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.
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.
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.
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.
I am remodeling a 100-year-old house and have calculated area when ordering tile or flooring or sod by the square foot. I’ve measured perimeter and length for irrigation and partial perimeter when calculating area of backsplashes. It is a very important skill, but for a niche application.
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.
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.
Great drawer analogy.
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.
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?
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
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.
Sue,
I completely agree. I also make it a point to introduce perimeter with ‘secure the perimeter’ with my war enthused middle school boys. I introduce concrete, abstract, and tricky diagrams one often finds on the state tests, before moving on to other concepts. With my most math phobic students we work on all manner of problems as they learn other aspects of geometric figures, always adding that we need the perimeter as well. By this time perimeter is well embedded as a vocabulary word, and math process, usually addition, sometimes using algebra to find the missing side, etc. I introduce area as a completely different topic, with it’s own set of uses. I have groaned to myself many times as I followed curricula that introduced both in the same lesson. Just because they are mentioned closely together in a standard does not mean we must teach them side by side.
“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.
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.
Thanks.
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.
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.
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?
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.
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.
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.
I agree 100%. Same is true of fractions. Kids rarely make the mistake of adding denominators until they learn the algorithm for multiplication of fractions.
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.
Understood. Check out my reply to Jeremy. I think that expresses a similar thought.
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?
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.
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.
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!
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.
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?
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!
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?
Thanks Michael. Your two problems remind me a lot of these on open middle, albeit without context. http://www.openmiddle.com/rectangles-maximizing-perimeter/ (others here: http://www.openmiddle.com/?s=perimeter)
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!
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.
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!
Interesting. I’d love for you to check back in with what you learn.
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.
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!
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 familiar to them. For example, my GED classes are located in a rural area. The same student who struggles to find area and perimeter can easily compute the average of a parcel of land, as well as calculate the amount of fencing needed. By connecting to their experience, it makes it far easier to then introduce the vocabulary of mathematics once that context is applied.
Thanks Charles. It’s pretty hard to think about concepts abstractly when you barely understand them concretely.
I used to get mixed up when I was in high school, now we have third graders trying to solve the same kind of math problems. Why are third graders learning about area and perimeter? What’s the context in their lives so that it makes sense? As an adult I only really understood perimeter, area and volume from doing hours of gardening and spreading fertilizer and figuring out mulch. (Do you know how many adults can’t figure out cubic yards/feet to apply mulch!!) Probably not as pertinent for third graders !! Somehow we need to make it relevant to satisfy the standards.
Hi Penny. I believe that some of the reason area and perimeter are taught in third and fourth grade is because of their connection to multiplying and adding strategies. For example, 3 x 5 can be represented as:
X X X
X X X
X X X
X X X
X X X
The idea of teaching a concept when it’s relevant would be a tricky rule to enforce, but I do understand where you’re coming from.
Hi Robert and Penny,
I also think that we have discovered just how useful rectangular and array models are in both arithmetic and algebra that it is definitely worth introducing this concept early. For example, multiplication of two and three-digit numbers can be demonstrated with a rectangular area model before teaching the algorithm. This then extends to using the same model for multiplying binomials in introductory algebra. I work with Community college students taking developmental math courses and I am amazed by how the rectangular area model helps them to understand multiplying binomials and then factoring trinomials! I wish I’d have learned with these models in elementary school!
One thing I have wondered about the confusion between perimeter and area and whether we have added to the confusion is the use of using one-dimensional units when we talk about buying flooring for example. You’ll hear someone say for example that they need 40 ft. of flooring when they really mean square feet but it seems to be an accepted practice. Mulch is another, you buying “2 yards” of mulch is in reality buying 2 cubic yards of mulch. I would appreciate others’ ideas on this.
Hmm. I haven’t thought much about this, probably because I had a different experience. During the one (AND ONLY) time I installed hardwood flooring, we definitely talked about it in terms of square feet. This might be because they were non-uniform lengths and widths as opposed to tiles that were square feet.
