I think maybe one (implied) step I might insert before the "large addend" problems might be to have students practice just noting whether the solution will be positive/negative/zero. So at this point, the point isn't to get the actual value, but to practice their "is this answer reasonable/does it make sense?" muscle. -1000+200 = ? What about 300 + -500?
I find the "is this answer reasonable?" muscle is a good approach to help students micro-check their answers throughout the process... for example, if you're working on equations of lines and you get a negative slope from two given points, you might ask yourself whether or not it is reasonable that the slope is negative (quickly plot and see if it visually matches) before continuing to calculate the y-intercept. The students who lack conceptual understanding really struggle with this.
Yea that's a good one. I do some of that but mixed in with the integer chips/floats and anchors representation, it would be good to break it out and spend more time on it alone.
1.) I independently came to all of the same breakdown steps for the number line. I haven't incorporated commutativity yet; that's a great point. I also make sure to mix in, at each step, problems that couldn't be on their number line. -65 + 2. 78 + -3 - 100 - 2
2.) I've gone to 100% vertical number lines. I would encourage it. You don't have that extra stress of wondering whether to move left or right. Subtracting a positive = down. Adding a negative = down. More intuitive.
3.) I read "Make it Stick" over the summer and it has really informed my thinking about the relationship between fluency and conceptual understanding.
re: 1) I always teach the commutativity but students don't generally take me up on it. I don't know if I should spend more time on that skill, I use it all the time but for whatever reason I've had trouble getting students to see it as helpful.
I should definitely consider more vertical number lines. We use some. I have a big horizontal one on the wall at the front of my room and I see students using it a lot, I don't know where I would put a big vertical one that would be as visible. But that makes sense, I have a few students who keep making right/left mistakes and vertical would help.
Yes. I've written about this, curious your take here.
Conceptual Understanding is often presented as a binary.
Better to use 4 phrases. Something like:
1. Don’t Know
2. Sort of Know
3. Full Know
4. Mega Know (this includes “The Why”)
Example: Find the Area of a Triangle with base 9 meters and height 12 meters.
1. Don’t Know:
Kid has no idea what to do. Even if you wait, or encourage, kid can’t start.
2. Sort of Know:
Maybe kid may remember the formula A = 1/2*B*H but doesn’t know how to multiply by half.
Maybe thinks the formula it’s B*H (forgets the 1/2 entirely).
Maybe able to trace the triangle and explain the Area means “The inside space” – so conceptually understands area – but can’t multiply by twelves very well.
3. Full Know:
Kid gets the problem right.
Then asked: Draw 3 very different triangles that all have base 9 and height 12. Struggles.
Then asked: “Why do we say 84 Square Meters as the answer…why that word Square”? Struggles.
4. Mega Know:
Can do all these natural extensions - not just the nth plus 1 problem of a procedure they've practiced n times.
In short: there are multiple types of conceptual understanding. There are also often multiple strands within a topic - i.e. number lines vs the "how much more negative/positive" strategy. And each topic is different, some have more strands, some require more flexibility, some are more about knowing when an idea applies. So there's no easy answer. This is where knowing of math for teaching is really important.
To address your example: there's flexibility in recognizing triangle area in lots of different contexts. There's the "why" that helps to apply that knowledge to other shapes, composite shapes, etc. There's intuition for what area means to make sure the answer makes sense. Those are a bunch of different things that build toward "mega know."
To what extent - let's say with training tutors - do you think there's utility in creating an Easy To Remember typology about making the "Do you know it/grasp it/get it" question less binary?
That's what I was going for with Don't/Sorta/Full/Mega.
I think that's helpful. It helps to avoid teaching a shallow version of a skill and then moving on, which can become a vicious cycle.
One approach is often to provide some questions on an assessment or whatever that we want students to solve with "mega." The issue then becomes that the teacher teaches to those specific problems, which is not really in the spirit of mega knowing something.
The other thing that's always been helpful for me when I'm teaching a new topic is talking to someone who knows how to break it down into pieces. Often getting to the mega level is more about doing the first few levels well by breaking things down. That sets me up with time and students up with knowledge do to the more ambitious stuff.
YES to slow-motion understanding. (I’ve always thought that “Teaching, Fast and Slow” would make a great title for a book.) Over a decade, as a math teacher and tutor, I built a tool that let students take control of this power themselves. I’ve explained it here, in case anyone wants to steal it: https://losttools.substack.com/p/how-to-build-a-deep-practice-book
I like "Teaching, Fast and Slow." The deep practice book is interesting! I agree about the assembly-line focus. Math is lots of connected ideas that we want students to retain, and the assembly line doesn't do that. I have trouble imagining that structure in middle school, but it makes a lot of sense to me for something self-motivated.
I agree with that. I also think conversations about conceptual understanding get hung up because it really depends on the content. A conceptual understanding of fraction multiplication is really challenging. Conceptual understanding of integer addition is much less challenging. So it depends on the content, and vague, fuzzy statements about conceptual-first lose sight of that.
I think maybe one (implied) step I might insert before the "large addend" problems might be to have students practice just noting whether the solution will be positive/negative/zero. So at this point, the point isn't to get the actual value, but to practice their "is this answer reasonable/does it make sense?" muscle. -1000+200 = ? What about 300 + -500?
