I completely agree with what you've written! I also think that students feel "dumb" more in math class compared to other subjects, because there is an impression of math as being a more "objective" subject than others... Some students feel that that there is only one correct answer and one correct method for every concept/problem and if you can't solve it (in your head) in the fastest way possible, you are "dumb." To compound this pervasive impression, some *teachers* teach in such a way to reinforce this perception by not celebrating different ways to reach a solution because it's not the conventional method, because it's a more inefficient method, or because the math teacher themselves don't understand why/how it works. When I was teaching 9th grade Algebra 1 students, I always felt that one of my biggest responsibilities was to turn my students' self-doubt into self-confidence, so that they could actually be open to learning in their remaining years in school.
That being said, I do have one worry - when the main conversation in a school becomes primarily about "helping students believe that they can learn" (i.e. when that becomes the focus of school-wide PD), I often find that it starts to skew towards *just* supporting students' social emotional health, and not about examining the structural and long-term changes that may need to happen in the school. For example, changes may be made to include discussions about growth mindset in each class or helping students combat negative self-talk or maybe techniques are taught for time management or creating SMART goals, etc. etc.. But rarely is that conversation expanded to include discussions about the need for a more vertically aligned math curriculum or critical examination of how we support students who don't have the prior knowledge to be successful in the current math classes. (Or other equivalent conversations for other subjects.)
In other words, I think that in order for schools to become spaces where we can help nurture students who believe they are "capable" learners, the time spent on understanding, building, and collaborating on the curriculum (i.e. the background stuff that doesn't seem as "student-facing") needs to be seen as valuable as discussing seemingly more immediate in-class issues. This is probably why more affluent schools where teachers have the time and mind-space to fully engage in these curricular discussions are often in a virtuous cycle of being able to nurture and benefit from inquisitive learners, as opposed to schools where teacher may often have to wear a gazillion other hats *in addition to* being just a teacher. (I say this as somebody who went from teaching in an inner-city public school to a very affluent private school. I think I was already a good teacher, in terms of being able to connect with students and helping them feel seen and valued... but I definitely became a much better *math* teacher and thus able to help students feel explicitly more successful in *math* once I didn't have to be everything for everyone all the time.)
I totally agree. For me the basic distinction is that I think academic success often precedes motivation. If a student isn't very motivated, one strategy to help is to get them some chances to feel successful academically, and the support to sustain that success. The default belief in many schools is the opposite: we see students who aren't motivated (and often aren't successful academically) and we try to convince them to be motivated through lots of conversations, social emotional stuff, etc.
Prioritizing academic success can solve lots of problems in schools. Not everything, but it's a huge lever. And if you spend a bunch of time and energy supporting students academically and it doesn't solve all of your behavior and motivation and attendance problems, well hey you still helped kids learn things and that's the core job of schools. If you spend a ton of time and energy on social-emotional well-being and that initiative doesn't move the needle (as is often the case) then that's a whole lot of wasted resources.
I agree with all of this! Great discussion too. A lot of the students who act out in school are also struggling academically -- and are often quite smart too.
The thing is, how do you get those students to experience successful moments? You have to go meet them where they are, and you have to use techniques that work for them. Right now, the prevailing wisdom tells us that we can only teach grade level standards. So, what do we do when kids don't have the background knowledge to learn those standards? We dumb down the standards so that they are "simple," with just memorization and mnemonics. That doesn't work. I think there is more learning and rigor involved in letting a student figure out how to solve a division word problem on their own (a 4th grade skill) than in teaching them steps to solve a linear equation (8th grade standard). And once they solve the problem on their own, they will know that they can do math, and they will build the motivation to learn the 8th grade skill.
Yea you point out a central contradiction in a lot of rhetoric about teaching today. Are we supposed to meet students where they are, or are we supposed to maintain grade-level rigor? People often say one or the other as if it's obvious, without wrestling with the challenges of trying to do both at the same time.
I am a tutor of 4th and 5th grade math. I work in an elementary school. The students I work with test below grade level. I have begun to question the need to "learn" math in the age of AI. Yuval Noah Harari has an intertering take. https://www.youtube.com/watch?v=h5_bullnfk0 Harari advocates the need for education to help develop mental flexibility and critical thinking skills as opposed to its traditional focus on rote learning and memorization. Thoughts?
I do think math is still worth learning. I don't think humans can learn mental flexibility or critical thinking without something to think or be flexible about. There's no easy shortcut to teach critical thinking directly, we teach critical thinking by teaching kids to think critically in all sorts of different contexts. I also don't think there's a sharp division between critical thinking and memorization. We begin with a shallow understanding of a concept, and then that understanding gradually gets deeper. We can't skip past those shallow levels of understanding.
