> Even more worrisome is the plan to replace an algebra-centered curriculum with one emphasizing data science, described as “21st century math.”
I've taught all sorts of subjects ranging from Linear/Nonlinear Programming to Stats to Economic Theory at levels ranging from Intro to Ph.D.
I have no idea how one does "data science" (something like very basic Intro Stats at this level, I presume) without a good grounding in algebra in a way that has any meaning.
Sure, it is possible to teach the students to add 10 numbers and divide by 10 on a calculator and say they did "Data Schmeince" or some such. Or maybe they look at pictures and try to discern associations.
Does anyone know what this "data science" approach entails?
Your second link never says the last two quotes. In fact, seems to me to support showing your work
> Today, getting the correct answer to a math problem is still important. But teachers are also showing students that being “good at math” means applying that correct answer in service of a larger goal ...
> In math classes, an answer becomes a launching point for conversations about how and why people solved the problem in different ways. As we expand our understandings, teachers often welcome the problem-solving strategies that students bring from their homes and cultures in addition to familiar formal U.S. strategies. This means teachers must make space for students to explore why solutions work.
> To accomplish this, we encourage teachers to focus on math problems that invite curiosity and discussion ...
Quote are from the first link only. I agree second link does not seem as bad.
If the goal is on letting the students use a different (procedure/algorithm) that they learned a home instead of using the common core algorithm then I support it 100%
Take the example of a geometry problem in Chapter Two:
“A farmer has 36 individual fences, each measuring one meter in length … the farmer wants to put them together to make the biggest possible area.”
This problem is completely inappropriate: It cannot be solved without very sophisticated math, rarely studied even in college.
Really? It's been a while since I've had high school math, new or otherwise, but how hard can it be? Obviously you arrange the fences in a 36-sided polygon that's as close as possible to a circle.
That can be represented as 36 pointy isosceles triangles, each with the pointy bit having 10 degrees angle and the base having one unit of length. High school trigonometry to work out the area of one of these, multiply by 36, and done. Easily within the skills of the high school curriculum I had.
You are a missing a proof of max area. Exactly what author complains about. According her (and not only) teaching math problems just to memorize is not very useful.
Its pointless in the context of understanding. The capital is the capital due to historical reasons/declared by the government of the country. On the other hand the circle has the maximum area not because some one declared it but because its deducible from base principles. Teaching children to learn these by rote memory prevents them from learning them from first principles. Which is the point the author is trying to make. Her point is Maths != trivia and even if Maths can be learnt by memory like trivia, it serves no purpose because the same problem might comes back in a different form and then the students wont be in a position to solve for it. For example, Draw the shape that has maximum area when all fence links are joined at and angle that's between 40 and 80 degree (all these problems have practical relevance for example while maximizing the usable area for a odd shaped plot).
Fields Medalist Terence Tao famously said: "There is more to mathematics than rigour and proofs".[1]
Look, I love proofs. But realistically, you can't expect most people to ever learn multidimensional calculus before you teach them the basic fact that a circle maximises the area: something that can be intuited by drawing polygons of increasing degree. Discovering this fact through inductive reasoning, even if mathematically unsound, is still a good teaching method (and will stick well in memory) and is not just "memorising trivia".
Incidentally, working inductively is also crucially important for working mathematicians, it's just that the insights gained by doing that need to be verified (and refined) by constructing a rigorous proof; but working towards a proof when you don't even know what you want to prove yet is often difficult, it might work in an optimisation case like this, but it won't in many other areas of mathematics.
Also, if you compare how other mathematics is taught in high school (and in the US, even in undergraduate university classes), it's not like there is a heavy emphasis on proof either. You're not deriving calculus from the real number axioms, you're handwaving the existence of derivatives and integral and the formulas to compute them. I'll be the first to say that knowing how this actually works behind the scenes is fascinating, but realistically, that's not what most people will need. I'm not sure why kids discovering that circles maximise the area through inductive reasoning is worse somehow than them memorising calculus rules and applying them to find the result.
I feel we are stating the same thing but may be using different terms:
> something that can be intuited by drawing polygons of increasing degree
That's not same as just remembering that the circle has maximum area. Its a good starting point to prove it in a more rigorous way.
