CHE: "The Divider" (math education)

[jimbogumbo]

Jo Boaler is very good, and much of what has been discussed on this thread has little to do with her. There is a in fact a rich literature on small  group learning in physics and math, and it is highly underutilized in the US. That literature applies in both tracked and untracked settings.

mleok, much of what mathematicians claim about math educators in fact has little to do with what they study and write about. I'm confining my remarks here to those who study cognitive aspects of the field, and classroom efforts at effecting change. The cognitive work to me is compelling to the point of being settled.

Anyone who really wants to learn about this should utilize the resources of the Dana Center at the University of Texas. Headed by Uri Treisman (a mathematician), you can learn wonders about the available research om math education in school settings. For the cognitive side, I'd recommend the work of Alan Schoenfeld at Berkeley.

[dismalist]

Everything discussed on this thread has to do with Jo Boaler. From the Wikipedia entry under her name:

Boaler is the primary author of the California Department of Education's controversial mathematics draft framework.[46][47][48] The draft framework seeks to refocus mathematics education towards equity.[49][50] The draft framework recommends that all students take the same fixed set of math courses until their junior year of high school, which critics, including some leading mathematicians, say will hold back students.[51][52][53] Berkeley Professor Jelani Nelson found the framework worrying, saying it removed rigor and created a lower track of study, which would negatively impact diversity in STEM careers.[54]

Googling the Dana Center leads to invitations to buy stuff. Zero information. The Wikipedia link for Uri Treisman is criticized by Wikipedia itself.The Wikipedia entry for Alan Schoenfeld

https://gsi.berkeley.edu/programs-services/hsl-project/hsl-speakers/schoenfeld/

is, well, a tad weird.

[jimbogumbo]

Everything discussed on this thread has to do with Jo Boaler.

Sorry, no. Boaler was tasked with the California Framework (along with others). She has written about far more nuanced aspects of learning. I was a primary member of a state task force. Saying that is what I am is inaccurate; I researched far more than that.

If you pm me I can send you a couple of Shoenfeld's accessible articles. Don't buy anything from the Dana Center; try to find links to the research compendiums and databases.

I'm also confused as to what you find weird about the link you sent? That describes one of his talks; the Wikipedia link about how is below that one in a Google search.

[dismalist]

No eighth grade algebra. No rush to calculus.

That's what I need to know.

[jimbogumbo]

No eighth grade algebra. No rush to calculus.

That's what I need to know.

That's about the framework, right? Not Schoenfeld. FWIW, I helped create algebra courses for 6th graders in several districts. I'm a firm believer in advancing students who have learned material, and want to move on.

[mleok]

Everything discussed on this thread has to do with Jo Boaler.

Sorry, no. Boaler was tasked with the California Framework (along with others). She has written about far more nuanced aspects of learning. I was a primary member of a state task force. Saying that is what I am is inaccurate; I researched far more than that.

If you pm me I can send you a couple of Shoenfeld's accessible articles. Don't buy anything from the Dana Center; try to find links to the research compendiums and databases.

I'm also confused as to what you find weird about the link you sent? That describes one of his talks; the Wikipedia link about how is below that one in a Google search.

Fair enough, but to what extent, if any, is the California Framework backed up by rigorous research, and what are the metrics for which it is optimized for? In particular, to what extent does the Framework deemphasize the learning outcomes of the most advanced students in favor of more "egalitarian" outcomes. I have a hard time with any approach that favors holding back the best students with a view towards creating a more uniform outcome.

[jimbogumbo]

Fair enough, but to what extent, if any, is the California Framework backed up by rigorous research, and what are the metrics for which it is optimized for? In particular, to what extent does the Framework deemphasize the learning outcomes of the most advanced students in favor of more "egalitarian" outcomes. I have a hard time with any approach that favors holding back the best students with a view towards creating a more uniform outcome.

I'll depart after this one.

Our posts crossed, so I lust modified this to include the above. I clearly agree with mleok's last sentence, and don't have enough deep familiarity with California to answer the questions re research and metrics.

Boaler did a longitudinal study (three years) comparing a reform middle school curriculum with no algebra versus a traditional one which did. I think her best work is on mindsets in a discipline (math) and helping students develop them.

Here are statements regarding the latest Framework (which is the 2nd revision of the 2005 framework) and what it says about "acceleration" ( a very mild version of acceleration fr those of us who have have worked with students learning Calc I-III in hs):

What does the draft Mathematics Framework say about middle school mathematics acceleration programs?
The IQC discussions from the May 2021 meeting underscored that the decision about acceleration/honors and AP courses is a local one and requested that the updated draft include specific guidance on acceleration (including middle school acceleration) and serving high achievers and gifted students. Those changes are reflected in the draft that is posted for the second 60-day public comment period.
Chapter 8 of the draft Mathematics Framework notes that: "Some students will be ready to accelerate into Algebra I or Integrated Mathematics I in eighth grade, and, where they are ready to do so successfully, this can support greater access a broader range of advanced courses for them."
The framework also notes that successful acceleration requires a strong mathematical foundation, and that earlier requirements that all students take eighth grade Algebra I were not optimal for all students.

