Really interesting to see how this prediction contest frames parent communication as a "non-interpersonal" activity! Of course some parent communication teachers do is pretty mechanical. But I think it also almost always involves some social, emotional, or relationship-building aspect, which they go on to say would be excluded. Caregivers are humans, too!
Speaking of interpersonal communication, I've seen a big increase in student emails that start with "I hope this email finds you well." An interesting phrase, and apparently quite popular with young people today, maybe it's like that 6/7 thing, or maybe something else is going on.
Loving Rohr's quotes here. Thanks to you and Fawn continually reminding those of us who are math leaders far removed from classrooms of the power of spending regular time teaching in classrooms, I am now doing it quite regularly. I need dust off my blog to write publically about it so help hold me accountable to that.
This question right here is the chef's kiss (as my 11 year old would say): "How do you “press” 30 kids to develop important conceptual knowledge when the easy operational knowledge is right there?"
I'm also left wondering if that building of conceptual knowledge must always intersect with students better understanding the algorithm. For example, proving the quadratic formula. I'm not sure if in a pinch I could rederive that proof right here as I type, but I have plenty of conceptual knowledge developed and stored away in long-term memory that I can solve any quadratic-related problem you throw at me without quite remembering the proof or why each coefficient and operation in the formula is what it is. Heck. Even if I have forgotten the quadratic formula, I have a whole toolbox of conceptual knowledge that would help me solve anything related to the topic.
I've found that conceptual knowledge always has a far more long-term lasting effect than procedural knowledge and that having a deep conceptual understanding often helps you do math when the formula or procedure may be long-forgotten. This happens to me often where kids (often my own high school kid) ask me for help and I have completely forgot the procedures on how to do the problem, but can figure it out because I retain the conceptual knowledge. It doesn't always mean I can re-derive the formula, but that the conceptual understanding I have retained endures so I can still solve the problem. Annnnnnnd....in the moment, with 30 students (more like 35+ in many public high schools) are awaiting your instruction, I know why pressing on with subtract, subtract, divide is where many turn. Thanks for that chef's kiss of a question to continue to ponder.
I think you're absolutely right to avoid subtract, subtract, divide. And I think it's worth avoiding for a very specific reason. Learning works best when new ideas are securely attached to solid existing knowledge. So what's the existing knowledge we want to attach it to?
I want students to know that slope is rise / run. I want students to know that subtracting is a way of finding the distance between two points. I want students to be able to picture a coordinate plane in their mind when they hear an x,y pair. I want students to intuit a rough estimate for a slope before calculating.
So here are a bunch of questions I would ask students along the way that I think can help create some of those connections.
Warmup - write a bunch of points on a plane given coordinates
How far apart are these points? (horizontal and vertical only, given a picture)
How far apart are these points? (horizontal and vertical only, no picture, start with points that are very close and then move them apart)
How far apart are these points horizontally? Vertically? (picture, then no picture, same thing start close then move farther apart)
Here are two points, with the horizontal and vertical distances labeled. What's the slope?
Here are two points. Will the slope be positive or negative? What's a good estimate? Would it be closer to 2 or 1/2?
Here are two points, picture given, very close together. What's the slope?
(It's been a while since I taught 8th grade. I bet if I had to teach this again, I would have a bunch more little micro questions to add in there afterwards.)
All of that work prepares students for the subtract, subtract, divide. The algorithm makes much more sense and sticks better because it's connected to all this other knowledge students have that we want to build on.
I think something you get at that's right on is that this is a very individual process. In my teaching, I'm a huge fan of giving students sequences of problems, building from simple to complex along the lines of what I just described. I want to launch with some framing and practice/review of the steps we've seen so far, support each student in making those new connections, and then bring the class together to share out some of the connections they made.
The catch -- I don't think it's possible to get through the entire sequence I lay out above in a day or even two days. My goal is to move a few more steps along that ladder each day. Review what we did the day before, practice a bit, then try to climb a few more rungs. Summarize, consolidate, then get a little further the next day. It's a very patient type of teaching, and a huge part is identifying small steps that I want students to take along the way.
I’ve always used tables to get to the algorithm, and I think the same approach could be adapted graphically.
I give students tables to fill out- “Find the pattern!” The first tables have three or four pairs filled in and ask for the next one or two or four. The X-values are filled in. There is a spot to write the slope “grow-by” below the table. You can be as creative as you like- negative slope, fraction slope with price denominator, negative X-values. Some slopes like 31 where a student needs to subtract. Maybe a slope like 29 where a student definitely has to subtract. The last problem in this sequence might have the missing y-value at the beginning instead of the end. These all provide consecutive x-value number pairs.
The next set of tables provides two non-consecutive y-values (again, all c-values provided). (1,3) (2, ___), (3, 7). Make bigger gaps, use fractions, negative x’s and y’s, negative slopes. Eventually, depending on student ability, make some that make subtraction and division by two clear, in advance for many kids but maybe only with a reflection question after for others: (7, 22) (8, ___) (9, 32)
Next, open up the gaps. Still a fun puzzle. Easy and hard. Guess and check. Eventually something like,
(4, 20) (5, ___) (it is a table, so still including every x-value with blank y) (14, 20).