So, you could be right, but I wonder how much experience others have too. Clearly, being precise with units is important though.
I teach SPED math, and my students definitely have a hard time with this. Most of my students struggle with reading and language as well, so I’ve often thought it was a vocabulary issue. Concrete models help tremendously, as well as real world problems. Many of my 5th graders have experience helping with construction projects. If I can frame my questions to match their experience, they understand much better.
Thanks Stacey. I can’t say that I’ve got this figured out completely, but I think it’s something that’s worth exploring in more detail. We can’t overstate the importance of context (or the problems with artificially creating it).
I think part of being educated is understanding what words mean. Perimeter and area are actually useful terms to understand. I agree that they should be taught contextually, but but we also need to teach the words more intentionally. I am guilty of assuming that high school students know those words. I am much more intentional about teaching and reviewing other math vocabulary. It would be easy to include a question such as, “Explain the meaning of the word “perimeter” in the context of this problem.”
Thanks Jane. Using precise language is very important for communicating ideas. I wonder though if it is better to start with vocabulary or context. Is it better to begin with the context and get students to a place where they realize that they don’t have the language to describe something (and therefore want it)? Or, is it better to give them the vocabulary first and teach them what it means? I’m more a fan of the former.
I may have missed this in the previous comments but I have two wonders. First wonder – I wonder what if we taught the context first and had the students think about what they might name it themselves. What gives it meaning to them. Second wonder – I wonder what if we taught the origins of words. For example, The word perimeter means a path that surrounds an area. It comes from the Greek word ‘peri,’ meaning around, and ‘metron,’ which means measure. Area comes from the Latin ārea, open space.
Interesting wonders April. I’m not certain about either of these, but I love how Polygraph from Desmos helps develop the need for more precise vocabulary. So while they may not necessarily name it themselves, they realize that it is something that needs to be properly defined.
I’ve been teaching elementary mathematics for over 25 years and I work hard to separate perimeter and area, as much as possible teach them at different times.
Learning in context is critical in building understanding. So I wonder how many students connect to turf and fences?
Instead, what context do they connect with? We could use the area of legos, or the area in a Mindcraft scene. Imagine building a wall around a structure they have built and coming up with the perimeter.
In addition, students themselves need to repeatedly use the terms so that it becomes part of the classroom conversation. I know many teachers who ask their students to sit on the perimeter of the gathering rug, no doubt this year-long set of instruction builds memory.
Last, I used to have my students run the perimeter of the playground, saying “Perimeter, Perimeter, Perimeter” over and over again! –trying to connect the kinetic to the experience, though I’m not sure what the long-term payoff was!
Multiple meaningful contexts!
Thanks Judith. I’m not sure I come to the same conclusion as you about teaching things separately. For example, when you play with Legos, you’ve got a great opportunity to them both simultaneously. I am less concerned with having the proper vocabulary than most, perhaps.
I have taught the topics with fencing or even using the baseboard square tiles around the room versus the tiles on the floor to give real life context. We’ve talked about carpeting versus baseboards as well. Actually, I think the real problem is teaching them too close together. I found that when Area was taught as an application of multiplication in third grade as Eureka sequences it, and when I didn’t touch perimeter at that time but REALLY focused on this new unit of measurement called area and how different it is from length in how the units look, my students were able to own it as its own concept. Later in the year, perimeter is introduced as an extension of length. Once both those ideas are internalized, THEN we can play around with the single room that has both of these measurement attributes.
I think we as teacher got used to introducing them almost as one – and then wonder about the confusion. Of course it’s confusing when two new concepts are introduced on top of each other. B-)
Hmm. I appreciate the pushback but don’t come to the same conclusion. I don’t have any evidence to support this as it is just based on my experiences. I’ll keep thinking.
