1. 'Kids considered to be “gifted” suffer from ability grouping the most because they develop the ultimate fixed mindset. They become terrified that if they struggle they’ll no longer be considered smart.'
2. 'Removing the time pressure from math is another important issue for Boaler. Neuroscience research... has shown that time pressure often blocks the brain’s working memory from functioning. This is particularly bad for kids with test anxiety. “The irony of this is mathematicians are not fast with numbers,” Boaler said. “We value speed in math classrooms, but I’ve talked with lots of mathematicians who say they’re not fast at all.” '
I have experienced the truth of both of these points during my math education, and have recently started talking with professors (I'm an undergrad stat major) about the idiocy of timed exams in math. Timed exams test for speed, which is not something that matters in real mathematics (if you can prove a theorem in one week vs. two it doesn't really matter), and this can push people out of the discipline who would otherwise stay. As people discuss retention rates in stem fields, particularly with under-represented groups, I hope they will consider getting rid of time limits as an avenue of effective policy change.
Test anxiety is a multiplier for stereotype threat. It is just the worst for performance. I am a good test-taker because I can put it aside, but it is not fair.
I think STEM students should be writing more papers across the board. You shouldn't just solve problems and write down the solutions, you should be writing exposition on the methods you use, why they work, and what other options are available.
My writing ability developed in high school and college due to a few specific causes. 1: Being forced to write for my AP European History course in HS (the regular assignments were just lists of questions/prompts to be answered with a few sentences or a paragraph or so, it's amazing how valuable that experience was). 2: Writing proofs in advanced math courses (elementary analysis, abstract algebra, etc.) 3: Participating in usenet newsgroups (by far the biggest contributor).
P.S. I meant to point out the irony that English classes contributed comparatively little. Actually writing about something and putting in the effort to string together something coherent is what really exercises and builds writing abilities. It's nice to have some of the groundwork laid, but practice is by far the most important component.
I got the impression that the article was about K-12 education, where the kids aren't yet completely separated into STEM and non-STEM tracks. (Some kids get onto an accelerated math track). If the schools that my kids attend are typical, they do extensive amounts of writing in all of their subjects. The kind of writing that you describe was pretty common, through 5th grade.
In 6th grade and beyond, the school went to a more traditional math curriculum, I suspect due to milestones such as getting through algebra in 8th grade, etc. For the kids to do more writing in math class requires a choice of what they will spend less time on. Do they learn fewer math topics, get fewer assignments in Social Studies, or get less sleep?
Maybe I'm a freak, but as a student I greatly enjoyed the "pure" nature of math. In other subjects, I found that I could get good grades by basically filling pages with drivel. On the other hand I didn't just write down solutions. I was expected to show a derivation, and in many cases, a proof. I have a copy of my high school pre-calc textbook, and a large fraction of the chapter problems are proofs, which I loved. That's what motivated me to become a math major in college. Today, the proofs have disappeared.
These are two really good points, but #2 seems so obvious yet has never occurred to me.
I believe a property of the "good at math" persona is that the individual can just "magically" solve a math problem _quickly_ in there heads, or more precisely, this individual already possesses the answer, but just needs to recall it, akin to recalling a history fact.
Yes, by practicing maths, you will eventually develop some "muscle memory" for certain types of problems, but understanding or solving a math equation does not inherently have time constraints associated with it, yet we somehow believe that to be part of what it means to be proficient in maths.
This false belief has held me back from believing I could achieve more in mathematics - the idea that if I can't find the solution to a problem in under 10 seconds, I obviously don't know or can't figure out the answer.
Thank you for making this point that is quite obvious, but has enlightened me.
This is how I felt about the Math GRE. On the practice test, when I gave myself plenty of time to think, I was able to answer almost every question correctly. But on the real timed test, I scored very poorly. They gave you so little time to solve the problem there was no "thinking" involved, it was all rote algorithmic manipulation.
I remember that in elementary school a lot of the early math education was especially focused on speed. The same quizzes were basically given to us repeatedly until our speed improved for the entire year. Later on, they intended for this to serve as a sort of placement test for math. I was pretty much always the slowest one, because I would get more stressed when a time limit was given (something around 2 minutes for 60 questions) and I didn't approach it as memorization like many of the other students did. My teachers were convinced that I was unable to learn math because my speeds at answering those questions were not improving as much as those of other students and most of the time I was not able to finish the quizzes. But as soon as my parents requested for the school to give me a normal test they found that none of those problems even existed for me. Afterwards, they ended up moving me to a class that was a lot less focused on just doing the work quickly and that put me ahead of where many of my peers were. I think that if everybody else had been taught in an environment like that were time was not a factor in your grade they would have a much better understanding of the concepts of math and would also find it easier to learn how to do the actual math than how to memorize the answers to repetitive problems.