I guess it depends on the approach to solving the problem (which as per the article is stated plainly as 'Circle' without going through the mental exercise above).
Yeah I don't know what the curriculum actually argues or how it would be put into practice (which are two quite different things anyway), but I just felt like the justification from the article - "you can't solve this problem in high school because it involves too many advanced concepts" - not particularly convincing.
Yo can show it physically by inflating a balloon or pushing things in a weak cardboard box (or marbles on a table and a cord, if you insist on circles). Intuition building is very useful to build concepts, even perhaps more than proofs. See https://betterexplained.com/articles/adept-method/ for a discussion of this.
You can also introduce the idea of problem decomposition: first identify target shape, then argue how to approach it with 36 fences, then try to derive the formula for the area.
At the end of the day, a good teacher will try to do their best with the curriculum, and a bad one will find a way to proselitize wokeness if they chisse to do so; in my opinion the good direction is to empower teachers with resources and curriculum discretion, and in turn require them better results.
I don't really understand the author's comment that "it cannot be solved without very sophisticated math, rarely studied even in college". That would be true if the lengths of the fences were arbitrary or the fences were potentially curved, but it's relatively straightforward to develop a proof of max area given that all the fences are the same length.
Step 1 would be to demonstrate that the shape with max area has to be convex. This is pretty straightforward, and a pretty decent example for advanced middle school kids.
Once you've got convexity proven, then the total area covered becomes the sum of triangular wedges. There are a variety of ways here to arrive at the maximum, either by a sequence of replacements (probably most appropriate for advanced middle schoolers) or a formulation in terms of Langrange multipliers (for advanced high school students).
I think a lot of students would find the proof pretty cool.
It's fairly obvious: first, it's pretty clear that you need to connect all of the fences together to form a single continuous fence (no branches). Second, if you think of it physically, you can take any configuration and pour more "space" (or animals or whatever it's containing) into the enclosed area so that they apply pressure. It's pretty clear that they'll all push outward, and thus that they'll tend towards a circular configuration as you increase the area enclosed.
But it's obvious, and only obvious. I have no clue how to prove it. I could prove that if you're close to a circular shape, that shifting to more circular will increase the area. But that's not a complete proof.
Still, I have to say that this seems obvious enough to me that the difficulty in proving it doesn't really bother me too much. I would have to read the actual proposed standards to know whether these complaints are cherry-picking.
There's long been a battle between conceptual understanding and rote learning, and I lean towards conceptual understand and yet don't want it to go too far -- you need practice in order to cement things and to provide an opportunity to cover all the little subtle gaps that you won't fill in with a top-down approach. A student has to work actual concrete problems no matter how well they grasp the general concepts.
You're getting downvoted because the same thing happens every time someone wants to fix mathematics education: People who learned it "the old way" insist that there can be no other way.
Somehow people think that applying calculus rules is "real maths" and "experimenting with shapes and intuiting a result" is not, but that's not true, and it's especially not true when those calculus rules were arbitrarily memorised at some point (because most people never derive calculus from the real number axioms).
I think it's because the circle is intuitive to you, but other likely commonly guessed shapes are probably rectangles and stars.
So like, from there you're faced with "well, OK given any polygon how do I calculate the area", and then "without measuring the area, how can I say that one polygon has a greater area than another"
All of that is pretty tricky IMO. I think if you took your average SWE and asked them to work this out, they'd either google it or futz around in a buffer for a couple hours. To me that means "too hard for your average 15-16 year old", although maybe the US has super lowered my standards.
It's and old joke but: You can enclose a larger area by having the farmer take 4 pieces of fencing, constructing a square around himself, and defining him to be outside the fence.
