I don’t know if other educators have seen a segment called, “My favorite ‘no’”. It discusses wrong answers of students and addresses their misconception. In the segment, the teacher chooses her favorite student misconception and projects it to a screen via document camera. She then proceeds to share with the students as to why she selected such a response as her “favorite no”. The students are allowed to comment as to why the answer is wrong, and volunteer information as to why the student might have made the error he, or she made. This blog entry reminded me of that segment. “My favorite ‘no’” is a great way to build knowledge of student misconceptions and create a classroom discussion regarding such misconceptions.
I LOVE "My Favorite 'No""! She has done one or two other videos as well, and I was hoping she would do them all the time. I'd follow her on YouTube if she made more.
Interesting idea but is there data to show it changes student thinking. Student ideas are often deeply help and not easy to change. However doing this often would create a good culture of metacognitive reflection. It would have to be done systematically not just once in a while
As a former adult student teacher I learned how stressful it can be when the adult students would get up and walk out during my lecture after telling me and the class that remained what they thought of my teaching approach. That experience caused me to search for what makes an effective teacher. I looked to Richard Feynman and what he wrote about the difference between knowing and understanding. And then I read what he wrote about teaching. I realized that there was so much that I thought that I knew but realized that I didn't understand what I was teaching at the time. I changed my entire approach. I asked myself why I would stay in a class that I was teaching. I used what I learned from that experience to change how I engaged my students and what they wanted to "get" out of the program. What followed changed my life forever. And I learned from Dr. Feynman that there is truly a difference between knowing and understanding and that that chasm is as wide as the Grand Canyon is long. Thank you Dr. Feynman. Thank you to my director for your patience at the time with me... for letting me continue and not terminating me when students walked out. And thank you Dan Meyer for the excellent work that you do.
I recently had the chance to serve as a sub (long story, but the district was dragging its heels on doing the paperwork for the full-time teacher) for a week for my daughter's grade 6 math class. As background, I'm a former secondary math educator who is currently a full-time doc student in education, so it's been a few years since I was in a math classroom, and I'd never taught a grade below 9th grade. Given that I had never taught 6th grade and I was only going to be there for a week, I needed *immediate* insight into how the 6th graders thought.
One of the most useful teaching strategies I used was asking students to purposely give a "wrong" answer to a question, and then reflect on why and how their colleagues may have arrived at that answer. I feel like this gave me important (and quick) insights into *how* students were thinking.
When I dug into physics education research, I didn't expect to relearn something I'd learned in conflict mediation training: people often can't soften their position until they feel heard.
It makes sense to me for teachers to study common misconceptions (calling them initial ideas has some advantages). I think it also makes sense for teachers to learn how to find out from their specific students, what those specific student think and, importantly, what experiences students have that lead them to think that way (in other words, to provide to the students the experience of being heard). Of course the proportion of "initial ideas" with reasons behind them isn't 100%; sometimes people guess in an exploratory way, or say things that don't make sense because they're tired or distracted. But it's not 0% either, and I'm pretty convinced that a major part of shifting those ideas lies in someone hearing them out, sometimes because that's the fastest way for the thinker themselves to more fully notice their own mind.
(Mostly I believe this because when I studied conflict mediation techniques designed for reducing sectarian violence, I started seeing major shifts in my students' understanding of atoms).
Learning what students "usually" think surely can of course be helpful. It can help us strengthen our ability to explore what "these" students think. Or it can become another list of rigid rules and tricks that we regurgitate in the very way we despair when students do. We as teachers will have to apply, to our own learning, the techniques we use to encourage students to explore alternatives to rule-seeking.
I think we also need to learn how to let go of fixing misconceptions in a single context, and instead work with the student to find new homes for the ideas. (This is one of my problems with rebranding corrections as "celebrating mistakes" without also doing some further digging). Most good-faith initial ideas that don't work in a certain context, the ones that have links to students' embodied experiences (and therefore the ones that are persistent and widespread, like, there is no gravity outside the atmosphere, or the earth is closest to the sun during the summer) are in fact good answers to other questions. We as teachers can practice understanding the idea well enough to find it a new home -- instead of killing it and dealing with its ghost haunting the old home (or repeatedly coming back to life in new clothes). We can also help students develop compassion for their own thinking (what question is this a good idea to? why might a reasonable person think this?) and, hopefully, for the thinking of others.
