Tuesday, April 30, 2019

But Seriously... Is Algebra Necessary?

A friend tagged me in a FB post with a link to a New York Times article by Andrew Hacker Is Algebra Necessary? I need to process this, so here goes....

I saw the article as having two distinct parts. Here are my reactions to each part.

Part The First: Math Is a Useless Pain

The first half or so of the article indicates that math causes high dropout rates and is basically useless to everyone not bound for MIT or Caltech. Well, if math causes high dropout rates, then I think there is a reasonable discussion to be had about how we address the issue. What should we do?
  1. Prepare students more effectively from the beginning. That is, should we fix our elementary and middle school math programs so that students can have more success later in their math careers.
  2. Fix the high school math curriculum so more students can find success and gain the math skills they need for life and for later mathematical study.
  3. Give up. Stop requiring so much math.
The eye-grabbing headline and some of the arguments in this first part seem to imply that Hacker is suggesting a focus on option 3, but after reading the second half of the article, I think he is making more of an argument for option 2. I believe that we need to focus on options 1 and 2.

By the way, the Common Core standards should  help with option 1. A coherent curriculum that focuses on important mathematical practices and covering fewer topics, but with more depth is a great place to start. It isn't a panacea, but spending more time on fewer topics can help everyone out. I think the Common Core standards do a pretty good job of mapping out a coherent mathematical progression from elementary school through the end of grade 8. Then, we come to...

Part the Second: The Traditional Algebra 1-Geometry-Algebra 2 High School Math Sequence Is Mostly Useless and Should Be Improved

OK. This is where Hacker and I agree. In an earlier post, I said that we drive too many kids (like cattle) on the pathway that leads to Calculus. As I said in Math Illiteracy, everyone needs to master some math skills and understand some math concepts, but these don't necessarily map well to the cattle drive to calculus that I think is over-enrolled. By pushing just about every kid through the same cattle drive, we are driving kids away from math and putting up barriers to later pursuits that should not be dependent upon relatively esoteric skills such as learning how to factor cubic polynomials.

Maybe most kids should take a (perhaps more applications-oriented) course in Algebra, but maybe fewer kids need to take geometry and Algebra 2. Maybe a course in Practical Math (the actual title of a course on which I am working) could help. This sort of course can help students see how math can be a tool for solving problems. It can still be a pretty rigorous course, though it certainly wouldn't prepare anyone for calculus. But then again, not everyone needs to take calculus.

From my perspective, the tricky part centers around mobility. If we have a math pathway that prepares potential engineers, scientists, and mathematicians with the math skills they need, and we create a separate pathway that helps the general non-mathy population acquire the skills they need, then how do we ensure that a student who starts on a less rigorous pathway can change his or her mind and switch to the mathy pathway? How do we make sure that the less rigorous pathway doesn't become a dumping ground for students who are ill-prepared for the more rigorous pathway? These are very real concerns. Perhaps I'm being a paranoid Black man, but I know that this sort of dumping ground mentality has been wide-spread in the past, and I'd hate to create a two-tiered curriculum that institutionalizes it.

The Bottom Line

As Hacker says:
Mathematics is used as a hoop, a badge, a totem to impress outsiders and elevate a profession’s status.
Here, he is talking about how medical schools (and other post-graduate programs) use calculus as a way to identify students who have the right academic stuff. Honestly, I can't see that changing any time soon. Also, I'd be interested in a study about this. Maybe being able to learn calculus has a high correlation with the ability to survive and thrive in medical school. Maybe this extends to undergraduate studies as well. As the author of the A Mind for Madness blog says in Thoughts on Nicholson Baker’s Case Against Algebra II:
... if a high school diploma is meant to indicate some level of readiness for college, then algebra is probably a good indicator. This does not mean that you will use it, but will just point out that you have some ability to do some abstract things. I’m not saying it is the only way to test this, but it is probably a pretty good one.
Regardless, I think we need to:
  • Prepare more students for success in mathematics more effectively. We shouldn't focus solely on the cattle drive to calculus, but then again, we shouldn't ignore it either.
  • Do a better job of creating math courses that engage all students. We need to have students wrestle with more authentic real-world applications and see how math can help solve real problems.
  • Help students see the beauty in math in addition to the utility it affords in problem-solving contexts.

Tuesday, April 23, 2019

The Cattle Drive: Too Much Calculus

Reading Pamela Burdman's piece at Hechinger Report: Numbers evoke joy and wonder, why doesn’t math class? got me thinking.

I have already argued that we push too many kids to Algebra too early. Now to complain about the other extreme.

High school students are often driven like cattle through their math courses. Those who can withstand the cattle drive make it to the fertile pastures of calculus. Sadly, many of the herd don’t make it and are left along the side of the trail.

Not every student belongs in calculus. Though students who plan to be engineers, physicists, mathematicians, or economists need calculus, many others have abilities or interests that make them better suited to other mathematical destinations. The cattle drive hurts kids who lose interest in a destination they don’t care about.

