In elementary classes we consider it a good thing to be able to move from the abstract to the concrete. We ask students to count and perform arithmetic on objects, even contrived ones. We ask students to group socks, slice pizzas, and describe snowballs. A critical person might suggest these are all examples of pseudo-context, and they’d be right! These are more-or-less contrived scenarios that don’t really require the context to get at the math involved. Why do we provide such seemingly inessential context? I’m venturing a little far away from my area of expertise here, but I’d guess it’s because it helps kids understand the math concept to have a concrete model of that concept in their heads.

My question is this: in secondary classrooms, is there inherent value in linking an abstract concept to an actualized context? Even if the context is contrived?

I mean, yeah that’s bad. Comically and tragically bad. It doesn’t do anything to enhance understanding. I’d say the context actually hinders understanding. The thickness of an ice sculpture dragon’s wing? That’s about three bridges too far.

But what about a slightly less convoluted, but also-contrived, example. Say:

This problem is still most certainly contrived: dimensions of tanks aren’t often given in terms of x. I’m not even suggesting this problem will engender immediate, massive engagement, but it might help students create a mental model of what’s going on with a third degree polynomial. Or at least the context allows students to affix understanding of the x- and y-axes once they create the graph of the volume of this tank.

We provide similarly pseudo-contextual in elementary classrooms in order to enhance understanding of arithmetic and geometry.

Of course, we also compliment such problems with manipulatives, games, play, and discourse, which secondary math classrooms often lack. In the best elementary classrooms we don’t just provide students with that single task. We provide others, in addition to the pure abstract tasks such as puzzles or number talks.

Perhaps the true sin of pseudo-context is that it can be the prevailing task model, rather than one tool in a teacher’s task toolbox. In secondary math classrooms pseudo-contextual problems are offered as the motivation for the math, instead of exercises to create models and nothing more.

“In elementary classes we consider it a good thing to be able to move from the abstract to the concrete. ”
Don’t you think that moving from the concrete to the abstract is a better idea? Especially with word numbers before symbols.

In elementary, I really like notice & wonder and 3-act lessons for, among other things, creating a legitimacy to the investigation, because the kids explore questions that they (or other kids) have actually asked.

With this perspective on (pseudo) context, I would take mild exception to your characterization of the Ramona the Pest activity. This is something I’ve frequently thought about while reading favorite books from my childhood to my own children. I bet it is something many adults wonder. I doubt it is something many kids wonder.

The second purpose for context is to make problems more interesting. As far as I’m concerned, Dan Meyer has effectively refuted their value for interest; in practice, most contexts are pretty obviously uninteresting to the students, though they smack of “things adults could be forgiven for thinking kids might like.” I’m lucky that I can mostly coast on teaching math that I find interesting, so the reasons I find it interesting and my obvious feelings toward the material help get the kids into it.

A third potential purpose is to enlist the students’ own experience and intuition to either help them find a solution, to sense check the answer, or to intentionally challenge intuition-based answers. “Share 6 cookies fairly among 3 kids” is something they’ve seen before, while “6 divided by 3” may be new. Pseudo-context doesn’t do any of these and sometimes (often?) can use a student’s knowledge of the context against them (see Crocodile Exam Question.)

My understanding is that concrete and pictorial (in the triumvirate of concrete-pictorial-abstract modes) are meant to be models for the abstract concepts. In your fish tank example, I would say the graph and the rectangular prism diagram are pictorial models of the polynomial; the context doesn’t add anything. For that example, I’m not sure the context is bad enough to be a major problem, but I’d potentially get stuck on:
(1) isn’t it usually easier to measure the height than to know the volume?
(2) can’t we estimate 3 feet (vs 2 feet) accurately enough to know this tank isn’t sufficient?
(3) why are the dimensions related in that way?
(4) what’s with this fish that height of the tank (and not volume) is the key issue anyway?

Really interesting stuff, Joshua. Perhaps I’m not clear on what pseudo-context really is. How is the Ramona example not an example of pseudo-context? I’m not criticizing the task, mind you. As I said, I think it’s (sometimes) helpful for students to affix their understanding of a context to mathematical content. I have a sneaking suspicion if the author of the Ramona task weren’t Burns – whom we all love – and it was, say, in a book from Pearson, folks would be critical of it as a pointless context as an entry point into subtraction. I see little difference between this and some hokey secondary task about, say, skateboarding.

