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If you’ve ever taught students how to find the area or perimeter of a shape, you won’t be surprised to read that students commonly confuse the two measurements.  For example, if you ask a student to find the perimeter of a rectangle, they will often give you the rectangle’s area.  Based on my experiences, this seems to be a pretty typical outcome for all math educators.

I realize that we’ve come to accept this as normal, but have you ever thought about why it happens?  Does this also happen in real life? Could this possibly be a problem of our own creation? After all, when a person is buying grass turf and fencing for their home, does that person ever get confused as to which measurement is which?  I can’t imagine that happening often.

This makes me wonder about whether it’s possible that the reason students confuse area and perimeter is because we often present problems without context or with fake/trivial contexts.  As a result, the terms “area” and “perimeter” remain abstract labels rather than something attached to a relatable meaning.

To better illustrate what I mean, consider the problem below that comes from the outgoing California standardized test prior to the Common Core State Standards.

To me, this is an example of a fake context. While the problem involves a basketball court, there is no context for what part of the court we are talking about.  The only hope students have is to have memorized the terms area and perimeter and know which one to apply.

What if the problem’s context was less fake and more useful? For example, if it was about a backyard, students could then be asked to find out how much fencing they would need. That would simultaneously provide the kind of context they would expect to find in real life and would better support conceptual understanding.

What’s important to realize is that in both cases, students are being asked to demonstrate the same skill: finding the perimeter of a rectangle.  My concern is that we penalize students for not knowing their vocabulary words in situations devoid of context when in reality, this never happens.

What do you think about this?  Do you see this happening in other contexts?  What am I right about and where am I mistaken?  Please let me know in the comments.

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1. LMcAnanana says:

I agree. In the English language, in general, words can have different meanings depending on the context. In math, we talk about using math word walls and building math vocabulary which is done without context. Do we want memorization of words or thinking about solving problems? When finishing our basement and determining materials needed we did not ever use the words area, perimeter or volume.

• Robert Kaplinsky says:

Thanks. It definitely makes me question the purpose of vocabulary. I believe that we should use it to precisely communicate ideas that already exist in context, but sometimes it feels like it’s the other way around.

• Kristyn says:

English vocabulary is a very tricky and complex thing. We must often rely heavily on context to get it right. We have multiple meaning words (right= correct and the opposite of left), words that are spelled the same spelling but different pronunciations and meanings (lead = to go first or a soft metal), and we even have contranyms – words that are antonyms of themselves (garnish – to add something to a plate of food, or to remove money from your wages; off – turn the light off and it stops working or the alarm goes off and it starts working; fast – to move quickly, or hold fast to not move at all). With a language like this, even with terms that are different from each other we owe it students to provide context and remove ambiguity whenever possible.

2. Tricia says:

True, I am over 30 and the way I remember being taught perimeter, area and volume was to visualize a drawer: If we trace around the drawer you have perimeter, if you rub the bottom section inside the drawer you have area while if you fill your drawer with clothing you were measuring volume hence the inclusion of height (I guess more two dimensional for perimeter and area and three dimensional for volume). Now as a past educator I can identify with teaching the two without context since that was a simple go to as a teacher to get kids moving through the standards. As bad as that seems it is the truth. I think today’s conversation about teaching real, authentic context based math questions focused on math concepts is a step in the right direction. But I also think that authentic and real math problems would be lacking without considering the abstract. For me, you can use the abstract to awaken or bring to light the real, authentic based in rich context. I think our mind can accommodate the two.

• Robert Kaplinsky says:

Hi Tricia. Thanks for your perspective. It isn’t my intention to say that we should never teach mathematics abstractly. What I am trying to say is that sometimes we lead with the abstract when I think it is more intuitive, whenever possible, to start with a context and THEN move to the abstract with the context being shed as bulky and unnecessary.

Then, you can return to the context once you’re done to give sense to the numbers you’ve calculated.

3. Jeremy says:

I agree that students would likely do better in a problem in a real context. That said, some definitions are worth knowing. I think a mix of problems would be appropriate. You could then separately report student understanding of the concept and mathematical vocabulary.

• Robert Kaplinsky says:

Thanks Jeremy. I totally agree. I think I didn’t explain myself clearly enough. I DEFINITELY think there is a need for precise mathematical vocabulary. I think it would go like this though:

1. Introduce the context.
2. Develop conceptual understanding through problem solving
3. Shed the context to work on the more abstract calculations.
4. Use precise vocabulary to communicate reasoning about the abstractions.

Did I do a better job explaining myself this time?