Similar with gaps of 2. Maybe graduate to (4,22) (9,72) or (10, 100) (50, 180) . Now I’m putting …. In the table instead of every x. Again, including negative slopes as well. Prompt some reflection questions about which operations they used. Work in variables gradually. Eventually: (x1, y1) and (x2, y2).
Voila, most kids solved a bunch of puzzles and ended up finding the formula/algorithm in their own. I value discovery more than others, because I’m fighting “I don’t understand! Where did this come from?!?” (It came from you!), but you could debate whether this gets you to deeper understanding or not.
"Learning works best when new ideas are securely attached to solid existing knowledge."
So true, I think with slope some existing knowledge you can connect to is rates, any phrase with the word "per" in it. Why are we dividing? What order you do divide in? Because I find when it's framed as a rate question ("I drive to Chicago, it was 200 miles and took me 3 hours, how fast did I drive, on average?") a couple things happen: Students seem to know they should divide, and students seem to know what order to divide (nobody does 7/200 and nearly everybody does 200/7) I always point that out: "YOU knew to divide, I didn't tell you, and YOU knew the correct order!" Gives us something to build on.
I have been thinking about this for a few weeks since I read the post. I have been using illustrative math problems primarily, so picture a graph of a line, showing a context from a word problem. Let’s use this warmup to show the context. https://accessim.org/6-8/grade-8/unit-3/section-b/lesson-6?a=student
Both the scales are different on the x and y axis. When I ask students what number plus 1 is 4? I will write on their desk with a whiteboard marker, 1 + x =4. And what number plus 70 is 160? 70 + y is 160. Does this conceptually help? I think so. I was so excited to figure out that this is one of those places to review one step equations before getting to the next unit, which is a nightmare for students who struggle with understanding inverse operations to access.
And yes. It’s far easier to teach, subtract, subtract, and divide, but what if the numbers are negative? What if the line is decreasing, what if the slope of the line is positive? Will this strategy help students to solve all problems? What about future problems? Does the context help understanding a rate of change?
I (coach) co-taught with a teacher the other day who was doing this exact topic. We used an Open Middle problem to start, which led to some great generalizations about slope, even though the students already knew the formula. I also think questioning plays a huge role in drawing out the conceptual ideas when the skill is pretty straightforward (credit to the math coordinator at this school who suggested this). What does the slope tell us about this situation? How did you start to figure this out? Is there another way you could have figured this out? What happens to the y-intercept when these are the two points you have? I'm still working on the questions, but I do think that's a key part here.
Math teachers: I stopped reading at slope because I wanted to reply: I recently was teaching slope to my "at-risk" high school math students. I was, once again, wondering what was so difficult about the concept. Suffice it say, I did come up with a nice exercise that seemed to help a few more students over the wall:
1. I asked if they realized that if you build stairs in your house that are too steep, it would be "illegal" because it could be dangerous. And that less steep stairs can be inefficient because they might take up too much space in a house.
2. I then gave each pair of students 12 identical Lego bricks.
3. I asked them to make 3 stair cases, using 3 bricks each, of different steepness.
4. After checking their stairs, I had them lay the 3 cases sideways on paper and had them trace it.
5. I then had them determine and record the rise and run of each of the stair cases. I checked those too.
6. Finally, they were to connect the top to the bottom and draw that line with a ruler. At that point, I asked them to record, "What do you notice about the lines you just drew and the slopes of the stairs?" We had conversations including the fact that some people's stairs sloped downward to the right and others up. So, we also made that connection to positive and negative slope. Some noticed that larger slopes signified steeper stairs (and lines). In following days, when students were working on slope we would refer back to our "Lego stairs" to conceptualize what was going on.
I would be asking questions like the ones you were asking, at the group or whole class level, giving them some time to think and discuss, and then sharing what they're thinking with the whole class (maybe that's what you *were* doing(?)). For this particular topic, I'd also be asking "What *does* happen if we divide the x-change by the y-change? Does that still seem to describe how "steep" the graph is?" "What's the difference between one that has a negative slope and one with a positive slope? Why?" (small differences in Desmos graphs would be great here).
But really, I think there isn't a shortcut on the conceptual work.
Thanks for the shoutout. It's a good exercise (for me) to figure out how to apply that here.
The first thing: what does it mean to develop an "intuition about slope"? I suggest in that post it means knowing generally true things about slope. So, what are some of those things? Let me try to tease out some of the ideas I think are floating around in the post and express them as generalizations:
* You can tell if a slope should be positive or negative by starting with the leftmost point and deciding if the next point goes up or down.
* The slope will be negative if when, after you subtract coordinates, one difference is positive and the other is negative.
* The "rise" is the change in the vertical height between points.
* The slope of a line will always be equal to a rate of change (ys per x).
We could definitely come up with questions or activities that would target any of these particular ideas.