I'm almost entirely indebted for these ideas to people whose teaching made some of my own initial ideas more malleable: the developers of Dialog for Peaceful Change (http://www.dialogueforpeacefulchange.org), and physics educator and education researcher Brian Frank, especially these articles.
- on how noticing what we ourselves don't know, and what nobody fully knows yet, and getting better at noticing the connections between teacher and student not-knowing, can invite students to join us in a disciplinary practice of engaging with the unknown with humility and curiosity (https://teachbrianteach.blogspot.com/2011/05/addressing-misconceptions-ii.html)
Dan, just wanted to point out that Mathematical Knowledge for Teaching DOES include knowledge of common errors and misconceptions, but does not only include that. I appreciate you digging into this specific subset of MKT and your thinking around how math curriculum programs can improve to help with KOSM, but wanted to be sure people understood how comprehensive MKT is and that teachers’ MKT actually has been shown to have some pretty significant effect sizes when compared to other predictors of student mathematics achievement -- presumably due in part to what you talk about in this post!
I'm thinking about the connection between this and the "anticipating" phase of 5 practices for orchestrating productive mathematics discussions. I wonder if anticipating has been taken up less than other practices or is just one of the most difficult to do.
Like I assume most experienced teachers, I have long held that this KOSM (thanks for the coinage) is critical to helping students. Nothing informs my lesson planning and execution more than my past experiences with student misunderstandings. As I teach, I feel these past experiences and they literally create greater emphasis in my speaking, as I strive to make sure this is something they won't miss. It affects which activities I do and which I drop. This has got to be true of all of us.
But the line that really stood out to me this morning was this:
"Create tools that give teachers visibility into that thinking."
I am interested in seeing what can be done in this regard. A while back I abandoned most online assignments because I couldn't get the kids to show me their thinking. So I'm back to paper and love how I can see their misunderstandings, some common and some not, and then I can bring these misunderstanding to my classes via Elmo and Promethean. Few things are as valuable as learning from mistakes, no matter whose they are. (And a side benefit of this is that my kids lose their fear of being seen making mistakes, it's just not a big deal in my class.)
There is a substantial literature on misconceptions in science. Modern lesson plans often list common misconceptions. Reviewing lists if common misconceptions helps teachers to prepare but may help them see their own misconceptions
It might be worthwhile to compile lists or common math misconceptions orgsnized by topic
Some have stopped using the eird misconception because ofvuts negative connotation in favor of alternative conception
I think it is important to distinguish errors which are local from misconceptions that are more global and impact a broader range of thinking
We find in science that misconceptions are often strongly held so simple quick corrections have little impact. Rather, carefully designed lessons with low stakes testing in several related contexts is needed
(If you are asking to fill in the missing wrong answer on the visual pattern in Problem 4.3 then my bet is on 5+3n.) I did some teaching experiments on growth patterns like this one, saw some surprising results, talked to a few people who published on the topic, and came to a sad conclusion.
First, the conclusion. We know from research and teaching that "misconceptions" take tremendous effort to change. But why? As your KOSM/MSK chart shows, some teachers know what those frequent mistakes/misconceptions even are, and others do not, but the researchers seem to know. As Joe mentions below, there are lists of common misconceptions. "Humanity" knows, even if the knowledge is unevenly distributed. However, behind these known frequent mistakes/misconceptions, some foundational cornerstones of mathematical understanding seem to be missing. Otherwise, the misconceptions wouldn't be so consistent and so hard to deal with.
A few researchers seem to be looking for those cornerstones. For rare examples, see conceptual fields; grounding in embodied cognition; and whatever the Feuerstein Institute does (sorry, I can't think of a representative enough term from their method). It's very hard to find references to those missing cornerstone pieces of knowledge by topic. These are unknown unknowns; are we even looking for them? What's missing in each person's mathematical foundations that prevents the person from doing well with fractions? Growth patterns? Linear equations?