I do think that we should rethink some of the exact content we teach. In the same way that we no longer teach how to use a slide rule, we should question exactly which computational skills are still worth learning. But if I was in charge, those changes would be relatively small.
Dylan, I agree completely that we need specific content to learn when we try to teach reasoning and critical thinking.
I'm curious what you mean when you say, "I also don't think there's a sharp division between critical thinking and memorization."
I certainly agree that these two things can be hard to distinguish on a test. Did the student reason their way through the problem, or did they memorize a series of steps to perform in a situation like this?
I also agree that memorizing a series of steps can be a useful jumping-off point for later critical thinking. When students have some sense of how to make their way through a problem, they can do things like compare methods, modify methods, and come up with new ways to apply those methods. I would even admit that some students will do some reasoning independently after they have been exposed to those series of steps to memorize. But none of this says that there is no sharp division between the two activities, but rather that they support each other. Agreed! They do!
Hmm, maybe what you are thinking is that when you introduce material, you do it in ways that shows students both how to think about it, and how to memorize it. So you don't need to distinguish in your own teaching, because you blend them. Or, maybe you what you really mean is that there is a blurry line between teaching procedures and teaching rich tasks designed for critical thinking? I agree with that. You can do a lot of critical reasoning around procedures.
However, when I see a class that is based entirely on rote memorization, I feel like that line is clear as day. "Yours is not to ask why. Just invert and multiply." If you stop there, I am sure that for 90% of the students, there will be no critical reasoning.
You make a bunch of great points. I think it's hard to make generalizations about memorization vs understanding because it depends so much on the context. For instance, I am fine with students memorizing how to multiply fractions because it's intuitively easy and a true understanding of why fraction multiplication works the way it does can develop slowly over time. With solving equations it can be a bit of both. Conceptual understanding often develops slowly, and as long as I don't rush things, memorizing how to solve one type of equation can grow into conceptual understanding over time. With lots of geometry topics, the conceptual understanding absolutely has to be strong first to connect procedures to the visual context. But the big idea is that conceptual understanding typically develops slowly, and memorizing some small pieces along the way can be helpful to contribute to the gradual development of conceptual understanding.
YES! YES! YES!
I completely agree with what you've written! I also think that students feel "dumb" more in math class compared to other subjects, because there is an impression of math as being a more "objective" subject than others... Some students feel that that there is only one correct answer and one correct method for every concept/problem and if you can't solve it (in your head) in the fastest way possible, you are "dumb." To compound this pervasive impression, some *teachers* teach in such a way to reinforce this perception by not celebrating different ways to reach a solution because it's not the conventional method, because it's a more inefficient method, or because the math teacher themselves don't understand why/how it works. When I was teaching 9th grade Algebra 1 students, I always felt that one of my biggest responsibilities was to turn my students' self-doubt into self-confidence, so that they could actually be open to learning in their remaining years in school.
That being said, I do have one worry - when the main conversation in a school becomes primarily about "helping students believe that they can learn" (i.e. when that becomes the focus of school-wide PD), I often find that it starts to skew towards *just* supporting students' social emotional health, and not about examining the structural and long-term changes that may need to happen in the school. For example, changes may be made to include discussions about growth mindset in each class or helping students combat negative self-talk or maybe techniques are taught for time management or creating SMART goals, etc. etc.. But rarely is that conversation expanded to include discussions about the need for a more vertically aligned math curriculum or critical examination of how we support students who don't have the prior knowledge to be successful in the current math classes. (Or other equivalent conversations for other subjects.)
In other words, I think that in order for schools to become spaces where we can help nurture students who believe they are "capable" learners, the time spent on understanding, building, and collaborating on the curriculum (i.e. the background stuff that doesn't seem as "student-facing") needs to be seen as valuable as discussing seemingly more immediate in-class issues. This is probably why more affluent schools where teachers have the time and mind-space to fully engage in these curricular discussions are often in a virtuous cycle of being able to nurture and benefit from inquisitive learners, as opposed to schools where teacher may often have to wear a gazillion other hats *in addition to* being just a teacher. (I say this as somebody who went from teaching in an inner-city public school to a very affluent private school. I think I was already a good teacher, in terms of being able to connect with students and helping them feel seen and valued... but I definitely became a much better *math* teacher and thus able to help students feel explicitly more successful in *math* once I didn't have to be everything for everyone all the time.)
I totally agree. For me the basic distinction is that I think academic success often precedes motivation. If a student isn't very motivated, one strategy to help is to get them some chances to feel successful academically, and the support to sustain that success. The default belief in many schools is the opposite: we see students who aren't motivated (and often aren't successful academically) and we try to convince them to be motivated through lots of conversations, social emotional stuff, etc.