Don't get me wrong: I love calculus. It is beautiful, useful, fun stuff, but it isn't for everyone. Everyone should master Algebra, but not everybody needs to master calculus. And honestly, many of the topics in Algebra II only exist to prepare students for calculus. This doesn't mean creating a dumbed-down track, but maybe there are other options for rigorous pathways.

Why not map out different mathematical pathways for different kids? After Algebra I, maybe some kids would be better served by courses in discrete math, probability, statistics, logic, or some other mathematical topics. Ideally, students would be able to switch back and forth between pathways if enough foundational skills were shared between parallel courses.

I know I'm tilting at windmills, but I'd like to see the cattle drive identify some other destinations; there is lots of interesting, useful math out there. Too many people think they hate math because they hated the cattle drive they were on in high school.

Tuesday, April 16, 2019

Early Algebra: Crushing Kids

Some states want to get all their students to take Algebra by the end of grade 8. The idea is that this will provide equal access to challenging curricula.

The Washington Post's Jay Matthews discussed the issue in Recalculating The 8th-Grade Algebra Rush and the original report is available here: The Misplaced Math Student: Lost in Eighth-Grade Algebra.

I'm a fan of Algebra and a fan of helping as many students as possible master it, but I am not a fan of aggressive time lines for when it has to get done. Pushing all 8th or even 9th graders into Algebra is a problem. Kids who are not ready for Algebra would be better served by shoring up their math foundations. They need number sense, especially when it comes to decimals and fractions. Students who are pushed into Algebra before they are ready are doomed to fail and are probably doomed to hate math forever. I don't want them to take silly math classes that lack any rigor, but I don't want to throw them into classes for which they are not prepared.

At the other end of the spectrum, there is also a push to get good math students to take Algebra in seventh grade or even earlier. Not every kid is ready for Algebra in eighth grade. Very, very few kids should be taking Algebra before eighth grade. If a kid is that good at math, why not provide a more rich mathematical curriculum for the kid instead of just having them rip through the same old courses more quickly?

As you might imagine, my reaction to the title of Jill Barshay's article in the Hechinger Report: Gifted classes may not help talented students move ahead faster was a resounding "good!"

Some of my issues with the "hurry up and go fast" approach to gifted math education:

  1. I have heard SO many stories of kids who took Algebra in seventh grade and ended up losing interest before their senior year. This is anecdotal, so I need to find solid data.
  2. There is solid evidence to support the existence of the Protege Effect: Students who teach their peers deepen their own understanding of skills and concepts. Many parents of precocious students complain that their kids are being held back and doing the teacher's work, but those peer teaching experiences are deepening both students' knowledge.
  3. Teachers at every level complain about students who have zoomed through prerequisites without really understanding everything they need to understand to prepare for later study. For almost all students, that zoom through leaves gaps.
  4. Emphasizing extrinsic outcomes can quash student motivation and long-term interest. Being accelerated can be one of those extrinsic outcomes that push students for a while at the cost of  their own intrinsic motivations to learn.

As Barshay indicates:
"... Research points to the lack of consensus on what the goals of gifted education should be. Many don’t think it should be about advancing students as quickly as possible. High quality instruction that helps kids who’ve already mastered the basics go deeper into the material may ultimately be beneficial. And annual state assessments may not do a good job of measuring this kind of depth, creativity or critical thinking."
Algebra is a good thing. I value its abstraction and generalization and problem solving, but let's make sure that the kids who take it have the right foundation and that it really drives deep learning and affection for math.

Thursday, April 11, 2019

My Education Research Obsession: Bloom's 2-Sigma Problem

Benjamin Bloom is best known for his taxonomy, but I think his most interesting work pivoted around the 2-Sigma Problem.

The 2 Sigma Problem: The Search for Methods of Group Instruction as Effective as One-to-One Tutoring

I have read this paper at least once a year for the past 12 years.

Bloom found that students who received 1-1 tutoring performed 2 standard deviations (2 sigmas) better than the control group, but he realized that 1-1 tutoring is financially infeasible, so he looked for a combination of scalable strategies that could help teachers with normal class sizes get to the same level of student learning as 1-on-1 tutoring.

Here are a couple things that bug me about how and why people cite or revisit Bloom's paper:

A: Mastery Learning isn't totally free form!

Many people use Bloom's paper to justify ideas for mastery learning that are quite different from what Bloom used. For some, "mastery learning" and "competency-based learning" are synonymous. The idea is that students should move at their own pace based on their ability to demonstrate mastery of the content. Bloom's idea of mastery learning was different. Here is a diagram from Thomas R. Guskey's Closing Achievement Gaps: Revisiting Benjamin S. Bloom’s “Learning for Mastery.”
Bloom's model assumed that all students were studying the same unit (maybe 2-weeks long?) at the same time. The different pathways through the unit were informed by Formative Assessment A. Students who didn't initially master the content would get more instruction on the basic content, while those who did master the unit's content after one attempt at instruction which would get enrichment. Notice that all students would move on to the next unit at the same time.