Is the teaching of any of these impaired by the "subtract, subtract, divide" procedure? I'm not sure. I can't really see how, to be honest.
Is it important to dig around with these things without getting specific and explicit? Is there a mucking around stage that is crucial for developing understanding before we introduce "subtract, subtract, divide"?
In this case, for these pieces of understanding, I don't think there is. We can teach these things, and knowing the slope formula is only an asset to that learning. I say, get the formula out of the way, then flesh it out with interesting challenges that get at these general principles to flesh out a full understanding of slope and its meaning.
> In this case, for these pieces of understanding, I don't think there is.
If students can look at a line and say "that slope is positive, that slope is negative, that slope is zero," I'm proposing this is useful knowledge to activate BEFORE "teaching subtract, subtract, divide." If that knowledge is active, then the numerical value for slope will have more meaning attached to it. When their predicted sign matches the calculated sign, that's a valuable connection. When those signs DON'T match, that's a valuable moment of disequilibrium.
I'm curious how this does or doesn't line up with how you see the situation.
There are two things we want kids to learn -- a method for computing slope and a method for visually classifying slopes into positive, negative, zero.
Now, quick aside: imo we should probably teach positive slopes first, then extend to zero and negative cases.
But supposing that you are teaching all at once...one way to think about this is the timescale. If you're going to teach the slope formula and then like weeks and weeks later you're going to teach kids how to tell if a slope is pos/neg/zero...that's a bad idea. Likewise, I don't see why there should be a huge gap between the "conceptual" lesson and the formula. Like, how well are you going to understand pos/neg slope without attaching some numbers? How well will you understand the numbers without calculating them on your own?
But if you want to argue about whether we should teach the formula on Monday and the "conceptual" lesson on Tuesday, or if the order makes more sense the other way...that's probably not worth stressing about. Teach both, around the same time. That's what "procedural and conceptual knowledge develop in tandem" means to me.
On the AI study mode inability to stop talking, I just saw the authors of the Eedi/Deepmind Human-in-the-loop tutoring paper that spoke to that. https://storage.googleapis.com/deepmind-media/LearnLM/learnLM_nov25.pdf... It's only sort of in the paper, but in the talk, they shared a figure which showed that a good chunk of substantive human intervention in their human-in-the-loop system was very late in a tutoring session. Basically, when the student first needed help, the AI offered some good questions (actually, the tutors reported they felt like they learned some good strategies from the AI suggestions). But once students figure it out, the AI keeps trying to suggest things, and then humans have to step in and cut the AI off to wrap up the session.
Thanks for the kind words on the Homework Machine, lots of your readers came over to listen! (And thanks to everyone listening!)
One of my favorite lessons: Give every pair of kids in the class a piece of rope about 2 feet long. Give them a table with the first column labeled “number of knots” and the second column labeled “length of rope”. Have the students measure the length of the rope and enter it next to zero for number of knots. Then ask them to tie a knot, measure the length of the rope again and record it. Tie another knot, measure the length of rope and record it. After tying 3 knots and recording the corresponding length of the rope, ask them to predict how long the rope will be if they tied 7 knots. How did they figure that out? Guided discussion or a good worksheet will lead them to discover that every knot reduces the length of the rope by a certain amount on average. Further guidance will lead to an equation with a slope and a Y intercept, the slope representing the amount of rope in each knot and the Y intercept the original length of the rope. I used this activity as a review of algebra one concepts when I was teaching algebra two. I loved seeing the lightbulbs go off as the kids figured out what they were doing. This could also be used as an introduction to linear equations in a lower level class. The amount of guidance needed for this activity will depend on previous knowledge of the students. But once the activity is completed and understood it provides a basis for understanding the meaning of slope and Y intercept for the rest of the year in class because there is a tangible thing to which to refer when discussing the meaning of slope and Y intercept.
Good morning Dan! So glad to see your Mathworlds back in action!
Something I do with my middle schoolers is I have them line up the ordered pairs like a subtraction problem. So:
(7, 9)
- (2, 3)
Even though it's still in (x,y) format, they've jumped on the concept of subtracting the change in the values and then flipping them for slope. For those who struggle with flipping into y/x after subtracting, they'll flip the ordered pairs first (so reverse order for ordered pairs above), and then subtract. I love it that they're thinking this way.
That's really interesting. I wonder what it does or doesn't do for a kid to see the rise & run kinda sorta in a coordinate pair. Seems like it'd do loads to get them to get the correct sign on the differences, though! Thanks for sharing, Kandace.
My thoughts as well. It seems like many students cannot see that rise over run when I introduce it to them, show them different ways from the y/x ratio to the ordered pairs, etc. Desmos has worked wonders though - they see it more visually in a program like that than they do in my handwriting on the board. I'm currently looking at different curriculum in hopes that my BOE agrees we need change. I noticed that enVision+ incorporates Desmos, 3-Act Maths, and Thinking Tasks (all my favs and strategies I use in the classroom). I'm looking at different curriculum because I have noticed that we have a big gap in learning - specifically grounded in foundational skills (Thank you, Covid). Do you think that could be part of the problem?