• Sarah says:

Yes 🙂 I was going to make that point but I couldn’t have said it better!

• Robert,

What is your opinion about teaching these two concepts together vs separately? Sometimes I think if we didn’t pair them from the start it and solidly developed one idea and then the other with the space of time in between we would be better off. Thoughts?

Also, the steps listed above echo the work of the Van Hiele levels in geometry: https://nrich.maths.org/2487

Vocabulary isn’t developed until Phase 3: Explication. Relevant research from 1959!
In some ways, we have been having the same conversation for decades!!

Thanks for this thoughtful topic of conversation – meeting with grade 3 teachers tomorrow and this is on the agenda.
Sue

• Robert Kaplinsky says:

Maybe neither? What if they arise out of a context? This isn’t great but it’s the first one that pops into my head. What if you showed them a picture of a lawn surrounded by a fence and you asked students what they noticed and wondered. From there they might say things like, “I notice it has fence.” or “I notice there is grass.” Then you could transition, eventually, to a conversation about how much of each you would need to build your own garden. There doesn’t need to be any labeling of area and perimeter until much later.

4. Xavier says:

“No todo en el monte es orégano” Not always the context is the only one cause. In my case, when we investigate how to calculate the cost of a picture framework some students thought they need an area but they calculated the perimeter.

• Robert Kaplinsky says:

Thanks Xavier. As an example of what I mean, if we ask kids to find the amount of red, blue, and white paint needed, I don’t think there will be any confusion. So, the confusion comes later on with the abstractions (I believe).
.

Later on we can introduce the vocabulary.

• Thanks, Robert, for you example. It’s definitively more easy to understand than I pretend with frame area. Perhaps you are on the point (is it right expression?): the difficult is the abstraction. In my lesson, I just ask “draw a frame for a phot… what’s the area of that framework?” I think this is more abstract question than “how much paint do you need to paint red zone?” (area) and “how much paint do you need to paint white lines?” (perimeter). How do you de-abstract my question?

• Maybe how much wood and how much canvas do you need? No estoy seguro. Es muy dependiente en el contexto.

• WENDY PARKER-Powell says:

I think that the basketball problem could still be used but within a different context. For instance, Jeremy was one minute late for basketball practice and had to run a lap. How many meters did he run? Some students can relate to both examples and some cannot. Students may not have any experiences with a fenced backyard. We have to use problems that are relevant to our population of students and their interests so that they understand then broaden their perspective with unfamiliar problems.

5. I agree, and I see the confusion even with contexts provided. It may be that some of the answer lies in the fact that we always teach these two concepts together, and that’s when they get all tangled up. I’m not sure if anyone has experimented with this, but exploring one concept, separated by some months from another, might alleviate the confusion.

• Robert Kaplinsky says:

Yeah, that’s a possibility. Check out the example I used in the reply to Xavier. I can’t see any confusion coming from that problem taught together. Thoughts?

• Caroline says:

Joe,
I agree with this completely. Our brain wires get crossed when we do not solidify the path before following a confusing path.
For example, Joe and Robert live in the same town. The directions to each mathematician’s house are very similar – they both head south along parallel roads with no cross streets. If I go to Joe’s house on Monday, Robert’s house on Tuesday then the following week, I need to remember the way Joe’s house, I will get it confused with the path to Robert’s. My brain has not solidified the neural pathways and connections that it needs to learn the way to Joe’s house.
If, however, I go to Joe’s house for a couple of days in a row, and only after I know the way like the back of my hand, I venture to Robert’s house, I will keep the two trips separate in my mind. I have made to separate neuro pathways, rather than two criss-crossing confusing methods.
Unfortunately, historically, we do not teach this way. We tend to think that teaching two confusing concepts together and asking kids to repeatedly discern between them. This often means that they will never keep the two concepts straight. (Think: inductive vs. deductive reasoning, factors vs multiples, perimeter vs. area.)
Curriculum needs to be written so that students learn one concept deeply before venturing into the confused concept.

• Robert Kaplinsky says:

Hmm. I do see your point Caroline, but this is a very I think that this is a very tricky situation. You could find yourself in a complex vs. complicated situation (more here: http://robertkaplinsky.com/is-problem-solving-complex-or-complicated/). I’m concerned that the extreme version of this is learn this and that and this and that and you don’t see the beautiful connections between the various standards.

So, what we agree with is that sometimes people have trouble learning something. One solution may be to bring context back. Another context might be to learn them separately. Worth considering both.

