. In fact, schools can become Friends of Core Knowledge on the first day of implementation. It's a way for us to welcome them into our growing network of schools. We require only that they complete a brief two-page profile that provides basic information about their school. We ask Friends of Core Knowledge Schools to update their profiles annually in order to remain in our database and on our website."
Thus...this doesn't mean much...although one is led to believe so, given that they bill themselves as a CK school! Tellingly, however, their website hedges: "In addition to solid language arts and math programs, students are taught the Core Knowledge Curriculum in the areas of American and world history, geography, music, and visual arts."
Phalen Lake Elementary is a true CK school(according to the CK website), but one wonders if this will continue now that they have been designated a Hmong magnet school.
About Everyday Math (when I asked) the CK people said this:
"While the Everyday Math program may not meet our highest standards, use of it is not grounds for rejection, as long as children are demonstrably learning a minimum percentage of the Core Knowledge Math content. The school itself, through assessments, should be able to evaluate the adequacy of this program."
Well, the CK website says their listed info on schools might not be up to date, and the RH website flat out says it's been "officially recognized", so I went with their claim--even more outrageous if it's false.
"2:05 pm-3:05 pm Core Knowledge Curriculum"
Well, if it isn't reading, writing, art, science, math, or social studies, what is it? Do they push all the factoids into that one hour?
Yes. Outrageous.
I was looking at Iowa's Model Core Curriculum Project
http://www.iowamodelcore.org/
I am struck by how vague it is. The usual mantras but nothing specific:
The intent of the Model Core Curriculum Project is twofold:
To ensure that all Iowa students have access to a rigorous and relevant curriculum to prepare them for success in post-secondary education and the emerging global economy, and
To provide a tool for Iowa educators to use to assure that essential subject matter is being taught and essential knowledge and skills are being learned.
[snip]
The Charge:
The charge given to the Project Lead Team was to define and collaborate with subcommittees in identifying the essential content and skills of a world-class core curriculum. The initial phase of model core curriculum work focused on the areas of literacy, mathematics, and science.
Important considerations in completing this work included the following:
The needs of students. These needs include not only legacy content like reading, writing, arithmetic, logical thinking, understanding the writings and ideas of the past, but also those Marc Prensky, author of "Digital Natives, Digital Immigrants" refers to as future content (2001). The "future" content is digital and technological, including software, hardware, robotics, nanotechnology, and genomics and the ethics, politics, sociology, and languages that come with them.
The needs of a changing workforce. According to the U.S. Department of Labor, jobs requiring science, engineering, and technical training will increase by 51 percent between 1998 and 2008, four times faster than overall job growth. By 2008, there will be six million job openings for scientists, engineers, and technicians.
The need to remain globally competitive. The sheer number of college graduates from other countries will change world dynamics. No longer do students from foreign countries have to come to the U.S. for higher education. No longer will the U.S. have enough engineers and scientists to fill the needs. Other countries will have the numbers that create new ideas, building companies that launch innovations, and produce goods wanted by the world.
The Outcome:
The most critical curriculum in literacy, mathematics, and science has been identified for Iowa educators. This is based upon a review of research and best practice literature; examination of national standards; and information from Iowa Testing Services, the National Assessment of Education Progress, ACT, and the College Board.
Iowa graduates who know these essential concepts and possess these essential skills should find success in any post-high school endeavor, whether that be in a classroom or the workplace.
The Model Core Curriculum encourages instructional practices that deeply engage students by requiring them to be active learners and critical thinkers who can apply their learning to new and unpredictable situations.
As a district determines the courses it will accept as part of the 4-3-3-3 graduation requirement, educators are encouraged to review local curriculum to ensure that these skills and concepts are part of the educational program of every graduate.
According to the U.S. Department of Labor, jobs requiring science, engineering, and technical training will increase by 51 percent between 1998 and 2008, four times faster than overall job growth. By 2008, there will be six million job openings for scientists, engineers, and technicians.
Tell this to all the scientists and software people I know that are looking for jobs right now!
And wow, what a boatload of educational jargon. This is not Core Knowledge, though, is it? It is just a "core" curriculum (for whatever that is worth).
Allison, as far as the Randolph Heights claim to be fully recognized as a Core Knowledge site, I guess you'd have to call them. As of Feb. 2007 this school was just a "Friend" so it might have changed--I do not know for sure. I'd be interested in what you find out, though, if you choose to pursue it!