I find the "is this answer reasonable?" muscle is a good approach to help students micro-check their answers throughout the process... for example, if you're working on equations of lines and you get a negative slope from two given points, you might ask yourself whether or not it is reasonable that the slope is negative (quickly plot and see if it visually matches) before continuing to calculate the y-intercept. The students who lack conceptual understanding really struggle with this.
Yea that's a good one. I do some of that but mixed in with the integer chips/floats and anchors representation, it would be good to break it out and spend more time on it alone.
Love this post!
1.) I independently came to all of the same breakdown steps for the number line. I haven't incorporated commutativity yet; that's a great point. I also make sure to mix in, at each step, problems that couldn't be on their number line. -65 + 2. 78 + -3 - 100 - 2
2.) I've gone to 100% vertical number lines. I would encourage it. You don't have that extra stress of wondering whether to move left or right. Subtracting a positive = down. Adding a negative = down. More intuitive.
3.) I read "Make it Stick" over the summer and it has really informed my thinking about the relationship between fluency and conceptual understanding.
re: 1) I always teach the commutativity but students don't generally take me up on it. I don't know if I should spend more time on that skill, I use it all the time but for whatever reason I've had trouble getting students to see it as helpful.
I should definitely consider more vertical number lines. We use some. I have a big horizontal one on the wall at the front of my room and I see students using it a lot, I don't know where I would put a big vertical one that would be as visible. But that makes sense, I have a few students who keep making right/left mistakes and vertical would help.
Yes. I've written about this, curious your take here.
Conceptual Understanding is often presented as a binary.
Better to use 4 phrases. Something like:
1. Don’t Know
2. Sort of Know
3. Full Know
4. Mega Know (this includes “The Why”)
Example: Find the Area of a Triangle with base 9 meters and height 12 meters.
1. Don’t Know:
Kid has no idea what to do. Even if you wait, or encourage, kid can’t start.
2. Sort of Know:
Maybe kid may remember the formula A = 1/2*B*H but doesn’t know how to multiply by half.
Maybe thinks the formula it’s B*H (forgets the 1/2 entirely).
Maybe able to trace the triangle and explain the Area means “The inside space” – so conceptually understands area – but can’t multiply by twelves very well.
3. Full Know:
Kid gets the problem right.
Then asked: Draw 3 very different triangles that all have base 9 and height 12. Struggles.
Then asked: “Why do we say 84 Square Meters as the answer…why that word Square”? Struggles.
4. Mega Know:
Can do all these natural extensions - not just the nth plus 1 problem of a procedure they've practiced n times.
What do you think, Dylan?
I wrote this post a while back trying to articulate some of these ideas and I would stand by it: https://fivetwelvethirteen.substack.com/p/what-is-conceptual-understanding
In short: there are multiple types of conceptual understanding. There are also often multiple strands within a topic - i.e. number lines vs the "how much more negative/positive" strategy. And each topic is different, some have more strands, some require more flexibility, some are more about knowing when an idea applies. So there's no easy answer. This is where knowing of math for teaching is really important.
To address your example: there's flexibility in recognizing triangle area in lots of different contexts. There's the "why" that helps to apply that knowledge to other shapes, composite shapes, etc. There's intuition for what area means to make sure the answer makes sense. Those are a bunch of different things that build toward "mega know."
To what extent - let's say with training tutors - do you think there's utility in creating an Easy To Remember typology about making the "Do you know it/grasp it/get it" question less binary?
That's what I was going for with Don't/Sorta/Full/Mega.
I think that's helpful. It helps to avoid teaching a shallow version of a skill and then moving on, which can become a vicious cycle.
One approach is often to provide some questions on an assessment or whatever that we want students to solve with "mega." The issue then becomes that the teacher teaches to those specific problems, which is not really in the spirit of mega knowing something.
The other thing that's always been helpful for me when I'm teaching a new topic is talking to someone who knows how to break it down into pieces. Often getting to the mega level is more about doing the first few levels well by breaking things down. That sets me up with time and students up with knowledge do to the more ambitious stuff.
YES to slow-motion understanding. (I’ve always thought that “Teaching, Fast and Slow” would make a great title for a book.) Over a decade, as a math teacher and tutor, I built a tool that let students take control of this power themselves. I’ve explained it here, in case anyone wants to steal it: https://losttools.substack.com/p/how-to-build-a-deep-practice-book
I like "Teaching, Fast and Slow." The deep practice book is interesting! I agree about the assembly-line focus. Math is lots of connected ideas that we want students to retain, and the assembly line doesn't do that. I have trouble imagining that structure in middle school, but it makes a lot of sense to me for something self-motivated.
See also: https://jemmaths.wordpress.com/2024/04/21/thinking-about-conceptual-understanding/
I agree with that. I also think conversations about conceptual understanding get hung up because it really depends on the content. A conceptual understanding of fraction multiplication is really challenging. Conceptual understanding of integer addition is much less challenging. So it depends on the content, and vague, fuzzy statements about conceptual-first lose sight of that.