From my data, for the topic of growth patterns, like your Problem 4.3, the missing cornerstone seems to be the very PREDICTABILITY of patterns. You can see which students are missing that foundational piece of knowledge with the following task. 1 - Show them a few examples of growth patterns (say, from Fawn Nguyen's collection at visualpatterns.org). 2 - Invite them to make four steps of a growth pattern of their own, using some counters or by drawing, but hide Step #4 from you so you can figure it out. The striking result is that many students create random or unpredictable sequences! When asked to build patterns, they build non-patterns. Even people with As in algebra do that. "Step #4 in my pattern is this, because it's my pattern and I said so. Guess what I made up for Step #5!" The cornerstone idea that a pattern is a predictable sequence and that we must build patterns to be predictable (rather than one person making it up as they go along and another guessing) - that idea just isn't altogether there. (You can teach that after the above task, by contrast. It takes some doing.)
> As Joe mentions below, there are lists of common misconceptions. "Humanity" knows, even if the knowledge is unevenly distributed. However, behind these known frequent mistakes/misconceptions, some foundational cornerstones of mathematical understanding seem to be missing. Otherwise, the misconceptions wouldn't be so consistent and so hard to deal with.
I think the frequency of these early ideas may depend a bit on the instructional methods of the teacher and the content of different curricula. (For example: if you don't show enough contrast in a particular lesson, students are more likely to overgeneralize in later lessons.) But let's go with this idea that there is a list of common early ideas! I'm ripping out my hair trying to think about effective ways to teach that list and what to do with it.
Here is one thing we can do. It's hard but doable, and it works. For every early idea on The List, find several corresponding mature ideas in existing mathematics. That is, work out promising pathways from that early idea into modern mathematics. As students explore the pathways, their early ideas grow nicely.
What does it mean to work out a pathway? I talk to teachers about roots and wings. Roots: embodied, grounded activities radically accessible without prerequisites. (My benchmark for accessibility is a five-year-old child.) Wings: connecting these accessible activities to the contexts of formal "higher" mathematics.
As an aside, modern mathematics does exist as a well-organized list: https://zbmath.org/static/msc2020.pdf As is, the list is useless for teaching, but it maps the territory.
Ironically, it takes relatively advanced mathematical ideas to create a robust grounding for early ideas. Even within the topics that look quite elementary! The wings inspire the roots.
For your blog post's example of explicit formulas for sequences, I tried this and that until something worked. I found that my students grow their early ideas decently well by grounding them in informal, embodied, accessible activities inspired by linear algebra (array and table operations), calculus (infinite sequences and series), and computer science (functions, iterations, and loops). I test the activities with literal five-year-olds to prototype the methods.
Daphne’s article shows one of the problems with AI in that it talks about moving the decimal point to help with scientific notation, which is incorrect teaching of what is really happening.
In addition to maybe not teaching math in ways that develop understanding, I just feel like it's so hit or miss whether or not the chatbot connects with the way the student is thinking about the math. "This is not how we did it in class" seems just as likely as "Ah I get it now."
When I interview prospective teacher candidates, I ask them to think like a test-maker and come up with the correct answer and 4 best wrong answers to build the multiple choices for several problems. Applicants' ability to think through misconceptions, common mistakes, and provide rationale into student thinking is part of our selection criteria.
When developing a new lesson or modifying and internalizing existing curriculum, teacher candidates learn to do the math multiple ways in an attempt to take on student perspectives and surface challenges. 2.0 is asking them to rehearse responses to different misconceptions and mistakes, and where those challenges aren't productive to plan in order to preempt them.
BTW, shout out to IM for including common misconceptions in their planning materials(!).
For additional research in the math space, check out this study, which replicates the Sadler study in math and found similar results with arguably much tighter methodology:
Hill, H. C., & Chin, M. (2018). Connections between teachers’ knowledge of students, instruction, and achievement outcomes. American Educational Research Journal, 55(5), 1076-1112.
I don’t know if other educators have seen a segment called, “My favorite ‘no’”. It discusses wrong answers of students and addresses their misconception. In the segment, the teacher chooses her favorite student misconception and projects it to a screen via document camera. She then proceeds to share with the students as to why she selected such a response as her “favorite no”. The students are allowed to comment as to why the answer is wrong, and volunteer information as to why the student might have made the error he, or she made. This blog entry reminded me of that segment. “My favorite ‘no’” is a great way to build knowledge of student misconceptions and create a classroom discussion regarding such misconceptions.
For those interested: here is the My Favorite No video Sarah references. https://www.youtube.com/watch?v=srJWx7P6uLE
I LOVE "My Favorite 'No""! She has done one or two other videos as well, and I was hoping she would do them all the time. I'd follow her on YouTube if she made more.