Wow!
This daily SAT problem is a challenge:
To make an orange dye, 3 parts of red dye are mixed with 2 parts of yellow dye. To make a green dye, 2 parts of blue dye are mixed with 1 part of yellow dye. If equal amounts of green and orange are mixed, what is the proportion of yellow dye in the new mixture?
Unless I've made a mistake, the answer is 11/30 yellow dye in the final mix. The answer is the mean of 1/3 and 2/5.
Nice to see an answer with a moderately ugly fraction.
11/30 is pretty damn ugly if you ask me
I got the right answer but definitely not as elegantly as Doug. I saw a 5 part mixture for orange and a 3 part mixture for green For some reason I felt the need to obtain the lowest common multiple (15). If I combine a 15 part mixture of orange (which contains 6 parts yellow) and a 15 part mixture of green (which contains 5 parts yellow) I end up with a 30 part mixture having 11 parts yellow, hence 11/30.
Clearly I've spent too much time working Everyday Math problems.
I got 11/30, too.
Got it quickly & did it by finding lowest common multiple.
I've never done an Everyday Math problem in my life, so that's not the explanation.
Bar models definitely move you in this direction, but the real explanation in my case is probably the fact that we put C. through a couple of days of power-cramming for the ISEE test, which has "common factor" problems like these.
(That's what I call them; don't know if there's a real term.)
Math Olympiads seems to have lots of problems like these; they're always used for enrichment.
Here's a fun problem from JMAPA_923_REGENTS_BOOK_AT_RANDOM.pdf (fantastic resource - amazing)
7. During a recent winter, the ratio of deer to foxes was 7 to 3 in one country of New York state. If there were 210 foxes in the country, what was the number of deer?
I don't understand why Doug's approach works, which means that instead of writing my wildlife chapter I will now obsess over this problem until I do understand.
I think I'll go obsess in the shower.
Kill 2 birds with one stone.
OK, I've got it.
I think the problem is, as usual, that I don't see fractions as "numbers."
I see the numerator as a number-on-the-number-line.
I see the denominator as a number-on-the-number-line.
I don't see the fraction as a number-on-the-number-line.
In this problem I see the fraction as a ratio, not a number.
In this problem I see the fraction as a ratio, not a number.
It is a ratio. You end up with two ratios when you find the common multiple of 15: 6 parts yellow to 15 parts orange dye, and 5 parts yellow to 15 parts green dye. You have ratios of 6/15 and 5/15 for the two dyes of equal amounts. Combining the dyes yields 30 parts of new dye, of which 11 parts is yellow or 11/30. A common mistake teachers (and students) make is to talk about ratios in these kind of problems as fractions, which then leads kids to think that 6/15 + 5/15/ = 11/30, so sometimes we can add fractions by adding numerators and denominators. No. We are not adding fractions. We are calculating a new ratio.
Why is it hard for me to see Doug's way of solving the problem but easy for me to see Vicky's way, do you think?
Also: I'm confused by the ratio/fraction issue...if you have orange paint made up of 3 parts red and 2 parts yellow, then 2/5 of the orange paint is yellow.
Is 2/5 a fraction or a ratio or both?
I'm thinking of 2/5 as a fraction in this case, which I think is a number-on-the-number-line... ???
I better go write a page on wildlife before my head explodes.
Yes, 2/5 can be either a ratio or a fraction (and even a division problem).
In the original problem of red, yellow, and orange paint, there is a ratio and a fraction. The ratio is 3 to 2, red paint to yellow paint, and is written 3/2. (Or 2 to 3, yellow paint to red paint, and written 2/3. Neither of these are fractions, though written as a fraction.
Now, when mixed you have 2 parts yellow out of a total of 5 parts, written 2/5. That is a fraction. There is even the possibility that, as Barry points out, the 2/5 fraction will become a ratio 2 to 5.
It is so critical for the teacher to teach the importance of looking at the way the question is worded. This needs to be "intentional" teaching. Students (and teachers) must be able to recognize what is actually going on in the problem.