I also want to add that not that long ago I had a class working on Turtle Trials so they could understand the connection between the constant rate and slope (um, the looks on their faces when I say "It's the same thing," so I've made a goal of not saying that to see if they can figure that out.)
Students went through the slides somewhat eagerly, putting in the values as prompted on tables and graphs - they really did not seem to have issues with it. Until THE moment happened.
When prompted to create their own Turtle Trials using up to four turtles and rates, I had one snarky student decide to enter one negative value for one of her turtles. It. Was. Awesome. The reaction of the students when we played it on the screen was one of the best experiences I've had as a teacher of witnessing student understanding. As we played the student's race over and over and observed it taking place on the grid, they understood the connection between rate and slope and that sometimes their mistakes - whether intentional or not - actually assist in their learning. The student actually said, "I wanted to know what would happen if I did this." If that's not subtle student questioning in disguise, I don't know what is.
Inspired by your comment which I just happened to read on the morning of the day I intended to work on graphs and equations of proportional relationships, I decided to use Turtle Trials with my 7th grade students. Though our standard only includes graphs and equations of proportional relationships, students were able to see and understand how and why turtle 1's graph and equation were "different." Yesterday, to prepare for our upcoming test, students were matching graphs and tables. I asked them to write equations for the relationships (without specifying that some weren't proportional). As I walked by a group, I overheard, "That's like the turtle that had the head start..."
I'm not sure that this fully answers the conundrum you are facing, but I think there's at least some benefit in ensuring kids have the algorithmic fluency, and then trying to address the conceptual matter.
It is much easier for a kid who can correctly calculate slope to then examine what they notice when it changes. Procedural fluency actually prepares them for conceptual understanding. Early in my teaching career I found there were a number of instances of catching myself saying "Oh that's why it's that way" because I had only done the calculations before, and then came back with a fluent understanding of the procedure it made it easier to process the "why". We can create those same experiences for students if we are intentional.
My favorite thing about it was not knowing entirely what to do next—the kids helped me craft the next steps. It was that magical back and forth between 30 kids and their teacher. It was 6th graders meeting an 8th-grade standard and it was fine. Kids are brilliant, and I'll never be tired of saying that.
Welcome back! This feels like the consistent different between the lesson was fine and great is how much it feels like the class got into this conceptual piece vs a handful got there through individual or small group questioning. Our department is leaning into making and the supporting and questioning claims as a class. "The slope of the line connecting two points is negative when..." or "If I have a point in quadrant II, the slope of the line is (always/sometimes/never) positive when a second point on the line is in ___ quadrant." Claims ranging from more obvious to more abstract being on the table have seemed to help kids be more flexible in how they see things and willingness to go out on a limb than in a space that feeds off of subtract/subtract/divide.
My suggestion for helping with conceptual understanding is just letting them think about and interpret all different kinds of graphs. My two go-to routines were the NTY "What's going on in this graph?" and (more often) "Slow Reveal Graphs." While these won't necessarily hit all of the particulars of linearity and slope, it will at least build some good intuition for what questions they should ask and what conclusions they can draw from points, lines, and axes. I would also argue focusing student attention to these kinds of routines is also more meaningful for growing a numerate citizenry.
I'm still cursing myself for not snatching and copying a "supplemental packet" that 2 students brought to me. THey had been doing *fine* in math but now they were hopelesslyl confused. It was graphing, and slope, and equation of the line. So I'll sort of reconstruct what I remember but some of what I "remember" might be other stuff in my head.
The packet started with the notion that oh, you have *two quantities* in a relationship on a graph. It can be like an address. If I say I'm on Bradley St. -- but where? If I say "2400" .... still, where?
2400 W. Bradley Ave. lands you right at the college.
Then, I *think,* it went to rates. So, miles and the distance traveled. Or maybe it was the cost of packages. Pretty sure there were similar graphs to compare and ask questions about. Which bicycle was going faster? I don't remember if it included "what if they had a 10 mile head start" and the "y-intercept" concept.
Re-building proportion ideas works well there, too.
A cool way to build in extra practice that our Math LIteracy course does is to change the scales. Sigh, it even has word problems where you use the distance formula but oh, each gridline is 4 inches so you have to multiply by 4.
I remember at Twitter Math Camp somebody with a neat image explaining why it's the change in y/ change in x, not the other way around, by showing 2 mountains and asking which one should have the *bigger* slope.
Really interesting to see how this prediction contest frames parent communication as a "non-interpersonal" activity! Of course some parent communication teachers do is pretty mechanical. But I think it also almost always involves some social, emotional, or relationship-building aspect, which they go on to say would be excluded. Caregivers are humans, too!
Speaking of interpersonal communication, I've seen a big increase in student emails that start with "I hope this email finds you well." An interesting phrase, and apparently quite popular with young people today, maybe it's like that 6/7 thing, or maybe something else is going on.