• Caroline says:

Ahhh, I agree. Nothing is ever black and white and the beauty is in the connections. And isn’t it wonderful when the kids discover the connections.
I think that it is possible, through the art of teaching and learning, to make sure that kids are allowed the time to solidify the pathways in the brain, create the appropriate schema for the body of mathematical knowledge, and create the learning experiences that allow them to discover the connections between the concepts in order to build larger understandings.
And yes, context is always critical.

6. Karen says:

Well, I agree; however, we still need to test vocabulary knowledge. These checks should probably be separate from context. Why make a child power through a problem, if she might be confused about the operations/formula she is to perform? That just makes a child more frustrated with math. Better to keep the work focused on real world situations.

• Robert Kaplinsky says:

Understood. Check out my reply to Jeremy. I think that expresses a similar thought.

7. Ben says:

I am currently grading a Geometry quiz I just gave about volume and surface area. These are sophomores in high school. There is an alarming number of them that don’t seem to understand the difference between volume and area, and I am not talking about struggling students. My honor roll, 4.0 students, that are 15 and 16 years old do not seem to be able understand the difference between between using three dimensions and two dimensions.

Here’s my question: Is it unreasonable to expect my honor roll students to know the difference, or do I need to provide context for them as well?

• Robert Kaplinsky says:

Let me answer your question with a question Ben. Check out this problem from Graham Fletcher: https://gfletchy.com/packing-sugar/.

Imagine asking students something like, “How many sugar cubes are inside the box? How much cardboard do you need (measured in sugar cubes)?”

Also imagine asking students something like, “What are the volume and surface area of the box (measured in sugar cubes)?”

If they correctly solve the first problem but not the second, then it seems like the issue is not about whether they conceptually understand volume and surface area but rather that they don’t remember the vocabulary. If they get them both wrong, then that’s a big problem.

As another example, I speak Spanish fairly well but I get some things mixed up. For example, I get the words for spoon and knife mixed up (cuchara and cuchillo). That’s not a big deal, because I have conceptual understanding of what each are.

While not ideal, personally I believe that in real life, the kids will be fine if they have conceptual understanding but forget the vocabulary.

8. Connie Rivera says:

I think we’ve skipped a different step which is more powerful than context. Why do we multiply to find area in the first place? Students need to experience arrays (made up of something like pennies or bingo chips) and realize that there is a shorter way to count all those [create a need for multiplying one row and one column]. Later they move to a measured area model where they can still see individual pieces (like on graph paper) before working with abstract area (like in your example). Concrete-representational-abstract.

• Robert Kaplinsky says:

Hi Connie. I think we are both on the same page and want the same thing. I think that an problem solving application like this will lead to a situation where students will be interested in more efficient methods for solving the problem. If students have that interest, then just put unreasonably big numbers like 120 x 47. Kids will realize that their current strategy doesn’t work well and will be open to developing conceptual understanding of alternative methods. Thanks!

9. Alecia Julian says:

I agree wholeheartedly–context greatly improves conceptual understanding. I’ve noticed that my 6th graders who recall what area and perimeter are can sometime get mixed up when using a model to determine area or perimeter. They understand that area is the space covered by the figure and can successfully count the squares to determine the area, but when counting perimeter they use a similar method but they only count each corner square once (as opposed to twice, for the two exposed sides), meaning that they give a value for perimeter that’s 4 units smaller than it should be. Composite figures also provide interesting insight into what students already know about area and perimeter. I have several students who will answer the perimeter as the sum of the given side lengths but not include any of the missing side lengths. Similarly, some students simply multiply all of the given measurements to find the area, rather than finding the area of individual shapes.

• Robert Kaplinsky says:

Interesting Alicia. I wonder what would happen if students were not given a grid overlay for counting the perimeter and area. Would they develop a model that avoids the possibility of incorrectly double/single counting corners?

• Alecia says:

It seems like that is the case, yes. Their homework last week covered area and perimeter and some problems had a grid and some did not. We’ll be having some intriguing discussions this week surrounding that!

10. Michael Sloan says:

For me it’s all about context. Problems that allow for multiple entry points, student discovery and are grounded in real world scenarios are ones that will allow for mastery of skills and retention. One of the things I do for perimeter and area is to talk about our school garden and building planters. How many designs can you find that have an area of 24 square units. Which one will need the least wood to frame?