Saying timed exams in math test for speed is a simplification. Perhaps at the underclassmen level, but I've found for a lot of upper level courses the tests were more about making sure you have certain concepts internalized and can make certain inferences without a secondary source. Oral examinations would work just as well, of course, but those are usually considered more stressful.
>Timed exams test for speed, which is not something that matters in real mathematics (if you can prove a theorem in one week vs. two it doesn't really matter),
Moreover, some conjectures don't get proved as theorems for decades or centuries, and even after such long periods of time, the conjecture might still sometimes be refuted or proved independent of the axiom framework.
I do not think 1 is necessarily true. You can group kids by skill, and still develop a culture where you can struggle and feel smart at the same time. In contrast, not segregating classes risks making the not gifted at math students feel not smart, and risks making the gifted students feel that they do not need to work at it.
A lot of problems here. (1) "This study shows that all kids can learn math when taught effectively". Well, as the definition of "teaching effectively" is that students learn from it, this observation really isn't all that startling. We could do a similar study to prove that all children arrive at school when driven there. What would be startling would be if the same technique worked equally well with all students (and this one won't). (2) Ah, it wouldn't be a paper about educational techniquese without some Gladwelling (which I define as using a vastly over-simplified and facile interpretation of scientific findings in order to advance a puerile generalization). Yes, research shows that connections in the brain are modified when learning. It would be shocking if learning didn't have an effect on the brain. By FMRIs and other imaging is far,far too blunt to detail what is happening in any useful detail. Taking a scientific finding that "people seem to use these particular areas of the brain when learning" as justification for an entire philosophy of pedagogy is absurd. And, for the umpteenth time --- multiplying two numbers is no more "Mathematics" than learning how to drive a screw is "Mechanical Engineering". And encouraging children to explore driving a screw with a hammer, a pair of pliers, their teeth, a teammate and then having them do a class presentation on their findings does not help them in their later careers as mechanics. Far better to show them the most effective technique, let them practice it, and move on. "Exploration" learning is only effective when applied to skills that already have mental infrastructure provided by evolution. You learn your first language by exploration. You learn to walk by exploration. Riding bikes and mathematics were never evolutionarily selected for, so we have to learn them by, well, learning them.
> Far better to show them the most effective technique, let them practice it, and move on.
This can work for the base levels of what most folks think as mathematics, namely arithmetic and algebra. But it starts falling apart as students get to more advanced algebra and mathematical topics. Memorizing a particular algorithm does not lend to understanding mathematics. In particular, the parallels between algebra (and some arithmetic) and geometry are non-obvious and not easily conveyed via a series of exercises and then jumping to the next topic. See the visualization examples of `18x5` in the linked article. Letting students see the correlation between the numbers and the objects (squares and rectangles) greatly helps their understanding of the subject. Discussing it reinforces this experience and gets them to see it beyond the exercises they've been doing, abstracting the concept beyond a handful of cases and to the general case.
> Riding bikes and mathematics were never evolutionarily selected for, so we have to learn them by, well, learning them.
There are many ways to learn and to teach. Teacher-tell, student-regurgitate is one method. And it works for some (many) subjects, at least in getting facts and figures into students heads. Teacher-tell, student-explain gets the students past thinking about facts and thinking about whys and hows of the world. (You can expand tell beyond spoken word to shown, demonstrated, etc.) Teachers can take it further by starting on a mathematical topic, and directing the student conversation and discussion towards understanding (I think this is the example I'm looking for, graph theory for eight-ear olds: http://jdh.hamkins.org/math-for-eight-year-olds/).
To supplement this comment, How to Solve It by Polya is a nice little handbook on the education of mathematics. I'm reading through this at the moment and I've learned quite a bit. It's especially useful to see how I have unknowingly applied some heuristic to a problem I've solved in the past.
As a side note, I'm finding both Calculus Made Easy and Concrete Mathematics incredibly useful. Thanks for the suggestions.
"Far better to show them the most effective technique, let them practice it, and move on."
Wasn't there a recent post on HN about something being expanded in Japan called "productive failing"? It's the opposite approach - let them struggle and then show them a/the way to solve the problem. The claim is that method is more effective.
Every child can easily learn all the math they would need in life.
Inability to understand math does not come naturally to any child. It is something that must be taught by specialists at intense effort and great expense.
> Boaler said a big problem is that math teachers themselves are math-traumatized. They came through a system very similar to the one in which they work. Elementary school teachers in particular often feel insecure about math.