Loving Rohr's quotes here. Thanks to you and Fawn continually reminding those of us who are math leaders far removed from classrooms of the power of spending regular time teaching in classrooms, I am now doing it quite regularly. I need dust off my blog to write publically about it so help hold me accountable to that.
This question right here is the chef's kiss (as my 11 year old would say): "How do you “press” 30 kids to develop important conceptual knowledge when the easy operational knowledge is right there?"
I'm also left wondering if that building of conceptual knowledge must always intersect with students better understanding the algorithm. For example, proving the quadratic formula. I'm not sure if in a pinch I could rederive that proof right here as I type, but I have plenty of conceptual knowledge developed and stored away in long-term memory that I can solve any quadratic-related problem you throw at me without quite remembering the proof or why each coefficient and operation in the formula is what it is. Heck. Even if I have forgotten the quadratic formula, I have a whole toolbox of conceptual knowledge that would help me solve anything related to the topic.
I've found that conceptual knowledge always has a far more long-term lasting effect than procedural knowledge and that having a deep conceptual understanding often helps you do math when the formula or procedure may be long-forgotten. This happens to me often where kids (often my own high school kid) ask me for help and I have completely forgot the procedures on how to do the problem, but can figure it out because I retain the conceptual knowledge. It doesn't always mean I can re-derive the formula, but that the conceptual understanding I have retained endures so I can still solve the problem. Annnnnnnd....in the moment, with 30 students (more like 35+ in many public high schools) are awaiting your instruction, I know why pressing on with subtract, subtract, divide is where many turn. Thanks for that chef's kiss of a question to continue to ponder.
Here's my thought:
I think you're absolutely right to avoid subtract, subtract, divide. And I think it's worth avoiding for a very specific reason. Learning works best when new ideas are securely attached to solid existing knowledge. So what's the existing knowledge we want to attach it to?
I want students to know that slope is rise / run. I want students to know that subtracting is a way of finding the distance between two points. I want students to be able to picture a coordinate plane in their mind when they hear an x,y pair. I want students to intuit a rough estimate for a slope before calculating.
So here are a bunch of questions I would ask students along the way that I think can help create some of those connections.
Warmup - write a bunch of points on a plane given coordinates
How far apart are these points? (horizontal and vertical only, given a picture)
How far apart are these points? (horizontal and vertical only, no picture, start with points that are very close and then move them apart)
How far apart are these points horizontally? Vertically? (picture, then no picture, same thing start close then move farther apart)
Here are two points, with the horizontal and vertical distances labeled. What's the slope?
Here are two points. Will the slope be positive or negative? What's a good estimate? Would it be closer to 2 or 1/2?
Here are two points, picture given, very close together. What's the slope?
(It's been a while since I taught 8th grade. I bet if I had to teach this again, I would have a bunch more little micro questions to add in there afterwards.)
All of that work prepares students for the subtract, subtract, divide. The algorithm makes much more sense and sticks better because it's connected to all this other knowledge students have that we want to build on.
I think something you get at that's right on is that this is a very individual process. In my teaching, I'm a huge fan of giving students sequences of problems, building from simple to complex along the lines of what I just described. I want to launch with some framing and practice/review of the steps we've seen so far, support each student in making those new connections, and then bring the class together to share out some of the connections they made.
The catch -- I don't think it's possible to get through the entire sequence I lay out above in a day or even two days. My goal is to move a few more steps along that ladder each day. Review what we did the day before, practice a bit, then try to climb a few more rungs. Summarize, consolidate, then get a little further the next day. It's a very patient type of teaching, and a huge part is identifying small steps that I want students to take along the way.
Thanks for your thoughts here, Dylan. This lines up with your post from yesterday, which I wish I had read before heading into the lesson.
For bystanders 👇
https://fivetwelvethirteen.substack.com/p/an-alternative-to-iweyou
I’ve always used tables to get to the algorithm, and I think the same approach could be adapted graphically.
I give students tables to fill out- “Find the pattern!” The first tables have three or four pairs filled in and ask for the next one or two or four. The X-values are filled in. There is a spot to write the slope “grow-by” below the table. You can be as creative as you like- negative slope, fraction slope with price denominator, negative X-values. Some slopes like 31 where a student needs to subtract. Maybe a slope like 29 where a student definitely has to subtract. The last problem in this sequence might have the missing y-value at the beginning instead of the end. These all provide consecutive x-value number pairs.
The next set of tables provides two non-consecutive y-values (again, all c-values provided). (1,3) (2, ___), (3, 7). Make bigger gaps, use fractions, negative x’s and y’s, negative slopes. Eventually, depending on student ability, make some that make subtraction and division by two clear, in advance for many kids but maybe only with a reflection question after for others: (7, 22) (8, ___) (9, 32)
Next, open up the gaps. Still a fun puzzle. Easy and hard. Guess and check. Eventually something like,
(4, 20) (5, ___) (it is a table, so still including every x-value with blank y) (14, 20).