11. From my experience and perspective, the underlying issue is less “math concept” and more language/terminology. Of course, there’s probably an interplay, and various students will be on different points on a continuum regarding this confusion. But I suspect that for many students for whom mathematics is just one big ball of confusion (cue THE TEMPTATIONS), their motivation is so low when they get to area and perimeter that they never make a point of processing the difference between the words, hence they don’t distinguish between the underlying concepts, and so “which formula should I use?” becomes guesswork. The area formula is easier to remember: just multiply the two different numbers together. So that is the one they might prefer to apply to ALL problems entailing either concept. Or for some, it might be that adding together any two numbers is their go-to strategy, so the addition part of perimeter sticks, but the multiplication part 2(l + w) goes out the window, and they miss that this is the same as 2l + 2w because math is really just a bunch of meaningless symbols (David Hilbert sort of said the same thing, though I don’t think he meant “meaningless” in the same way students do). In any of these cases, there’s really no thinking about what it means to find area or perimeter, so any calculation is as good as another.

I always try to unpack mathematical terminology whenever I can. I was, after all, an English teacher long before I became a mathematics teacher, and in my own mathematics learning, I hate when I can’t suss out some core linguistic residue in the terminology that helps me remember what something means conceptually. Some terminology seems unfortunate (e.g., “real” and “imaginary” might be amongst the worst and most misleading choices of all time), and some seems opaque (I’ve never been fond of the Bourbaki convention of “injection,” “surjection,” and “bijection,” but I can actually make sense of them. “Coefficient” seems a bit obscure and tortured, though not completely hopeless. “Scalene,” however, is beyond my linguistic paygrade. I just know what it means, but not how it was chosen for triangles with three different side lengths.

But “perimeter” literally means “around” + “measure,” and once you see that, if you have the vaguest interest in learning mathematics (no guarantee that any given student does, of course), that should seem pretty dramatically different from pretty much ANY understanding or concept you have of “area” from everyday language, let alone mathematics. Similarly, “circumference” (a word I wish were abandoned so that we said “perimeter” for circles, too) literally means, “around” + “carry”; perhaps that’s not quite the slam-dunk that “perimeter” is, but the “around” part should again remind students that we’re talking about the border, boundary, or outside of a shape, not a measure of what it contains within it (this gets really fun when “volume” enters the story, and we have “capacity” or 3D-enclosure rather than “surface” enclosure of space on planes of the faces of 3D figures that aren’t curved!)

And so dimensional thinking is another issue that might be relevant to separating “perimeter” from “area” so that never the twain shall meet in the mind of a given student. I think this all argues for a more “interdisciplinary” and multi-sensory presentation of these ideas. Manipulatives and/or good computer graphics, and/or really good illustrations strike me as fundamental for many students. Expecting them to “get it” in any meaningful way from a one-size-fits-all approach, particularly one that is highly abstract, is a guarantee that many will be left in the dust. It’s a way to reinforce every negative experience students have already had with mathematics as a bunch of painful and irrelevant bilge.

There’s much more to be said about all of this, of course. You’ve given me motivation for a blog post for June!

• Robert Kaplinsky says:

I’m glad that this post got you thinking so much! There really are so many ways that this could be approached. I intuitively believe that sometimes we create issues that shouldn’t really exist in real life. So, I’m interested in seeing your take on this.

12. Sarah says:

To make a connection to standard good practice (teach concept before procedure) perhaps it follows to also teach concept before vocabulary… in examining my own practice, I realize that this is how I already teach my students mean, median, and mode. We do a substantial amoubt of work figuring out how much each if evenly distributed (mean) before I ever mention the word ‘mean’. However, I never took specific notice on how often students still confused the terms compared to how I used to teach it. I will be taking note of that from now on!

• Robert Kaplinsky says:

Interesting. I’d love for you to check back in with what you learn.

13. When I teach area and volume to my high school geometry students, I start the lesson with this Do Now question.
Draw a sketch of 1 inch, 1 square inch and 1 cubic inch.
Draw a sketch of 6 inches, 6 square inches and 6 cubic inches (there are multiple answers).
I think this gives students a visual representation of dimensions. You are teaching vocabulary without giving a formal definition. This interactive from illuminations http://illuminations.nctm.org/activity.aspx?id=4095 is great.
Afterwards, I review the uses of each dimension and give each dimension a context.

• Robert Kaplinsky says:

That is a really neat idea Joan! I would imagine you would get everything from kids who are unsure of how an inch, square inch, and cubic inch are different to people unsure how to represent it to people unsure if there are multiple answers.

Great way to formatively assess them and get conversations going!

14. Charles says:

In my work with adult learners, context is everything. Often, when presented with abstract concepts, they will shut down until I connect the abstract concept to something that is fam