Lori, a high school geometry teacher, introduces a problem to students. Lori explains that a farmer has 36 individual fences, each measuring one meter in length, and that the farmer wants to put them together to make the biggest possible area. Lori takes time to ask her students about their knowledge of farming, making reference to California’s role in the production of fruit, vegetables, and livestock. The students engage in an animated discussion about farms and the reasons a farmer may want a fenced area. While some of Lori’s long-term English learners show fluency with social/conversational English, she knows some will be challenged by forthcoming disciplinary literacy tasks. To support meaningful engagement in increasingly rigorous course work, she ensures images of all regular and irregular shapes are posted and labeled on the board, along with an optional sentence frame, “The fence should be arranged in a [blank] shape because [blank].” These support instruction when Lori asks students what shapes they think the fences could be arranged to form. Students suggest a rectangle, triangle, or square. With each response, Lori reinforces the word with the shape by pointing at the image of the shapes. When she asks, “How about a pentagon?” she reminds students of the optional sentence frame as they craft their response. Lori asks, the students think about this and talk about it as mathematicians. Lori asked them whether they want to make irregular shapes allowable or not.
After some discussion, Lori asks the students to think about the biggest possible area that the fences can make. Some students begin by investigating different sizes of rectangles and squares, some plot graphs to investigate how areas change with different side lengths.
Susan works alone, investigating hexagons––she works out the area of a regular hexagon by dividing it into six triangles and she has drawn one of the triangles separately. She tells Lori that she knew that the angle at the top of each triangle must be 60 degrees, so she could draw the triangles exactly to scale using compasses and find the area by measuring the height.
Niko has found that the biggest area for a rectangle with perimeter 36 is a 9 x 9 square—which gave him the idea that shapes with equal sides may give bigger areas and he started to think about equilateral triangles. Niko was about to draw an equilateral triangle when he was distracted by Jaden who told him to forget triangles, he had found that the shape with the largest area made of 36 fences was a 36-sided shape. Jaden suggested to Niko that he find the area of a 36-sided shape too and he leant across the table excitedly, explaining how to do this. He explained that you divide the 36-sided shape into triangles and all of the triangles must have a one-meter base, Niko joined in saying, “Yes, and their angles must be 10 degrees!” Jaden said, “Yes, and to work it out we need tangent ratios which Lori has just explained to me.”
Jaden and Niko move closer together, incorporating ideas from trigonometry, to calculate the area.
As the class progressed many students started using trigonometry, some students were shown the ideas by Lori, some by other students. The students were excited to learn about trig ratios as they enabled them to go further in their investigations, they made sense to them in the context of a real problem, and the methods were useful to them. In later activities the students revisited their knowledge of trigonometry and used them to solve...
What's with the expectation that students must discover by themselves that a regular n-polygon maximizes the area and thus solves the problem as stated? Is this some super grotesque bizarro version of the https://en.wikipedia.org/wiki/Moore_method ? It certainly sticks to the adage that "the student is taught best who is told the least", but that's not generally thought of as a very "inclusive" teaching method. It's letting the students sink or swim, and most of them will just sink to the bottom and never catch up.
There are aspects of the Oakland workbook that I could agree with ("Contrived word problems are valued over the math
in students' lived experiences," etc.)
But I'm at a complete loss for how this is an example of white supremacy. As a black guy who like math, I think I would hate having this cultural viewpoint forced on me in my otherwise race-agnostic class material. I'm completely turned off by the framing, and therefore have little to no trust in the writers.
My mom thought American math was already dumb and watered down in the 1990s when we came here from Bangladesh. Apparently it just keeps getting worse.
The sad thing about all this is that the elites will just bypass this damage in the system. We just enrolled our 8 year old daughter in outside of school math tutoring. This will just hurt middle class kids who will get a watered down education.
If read an article that parents in the US sends their children into "russian-math-classes" after school, but maybe gender/social studies are more important atm in the US ;)
Public schools in USA are more about babysitting vs teaching kids to critically think and learn.
I don't see how people in the USA argue over specific spots of land/housing stock just to send their kid to a "good" school when the standards are so bad across the board.
I went to public but got math tutoring from an Indian guy. It costs money but it’s not some outrageous expense. The best tutors are educated but underemployed Indians and Russians. My uncle was an electrical engineer “back in old country.” His degree didn’t carry over in Canada, so he does math tutoring.
> Another example is a 5th-grade lesson, where a problem with Ms. Hernandez knitting a scarf for her grandson is labeled noninclusive and then modified to one where Mr. Hernandez is knitting a scarf because “guys can knit too.” Unless the authors intend to imply that women should never knit, it is a crime against fundamental logic.