Similar with gaps of 2. Maybe graduate to (4,22) (9,72) or (10, 100) (50, 180) . Now I’m putting …. In the table instead of every x. Again, including negative slopes as well. Prompt some reflection questions about which operations they used. Work in variables gradually. Eventually: (x1, y1) and (x2, y2).
Voila, most kids solved a bunch of puzzles and ended up finding the formula/algorithm in their own. I value discovery more than others, because I’m fighting “I don’t understand! Where did this come from?!?” (It came from you!), but you could debate whether this gets you to deeper understanding or not.
"Learning works best when new ideas are securely attached to solid existing knowledge."
So true, I think with slope some existing knowledge you can connect to is rates, any phrase with the word "per" in it. Why are we dividing? What order you do divide in? Because I find when it's framed as a rate question ("I drive to Chicago, it was 200 miles and took me 3 hours, how fast did I drive, on average?") a couple things happen: Students seem to know they should divide, and students seem to know what order to divide (nobody does 7/200 and nearly everybody does 200/7) I always point that out: "YOU knew to divide, I didn't tell you, and YOU knew the correct order!" Gives us something to build on.
I have been thinking about this for a few weeks since I read the post. I have been using illustrative math problems primarily, so picture a graph of a line, showing a context from a word problem. Let’s use this warmup to show the context. https://accessim.org/6-8/grade-8/unit-3/section-b/lesson-6?a=student
Both the scales are different on the x and y axis. When I ask students what number plus 1 is 4? I will write on their desk with a whiteboard marker, 1 + x =4. And what number plus 70 is 160? 70 + y is 160. Does this conceptually help? I think so. I was so excited to figure out that this is one of those places to review one step equations before getting to the next unit, which is a nightmare for students who struggle with understanding inverse operations to access.
And yes. It’s far easier to teach, subtract, subtract, and divide, but what if the numbers are negative? What if the line is decreasing, what if the slope of the line is positive? Will this strategy help students to solve all problems? What about future problems? Does the context help understanding a rate of change?
Love those last questions.
I (coach) co-taught with a teacher the other day who was doing this exact topic. We used an Open Middle problem to start, which led to some great generalizations about slope, even though the students already knew the formula. I also think questioning plays a huge role in drawing out the conceptual ideas when the skill is pretty straightforward (credit to the math coordinator at this school who suggested this). What does the slope tell us about this situation? How did you start to figure this out? Is there another way you could have figured this out? What happens to the y-intercept when these are the two points you have? I'm still working on the questions, but I do think that's a key part here.
Math teachers: I stopped reading at slope because I wanted to reply: I recently was teaching slope to my "at-risk" high school math students. I was, once again, wondering what was so difficult about the concept. Suffice it say, I did come up with a nice exercise that seemed to help a few more students over the wall:
1. I asked if they realized that if you build stairs in your house that are too steep, it would be "illegal" because it could be dangerous. And that less steep stairs can be inefficient because they might take up too much space in a house.
2. I then gave each pair of students 12 identical Lego bricks.
3. I asked them to make 3 stair cases, using 3 bricks each, of different steepness.
4. After checking their stairs, I had them lay the 3 cases sideways on paper and had them trace it.
5. I then had them determine and record the rise and run of each of the stair cases. I checked those too.
6. Finally, they were to connect the top to the bottom and draw that line with a ruler. At that point, I asked them to record, "What do you notice about the lines you just drew and the slopes of the stairs?" We had conversations including the fact that some people's stairs sloped downward to the right and others up. So, we also made that connection to positive and negative slope. Some noticed that larger slopes signified steeper stairs (and lines). In following days, when students were working on slope we would refer back to our "Lego stairs" to conceptualize what was going on.
. . . for what it's worth.
Oh I love this approach! Thank you for sharing!
Oops, I meant to say 4 bricks for each of the 3 stair cases! 😏
Nice!
Obviously you've seen it, but Pershan's recent post about conceptual understanding is a great one: https://pershmail.substack.com/p/understanding-shouldnt-be-vague-or
I would be asking questions like the ones you were asking, at the group or whole class level, giving them some time to think and discuss, and then sharing what they're thinking with the whole class (maybe that's what you *were* doing(?)). For this particular topic, I'd also be asking "What *does* happen if we divide the x-change by the y-change? Does that still seem to describe how "steep" the graph is?" "What's the difference between one that has a negative slope and one with a positive slope? Why?" (small differences in Desmos graphs would be great here).
But really, I think there isn't a shortcut on the conceptual work.
Thanks for the shoutout. It's a good exercise (for me) to figure out how to apply that here.
The first thing: what does it mean to develop an "intuition about slope"? I suggest in that post it means knowing generally true things about slope. So, what are some of those things? Let me try to tease out some of the ideas I think are floating around in the post and express them as generalizations:
* You can tell if a slope should be positive or negative by starting with the leftmost point and deciding if the next point goes up or down.
* The slope will be negative if when, after you subtract coordinates, one difference is positive and the other is negative.
* The "rise" is the change in the vertical height between points.
* The slope of a line will always be equal to a rate of change (ys per x).
We could definitely come up with questions or activities that would target any of these particular ideas.