A teacher must tell the students: "OK, I'm going to show you how you can know if it's a fraction of the whole or if it is a ratio." Then he/she must point out key words to look for, must demonstrate, must give students opportunities to pick the key words and then solve several problems under your guidance. Teacher-guided activities should be a part of every lesson. And it should be repeated several days in a row.
In my 5th grade class, my students are just realizing that the appearance of a fraction doesn't mean it is a fraction. They are getting it!!!
Also: I'm confused by the ratio/fraction issue...if you have orange paint made up of 3 parts red and 2 parts yellow, then 2/5 of the orange paint is yellow. Is 2/5 a fraction or a ratio or both?
It can be thought of as both. If you operate with 2/5 as a ratio then you solve it as Vicky did.
If you want to solve the problem with 2/5 and 1/3 as fractions, then we can let x = the respective amount of green and orange paint that is to be mixed together. (I.e., it said equal amounts of green and orange paint are mixed, so x is the amount of each of the paints).
Then we know that the amount of yellow paint will be (2/5)*x +(1/3)*x, or 2x/5 + x/3. Since we want to know the fraction of yellow paint to the entire mixture, we divide 2x/5 + x/3 by the amount of the entire mixture which is x + x = 2x (orange + green paint).
Now we have (6x + 5x)/(15*2x). Simplifying yields 11x/30x. The x's cancel, leaving 11/30.
OR: Taking it in slow motion:
(6x + 5x)/(15*2x) = x(6 + 5)/15*2x. Cancelling the x's yields (6+5)/15*2, which is (6/15 + 5/15)/2, or (2/5 + 1/3)/2. Otherwise known as the mean of 2/5 + 1/3.
I could also benefit from more clarity about the fraction/ratio distinction.
The way I understand it, a fraction is limited to expressing a part-to-whole relationship, whereby the part is always indicated by the numerator and the whole by the denominator.
In contrast, ratios can express additional relationships, e.g. part to part, whole to part.
So I'll go out on a limb and claim that in the dye problem it's possible to find a fraction and ratio in this case: If a mixture consists of five cups (both red and yellow dye) of which two cups are yellow dye, then 2/5 is both a fraction and ratio.
Upon further thought it seems that is wrong. A fraction can't compare apples and oranges at it were (red and yellow dye). The two in the 2/5above can only refer to two cups of the mixture, not a component of the mixture. So it can only be a ratio.
There is more to this than meets the eye.
I found a few practice problems on this distinction:
Directions: Analyze each of the following statements to
determine whether a ratio concept or fraction concept is
present. Write down the reasoning behind your answers in
the space provided.
1. There are nine women for every two men in this class.
Circle One: Ratio Fraction Both
2. Two out of every five students in this class plan to be
middle school teachers.
Circle One: Ratio Fraction Both
3. Brand A orange juice costs 7 cents per ounce while
Brand B orange juice only costs 6.5 cents per ounce.
Circle One: Ratio Fraction Both
4. One fourth of the marbles in the jar are blue.
Circle One: Ratio Fraction Both
5. My average speed while driving to work this morning
was 35 mph.
Circle One: Ratio Fraction Both
6. Jane drank ¾ cup of milk.
Circle One: Ratio Fraction Both
7. According to the Kool-Aid directions, for each quart
of water you should add one scoop of Kool-Aid and 2
scoops of sugar.
Circle One: Ratio Fraction Both
The two in the 2/5 above can only refer to two cups of the mixture, not a component of the mixture. So it can only be a ratio.
Sounds like Instructivist is correct here.
So the Jan and Dean song "Surf City" (2 girls for every boy) was really a song about ratios.
[Now, when mixed you have 2 parts yellow out of a total of 5 parts, written 2/5. That is a fraction. There is even the possibility that, as Barry points out, the 2/5 fraction will become a ratio 2 to 5.]
It's definitely a ratio. I am wracking my brain to figure out if it is also a fraction. I tend to think it is not. In a fraction, the part has to be of the same substance or material or nature or quality as the whole.
This has the potential of becoming an ontological or scholastic debate a la pins and heads and angels.
A ratio problem I like giving is to make as many ratios as possible out of a figure with six squares of which four are shaded. Answer: six ratios.
Google is constructivist. It doesn't like minds-on activities. If you search for minds-on activities, it will suggest hands-on activities.
Try searching for:
why use hands on minds on activities in any content class
to see what I mean.