What does the draft Mathematics Framework say about access to calculus in high school?
The draft Mathematics Framework includes calculus in the possible high school pathways, and also suggests ways to enable more students to get access to calculus. It notes that many high schools currently organize their coursework in a manner that requires eighth grade acceleration in order to reach calculus or other advanced mathematics courses by senior year. While some students succeed with this approach, acceleration has proved a problematic option for other students who could reach higher level math courses but would benefit from a stronger foundation in middle school mathematics.
In chapter 8, the draft framework notes: "Since achieving a solid foundation in mathematics is more important for long-term success than rushing through courses with a superficial understanding, it would be desirable to consider how students who do not accelerate in eighth grade can reach higher level courses, potentially including Calculus, by twelfth grade. One possibility could involve reducing the repetition of content in high school, so that students do not need four courses before Calculus. Algebra 2 repeats a significant amount of the content of Algebra 1 and Pre-calculus repeats content from Algebra 2. While recognizing that some repetition of content has value, further analysis should be conducted to evaluate how high school course pathways may be redesigned to create a more streamlined three-year pathway to pre-calculus / calculus or statistics or data science, allowing students to take three years of middle school foundations and still reach advanced mathematics courses."
While encouraging greater access to calculus, the framework also presents research that the "rush to calculus" without depth of understanding is not helpful to students' long-term mathematics preparation. Data shows that about one-half of all high school students who take calculus repeat the course in college or take pre-calculus in college.
The Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) issued a joint statement that included the premise: "Although calculus can play an important role in secondary school, the ultimate goal of the K–12 mathematics curriculum should not be to get students into and through a course in calculus by twelfth grade but to have established the mathematical foundation that will enable students to pursue whatever course of study interests them when they get to college." (See MAA and NCTM Joint Statement ).
Similarly, the University of California's board of admissions "strongly urges students not to race to calculus at the cost of full mastery of the earlier math curriculum. A strong grasp of these ideas is crucial for college coursework in many fields, and students should be sure to take enough time to master the material. Choosing an individually appropriate course of study is far more important than rushing into advanced classes without first solidifying conceptual knowledge."

[ciao_yall]

Do you think being "streamed" is what caused you to be singled out, and dehumanized by the kids who resented having been labeled "not so smart" and, as a result, decided to beat you up because they didn't have a chance to get to know you as a person?

That misses the point. In that Junior High, everybody beat on everybody else except within classes of the better students. Streaming is a way of minimizing pollution in such environments, pollution that promotes a race to the bottom.

Yikes. What a word.

It's about all instruction, which includes math.

Don't cry for me. I got my character built. How to resist violence and such. But survivorship bias, as I said.

I'm for segregation in schools based on pollution. Mixing different intellectual speeds and capacities is one form of pollution -- in both directions! Violence is another. Differing cultural attitudes toward learning is a further source of pollution.

Yikes.

I repeat, too, that mixing intellectual capacities in school does no favors for the slower among us.

You are acting as though capacity and learning is fixed and only relies on a few dimensions. People are smart in many different ways.

I have been the fast learner in class and enjoyed taking leadership among my peers when that happened. I have been the slow learner in class and learned to take advice and help from others, learned to think about problems in different ways, and developed appreciation and admiration for people in unexpected ways.

[marshwiggle]

Here are statements regarding the latest Framework (which is the 2nd revision of the 2005 framework) and what it says about "acceleration" ( a very mild version of acceleration fr those of us who have have worked with students learning Calc I-III in hs):

What does the draft Mathematics Framework say about middle school mathematics acceleration programs?
The IQC discussions from the May 2021 meeting underscored that the decision about acceleration/honors and AP courses is a local one and requested that the updated draft include specific guidance on acceleration (including middle school acceleration) and serving high achievers and gifted students. Those changes are reflected in the draft that is posted for the second 60-day public comment period.
Chapter 8 of the draft Mathematics Framework notes that: "Some students will be ready to accelerate into Algebra I or Integrated Mathematics I in eighth grade, and, where they are ready to do so successfully, this can support greater access a broader range of advanced courses for them."
The framework also notes that successful acceleration requires a strong mathematical foundation, and that earlier requirements that all students take eighth grade Algebra I were not optimal for all students.

This sounds incredibly weasely to me; without being clear on how they determine when a student is "ready" for acceleration it's quite possible the practical result may be to assume virtually no-one is.

What does the draft Mathematics Framework say about access to calculus in high school?
The draft Mathematics Framework includes calculus in the possible high school pathways, and also suggests ways to enable more students to get access to calculus. It notes that many high schools currently organize their coursework in a manner that requires eighth grade acceleration in order to reach calculus or other advanced mathematics courses by senior year. While some students succeed with this approach, acceleration has proved a problematic option for other students who could reach higher level math courses but would benefit from a stronger foundation in middle school mathematics.

It's stating the obvious to point out that some (actually many) students struggle under the current approach. What's not at all clear (again) is how they identify students who are perfectly capable of handling the path to complete calculus by the end of high school.

I have no problem with assuming most students may benefit from a slower pace; I have big problems with making it difficult for good students to establish their ability to go at an "accelerated" pace.