Interesting idea but is there data to show it changes student thinking. Student ideas are often deeply help and not easy to change. However doing this often would create a good culture of metacognitive reflection. It would have to be done systematically not just once in a while
She does do it systematically, and while it is only anecdotal, I think it at least changes their attitude about mistakes, reducing their fear.
I loved Daphne's observation: "So obviously you learn more from the teacher, but occasionally the teacher can ramble on."
As a former adult student teacher I learned how stressful it can be when the adult students would get up and walk out during my lecture after telling me and the class that remained what they thought of my teaching approach. That experience caused me to search for what makes an effective teacher. I looked to Richard Feynman and what he wrote about the difference between knowing and understanding. And then I read what he wrote about teaching. I realized that there was so much that I thought that I knew but realized that I didn't understand what I was teaching at the time. I changed my entire approach. I asked myself why I would stay in a class that I was teaching. I used what I learned from that experience to change how I engaged my students and what they wanted to "get" out of the program. What followed changed my life forever. And I learned from Dr. Feynman that there is truly a difference between knowing and understanding and that that chasm is as wide as the Grand Canyon is long. Thank you Dr. Feynman. Thank you to my director for your patience at the time with me... for letting me continue and not terminating me when students walked out. And thank you Dan Meyer for the excellent work that you do.
I recently had the chance to serve as a sub (long story, but the district was dragging its heels on doing the paperwork for the full-time teacher) for a week for my daughter's grade 6 math class. As background, I'm a former secondary math educator who is currently a full-time doc student in education, so it's been a few years since I was in a math classroom, and I'd never taught a grade below 9th grade. Given that I had never taught 6th grade and I was only going to be there for a week, I needed *immediate* insight into how the 6th graders thought.
One of the most useful teaching strategies I used was asking students to purposely give a "wrong" answer to a question, and then reflect on why and how their colleagues may have arrived at that answer. I feel like this gave me important (and quick) insights into *how* students were thinking.
When I dug into physics education research, I didn't expect to relearn something I'd learned in conflict mediation training: people often can't soften their position until they feel heard.
It makes sense to me for teachers to study common misconceptions (calling them initial ideas has some advantages). I think it also makes sense for teachers to learn how to find out from their specific students, what those specific student think and, importantly, what experiences students have that lead them to think that way (in other words, to provide to the students the experience of being heard). Of course the proportion of "initial ideas" with reasons behind them isn't 100%; sometimes people guess in an exploratory way, or say things that don't make sense because they're tired or distracted. But it's not 0% either, and I'm pretty convinced that a major part of shifting those ideas lies in someone hearing them out, sometimes because that's the fastest way for the thinker themselves to more fully notice their own mind.
(Mostly I believe this because when I studied conflict mediation techniques designed for reducing sectarian violence, I started seeing major shifts in my students' understanding of atoms).
Learning what students "usually" think surely can of course be helpful. It can help us strengthen our ability to explore what "these" students think. Or it can become another list of rigid rules and tricks that we regurgitate in the very way we despair when students do. We as teachers will have to apply, to our own learning, the techniques we use to encourage students to explore alternatives to rule-seeking.
I think we also need to learn how to let go of fixing misconceptions in a single context, and instead work with the student to find new homes for the ideas. (This is one of my problems with rebranding corrections as "celebrating mistakes" without also doing some further digging). Most good-faith initial ideas that don't work in a certain context, the ones that have links to students' embodied experiences (and therefore the ones that are persistent and widespread, like, there is no gravity outside the atmosphere, or the earth is closest to the sun during the summer) are in fact good answers to other questions. We as teachers can practice understanding the idea well enough to find it a new home -- instead of killing it and dealing with its ghost haunting the old home (or repeatedly coming back to life in new clothes). We can also help students develop compassion for their own thinking (what question is this a good idea to? why might a reasonable person think this?) and, hopefully, for the thinking of others.
I'm almost entirely indebted for these ideas to people whose teaching made some of my own initial ideas more malleable: the developers of Dialog for Peaceful Change (http://www.dialogueforpeacefulchange.org), and physics educator and education researcher Brian Frank, especially these articles.