What the hell is up with that weird example?
The link sends me to a YouTube video completely devoid of context and in very poor quality. I don't know who made that video and in what way (if in any way) it is related to this proposed curriculum. If it is, the article should make this connection clear.
Moreover, as much as it may be pointless to replace "Mrs. Hernandez" with "Mr. Hernandez" in that question, it's certainly not "defying logic", it's just arbitrarily redefining the (irrelevant) context for a maths problem. I find people who complain about such a change at least as annoying as the ones arguing incessantly for these kinds of changes.
As the article is otherwise very light on details about what's wrong with the curriculum (the example with the fence is the only concrete thing that is being referred to that is not ridiculous, and even that is arguable), this just looks to me like some opinion piece that tries to fan the flames of some culture war.
It's quite possible that there are problems with this proposed curriculum (I don't live in California, I know nothing about it), but this article is not making a very clear case.
> Even more worrisome is the plan to replace an algebra-centered curriculum with one emphasizing data science
As someone who never jumped on or even cared to investigate the whole “data science” hype train, what does that look like at a primary/secondary education level?
37 comments
[ 2.7 ms ] story [ 90.5 ms ] threadI've taught all sorts of subjects ranging from Linear/Nonlinear Programming to Stats to Economic Theory at levels ranging from Intro to Ph.D.
I have no idea how one does "data science" (something like very basic Intro Stats at this level, I presume) without a good grounding in algebra in a way that has any meaning.
Sure, it is possible to teach the students to add 10 numbers and divide by 10 on a calculator and say they did "Data Schmeince" or some such. Or maybe they look at pictures and try to discern associations.
Does anyone know what this "data science" approach entails?
Update: Drafts are here in Word format: https://www.cde.ca.gov/ci/ma/cf/
- High School 52
- Data Science for Equity and Inclusion 53
- Data for All Students: Living in a World Overloaded with Information 56
- Advanced high school data science 66
- Content Learning Outcomes 74
- Sample Courses 79
- High School Tools and Resources 81
Where's the math?
They seem to think that math is not "inclusive" and the solution is to teach less math.
its very similar to those 2 documents:
[1]:https://equitablemath.org/wp-content/uploads/sites/2/2020/11...
"The concept of mathematics being purely objective is unequivocally false" is on p. 65.
"The focus is on getting the correct answer is White Supremacy"
"requiring student to 'show their work is racist"
[2] https://www.oregonlive.com/opinion/2021/03/opinion-antiracis...
> Today, getting the correct answer to a math problem is still important. But teachers are also showing students that being “good at math” means applying that correct answer in service of a larger goal ...
> In math classes, an answer becomes a launching point for conversations about how and why people solved the problem in different ways. As we expand our understandings, teachers often welcome the problem-solving strategies that students bring from their homes and cultures in addition to familiar formal U.S. strategies. This means teachers must make space for students to explore why solutions work.
> To accomplish this, we encourage teachers to focus on math problems that invite curiosity and discussion ...
Take the example of a geometry problem in Chapter Two:
“A farmer has 36 individual fences, each measuring one meter in length … the farmer wants to put them together to make the biggest possible area.”
This problem is completely inappropriate: It cannot be solved without very sophisticated math, rarely studied even in college.
Really? It's been a while since I've had high school math, new or otherwise, but how hard can it be? Obviously you arrange the fences in a 36-sided polygon that's as close as possible to a circle.
That can be represented as 36 pointy isosceles triangles, each with the pointy bit having 10 degrees angle and the base having one unit of length. High school trigonometry to work out the area of one of these, multiply by 36, and done. Easily within the skills of the high school curriculum I had.
What am I missing?
To truly understand it, students have to go beyond and those concepts aren't available for them yet.
Similarly, remembering the capital of a country is useful when you hear that capital mentioned in a discussion.
Both of these can be scaffolding that enables you to learn the deeper concepts.
A far greater problem is students faced with nobody willing or able to teach the deeper concepts when they want to know about them.
Look, I love proofs. But realistically, you can't expect most people to ever learn multidimensional calculus before you teach them the basic fact that a circle maximises the area: something that can be intuited by drawing polygons of increasing degree. Discovering this fact through inductive reasoning, even if mathematically unsound, is still a good teaching method (and will stick well in memory) and is not just "memorising trivia".