Is the teaching of any of these impaired by the "subtract, subtract, divide" procedure? I'm not sure. I can't really see how, to be honest.
Is it important to dig around with these things without getting specific and explicit? Is there a mucking around stage that is crucial for developing understanding before we introduce "subtract, subtract, divide"?
In this case, for these pieces of understanding, I don't think there is. We can teach these things, and knowing the slope formula is only an asset to that learning. I say, get the formula out of the way, then flesh it out with interesting challenges that get at these general principles to flesh out a full understanding of slope and its meaning.
> In this case, for these pieces of understanding, I don't think there is.
If students can look at a line and say "that slope is positive, that slope is negative, that slope is zero," I'm proposing this is useful knowledge to activate BEFORE "teaching subtract, subtract, divide." If that knowledge is active, then the numerical value for slope will have more meaning attached to it. When their predicted sign matches the calculated sign, that's a valuable connection. When those signs DON'T match, that's a valuable moment of disequilibrium.
I'm curious how this does or doesn't line up with how you see the situation.
There are two things we want kids to learn -- a method for computing slope and a method for visually classifying slopes into positive, negative, zero.
Now, quick aside: imo we should probably teach positive slopes first, then extend to zero and negative cases.
But supposing that you are teaching all at once...one way to think about this is the timescale. If you're going to teach the slope formula and then like weeks and weeks later you're going to teach kids how to tell if a slope is pos/neg/zero...that's a bad idea. Likewise, I don't see why there should be a huge gap between the "conceptual" lesson and the formula. Like, how well are you going to understand pos/neg slope without attaching some numbers? How well will you understand the numbers without calculating them on your own?
But if you want to argue about whether we should teach the formula on Monday and the "conceptual" lesson on Tuesday, or if the order makes more sense the other way...that's probably not worth stressing about. Teach both, around the same time. That's what "procedural and conceptual knowledge develop in tandem" means to me.
On the AI study mode inability to stop talking, I just saw the authors of the Eedi/Deepmind Human-in-the-loop tutoring paper that spoke to that. https://storage.googleapis.com/deepmind-media/LearnLM/learnLM_nov25.pdf... It's only sort of in the paper, but in the talk, they shared a figure which showed that a good chunk of substantive human intervention in their human-in-the-loop system was very late in a tutoring session. Basically, when the student first needed help, the AI offered some good questions (actually, the tutors reported they felt like they learned some good strategies from the AI suggestions). But once students figure it out, the AI keeps trying to suggest things, and then humans have to step in and cut the AI off to wrap up the session.
Thanks for the kind words on the Homework Machine, lots of your readers came over to listen! (And thanks to everyone listening!)
One of my favorite lessons: Give every pair of kids in the class a piece of rope about 2 feet long. Give them a table with the first column labeled “number of knots” and the second column labeled “length of rope”. Have the students measure the length of the rope and enter it next to zero for number of knots. Then ask them to tie a knot, measure the length of the rope again and record it. Tie another knot, measure the length of rope and record it. After tying 3 knots and recording the corresponding length of the rope, ask them to predict how long the rope will be if they tied 7 knots. How did they figure that out? Guided discussion or a good worksheet will lead them to discover that every knot reduces the length of the rope by a certain amount on average. Further guidance will lead to an equation with a slope and a Y intercept, the slope representing the amount of rope in each knot and the Y intercept the original length of the rope. I used this activity as a review of algebra one concepts when I was teaching algebra two. I loved seeing the lightbulbs go off as the kids figured out what they were doing. This could also be used as an introduction to linear equations in a lower level class. The amount of guidance needed for this activity will depend on previous knowledge of the students. But once the activity is completed and understood it provides a basis for understanding the meaning of slope and Y intercept for the rest of the year in class because there is a tangible thing to which to refer when discussing the meaning of slope and Y intercept.
Good morning Dan! So glad to see your Mathworlds back in action!
Something I do with my middle schoolers is I have them line up the ordered pairs like a subtraction problem. So:
(7, 9)
- (2, 3)
Even though it's still in (x,y) format, they've jumped on the concept of subtracting the change in the values and then flipping them for slope. For those who struggle with flipping into y/x after subtracting, they'll flip the ordered pairs first (so reverse order for ordered pairs above), and then subtract. I love it that they're thinking this way.
Hope this helps for some.
That's really interesting. I wonder what it does or doesn't do for a kid to see the rise & run kinda sorta in a coordinate pair. Seems like it'd do loads to get them to get the correct sign on the differences, though! Thanks for sharing, Kandace.
My thoughts as well. It seems like many students cannot see that rise over run when I introduce it to them, show them different ways from the y/x ratio to the ordered pairs, etc. Desmos has worked wonders though - they see it more visually in a program like that than they do in my handwriting on the board. I'm currently looking at different curriculum in hopes that my BOE agrees we need change. I noticed that enVision+ incorporates Desmos, 3-Act Maths, and Thinking Tasks (all my favs and strategies I use in the classroom). I'm looking at different curriculum because I have noticed that we have a big gap in learning - specifically grounded in foundational skills (Thank you, Covid). Do you think that could be part of the problem?