Robert, love this post. It reminded me of when I taught second grade many years ago. For the longest time I couldn’t figure out why students couldn’t get the difference between pronouns and proper nouns. It was amazing. It happened every year. Finally it occurred to me that those were taught back to back and with the pacing a week apart. Those two terms sounded alike, were crammed into a week of study, and after all, these were 7 year olds! So transferring that thought over to area and perimeter, those two concepts are often taught together. Most likely, students haven’t fully internalized the idea of one, before the second concept is introduced.
Thanks again for all you do.
Interesting Fran. I guess this problem pervades many areas. I do come to a different conclusion though as to what the problem might be. I am less concerned about the two ideas being taught together and much more concerned about teaching it out of context or with an artificial context. I appreciate your feedback though as it makes me think.
Should area and perimeter be taught at the same time? Some say “yes”, while others disagree. Regardless of the sequence, the fact remains that our students are constantly searching for the shortcut. They want to take a quick look at the picture, decide what to do, mark an answer, and move away at record speed! A rectangle with dimensions identified could be used to find both the area and perimeter, which gives students at best a 50-50 chance. While teaching students that the area refers to the flat surface, while the perimeter is talking about the outer edge, this negates the fact that both area and perimeter are measurements….not locations!
We just finished our unit on area and perimeter in 3rd grade. When it was time to begin the unit I had just finished watching several of Dan Meyer’s presentations on creating “headaches” for students that the math becomes the aspirin to heal. (https://blog.mrmeyer.com/2015/if-math-is-the-aspirin-then-how-do-you-create-the-headache/). As the math coach, I wanted to try to use this idea in one of my classrooms so I choose our remedial 3rd graders who often lack motivation. The teacher and I each cut a shape out of a piece of card stock. We then began to have an argument about whose shape was the biggest. Her shape had many angles while mine had many curves. We asked the students to settle the argument for us – but to do so using facts that would convince a skeptic (an idea I got from attending an amazing workshop with Jo Baoler before Christmas). We explained what a skeptic was and asked one person in each group to be the skeptic and to decide if the arguments were convincing enough. The students really got into arguing for one shape or another. They even tried to use logical reasons ( the shape that has the most angles must be the largest, the shape with the least angles must be the largest and then they would defend why they thought their reason was best.) Each time the skeptics said they were not completely convinced. So then we talked about SMP #5 Using Appropriate Math Tools Strategically and I asked if there was a tool on their desk that they could maybe use to help solve the argument. In their desk buckets they have color tiles, place value disks, and counters at any given time (I was careful to make sure a ruler wasn’t readily available – I would have given it to anyone who asked for it but since I was trying to encourage tiling – I didn’t want to give a visual cue for it.) Each group immediately went for the tiles. We made duplicates of our shapes for them to play with and they began to explore. They came up with many different ideas on how to prove how big each shape was – one group tiled but not very neatly, one group tiled but didn’t cover the entire shape, one group made a tile line that measured the length and width of the longest point of each shape, one group placed pairs of tiles all over the shape being careful not to go over the lines. We then walked around and looked at each others ideas and talked about what was good about each one. We then came together as a group and talked about what we thought would be the best way to make a definite decision. The kids decided we should neatly tile the surface of each shape, covering the entire thing, without going over, and then count which shape had the most tiles and it would be the biggest shape. What amazed me was that they also asked if they could have smaller tiles so they could be more precise.
We went on from there to explore area in many other ways – trying to come up with shapes that had a specific “size” (number of tiles – area), then we worked just with rectangles and looked for patterns in finding the size of the rectangle. Eventually we had to changed our words because we introduced a new problem to the kids : which shape would need the biggest ribbon to cover it’s edge completely? So then we knew size was too general of a word. We decided as a class that were at least two ways we could think of to measure size (the tiling of the inside and the ribbon of the outside) – and then there was a need for vocabulary. The lesson continued over the course of a week and the kids had a blast and did amazingly well on the assessment (even better than some of the “regular” classes). Confusion seemed much less this year and truthfully we all had more fun. Teaching the kids with a context – even one that was contrived allowed them to make sense of area in an amazing way! …And for the record, my shape had the largest area!!!!! 🙂
Thank you for describing your journey, Laurie. I soooo love Dan’s headache and aspirin metaphor as it really helps you sort out the WHY behind mathematics. I think you nail it here: “Teaching the kids with a context – even one that was contrived allowed them to make sense of area in an amazing way! ”
That context really is important.