Incidentally, working inductively is also crucially important for working mathematicians, it's just that the insights gained by doing that need to be verified (and refined) by constructing a rigorous proof; but working towards a proof when you don't even know what you want to prove yet is often difficult, it might work in an optimisation case like this, but it won't in many other areas of mathematics.
Also, if you compare how other mathematics is taught in high school (and in the US, even in undergraduate university classes), it's not like there is a heavy emphasis on proof either. You're not deriving calculus from the real number axioms, you're handwaving the existence of derivatives and integral and the formulas to compute them. I'll be the first to say that knowing how this actually works behind the scenes is fascinating, but realistically, that's not what most people will need. I'm not sure why kids discovering that circles maximise the area through inductive reasoning is worse somehow than them memorising calculus rules and applying them to find the result.
[1]: https://terrytao.wordpress.com/career-advice/theres-more-to-...
> something that can be intuited by drawing polygons of increasing degree
That's not same as just remembering that the circle has maximum area. Its a good starting point to prove it in a more rigorous way.
I guess it depends on the approach to solving the problem (which as per the article is stated plainly as 'Circle' without going through the mental exercise above).
You can also introduce the idea of problem decomposition: first identify target shape, then argue how to approach it with 36 fences, then try to derive the formula for the area.
At the end of the day, a good teacher will try to do their best with the curriculum, and a bad one will find a way to proselitize wokeness if they chisse to do so; in my opinion the good direction is to empower teachers with resources and curriculum discretion, and in turn require them better results.
Does it? I have never heard of evidence of this.
Step 1 would be to demonstrate that the shape with max area has to be convex. This is pretty straightforward, and a pretty decent example for advanced middle school kids.
Once you've got convexity proven, then the total area covered becomes the sum of triangular wedges. There are a variety of ways here to arrive at the maximum, either by a sequence of replacements (probably most appropriate for advanced middle schoolers) or a formulation in terms of Langrange multipliers (for advanced high school students).
I think a lot of students would find the proof pretty cool.
But it's obvious, and only obvious. I have no clue how to prove it. I could prove that if you're close to a circular shape, that shifting to more circular will increase the area. But that's not a complete proof.
Still, I have to say that this seems obvious enough to me that the difficulty in proving it doesn't really bother me too much. I would have to read the actual proposed standards to know whether these complaints are cherry-picking.
There's long been a battle between conceptual understanding and rote learning, and I lean towards conceptual understand and yet don't want it to go too far -- you need practice in order to cement things and to provide an opportunity to cover all the little subtle gaps that you won't fill in with a top-down approach. A student has to work actual concrete problems no matter how well they grasp the general concepts.
Somehow people think that applying calculus rules is "real maths" and "experimenting with shapes and intuiting a result" is not, but that's not true, and it's especially not true when those calculus rules were arbitrarily memorised at some point (because most people never derive calculus from the real number axioms).
So like, from there you're faced with "well, OK given any polygon how do I calculate the area", and then "without measuring the area, how can I say that one polygon has a greater area than another"
All of that is pretty tricky IMO. I think if you took your average SWE and asked them to work this out, they'd either google it or futz around in a buffer for a couple hours. To me that means "too hard for your average 15-16 year old", although maybe the US has super lowered my standards.
Professors and professionals at Berkeley/Stanford and many major universities have written an open letter highlighting concerns to the California Governor: https://www.independent.org/news/article.asp?id=13658
Apparently, this was heard and they’re postponed the proposal for a vote to next year: https://www.cde.ca.gov/be/ag/ag/yr21/agenda202107.asp
I’ll leave my personal opinion out of this one, make what you will.
