I was watching a recent episode of Amazing Race where the contestants had to learn part of the Riverdance. I was thinking about explicit instruction, which all their tutors used. But, If their tutor said 'we'll let's figure it out the perfect configuration without me telling you' what would have happened? It would have been absolutely impossible and preposterous (though admittedly fun to watch). So why do we do the same for kids?
This reminds me of a recent post from Ben Orlin's blog Math with Bad Drawings, titled "Should math class be hard?" (https://mathwithbaddrawings.com/2023/12/11/should-math-class-be-hard/). Orlin's observation, which jibes with my own, is that more traditional mathematics education often favours a higher level of difficulty, while more progressive mathematics education tends to eschew this difficulty, but that given the philosophical underpinnings of traditional and progressive education, it should probably be the opposite. This opposite is what Greg illustrates here.
I'll admit that I don't mind it too much if the classes that I teach are harder than they could have been, but I teach at the postsecondary level and not at the primary or secondary level, and I also mostly teach future secondary school mathematics teachers. My role, as I see it, is to teach them as much mathematics as I possibly can, and to transmit to them the culture of doing mathematics. Given this goal, I don't think I can do "too much". But I will admit that it would be good to have actual research on what mathematics do future secondary school teachers actually need, and to develop a program to teach this material as seamlessly as possible. As for why progressive mathematics education often values the avoidance of difficulty (if we disregard the idea of progressive struggle), if I had to I would answer with something else Greg observed: many mathematics education specialists seem not to like mathematics very much, and their progressive orientation is part of their trying to do as little mathematics as possible.
A group I teach this year is mainly composed of those students who have struggled with maths for 9 years of education. Too often they conflate familiarity with mastery "I've done this before sir its too easy" - And yet you cannot do it accurately now. I think it is perhaps a reflection of the attitude of their previous teachers / the focus on "coverage" at the expense of mastery. Perversely, they sometimes complain when they succeed rather than fail as they are accustomed to.
I would argue that perhaps traditionalists expect more _rigour_ which I would not equate with difficulty. Correct use of terminology, notation etc. It's impossible to discover them yourself as they are often arbitrary conventions.
Greg. If you reframe this as a question of the value of challenging math problems once concept mastery is achieved how would your answer change?
This often seems to be missing in the discussion. In any game like endeavour, which math problems can be, the optimal motivational level is neither too easy or too hard. As you suggest there is a benefit to learning that with effort you can solve challenging problems. Whether there is a benefit in terms of retention or otherwise would need to be tested. But optimum motivation would be beneficial for long term interest in the subject.
Once you have gained a certain amount of expertise, you are more likely to view solving the problem as achievable and this may be one reason why problem solving is effective for relative experts. However, if they are ‘struggling’ — depending on how you define that word — it might imply they don’t yet have enough expertise. My maths students do eventually need to solve complex problems on their own, but I provide a highly scaffolded route to that end point. When we first encounter complex problem types, we break them down as a whole class.
I’d look at the world of math competitions where you have people who study enough to be knowledgeable but still take many minutes to solve some of the problems and won’t solve them all. There is clearly struggle. At any level there will be people with more expertise who have less struggle. But someone who had no struggle would be bored.
I’d say math competitions get the balance correct or would die out.
Clearly they provide motivation for some to learn more.
So can be given as an existence proof that some struggle is a good thing.
But I think this is what the people you are debating miss. Any error in the too much struggle direction would end the party.
I thought that productive struggle is about students who find patterns and algorithms learned in class not challenging enough. Having more difficult problems to solve makes classes more interesting. At college I was taught that time boxed approach works: set yourself a fixed amount of time (say 15 minutes) to struggle on a problem before looking for assistance (from a fellow student). The emphasis was on importance of not spending more than a certain amount of time on the assigned homework problems. Now I would strongly advise young myself to look through a lecture topic before attending it. Such planned struggle on my own should help to better appreciate the lecture. I understand how providing enough struggling for one student can be unethical wrt to others. But if resources allow (probably in after school class context) productive struggle can help advanced students to appreciate the beauty of math. Am I missing something? I’m also skeptical about timed tests where somebody needs to “solve” 20 standard problems for 40 minutes or less. When I was a student tests were with at least 10 minutes for a problem so a student had a plenty of time to struggle to combine available techniques and even to check for obvious mistakes.
There is evidence that when students are relatively expert, spending time tackling complex problems is worthwhile. So, if productive struggle is going to work with anyone, it will be more advanced students.
Thanks for this.
I was watching a recent episode of Amazing Race where the contestants had to learn part of the Riverdance. I was thinking about explicit instruction, which all their tutors used. But, If their tutor said 'we'll let's figure it out the perfect configuration without me telling you' what would have happened? It would have been absolutely impossible and preposterous (though admittedly fun to watch). So why do we do the same for kids?