- on how teacher thinking about misconceptions can be a springboard to talk moves that get traction (https://teachbrianteach.blogspot.com/2011/04/i-said-i-didnt-want-to-talk-about.html)
- on problems that can arise when we frame things as misconceptions https://teachbrianteach.blogspot.com/2011/04/i-guess-i-do-want-to-talk-about.html
- on how exploring misconceptions can help us discover what students are intrigued about (http://teachbrianteach.blogspot.com/2011/03/misconceptions-misconceived-example-of.html)
- on how noticing what we ourselves don't know, and what nobody fully knows yet, and getting better at noticing the connections between teacher and student not-knowing, can invite students to join us in a disciplinary practice of engaging with the unknown with humility and curiosity (https://teachbrianteach.blogspot.com/2011/05/addressing-misconceptions-ii.html)
Dan, just wanted to point out that Mathematical Knowledge for Teaching DOES include knowledge of common errors and misconceptions, but does not only include that. I appreciate you digging into this specific subset of MKT and your thinking around how math curriculum programs can improve to help with KOSM, but wanted to be sure people understood how comprehensive MKT is and that teachers’ MKT actually has been shown to have some pretty significant effect sizes when compared to other predictors of student mathematics achievement -- presumably due in part to what you talk about in this post!
I'm thinking about the connection between this and the "anticipating" phase of 5 practices for orchestrating productive mathematics discussions. I wonder if anticipating has been taken up less than other practices or is just one of the most difficult to do.
Like I assume most experienced teachers, I have long held that this KOSM (thanks for the coinage) is critical to helping students. Nothing informs my lesson planning and execution more than my past experiences with student misunderstandings. As I teach, I feel these past experiences and they literally create greater emphasis in my speaking, as I strive to make sure this is something they won't miss. It affects which activities I do and which I drop. This has got to be true of all of us.
But the line that really stood out to me this morning was this:
"Create tools that give teachers visibility into that thinking."
I am interested in seeing what can be done in this regard. A while back I abandoned most online assignments because I couldn't get the kids to show me their thinking. So I'm back to paper and love how I can see their misunderstandings, some common and some not, and then I can bring these misunderstanding to my classes via Elmo and Promethean. Few things are as valuable as learning from mistakes, no matter whose they are. (And a side benefit of this is that my kids lose their fear of being seen making mistakes, it's just not a big deal in my class.)
There is a substantial literature on misconceptions in science. Modern lesson plans often list common misconceptions. Reviewing lists if common misconceptions helps teachers to prepare but may help them see their own misconceptions
It might be worthwhile to compile lists or common math misconceptions orgsnized by topic
Some have stopped using the eird misconception because ofvuts negative connotation in favor of alternative conception
I think it is important to distinguish errors which are local from misconceptions that are more global and impact a broader range of thinking
We find in science that misconceptions are often strongly held so simple quick corrections have little impact. Rather, carefully designed lessons with low stakes testing in several related contexts is needed
"I think it is important to distinguish errors which are local from misconceptions that are more global and impact a broader range of thinking"
Excellent point.
(If you are asking to fill in the missing wrong answer on the visual pattern in Problem 4.3 then my bet is on 5+3n.) I did some teaching experiments on growth patterns like this one, saw some surprising results, talked to a few people who published on the topic, and came to a sad conclusion.
First, the conclusion. We know from research and teaching that "misconceptions" take tremendous effort to change. But why? As your KOSM/MSK chart shows, some teachers know what those frequent mistakes/misconceptions even are, and others do not, but the researchers seem to know. As Joe mentions below, there are lists of common misconceptions. "Humanity" knows, even if the knowledge is unevenly distributed. However, behind these known frequent mistakes/misconceptions, some foundational cornerstones of mathematical understanding seem to be missing. Otherwise, the misconceptions wouldn't be so consistent and so hard to deal with.
A few researchers seem to be looking for those cornerstones. For rare examples, see conceptual fields; grounding in embodied cognition; and whatever the Feuerstein Institute does (sorry, I can't think of a representative enough term from their method). It's very hard to find references to those missing cornerstone pieces of knowledge by topic. These are unknown unknowns; are we even looking for them? What's missing in each person's mathematical foundations that prevents the person from doing well with fractions? Growth patterns? Linear equations?
From my data, for the topic of growth patterns, like your Problem 4.3, the missing cornerstone seems to be the very PREDICTABILITY of patterns. You can see which students are missing that foundational piece of knowledge with the following task. 1 - Show them a few examples of growth patterns (say, from Fawn Nguyen's collection at visualpatterns.org). 2 - Invite them to make four steps of a growth pattern of their own, using some counters or by drawing, but hide Step #4 from you so you can figure it out. The striking result is that many students create random or unpredictable sequences! When asked to build patterns, they build non-patterns. Even people with As in algebra do that. "Step #4 in my pattern is this, because it's my pattern and I said so. Guess what I made up for Step #5!" The cornerstone idea that a pattern is a predictable sequence and that we must build patterns to be predictable (rather than one person making it up as they go along and another guessing) - that idea just isn't altogether there. (You can teach that after the above task, by contrast. It takes some doing.)
That hidden belief that patterns don't have to be predictable is one of the most bizarre cognitive glitches I've ever encountered. It's like the Wason selection task https://en.wikipedia.org/wiki/Wason_selection_task or "Heavy boots on the Moon" http://www.phys.ufl.edu/~det/phy2060/heavyboots.html (I tried both with real students; the tasks confuse people just as described.)
> As Joe mentions below, there are lists of common misconceptions. "Humanity" knows, even if the knowledge is unevenly distributed. However, behind these known frequent mistakes/misconceptions, some foundational cornerstones of mathematical understanding seem to be missing. Otherwise, the misconceptions wouldn't be so consistent and so hard to deal with.
I think the frequency of these early ideas may depend a bit on the instructional methods of the teacher and the content of different curricula. (For example: if you don't show enough contrast in a particular lesson, students are more likely to overgeneralize in later lessons.) But let's go with this idea that there is a list of common early ideas! I'm ripping out my hair trying to think about effective ways to teach that list and what to do with it.
Here is one thing we can do. It's hard but doable, and it works. For every early idea on The List, find several corresponding mature ideas in existing mathematics. That is, work out promising pathways from that early idea into modern mathematics. As students explore the pathways, their early ideas grow nicely.
What does it mean to work out a pathway? I talk to teachers about roots and wings. Roots: embodied, grounded activities radically accessible without prerequisites. (My benchmark for accessibility is a five-year-old child.) Wings: connecting these accessible activities to the contexts of formal "higher" mathematics.
As an aside, modern mathematics does exist as a well-organized list: https://zbmath.org/static/msc2020.pdf As is, the list is useless for teaching, but it maps the territory.
Ironically, it takes relatively advanced mathematical ideas to create a robust grounding for early ideas. Even within the topics that look quite elementary! The wings inspire the roots.
For your blog post's example of explicit formulas for sequences, I tried this and that until something worked. I found that my students grow their early ideas decently well by grounding them in informal, embodied, accessible activities inspired by linear algebra (array and table operations), calculus (infinite sequences and series), and computer science (functions, iterations, and loops). I test the activities with literal five-year-olds to prototype the methods.
Daphne’s article shows one of the problems with AI in that it talks about moving the decimal point to help with scientific notation, which is incorrect teaching of what is really happening.
In addition to maybe not teaching math in ways that develop understanding, I just feel like it's so hit or miss whether or not the chatbot connects with the way the student is thinking about the math. "This is not how we did it in class" seems just as likely as "Ah I get it now."
When I interview prospective teacher candidates, I ask them to think like a test-maker and come up with the correct answer and 4 best wrong answers to build the multiple choices for several problems. Applicants' ability to think through misconceptions, common mistakes, and provide rationale into student thinking is part of our selection criteria.
When developing a new lesson or modifying and internalizing existing curriculum, teacher candidates learn to do the math multiple ways in an attempt to take on student perspectives and surface challenges. 2.0 is asking them to rehearse responses to different misconceptions and mistakes, and where those challenges aren't productive to plan in order to preempt them.
BTW, shout out to IM for including common misconceptions in their planning materials(!).
For additional research in the math space, check out this study, which replicates the Sadler study in math and found similar results with arguably much tighter methodology:
Hill, H. C., & Chin, M. (2018). Connections between teachers’ knowledge of students, instruction, and achievement outcomes. American Educational Research Journal, 55(5), 1076-1112.
Love that interview protocol. Thanks for lit reference also.