Prioritizing academic success can solve lots of problems in schools. Not everything, but it's a huge lever. And if you spend a bunch of time and energy supporting students academically and it doesn't solve all of your behavior and motivation and attendance problems, well hey you still helped kids learn things and that's the core job of schools. If you spend a ton of time and energy on social-emotional well-being and that initiative doesn't move the needle (as is often the case) then that's a whole lot of wasted resources.
I agree with all of this! Great discussion too. A lot of the students who act out in school are also struggling academically -- and are often quite smart too.
The thing is, how do you get those students to experience successful moments? You have to go meet them where they are, and you have to use techniques that work for them. Right now, the prevailing wisdom tells us that we can only teach grade level standards. So, what do we do when kids don't have the background knowledge to learn those standards? We dumb down the standards so that they are "simple," with just memorization and mnemonics. That doesn't work. I think there is more learning and rigor involved in letting a student figure out how to solve a division word problem on their own (a 4th grade skill) than in teaching them steps to solve a linear equation (8th grade standard). And once they solve the problem on their own, they will know that they can do math, and they will build the motivation to learn the 8th grade skill.
Yea you point out a central contradiction in a lot of rhetoric about teaching today. Are we supposed to meet students where they are, or are we supposed to maintain grade-level rigor? People often say one or the other as if it's obvious, without wrestling with the challenges of trying to do both at the same time.
I am a tutor of 4th and 5th grade math. I work in an elementary school. The students I work with test below grade level. I have begun to question the need to "learn" math in the age of AI. Yuval Noah Harari has an intertering take. https://www.youtube.com/watch?v=h5_bullnfk0 Harari advocates the need for education to help develop mental flexibility and critical thinking skills as opposed to its traditional focus on rote learning and memorization. Thoughts?
I do think math is still worth learning. I don't think humans can learn mental flexibility or critical thinking without something to think or be flexible about. There's no easy shortcut to teach critical thinking directly, we teach critical thinking by teaching kids to think critically in all sorts of different contexts. I also don't think there's a sharp division between critical thinking and memorization. We begin with a shallow understanding of a concept, and then that understanding gradually gets deeper. We can't skip past those shallow levels of understanding.
I do think that we should rethink some of the exact content we teach. In the same way that we no longer teach how to use a slide rule, we should question exactly which computational skills are still worth learning. But if I was in charge, those changes would be relatively small.
Dylan, I agree completely that we need specific content to learn when we try to teach reasoning and critical thinking.
I'm curious what you mean when you say, "I also don't think there's a sharp division between critical thinking and memorization."
I certainly agree that these two things can be hard to distinguish on a test. Did the student reason their way through the problem, or did they memorize a series of steps to perform in a situation like this?
I also agree that memorizing a series of steps can be a useful jumping-off point for later critical thinking. When students have some sense of how to make their way through a problem, they can do things like compare methods, modify methods, and come up with new ways to apply those methods. I would even admit that some students will do some reasoning independently after they have been exposed to those series of steps to memorize. But none of this says that there is no sharp division between the two activities, but rather that they support each other. Agreed! They do!
Hmm, maybe what you are thinking is that when you introduce material, you do it in ways that shows students both how to think about it, and how to memorize it. So you don't need to distinguish in your own teaching, because you blend them. Or, maybe you what you really mean is that there is a blurry line between teaching procedures and teaching rich tasks designed for critical thinking? I agree with that. You can do a lot of critical reasoning around procedures.
However, when I see a class that is based entirely on rote memorization, I feel like that line is clear as day. "Yours is not to ask why. Just invert and multiply." If you stop there, I am sure that for 90% of the students, there will be no critical reasoning.
You make a bunch of great points. I think it's hard to make generalizations about memorization vs understanding because it depends so much on the context. For instance, I am fine with students memorizing how to multiply fractions because it's intuitively easy and a true understanding of why fraction multiplication works the way it does can develop slowly over time. With solving equations it can be a bit of both. Conceptual understanding often develops slowly, and as long as I don't rush things, memorizing how to solve one type of equation can grow into conceptual understanding over time. With lots of geometry topics, the conceptual understanding absolutely has to be strong first to connect procedures to the visual context. But the big idea is that conceptual understanding typically develops slowly, and memorizing some small pieces along the way can be helpful to contribute to the gradual development of conceptual understanding.
I think we are mostly in agreement. But I wrote this blog post to say more about what I'm thinking :)
https://simplyconnectedorg.wordpress.com/wp-admin/post.php?post=176&action=edit