This is incredibly important to understand. I know a number of middle grades teachers (not math or science) who "aren't math people". Their anxieties significantly hamper their ability to help their students when math-related things come up in class (statistics in a social sciences class, for instance). Or if a student wants to go to a trusted teacher for help outside that teacher's subject area.
Similarly, and this is really bad, they can't calculate students' grades! I've witnessed this one several times with a friend, and wish I could sit with her every time she worked out her grades so it will stop happening:
Test is out of 100 points. 10 points of extra credit. Calculate the students percentage as X/110. WTF, it's no longer extra credit, the top score (110) is still 100% (instead of 110%), and now 90 points (should be 90%, an A-) becomes an 82% (B-)!
I absolutely refused to learn the times-table in school. Even to this day I struggle to add seven and eight. That one seems to confuse me every time. I think the long division algorithm is one of the most difficult things in mathematics, and they teach this to ten year olds! (or they did when i was ten.)
So, fast forward many years, and I started doing really well studying advanced mathematics in my undergrad. But somewhere along the line I hit a wall: most math textbooks are written ass-backwards for my brain. I really don't get maths by looking at equations. It has taken another 20 some years to move beyond this, and to not be shy about being a disaster when it comes to algebra. Because when it's about shape or process that is when I really start to move.
My research today is in theoretical quantum physics (finally doing my postgrad.) I've started to see that there are others like me out there: we often sound like nutjobs when talking about maths, because it's all a confused mess internally. But somehow from the chaos miracles emerge. This is in contrast to the heavily-linear-thinking types: these people produce sparkling sentences one after another, but often get lost seeing the big picture. The "nutjobs" are a minority, and are so easily criticized. But I sure wish there were more of us doing this hard stuff.
One of my favourite math lessons in school occurred when I was about six years old. The teacher had some coloured shapes scattered across the floor, some yellow, some red, some triangles, some discs, etc. She placed a big hoop around all the yellow shapes, and another separate hoop around all the triangles. But there was a problem with this scheme: what to do about the yellow triangles? She let us all stew on this for a few moments, and then the magic happened. If you place one of the hoops overlapping the other, the yellow triangles could inhabit both hoops at the same time.
Thanks so much for saying this. I gave up on ever being like my peers, the heavily-linear-thinking types, in maths (deciding factor for not pursuing maths post-grad). Any recommendations for math texts that take this visual or different approach?
23 comments
[ 2.4 ms ] story [ 65.7 ms ] thread2. 'Removing the time pressure from math is another important issue for Boaler. Neuroscience research... has shown that time pressure often blocks the brain’s working memory from functioning. This is particularly bad for kids with test anxiety. “The irony of this is mathematicians are not fast with numbers,” Boaler said. “We value speed in math classrooms, but I’ve talked with lots of mathematicians who say they’re not fast at all.” '
I have experienced the truth of both of these points during my math education, and have recently started talking with professors (I'm an undergrad stat major) about the idiocy of timed exams in math. Timed exams test for speed, which is not something that matters in real mathematics (if you can prove a theorem in one week vs. two it doesn't really matter), and this can push people out of the discipline who would otherwise stay. As people discuss retention rates in stem fields, particularly with under-represented groups, I hope they will consider getting rid of time limits as an avenue of effective policy change.
P.S. I meant to point out the irony that English classes contributed comparatively little. Actually writing about something and putting in the effort to string together something coherent is what really exercises and builds writing abilities. It's nice to have some of the groundwork laid, but practice is by far the most important component.
In 6th grade and beyond, the school went to a more traditional math curriculum, I suspect due to milestones such as getting through algebra in 8th grade, etc. For the kids to do more writing in math class requires a choice of what they will spend less time on. Do they learn fewer math topics, get fewer assignments in Social Studies, or get less sleep?
Maybe I'm a freak, but as a student I greatly enjoyed the "pure" nature of math. In other subjects, I found that I could get good grades by basically filling pages with drivel. On the other hand I didn't just write down solutions. I was expected to show a derivation, and in many cases, a proof. I have a copy of my high school pre-calc textbook, and a large fraction of the chapter problems are proofs, which I loved. That's what motivated me to become a math major in college. Today, the proofs have disappeared.
I believe a property of the "good at math" persona is that the individual can just "magically" solve a math problem _quickly_ in there heads, or more precisely, this individual already possesses the answer, but just needs to recall it, akin to recalling a history fact.
Yes, by practicing maths, you will eventually develop some "muscle memory" for certain types of problems, but understanding or solving a math equation does not inherently have time constraints associated with it, yet we somehow believe that to be part of what it means to be proficient in maths.