I also want to add that not that long ago I had a class working on Turtle Trials so they could understand the connection between the constant rate and slope (um, the looks on their faces when I say "It's the same thing," so I've made a goal of not saying that to see if they can figure that out.)
Students went through the slides somewhat eagerly, putting in the values as prompted on tables and graphs - they really did not seem to have issues with it. Until THE moment happened.
When prompted to create their own Turtle Trials using up to four turtles and rates, I had one snarky student decide to enter one negative value for one of her turtles. It. Was. Awesome. The reaction of the students when we played it on the screen was one of the best experiences I've had as a teacher of witnessing student understanding. As we played the student's race over and over and observed it taking place on the grid, they understood the connection between rate and slope and that sometimes their mistakes - whether intentional or not - actually assist in their learning. The student actually said, "I wanted to know what would happen if I did this." If that's not subtle student questioning in disguise, I don't know what is.
Awesome. The Platonic ideal, right down to the kid being the snarky sort. "Hey ... that snark of yours ... it's actually MATH."
Inspired by your comment which I just happened to read on the morning of the day I intended to work on graphs and equations of proportional relationships, I decided to use Turtle Trials with my 7th grade students. Though our standard only includes graphs and equations of proportional relationships, students were able to see and understand how and why turtle 1's graph and equation were "different." Yesterday, to prepare for our upcoming test, students were matching graphs and tables. I asked them to write equations for the relationships (without specifying that some weren't proportional). As I walked by a group, I overheard, "That's like the turtle that had the head start..."
I'm not sure that this fully answers the conundrum you are facing, but I think there's at least some benefit in ensuring kids have the algorithmic fluency, and then trying to address the conceptual matter.
It is much easier for a kid who can correctly calculate slope to then examine what they notice when it changes. Procedural fluency actually prepares them for conceptual understanding. Early in my teaching career I found there were a number of instances of catching myself saying "Oh that's why it's that way" because I had only done the calculations before, and then came back with a fluent understanding of the procedure it made it easier to process the "why". We can create those same experiences for students if we are intentional.
Hi Dan! Welcome back.
10% of the time I get lucky and a lesson goes well enough that I'd repeat it as is. My lesson on slope landed in that spot: https://fawnnguyen.substack.com/p/staircase-and-steepness. And part 2 is here: https://fawnnguyen.substack.com/p/staircases-and-steepness-continued?r=hbzf
My favorite thing about it was not knowing entirely what to do next—the kids helped me craft the next steps. It was that magical back and forth between 30 kids and their teacher. It was 6th graders meeting an 8th-grade standard and it was fine. Kids are brilliant, and I'll never be tired of saying that.
Great to have you(r emails) back Dan!
Welcome back! This feels like the consistent different between the lesson was fine and great is how much it feels like the class got into this conceptual piece vs a handful got there through individual or small group questioning. Our department is leaning into making and the supporting and questioning claims as a class. "The slope of the line connecting two points is negative when..." or "If I have a point in quadrant II, the slope of the line is (always/sometimes/never) positive when a second point on the line is in ___ quadrant." Claims ranging from more obvious to more abstract being on the table have seemed to help kids be more flexible in how they see things and willingness to go out on a limb than in a space that feeds off of subtract/subtract/divide.
My suggestion for helping with conceptual understanding is just letting them think about and interpret all different kinds of graphs. My two go-to routines were the NTY "What's going on in this graph?" and (more often) "Slow Reveal Graphs." While these won't necessarily hit all of the particulars of linearity and slope, it will at least build some good intuition for what questions they should ask and what conclusions they can draw from points, lines, and axes. I would also argue focusing student attention to these kinds of routines is also more meaningful for growing a numerate citizenry.
I'm still cursing myself for not snatching and copying a "supplemental packet" that 2 students brought to me. THey had been doing *fine* in math but now they were hopelesslyl confused. It was graphing, and slope, and equation of the line. So I'll sort of reconstruct what I remember but some of what I "remember" might be other stuff in my head.
The packet started with the notion that oh, you have *two quantities* in a relationship on a graph. It can be like an address. If I say I'm on Bradley St. -- but where? If I say "2400" .... still, where?
2400 W. Bradley Ave. lands you right at the college.
Then, I *think,* it went to rates. So, miles and the distance traveled. Or maybe it was the cost of packages. Pretty sure there were similar graphs to compare and ask questions about. Which bicycle was going faster? I don't remember if it included "what if they had a 10 mile head start" and the "y-intercept" concept.
Re-building proportion ideas works well there, too.
A cool way to build in extra practice that our Math LIteracy course does is to change the scales. Sigh, it even has word problems where you use the distance formula but oh, each gridline is 4 inches so you have to multiply by 4.
I remember at Twitter Math Camp somebody with a neat image explaining why it's the change in y/ change in x, not the other way around, by showing 2 mountains and asking which one should have the *bigger* slope.