I was seeing it this way:
3 parts red mixed with 3 parts yellow is a ratio: 3/2
2/5 of the orange dye is yellow, and in this case 2/5 is a fraction
BUT even though I switched from ratios to fractions (or at least I think I did) I still couldn't come up with Doug's way of adding the two fractions (1/3 and 2/5) and dividing by 2 to get their average.
I had to find a common denominator of 15 and then find out what fraction yellow would be of the whole.
(hmm...now I'm saying "fraction" to mean "proportion"-----)
This is why I say I think I'm not seeing fractions as numbers-on-the-number-line: I don't automatically see fractions as quantities you could average....
I am now utterly confused.
I have been reading KTM for a a while as a way to keep up with what other people are doing with math instruction. Let me chime in on the question of what 2/5 is. Two-fifths is a number. It does not matter whether certain people happen to be thinking about paint or apples at any given time; 2/5 existed before we did, and it has been a number all along.
A ratio is a number with an interpretaton attached to it. You can solve the paint problem thinking in terms of fractions or ratios, but the nature of 2/5 is undisturbed. I don't think this is a pedantic point; it seems to be at the heart of a lot of the difficulty people have with using arithmetic.
One thing I would exhort people to do with this kinds of problems is explain how they work them and not just report an answer. I know that on SAT-type tests they just want the answer, but before then, learning, you need to say what you are doing. As has been pointed out, to say that you "combine" your 15 parts orange containing 6 parts yellow with 15 parts green containing 5 parts yellow can make it seem like you are adding numerator and denominator (instead of adding paint, which is what was meant) and thus a student can come away thinking:
6/15 + 5/15 "=" 11/30
Here I quote the equal sign so you know I don't mean it! Make it clear that you mean you add the two paints, 15 parts each to get 30 parts, of which 6 + 5 parts are yellow.
Kids have got to be able deal with it on the nuts and bolts level and not be expected to learn to match certain computations with certain keywords. They need to learn to solve problems, not suss out the textbook writers.
For what it's worth, a very straightforward way to do this is to say that 2/5 of my orange paint is yellow; 1/3 of my green paint is yellow; and if I combine 1 part of each, I have 2/5 part + 1/3 part = 11/15 part yellow in a total of two parts, which means that the fraction of my new paint that is yellow is (11/15)/2 = 11/30 .
(This is exactly what Barry Garelick did, except that I am choosing to work in units (parts) so that I can set x = 1.)
[I am now utterly confused.]
It's like going from pilpul to bilbul.
I might be adding to the confusion, but consider this definition:
rational numbers: set of numbers formed by the ratio (i.e., fraction) of two Integers where zero in the denominator is not permitted (division by zero is undefined); frequently represented as Q = {a/b | a ∈ I, b ∈ I, b ≠ 0}
Hypatia
Thanks so much for all of this!
YET ANOTHER THREAD I NEED TO "PULL UP FRONT" OR SOMEHOW FLAG.... (actually, I know how to do that)
bky & hypatia: I think you're both new around here, so I'll mention that I've been trying to reteach myself math for about 3 years now partly so I can remediate and/or supplement my son's math ed and partly because I got interested in math as a result.
The other aspect of this project that interests me is "pedagogical content knowledge": what is it that confuses students and why?
I've been using myself a bit the way Piaget used his kids: as a subject.
What trips me up?
How come?
Anyway, in the interests of that project, here's where I go off the rails:
if I combine 1 part of each, I have 2/5 part + 1/3 part = 11/15 part yellow in a total of two parts
I have a great deal of trouble "seeing" this. ("seeing," "understanding," "grasping" -- any such word will do)
It's the same problem I had grasping that Doug's averaging approach was a good way to go about it.
I think that I've always had trouble with the idea of "1 part + 1 part" --- when I first encountered a mixture problem (as an adult, just a couple of years ago) I was flummoxed.
I think this is why I had to work with two batches of paint, each with a size of 15 units.
That made the 2/5 not 2/5 of 1, but 2/5 of 15.
In other words, I made a kind of u-turn; I started with 2/5 and 1/3; I moved to 2/5 of 15 (6) and 1/3 of 15 (5); I added 6 and 5 and got 11; then I created a new fraction (or ratio).
I did know there was "a way" just to work directly with 2/5 and 1/3, but I couldn't think what it was.