This false belief has held me back from believing I could achieve more in mathematics - the idea that if I can't find the solution to a problem in under 10 seconds, I obviously don't know or can't figure out the answer.
Thank you for making this point that is quite obvious, but has enlightened me.
Moreover, some conjectures don't get proved as theorems for decades or centuries, and even after such long periods of time, the conjecture might still sometimes be refuted or proved independent of the axiom framework.
This can work for the base levels of what most folks think as mathematics, namely arithmetic and algebra. But it starts falling apart as students get to more advanced algebra and mathematical topics. Memorizing a particular algorithm does not lend to understanding mathematics. In particular, the parallels between algebra (and some arithmetic) and geometry are non-obvious and not easily conveyed via a series of exercises and then jumping to the next topic. See the visualization examples of `18x5` in the linked article. Letting students see the correlation between the numbers and the objects (squares and rectangles) greatly helps their understanding of the subject. Discussing it reinforces this experience and gets them to see it beyond the exercises they've been doing, abstracting the concept beyond a handful of cases and to the general case.
> Riding bikes and mathematics were never evolutionarily selected for, so we have to learn them by, well, learning them.
There are many ways to learn and to teach. Teacher-tell, student-regurgitate is one method. And it works for some (many) subjects, at least in getting facts and figures into students heads. Teacher-tell, student-explain gets the students past thinking about facts and thinking about whys and hows of the world. (You can expand tell beyond spoken word to shown, demonstrated, etc.) Teachers can take it further by starting on a mathematical topic, and directing the student conversation and discussion towards understanding (I think this is the example I'm looking for, graph theory for eight-ear olds: http://jdh.hamkins.org/math-for-eight-year-olds/).
As a side note, I'm finding both Calculus Made Easy and Concrete Mathematics incredibly useful. Thanks for the suggestions.
Wasn't there a recent post on HN about something being expanded in Japan called "productive failing"? It's the opposite approach - let them struggle and then show them a/the way to solve the problem. The claim is that method is more effective.
Inability to understand math does not come naturally to any child. It is something that must be taught by specialists at intense effort and great expense.
This is incredibly important to understand. I know a number of middle grades teachers (not math or science) who "aren't math people". Their anxieties significantly hamper their ability to help their students when math-related things come up in class (statistics in a social sciences class, for instance). Or if a student wants to go to a trusted teacher for help outside that teacher's subject area.
Similarly, and this is really bad, they can't calculate students' grades! I've witnessed this one several times with a friend, and wish I could sit with her every time she worked out her grades so it will stop happening:
Test is out of 100 points. 10 points of extra credit. Calculate the students percentage as X/110. WTF, it's no longer extra credit, the top score (110) is still 100% (instead of 110%), and now 90 points (should be 90%, an A-) becomes an 82% (B-)!
I absolutely refused to learn the times-table in school. Even to this day I struggle to add seven and eight. That one seems to confuse me every time. I think the long division algorithm is one of the most difficult things in mathematics, and they teach this to ten year olds! (or they did when i was ten.)
So, fast forward many years, and I started doing really well studying advanced mathematics in my undergrad. But somewhere along the line I hit a wall: most math textbooks are written ass-backwards for my brain. I really don't get maths by looking at equations. It has taken another 20 some years to move beyond this, and to not be shy about being a disaster when it comes to algebra. Because when it's about shape or process that is when I really start to move.
My research today is in theoretical quantum physics (finally doing my postgrad.) I've started to see that there are others like me out there: we often sound like nutjobs when talking about maths, because it's all a confused mess internally. But somehow from the chaos miracles emerge. This is in contrast to the heavily-linear-thinking types: these people produce sparkling sentences one after another, but often get lost seeing the big picture. The "nutjobs" are a minority, and are so easily criticized. But I sure wish there were more of us doing this hard stuff.
One of my favourite math lessons in school occurred when I was about six years old. The teacher had some coloured shapes scattered across the floor, some yellow, some red, some triangles, some discs, etc. She placed a big hoop around all the yellow shapes, and another separate hoop around all the triangles. But there was a problem with this scheme: what to do about the yellow triangles? She let us all stew on this for a few moments, and then the magic happened. If you place one of the hoops overlapping the other, the yellow triangles could inhabit both hoops at the same time.
The teacher would explain something on the board. Everyone but me would say 'Ohhh, I see' or similar (think Bart Simpson and the 'RDRR' gag).
The teacher would explain the concept to me again. I still wouldn't get it. I got a 'Really?' look from the teacher, who then walked off.
So I wasn't a 'math person'.
Then halfway through a thesis I encountered some math that I had to understand and use, or give up the thesis.
I discovered I could some math after all.