I appreciate the three purposes of context you discuss above. I think it’s helpful to establish why one might ask a question using a fish tank rather than just giving a function. I’d agree that the difference between calling it a fish tank and just “a box” is negligible for student understanding. I can also see why given the choice, “a box” might be better in the long run. I’d also argue that providing the fish tank is significantly better than just providing the polynomial and asking students to factor it.

Consider Desmos’ Function Carnival activities, which is one of my favorite tasks in existence. What makes that not pseudo-context? My hunch is that it is pseudo-context, but the activity is just so delightful that we give it a pass, and we should! Being able to imagine a dude being shot out of a cannon is helpful for understanding the math, if not particularly useful for the understanding of cannon-shooting.

There’s no disagreement that really bad pseudo-context gets in the way of understanding and messages to kids that math was invented by math teachers to torture their students. I also think it might be helpful sometimes just to give students something with which to create mental models of what the math is actually doing.

One good way to “test” whether something is a pseudo-context is to ask yourself whether removing the context or replacing it with a different context changes the problem in any way. So, if I replace widgets with dolphins and the problem doesn’t change, the context is meaningless and therefore it’s a pseudo-context. The Ramona the Pest problem would be a different problem if you changed the context (still, a subtraction problem, but a different one) so the context actually serves the problem in some way.

I really like your thinking about how grade level plays into this. That’s a conversation I haven’t seen (maybe because I hang out with too many upper school teachers 😉 ) and which seems really rich. It is my understanding that students at the early grades are still forming their understanding of number, quantity and operation, so having a concrete model is not only helpful, but actually essential, to them building their understanding of these concepts. There’s some really great examples of this in Tracy Zager’s book, which I am in the midst of reading right now.

Quoting from a quote from Kathy Richardson (pp 197-8): “Children who deal almost exclusively with symbols begin to feel that the symbols exist in and of themselves, rather than as representations for something else…The number combinations and relationships children need to understand can only be learned through counting, comparing, composing and decomposing actual groups of objects.” And further down the page: “If teachers ignore these stages and just ask the children to memorize the words ‘three plus four equals seven,’ they are, in effect, asking them to learn a ‘song’ rather than learn the important relationships these words describe.”

I think you articulated this better than I did. This was intended to be the main question of the post: in elementary we find value in affixing numerical calculations to concrete “things” (even if the “things” are only in our minds). Is there similar value in upper grade levels? And when does that value (if there is any) become a detriment to understanding the math?

Thanks for the comment and a better articulation of my quandary.

I’ve been watching The Americans, and I often think about how old the characters would be today. This increases my enjoyment of the series, and it helps me to relate to the characters. For example, Paige is about the same age as me. I think that a child who was reading Ramona the Pest might like to know that one of their relatives grew up around the same time as Ramona. This might help them to relate the story to their own family history.

Yes, if the fish is a small one.

“In elementary classes we consider it a good thing to be able to move from the abstract to the concrete. ”

Don’t you think that moving from the concrete to the abstract is a better idea? Especially with word numbers before symbols.

Ah yes. Forgive my imprecision. I simply meant a fluidity back and forth.

My bias is to be very critical of weak contexts.

In elementary, I really like notice & wonder and 3-act lessons for, among other things, creating a legitimacy to the investigation, because the kids explore questions that they (or other kids) have actually asked.

With this perspective on (pseudo) context, I would take mild exception to your characterization of the Ramona the Pest activity. This is something I’ve frequently thought about while reading favorite books from my childhood to my own children. I bet it is something many adults wonder. I doubt it is something many kids wonder.

The second purpose for context is to make problems more interesting. As far as I’m concerned, Dan Meyer has effectively refuted their value for interest; in practice, most contexts are pretty obviously uninteresting to the students, though they smack of “things adults could be forgiven for thinking kids might like.” I’m lucky that I can mostly coast on teaching math that I find interesting, so the reasons I find it interesting and my obvious feelings toward the material help get the kids into it.