Actually, assuming I've described this process right, I think for the first time I see much more clearly why the circuit-through-whole-numbers is unnecessary -- THANK YOU!
Two-fifths is a number. It does not matter whether certain people happen to be thinking about paint or apples at any given time; 2/5 existed before we did, and it has been a number all along.
I love this.
Two-fifths is a number. It does not matter whether certain people happen to be thinking about paint or apples at any given time; 2/5 existed before we did, and it has been a number all along.
I love this.
A ratio is a number with an interpretaton attached to it. You can solve the paint problem thinking in terms of fractions or ratios, but the nature of 2/5 is undisturbed. I don't think this is a pedantic point; it seems to be at the heart of a lot of the difficulty people have with using arithmetic.
Well, this is what I would prefer to think - and I had thought it was Wu's point (though I've only read a few pages of his paper).
But the fact is that I can't resolve any of these issues for myself.
As much as I've learned (or think I've learned)....I'm still in the novice category.
A ratio is a number with an interpretaton attached to it. You can solve the paint problem thinking in terms of fractions or ratios, but the nature of 2/5 is undisturbed. I don't think this is a pedantic point; it seems to be at the heart of a lot of the difficulty people have with using arithmetic.
Well, this is what I would prefer to think - and I had thought it was Wu's point (though I've only read a few pages of his paper).
But the fact is that I can't resolve any of these issues for myself.
As much as I've learned (or think I've learned)....I'm still in the novice category.
hypatia - works for me!
Here's a brief description of five different ways that middle schoolers can interpret "fractions'.
http://io.uwinnipeg.ca/~jameis/New%20Pages/MYR21.html
Hypatia
"But the fact is that I can't resolve any of these issues for myself."
That's a very good point, Catherine. And the fact is, students can't either. They need direct instruction and maybe their teachers do, too. I know I did. Being a good student in math, doesn't make you a good teacher of math. I was fortunate to have worked with very good, experienced teachers when I first started teaching.
I'm a big fan of Wu's. I keep an icon of his home page at the top of my IE toolbar. From a pedagogical point of view, I particularly like his interpretation of "What is Math Education?"
http://math.berkeley.edu/~wu/C49.pdf
Hypatia
A visual way of looking at Barry's algebraic method of solving this mixture problem might be a way I call the beaker/beaker method I have used in first year algebra mixture problems. I didn't discover the method - I don't know who did or I would give credit.
It's a little hard to explain here, but looks sort of like this:
Draw three "beakers" (I'll use u's)
u + u = U
Inside the beaker enter the amounts; underneath, the labels (can be percents or fractional parts of the contents.)
In this problem the small beakers each have M inside - the large beaker has 2M. (because the actual amounts aren't necessary, only the fact that the mixture amounts are equal)
Underneath the first small beaker is 2/5; the second has 1/3. The large beaker has an x underneath.
The equation becomes
(2/5)M + (1/3)M = 2M * x
(11/15)M divided by 2M = x
The M's will cancel; x = 11/30
The method works for all mixture problems, although sometimes we're putting metal alloys into those beakers :)
It's easier to draw than to explain the drawing in this format.
Sorry.
Hypatia
Sorry for the 'Anonymous'. My Google Account wasn't working before - now it is!
What does the x stand for in the equation?
Let x =
Sorry, Heron. (I do like your name!)
You are absolutely correct. I didn't identify the unknown. From the last sentence in the original problem:
If equal amounts of green and orange are mixed, what is the proportion of yellow dye in the new mixture?
Let x = the 'proportion' of yellow dye in the new mixture.
Added note: I really would prefer a synomyn for the meaning of the word 'proportion' used here. It just adds to the confusion in a ratio problem.
I like the way you manipulate numbers and find unkowns, Hypatia. I am currently experimenting with steam and rotating balls, a contraption called an aeolipile. In my dreams I see it pulling wagons on rails to carry people and goods. Your way of finding unkowns can help me find the many unkowns in my work.
Beware of religious mobs!
Oh, it is you! I wasn't certain, Heron being a rather common name. But thank you for your kind words. I am familiar with your work and an admirer. -- I'm afraid your warning comes a bit late! But it's so nice that we can share these warm thoughts on KTM so many years later.
Post a Comment