This reminds me of a recent post from Ben Orlin's blog Math with Bad Drawings, titled "Should math class be hard?" (https://mathwithbaddrawings.com/2023/12/11/should-math-class-be-hard/). Orlin's observation, which jibes with my own, is that more traditional mathematics education often favours a higher level of difficulty, while more progressive mathematics education tends to eschew this difficulty, but that given the philosophical underpinnings of traditional and progressive education, it should probably be the opposite. This opposite is what Greg illustrates here.
I'll admit that I don't mind it too much if the classes that I teach are harder than they could have been, but I teach at the postsecondary level and not at the primary or secondary level, and I also mostly teach future secondary school mathematics teachers. My role, as I see it, is to teach them as much mathematics as I possibly can, and to transmit to them the culture of doing mathematics. Given this goal, I don't think I can do "too much". But I will admit that it would be good to have actual research on what mathematics do future secondary school teachers actually need, and to develop a program to teach this material as seamlessly as possible. As for why progressive mathematics education often values the avoidance of difficulty (if we disregard the idea of progressive struggle), if I had to I would answer with something else Greg observed: many mathematics education specialists seem not to like mathematics very much, and their progressive orientation is part of their trying to do as little mathematics as possible.
I'd push back a little; I often find that the progressivist camp lionises challenge as somehow being more "authentic" and requiring "deeper thinking". I am reminded of Greg's previous blog on "Path smoothing" https://gregashman.wordpress.com/2019/10/14/path-smoothing-or-challenging-maths/
A group I teach this year is mainly composed of those students who have struggled with maths for 9 years of education. Too often they conflate familiarity with mastery "I've done this before sir its too easy" - And yet you cannot do it accurately now. I think it is perhaps a reflection of the attitude of their previous teachers / the focus on "coverage" at the expense of mastery. Perversely, they sometimes complain when they succeed rather than fail as they are accustomed to.
I would argue that perhaps traditionalists expect more _rigour_ which I would not equate with difficulty. Correct use of terminology, notation etc. It's impossible to discover them yourself as they are often arbitrary conventions.
Greg. If you reframe this as a question of the value of challenging math problems once concept mastery is achieved how would your answer change?
This often seems to be missing in the discussion. In any game like endeavour, which math problems can be, the optimal motivational level is neither too easy or too hard. As you suggest there is a benefit to learning that with effort you can solve challenging problems. Whether there is a benefit in terms of retention or otherwise would need to be tested. But optimum motivation would be beneficial for long term interest in the subject.
Once you have gained a certain amount of expertise, you are more likely to view solving the problem as achievable and this may be one reason why problem solving is effective for relative experts. However, if they are ‘struggling’ — depending on how you define that word — it might imply they don’t yet have enough expertise. My maths students do eventually need to solve complex problems on their own, but I provide a highly scaffolded route to that end point. When we first encounter complex problem types, we break them down as a whole class.
I’d look at the world of math competitions where you have people who study enough to be knowledgeable but still take many minutes to solve some of the problems and won’t solve them all. There is clearly struggle. At any level there will be people with more expertise who have less struggle. But someone who had no struggle would be bored.
I’d say math competitions get the balance correct or would die out.
Clearly they provide motivation for some to learn more.
So can be given as an existence proof that some struggle is a good thing.
But I think this is what the people you are debating miss. Any error in the too much struggle direction would end the party.
I thought that productive struggle is about students who find patterns and algorithms learned in class not challenging enough. Having more difficult problems to solve makes classes more interesting. At college I was taught that time boxed approach works: set yourself a fixed amount of time (say 15 minutes) to struggle on a problem before looking for assistance (from a fellow student). The emphasis was on importance of not spending more than a certain amount of time on the assigned homework problems. Now I would strongly advise young myself to look through a lecture topic before attending it. Such planned struggle on my own should help to better appreciate the lecture. I understand how providing enough struggling for one student can be unethical wrt to others. But if resources allow (probably in after school class context) productive struggle can help advanced students to appreciate the beauty of math. Am I missing something? I’m also skeptical about timed tests where somebody needs to “solve” 20 standard problems for 40 minutes or less. When I was a student tests were with at least 10 minutes for a problem so a student had a plenty of time to struggle to combine available techniques and even to check for obvious mistakes.
There is evidence that when students are relatively expert, spending time tackling complex problems is worthwhile. So, if productive struggle is going to work with anyone, it will be more advanced students.
Thanks! Next question would be about low expectations… so for a school teacher it’s safer just to avoid any pre planned math struggles all together.