Here's the full vignette:
Vignette–36 Fences
Lori, a high school geometry teacher, introduces a problem to students. Lori explains that a farmer has 36 individual fences, each measuring one meter in length, and that the farmer wants to put them together to make the biggest possible area. Lori takes time to ask her students about their knowledge of farming, making reference to California’s role in the production of fruit, vegetables, and livestock. The students engage in an animated discussion about farms and the reasons a farmer may want a fenced area. While some of Lori’s long-term English learners show fluency with social/conversational English, she knows some will be challenged by forthcoming disciplinary literacy tasks. To support meaningful engagement in increasingly rigorous course work, she ensures images of all regular and irregular shapes are posted and labeled on the board, along with an optional sentence frame, “The fence should be arranged in a [blank] shape because [blank].” These support instruction when Lori asks students what shapes they think the fences could be arranged to form. Students suggest a rectangle, triangle, or square. With each response, Lori reinforces the word with the shape by pointing at the image of the shapes. When she asks, “How about a pentagon?” she reminds students of the optional sentence frame as they craft their response. Lori asks, the students think about this and talk about it as mathematicians. Lori asked them whether they want to make irregular shapes allowable or not.
After some discussion, Lori asks the students to think about the biggest possible area that the fences can make. Some students begin by investigating different sizes of rectangles and squares, some plot graphs to investigate how areas change with different side lengths.
Susan works alone, investigating hexagons––she works out the area of a regular hexagon by dividing it into six triangles and she has drawn one of the triangles separately. She tells Lori that she knew that the angle at the top of each triangle must be 60 degrees, so she could draw the triangles exactly to scale using compasses and find the area by measuring the height.
Niko has found that the biggest area for a rectangle with perimeter 36 is a 9 x 9 square—which gave him the idea that shapes with equal sides may give bigger areas and he started to think about equilateral triangles. Niko was about to draw an equilateral triangle when he was distracted by Jaden who told him to forget triangles, he had found that the shape with the largest area made of 36 fences was a 36-sided shape. Jaden suggested to Niko that he find the area of a 36-sided shape too and he leant across the table excitedly, explaining how to do this. He explained that you divide the 36-sided shape into triangles and all of the triangles must have a one-meter base, Niko joined in saying, “Yes, and their angles must be 10 degrees!” Jaden said, “Yes, and to work it out we need tangent ratios which Lori has just explained to me.” Jaden and Niko move closer together, incorporating ideas from trigonometry, to calculate the area.
As the class progressed many students started using trigonometry, some students were shown the ideas by Lori, some by other students. The students were excited to learn about trig ratios as they enabled them to go further in their investigations, they made sense to them in the context of a real problem, and the methods were useful to them. In later activities the students revisited their knowledge of trigonometry and used them to solve...
But I'm at a complete loss for how this is an example of white supremacy. As a black guy who like math, I think I would hate having this cultural viewpoint forced on me in my otherwise race-agnostic class material. I'm completely turned off by the framing, and therefore have little to no trust in the writers.
The sad thing about all this is that the elites will just bypass this damage in the system. We just enrolled our 8 year old daughter in outside of school math tutoring. This will just hurt middle class kids who will get a watered down education.
Public schools in USA are more about babysitting vs teaching kids to critically think and learn.
I don't see how people in the USA argue over specific spots of land/housing stock just to send their kid to a "good" school when the standards are so bad across the board.
https://www.youtube.com/watch?v=UIKGV2cTgqA
(To be honest, "new math" makes sense to me. But whatever, for us oldies this is as obligatory as a XKCD)
He had no idea about where the world would go with "woke", though.
What the hell is up with that weird example? The link sends me to a YouTube video completely devoid of context and in very poor quality. I don't know who made that video and in what way (if in any way) it is related to this proposed curriculum. If it is, the article should make this connection clear.
Moreover, as much as it may be pointless to replace "Mrs. Hernandez" with "Mr. Hernandez" in that question, it's certainly not "defying logic", it's just arbitrarily redefining the (irrelevant) context for a maths problem. I find people who complain about such a change at least as annoying as the ones arguing incessantly for these kinds of changes.
As the article is otherwise very light on details about what's wrong with the curriculum (the example with the fence is the only concrete thing that is being referred to that is not ridiculous, and even that is arguable), this just looks to me like some opinion piece that tries to fan the flames of some culture war.
It's quite possible that there are problems with this proposed curriculum (I don't live in California, I know nothing about it), but this article is not making a very clear case.
As someone who never jumped on or even cared to investigate the whole “data science” hype train, what does that look like at a primary/secondary education level?