A third potential purpose is to enlist the students’ own experience and intuition to either help them find a solution, to sense check the answer, or to intentionally challenge intuition-based answers. “Share 6 cookies fairly among 3 kids” is something they’ve seen before, while “6 divided by 3” may be new. Pseudo-context doesn’t do any of these and sometimes (often?) can use a student’s knowledge of the context against them (see Crocodile Exam Question.)

My understanding is that concrete and pictorial (in the triumvirate of concrete-pictorial-abstract modes) are meant to be models for the abstract concepts. In your fish tank example, I would say the graph and the rectangular prism diagram are pictorial models of the polynomial; the context doesn’t add anything. For that example, I’m not sure the context is bad enough to be a major problem, but I’d potentially get stuck on:

(1) isn’t it usually easier to measure the height than to know the volume?

(2) can’t we estimate 3 feet (vs 2 feet) accurately enough to know this tank isn’t sufficient?

(3) why are the dimensions related in that way?

(4) what’s with this fish that height of the tank (and not volume) is the key issue anyway?

Really interesting stuff, Joshua. Perhaps I’m not clear on what pseudo-context really is. How is the Ramona example not an example of pseudo-context? I’m not criticizing the task, mind you. As I said, I think it’s (sometimes) helpful for students to affix their understanding of a context to mathematical content. I have a sneaking suspicion if the author of the Ramona task weren’t Burns – whom we all love – and it was, say, in a book from Pearson, folks would be critical of it as a pointless context as an entry point into subtraction. I see little difference between this and some hokey secondary task about, say, skateboarding.

I appreciate the three purposes of context you discuss above. I think it’s helpful to establish why one might ask a question using a fish tank rather than just giving a function. I’d agree that the difference between calling it a fish tank and just “a box” is negligible for student understanding. I can also see why given the choice, “a box” might be better in the long run. I’d also argue that providing the fish tank is significantly better than just providing the polynomial and asking students to factor it.

Consider Desmos’ Function Carnival activities, which is one of my favorite tasks in existence. What makes that not pseudo-context? My hunch is that it is pseudo-context, but the activity is just so delightful that we give it a pass, and we should! Being able to imagine a dude being shot out of a cannon is helpful for understanding the math, if not particularly useful for the understanding of cannon-shooting.

There’s no disagreement that really bad pseudo-context gets in the way of understanding and messages to kids that math was invented by math teachers to torture their students. I also think it might be helpful sometimes just to give students something with which to create mental models of what the math is actually doing.

One good way to “test” whether something is a pseudo-context is to ask yourself whether removing the context or replacing it with a different context changes the problem in any way. So, if I replace widgets with dolphins and the problem doesn’t change, the context is meaningless and therefore it’s a pseudo-context. The Ramona the Pest problem would be a different problem if you changed the context (still, a subtraction problem, but a different one) so the context actually serves the problem in some way.

I really like your thinking about how grade level plays into this. That’s a conversation I haven’t seen (maybe because I hang out with too many upper school teachers 😉 ) and which seems really rich. It is my understanding that students at the early grades are still forming their understanding of number, quantity and operation, so having a concrete model is not only helpful, but actually essential, to them building their understanding of these concepts. There’s some really great examples of this in Tracy Zager’s book, which I am in the midst of reading right now.

Quoting from a quote from Kathy Richardson (pp 197-8): “Children who deal almost exclusively with symbols begin to feel that the symbols exist in and of themselves, rather than as representations for something else…The number combinations and relationships children need to understand can only be learned through counting, comparing, composing and decomposing actual groups of objects.” And further down the page: “If teachers ignore these stages and just ask the children to memorize the words ‘three plus four equals seven,’ they are, in effect, asking them to learn a ‘song’ rather than learn the important relationships these words describe.”

I think you articulated this better than I did. This was intended to be the main question of the post: in elementary we find value in affixing numerical calculations to concrete “things” (even if the “things” are only in our minds). Is there similar value in upper grade levels? And when does that value (if there is any) become a detriment to understanding the math?

Thanks for the comment and a better articulation of my quandary.

I’ve been watching The Americans, and I often think about how old the characters would be today. This increases my enjoyment of the series, and it helps me to relate to the characters. For example, Paige is about the same age as me. I think that a child who was reading Ramona the Pest might like to know that one of their relatives grew up around the same time as Ramona. This might help them to relate the story to their own family history.