In chapter 8, the draft framework notes: "Since achieving a solid foundation in mathematics is more important for long-term success than rushing through courses with a superficial understanding, it would be desirable to consider how students who do not accelerate in eighth grade can reach higher level courses, potentially including Calculus, by twelfth grade. One possibility could involve reducing the repetition of content in high school, so that students do not need four courses before Calculus. Algebra 2 repeats a significant amount of the content of Algebra 1 and Pre-calculus repeats content from Algebra 2. While recognizing that some repetition of content has value, further analysis should be conducted to evaluate how high school course pathways may be redesigned to create a more streamlined three-year pathway to pre-calculus / calculus or statistics or data science, allowing students to take three years of middle school foundations and still reach advanced mathematics courses."
While encouraging greater access to calculus, the framework also presents research that the "rush to calculus" without depth of understanding is not helpful to students' long-term mathematics preparation. Data shows that about one-half of all high school students who take calculus repeat the course in college or take pre-calculus in college.
The Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) issued a joint statement that included the premise: "Although calculus can play an important role in secondary school, the ultimate goal of the K–12 mathematics curriculum should not be to get students into and through a course in calculus by twelfth grade but to have established the mathematical foundation that will enable students to pursue whatever course of study interests them when they get to college." (See MAA and NCTM Joint Statement ).
Similarly, the University of California's board of admissions "strongly urges students not to race to calculus at the cost of full mastery of the earlier math curriculum. A strong grasp of these ideas is crucial for college coursework in many fields, and students should be sure to take enough time to master the material. Choosing an individually appropriate course of study is far more important than rushing into advanced classes without first solidifying conceptual knowledge."

Again, this is stating the obvious. But the bigger reality is that most students who struggle with math in school aren't likely to be heading for highly quantitative careers. (If there's evidence to the contrary, I'd love to see it.) Just because they go slower so they end up with a better grasp of it, (which is great), doesn't mean they're going to be much more keen on doing a lot more of it at a higher level.

[jimbogumbo]

Well known info: https://www.usatoday.com/story/news/education/2023/03/30/poor-rural-students-fewer-advanced-math-courses-stem/11565375002/

[Stockmann]

So what is the debate over?  Methods of math instruction?  Diversity and inclusion in math?

The article is really sketchy on details, but ti seems to be that her focus is on lots of group work, with students of all ability levels, and she claims it reduces math anxiety and makes weaker students do better at math. Her critics (reading between the lines) suggest that the better students are being held back by it. (Anyone who knows more feel free to correct me.) If improving the performance of the bottom 30% (arbitrarily chosen number) is at the expense of the top 10% (arbitrarily chosen number), is that an improvement or not?
Again, if that's kind of what's going on, there is lots of history of using good students to essentially tutor weak students, which may benefit the system, but may not be best for the good students.

Actually, tutoring weaker students helps the stronger students by making them learn the material more deeply in order to explain it to others, and in different ways.

It's not a zero-sum game.

That's been my anecdotal experience teaching formal logic. The strong students develop more fluency and facility with the material, which makes them better able to apply it to novel problems.

So yeah, marshwiggle, they don't learn more brute content. But there's a limit to how much new content they will get in any class, and they'll get there by the end of the course. If there was no cap to how much they'd get, then sure, it's be holding the ones who get it back somewhat. But that's not the case.

But it also benefits the students who aren't getting it to have them explain it to me/one another (with supervision). It helps them to identify and understand where their gaps are, and it is easier to fix the problem that way. If they don't understand where they're struggling, it just seems impossible to them.

The problem is it provides a perverse incentive.

In a high school, suppose you have 40 students in Grade 9. You can offer two classes of Algebra; you could offer one "remedial" and one "advanced" , or two "remedial". The perverse incentive is to go with two "remedial", because using the good students as tutors will improve the average grades in both sections. The primary benefit of this is to the school and its average. Putting the students first would require having the two different classes so that each class could focus on one group of students. (But, of course, it would show that a single teacher without unpaid tutors would have limits. This is a system problem, that shouldn't be "*solved" on the backs of the good students.)

*Hidden is more like it.

Surely the biggest perverse incentive is it creates an incentive for mediocrity. The top students are given every incentive to not apply themselves - there's no grade incentive, as A really is Average these days, there's no incentive of moving on to fresh material instead of lingering with the same old stuff they've already mastered, and instead they're basically punished by being used as unpaid tutors - on top of possible ostracism by their peers. If you have some talent, but don't apply yourself, you'll still get an A, you'll be a bit less bored with the same old material because you probably ignored it the first time around anyway, and the teacher will largely ignore you instead of asking you to do unpaid work that will help make you a social pariah. This is a system designed to dull diamonds (I don't think it's polishing pebbles at all - it's not even making everyone mediocre, it's setting mediocrity as the ceiling).

[kaysixteen]

For college students who do not need to take any given subject for their major and/or professional career objectives, which subjects do we feel that a liberal arts student should nonetheless be required to take a course in, and why?

[Wahoo Redux]

Are you asking what general education requirements lib arts students should take from outside their major?

Foreign languages
At least one freshman level science sequence