I think that we try to introduce the vocabulary too early and that’s why they get confused. If we use a familiar context and let them talk about the distance around the outside versus the amount square units covering the inside they better understand it. Contexts are tricky because all kids bring in their own background knowledge and cultural experiences. I have yet to find a context that all kids know because of the diversity of our students.
We use the designing a bumper card ride from the connected math project series from the covering and surrounding unit. The whole unit is fantastic. Even with that context we still showed all students a video of a bumper car ride and talked about the rails versus the tiles covering the floor because not all kids have been on a bumper car ride or even to an amusement part. As we get further into the unit we introduce the vocabulary and the kids go to a vocabulary activity where they write a tweet, draw a cartoon, Complete a crater model or a text message explaining the difference between the area versus the perimeter. This helps they to develop their own definition after having multiple exposures if seeing, hearing and using the word while solving problems throughout the unit.
Thanks for adding this perspective, Gina. I agree that we have to be careful with context. My broader point is that math is not present in real life without context. The people solving the problems are the ones who remove the context. So, presenting math in context really is important.
Hmmm… now I am thinking about numberless word problems and the removal of a question altogether. How would it change the way students think about area and perimeter if we game them more opportunities to grapple with statements like, “Man, my dog keeps running away. I need some fence!” or “Man, my basement floor is cold. I need some rug!” and then ask them to generate questions. What information do they need? How would they go about solving (acquiring) the problem (material needed for the specific purpose). At the same time, in the end, this is about a term and it’s meaning. Open ended discussions and explorations take a lot of time. At what point do we just have to assign some terminology and set about memorizing which terms go with which concepts? What do we give up if we push ahead? What do we give up if we devote all the time needed to cementing the conceptual understanding and matching terminology of both concepts?
I think the important thing is finding the balance between context and abstraction. I believe that historically, we’ve thought we’ve been doing students a favor by skipping everything straight to the formula. After all, if an easy way to do it exists, why not just give it to them? The problem with that though is it creates robots who only know how to solve problems in that specific situation.
So, i think we should aim to find a balance where students begin with the context, realize that it is burdensome to always have the context, and then abstract it so that they can focus more on efficiency. They need to have some role in that process.
Hi Robert,
If someone else has stated this in a previous comment, I apologize…
While I agree with you that context is so very important in having students “understand” that which we are asking them to do/solve, I also think that a lot of teachers (who are obviously adults) fail to understand that “context” for them is different from “context” for a young student. In the example presented, it is suggested that equating the amount of fencing required to surround a yard to perimeter gives the student context. Well… I would present that many students may have not concept (or context) for this idea in the sense that they have likely never had to go to a hardware store to purchase fencing to surround a back yard. What if they don’t have a back yard (or front)? Or have never stepped into a fenced bball court? And even if they have, have they considered or thought about the fence that surrounds it? All I am suggesting is that sometimes what we (as adults) think is a good context (because we understand or have experienced it) is not necessarily “good context” for the young student.
Great discussion!
Great point Ian. This gets into the whole mess that is “relevant” versus “real world”. There may not be any content that every student is familiar with. I don’t see that as a big deal. I just think that it’s a teacher’s job to make sure that everyone feels comfortable with a context.
Hallelujah! Would you please talk to textbook publishers?!! Why are area and perimeter in either the same or consecutive lessons every year? Why don’t we teach perimeter–in context–starting in a lower grade when students learn addition? Then we could teach area–in context–in fourth grade–with multiplication? