Thank you for sharing I loved hearing the student thinking as they made sense and accessed their resources to evaluate how they were matching cards. Your questions made me wonder if direct instruction was necessary or rather a facilitation to stamp co-constructing understanding based on their experience with the cards. Something like posting a symbol on the board and have students "list all the things they are expecting to see in a graph/statement of that symbol". Quickly generate from the students a list in an anchor chart/notes or a place for reference. I used to do summaries in my secondary math classrooms using this method.
Kids are awesome. I love how the student switches his answer when asked for it again. I only asked him to repeat it because I didn't hear, not because I was trying to press him!
I hear you on the value of co-construction. Sometimes it really feels to me like that's insufficient, though, and we need something quite a bit more authoritative.
FWIW the teacher whose class I was borrowing (excellent teacher IMO) put up a slide at the end that showed each of the four signs and their english language translation and said "put 'em in your notes." Loved that. It spoke to the difference between a tourist in the classroom and someone who's responsible for helping students develop more ideas tomorrow.
Regarding remembering the symbols for inequalities, I like to encourage students to "read" the comparison like a sentence - left to right. It seems like an easier way to connect the schema of the alligator that they may have been taught early on to the vocabulary of inequalities; the open side of the symbol is "bigger" or greater than the vertex, hence the left value is greater than the right value in the inequality statement and vice versa.
I'd love some other ideas too!
PS - Thanks for sharing the Amazon review. We'll be discussing this real world situation tomorrow!
They had a great time noticing all the misconceptions and debating how one may arrive at these conclusions. An excellent opportunity to remind ourselves of the value of math in the real world and the need for clarity in our thinking. Thanks again!
Thinking about the bigger side of the symbol indicating the bigger value also works with the equals sign — both sides are the same. But with an inequality, the lines of the equals sign tilt so that one side is bigger and the other is smaller.
Now don't tell this idea to my 5yo or he'll probably want to draw a symbol that's in between an equals and an inequality for numbers that are only a little bit different :P
A natural way to say this in words is "x is greater than 10 and less than 100." But in this case, the same symbol is standing for greater than and less then, we're just reading it in different directions.
Always reading left-to-right seems like it could be helpful to avoid initial confusion but I think we outgrow it as we become more comfortable with the notation. At least I don't think about it strictly as left-to-right. Maybe others do?
In this case, you're teaching something that's arbitrary, since mathematical notation is determined by convention rather than by fundamental truths about how the world works. I wonder if that category of knowledge is more efficiently taught via direct instruction. At the very least I think it requires a lot of modeling, so that students have enough exposure that they can pick up on the language through immersion.
Providing a reference for translating symbols into familiar language might be helpful — I'm imagining a poster-sized cheat-sheet that you write out and stick on the wall where it's easy for students to glance at if they need a reminder of which symbol is which.
I like how you're thinking here. It's very difficult to intuit our way to social conventions that are, by their nature, arbitrary. Though I like the idea that they should be intuitive in hindsight. (eg. Less than and greater than are opposite descriptions so it makes sense that their symbols have an opposite-ness to them.)
I had a student once say that = made sense because the two lines were equally far apart (parallel) but this symbol, <, meant that the number on the left was smaller because the two lines were now touching, meaning that the distance was less on the left than the right...and vice versa. I used that student's description often after that moment.
I think there is value in students being able to read inequalities in both directions. For instance, one of the cards had "0<x" which can be interpreted simultaneously as "0 is less than x" and equivalently "x is greater than 0." The value of this, in my opinion, not only helps students see relationship oriented statements meaningfully and flexibly, but also helps in other areas, such as why |2-x| is equivalent to |x-2|.
I like to ask kids “what have you heard about ways to read this symbol?” or “how many of you have heard ‘the alligator eats the bigger number’? What else?” All of these ways are good if they work for you consistently.
Then I tell them my favorite way to remember it is “the big end of the symbol points the the bigger/greater number AND THE SMALL END POINTS TO THE SMALLER NUMBER.” I like it because you can read it both directions.
I write on the board:
Smaller number < Bigger number
Bigger number > Smaller number
(or maybe I use lesser/greater?)
I tell kids we start with where x is and then explain what the symbol tells us about it.
Then I’d have them practice reading off examples like what 5<x says. It would be cool to have students try matching both ways to say this “x is greater than 5” and “5 is less than x”.
Student opinions and contribution to the convention come in asking which way sounds better to them, or is easier to use to reason about x, and which is easier to generate from looking at the inequality as written.
As Celeste points out in another comment, being able to read an inequality symbol in both directions is pretty important as we deal with more complex inequalities, even things written like 2 < x. We’re used to reading math sentences from left to right. But in this sort of situation we actually need to start with where x is and explain the stuff around it.
That reminds me of some article I read back in my credential program, about how students need to shift from an calculation/answer understanding of the = symbol (do the stuff on the left to get/put the answer on the right) to a relationship understanding of = (the value of the stuff on the left is the same as the value of all the stuff on the right).
Right now I’m teaching one variable stats and I realized that there are a lot of conventions that students draw on from other contexts that don’t apply here. Like normally you read a graph left to right, but with stats you start with where the data is, where it’s centered, and then expand out from there. Also kids are used to checking for exact symmetry in middle school geometry, but with stats it doesn’t have to be perfect. And we’ve learned about range as a crude measure of variability, but that word is going to mean something slightly different when we get to functions! These hidden conventions are important for me to signpost explicitly, otherwise I’m just giving advantage to kids who have picked up on those conventions implicitly. If I could get kids to realize that and share the strategy with the class, great! But I can just say it too, and I want everyone to have practice using it.
When learning the < or > symbols I rely on what the nuns taught more than 60 years ago. The less than symbol is < when you rotate this symbol it turns into an L which stands for LESS than. If you can't make an L with it then it is greater than > I know pretty simple but easy to remember.
On when to use direct instruction: my guide is usually on whether I would be willing to say "no" to a kid. If a student is reasoning through something and arrives at a wrong answer, I am never going to say "no" or correct them, but try to show them their own thinking and see what they can figure out with the class. If a student starts by saying there are 26 minutes when the question says 62, I might quickly offer a correction without feeling bad about it. Since I never want to shut down a kid's thinking, I feel like I know when something is direct instruction when I would be willing to correct them in the moment without feeling like I am shutting down their ideas. Reading x < 7 as "is more than 7" seems like vocabulary work that students need to be taught and then have more and more experiences with in scenarios where the context can help reinforce their budding facility of new language.
In your video, I really cannot say how I would have handled that situation with your students in your classroom but I can see you trying to leverage the understanding they comfort with the symbols that they currently have (and realizing that it was pretty variable). From the sidelines, I would have been tempted to have students give me english language sentences like "17 is less than 23" before I wrote a translation of "17 < 23." After I translated a couple for them, then asking kids to come up with their own translation and then asking the question you did about how they remember might leverage their wealth of knowledge a little more clearly once primed. As always, this feels like the least sexy part of any lesson but knowing they got so much experience with the Desmos representations makes a big difference.
Yeah like I'm sweating buckets of blood trying to draw out existing ideas, connect them, develop them, etc, and then I'm caught flatfooted by a moment that requires a different skill entirely, maybe even a skill that's much EASIER.
When I plan for math class, I'm often focused on the first skill set you named: drawing out existing ideas, connecting them, developing them, pushing student thinking forward. One reason I think those 'direct instruction' moments, like the grammar of mathematical inequalities, can feel stickier for me is because it goes against these instincts. I want to help students make connections to prior learning, and when it's fast and in the moment, I may not know what that learning is. I'm worried that I'll overwrite some great thinking with some shorthand I've developed, and that this might confuse some kids.
One of the least successful lessons I've facilitated in recent memory was when I was asked, ten minutes before students arrived, to teach the long division algorithm. Like... what? (And I am very fluent with long division! I swear! I'm so good!) I had not met these students before, and I knew effectively nothing about their background except what the teacher had taught the previous day. It was a full on nightmare for everyone in the room.
When I plan for that direct instruction -- e.g. knowing that students may struggle with the convention of symbols -- it goes down much smoother. It's okay to make a statement.
Yeah I like the idea of a lesson plan that helps teachers think about selecting and sequencing student work to drive discussion and develop big ideas around connecting representations with some fine print that says fyi at some point you’re going to have to tell kids exactly what the smooshed sideways L thing means when it faces in each direction and maybe include some reminders from time to time.
Thank you for sharing I loved hearing the student thinking as they made sense and accessed their resources to evaluate how they were matching cards. Your questions made me wonder if direct instruction was necessary or rather a facilitation to stamp co-constructing understanding based on their experience with the cards. Something like posting a symbol on the board and have students "list all the things they are expecting to see in a graph/statement of that symbol". Quickly generate from the students a list in an anchor chart/notes or a place for reference. I used to do summaries in my secondary math classrooms using this method.
Kids are awesome. I love how the student switches his answer when asked for it again. I only asked him to repeat it because I didn't hear, not because I was trying to press him!
I hear you on the value of co-construction. Sometimes it really feels to me like that's insufficient, though, and we need something quite a bit more authoritative.
FWIW the teacher whose class I was borrowing (excellent teacher IMO) put up a slide at the end that showed each of the four signs and their english language translation and said "put 'em in your notes." Loved that. It spoke to the difference between a tourist in the classroom and someone who's responsible for helping students develop more ideas tomorrow.
Regarding remembering the symbols for inequalities, I like to encourage students to "read" the comparison like a sentence - left to right. It seems like an easier way to connect the schema of the alligator that they may have been taught early on to the vocabulary of inequalities; the open side of the symbol is "bigger" or greater than the vertex, hence the left value is greater than the right value in the inequality statement and vice versa.
I'd love some other ideas too!
PS - Thanks for sharing the Amazon review. We'll be discussing this real world situation tomorrow!
Would love to know what your students do with the review. Circle back around here!
They had a great time noticing all the misconceptions and debating how one may arrive at these conclusions. An excellent opportunity to remind ourselves of the value of math in the real world and the need for clarity in our thinking. Thanks again!
Thinking about the bigger side of the symbol indicating the bigger value also works with the equals sign — both sides are the same. But with an inequality, the lines of the equals sign tilt so that one side is bigger and the other is smaller.
Now don't tell this idea to my 5yo or he'll probably want to draw a symbol that's in between an equals and an inequality for numbers that are only a little bit different :P
And to throw a wrench in the works:
10 < x < 100
A natural way to say this in words is "x is greater than 10 and less than 100." But in this case, the same symbol is standing for greater than and less then, we're just reading it in different directions.
Always reading left-to-right seems like it could be helpful to avoid initial confusion but I think we outgrow it as we become more comfortable with the notation. At least I don't think about it strictly as left-to-right. Maybe others do?
In this case, you're teaching something that's arbitrary, since mathematical notation is determined by convention rather than by fundamental truths about how the world works. I wonder if that category of knowledge is more efficiently taught via direct instruction. At the very least I think it requires a lot of modeling, so that students have enough exposure that they can pick up on the language through immersion.
Providing a reference for translating symbols into familiar language might be helpful — I'm imagining a poster-sized cheat-sheet that you write out and stick on the wall where it's easy for students to glance at if they need a reminder of which symbol is which.
I like how you're thinking here. It's very difficult to intuit our way to social conventions that are, by their nature, arbitrary. Though I like the idea that they should be intuitive in hindsight. (eg. Less than and greater than are opposite descriptions so it makes sense that their symbols have an opposite-ness to them.)
Agreed! Reminds me of https://xkcd.com/567/
I had a student once say that = made sense because the two lines were equally far apart (parallel) but this symbol, <, meant that the number on the left was smaller because the two lines were now touching, meaning that the distance was less on the left than the right...and vice versa. I used that student's description often after that moment.
I think there is value in students being able to read inequalities in both directions. For instance, one of the cards had "0<x" which can be interpreted simultaneously as "0 is less than x" and equivalently "x is greater than 0." The value of this, in my opinion, not only helps students see relationship oriented statements meaningfully and flexibly, but also helps in other areas, such as why |2-x| is equivalent to |x-2|.
I like to ask kids “what have you heard about ways to read this symbol?” or “how many of you have heard ‘the alligator eats the bigger number’? What else?” All of these ways are good if they work for you consistently.
Then I tell them my favorite way to remember it is “the big end of the symbol points the the bigger/greater number AND THE SMALL END POINTS TO THE SMALLER NUMBER.” I like it because you can read it both directions.
I write on the board:
Smaller number < Bigger number
Bigger number > Smaller number
(or maybe I use lesser/greater?)
I tell kids we start with where x is and then explain what the symbol tells us about it.
Then I’d have them practice reading off examples like what 5<x says. It would be cool to have students try matching both ways to say this “x is greater than 5” and “5 is less than x”.
Student opinions and contribution to the convention come in asking which way sounds better to them, or is easier to use to reason about x, and which is easier to generate from looking at the inequality as written.
As Celeste points out in another comment, being able to read an inequality symbol in both directions is pretty important as we deal with more complex inequalities, even things written like 2 < x. We’re used to reading math sentences from left to right. But in this sort of situation we actually need to start with where x is and explain the stuff around it.
That reminds me of some article I read back in my credential program, about how students need to shift from an calculation/answer understanding of the = symbol (do the stuff on the left to get/put the answer on the right) to a relationship understanding of = (the value of the stuff on the left is the same as the value of all the stuff on the right).
Right now I’m teaching one variable stats and I realized that there are a lot of conventions that students draw on from other contexts that don’t apply here. Like normally you read a graph left to right, but with stats you start with where the data is, where it’s centered, and then expand out from there. Also kids are used to checking for exact symmetry in middle school geometry, but with stats it doesn’t have to be perfect. And we’ve learned about range as a crude measure of variability, but that word is going to mean something slightly different when we get to functions! These hidden conventions are important for me to signpost explicitly, otherwise I’m just giving advantage to kids who have picked up on those conventions implicitly. If I could get kids to realize that and share the strategy with the class, great! But I can just say it too, and I want everyone to have practice using it.
I'd be interested in trying a teaching move where you introduce the concept of symbols in English.
Maybe put these symbols on the board or in a sentence, and start the discussion by having a student read the sentence:
I got the J & J vaccine.
I'm having a great day. #Fridays!
What's your phone #?
Can you call me @ home?
Then maybe give some explicit instruction on how < and > are just symbols that we use as we read math from left to right.
When learning the < or > symbols I rely on what the nuns taught more than 60 years ago. The less than symbol is < when you rotate this symbol it turns into an L which stands for LESS than. If you can't make an L with it then it is greater than > I know pretty simple but easy to remember.
On when to use direct instruction: my guide is usually on whether I would be willing to say "no" to a kid. If a student is reasoning through something and arrives at a wrong answer, I am never going to say "no" or correct them, but try to show them their own thinking and see what they can figure out with the class. If a student starts by saying there are 26 minutes when the question says 62, I might quickly offer a correction without feeling bad about it. Since I never want to shut down a kid's thinking, I feel like I know when something is direct instruction when I would be willing to correct them in the moment without feeling like I am shutting down their ideas. Reading x < 7 as "is more than 7" seems like vocabulary work that students need to be taught and then have more and more experiences with in scenarios where the context can help reinforce their budding facility of new language.
In your video, I really cannot say how I would have handled that situation with your students in your classroom but I can see you trying to leverage the understanding they comfort with the symbols that they currently have (and realizing that it was pretty variable). From the sidelines, I would have been tempted to have students give me english language sentences like "17 is less than 23" before I wrote a translation of "17 < 23." After I translated a couple for them, then asking kids to come up with their own translation and then asking the question you did about how they remember might leverage their wealth of knowledge a little more clearly once primed. As always, this feels like the least sexy part of any lesson but knowing they got so much experience with the Desmos representations makes a big difference.
Thanks as always!
Yeah like I'm sweating buckets of blood trying to draw out existing ideas, connect them, develop them, etc, and then I'm caught flatfooted by a moment that requires a different skill entirely, maybe even a skill that's much EASIER.
When I plan for math class, I'm often focused on the first skill set you named: drawing out existing ideas, connecting them, developing them, pushing student thinking forward. One reason I think those 'direct instruction' moments, like the grammar of mathematical inequalities, can feel stickier for me is because it goes against these instincts. I want to help students make connections to prior learning, and when it's fast and in the moment, I may not know what that learning is. I'm worried that I'll overwrite some great thinking with some shorthand I've developed, and that this might confuse some kids.
One of the least successful lessons I've facilitated in recent memory was when I was asked, ten minutes before students arrived, to teach the long division algorithm. Like... what? (And I am very fluent with long division! I swear! I'm so good!) I had not met these students before, and I knew effectively nothing about their background except what the teacher had taught the previous day. It was a full on nightmare for everyone in the room.
When I plan for that direct instruction -- e.g. knowing that students may struggle with the convention of symbols -- it goes down much smoother. It's okay to make a statement.
Yeah I like the idea of a lesson plan that helps teachers think about selecting and sequencing student work to drive discussion and develop big ideas around connecting representations with some fine print that says fyi at some point you’re going to have to tell kids exactly what the smooshed sideways L thing means when it faces in each direction and maybe include some reminders from time to time.