B: It's not just about Mastery Learning!

Mastery learning showed 1 sigma effect, but was not the whole story. The greatest impact (1.6 sigma) came when he coupled mastery learning with "enhanced prerequisites." Essentially, they figured out what specific skill gaps students had relative to the content they were about to learn and remediated those gaps before having students dive into mastery learning with the on-level content. It's important to note how targeted the enhanced prerequisite instruction was. This wasn't about remediating all of pre-algebra before taking algebra. It was about bridging specific foundational skill gaps that were critical to the content to come.

When I go on about student readiness or using formative assessment to inform instruction while keeping an entire class on the same unit of instruction, I'm basing my ideas on Bloom. I'll save my rants about formative assessment death spirals and catching kids up for another day.

Tuesday, April 9, 2019

Teach to One: Technology that Kills Learning Relationships

Teach to One Math is an exciting idea. What if computers could help students get a truly personalized learning experience? Their supporters include an amazing list of educational organizations for which I have great admiration and respect, including Gates, Chan Zuckerberg, New Profit, and Oak.

Hechinger Report published an interesting article a few years ago What happens when computers, not teachers, pick what students learn? that paints a picture of how Teach to One Math can look in a classroom. It's certainly innovative, and probably works for some kids and teachers, but I was skeptical.

When Open Culture published Trainwreck: The Teach to One Math Experiment in Mountain View, CA Is a Cautionary Tale About the Perils of Digital Math Education, more people took notice of the downside of Teach to One. Around the time the Open Culture article came out, I spoke with a teacher from a Teach to One school, and her comment was that she felt...
"... cut out of the process and overwhelmed at the same time."
I'm sure that computer-driven adaptivity has its place, but when those algorithms get in the way of effective teacher-student relationships, we have a problem.

Why mention this idea that is a few years old? Artificial Intelligence (AI) is getting more powerful and so people keep coming up ways that computers can improve teaching and learning. For instance, some of TeachThought's 10 Roles For Artificial Intelligence In Education have AI-driven systems re-framing the role of the teacher. Teach to One Math should be a cautionary tale that helps us evaluate huge shifts that could harm relationships between teachers and students.

Thursday, April 4, 2019

New Pathways to Success: The Swiss Apprenticeship Model

I am on the record as having an education crush on Quebec's Cegep system. Read the previous post (it's short) to get a sense of what Cegep is all about and why I love it so. Cegep is great, but completely unattainable for the U.S., which loves its college machine (perhaps a bit too much). Still, a guy can dream, can't he?

A glimmer of hope that U.S. school districts could create something really great comes from The 74: Robots, Inequality, Apprenticeships: If America Is to Usher In an ‘Age of Agility’ in Education, Experts Say We Must Talk Less About Schools — and More About Students. First, the problem:
... The traditional high-school-to-college continuum leaves too many talented people behind. About a third of Americans have a four-year college degree, yet an estimated 6.3 million jobs are going unfilled for lack of skilled candidates.
Yep. That describes the problem pretty well. The U.S. system has too many academic dead ends that leave tons of students feeling like failures and unprepared for the workforce. If only we could create a system that helps more students really prepare for careers while exploring possibilities. If only this system could have ramps that allow students to move from career preparation to college if they want to.

It turns out that the Swiss have a solution:
After nine years of compulsory schooling, ... every Swiss student has the opportunity to opt in to a national system of apprenticeships. 70 percent of Swiss teens participate, choosing one of 250 career pathways. They continue to go to high school part time, and many later earn a college degree.
OK, so you create fewer dead ends and help prepare students for careers, but how would it be paid for?
Swiss businesses contribute 60 percent of the $6 billion annual cost .... Business sees the expense as an investment, not an act of corporate responsibility.
So, Swiss businesses are seeing that supporting a program like this is an investment. I need to take some time to dig into this more, but I am pretty sure that I have a new education crush. Cegep is still dreamy, but so is Switzerland's apprenticeship program.

Monday, April 1, 2019

A New Adventure: Docent Learning

As of today, I am completely self-employed!

I've got an incorporated LLC, business cards, a website, and as of today, I am no longer an employee of any other company. My business has a couple contracts that should keep me going at least through July, so now I need to focus on my work and my business.

Yeah, it's a little scary, but this is something I've wanted to do for a long time. As the sole proprietor and employee of Docent Learning LLC, I am focusing on educational consulting and curriculum development. Right now, I am working on a high school computer science course and several high school math modules to support the new PISA 2021.

Now that I am Docent Learning, I will endeavor to post to this blog and get active on Twitter (I am @DocentLearning). Stay tuned and wish me luck.

Merit and Diversity in College Admissions

The recent Supreme Court ruling against race-conscious university admissions has everyone thinking about racism, privilege, equity, merit, ...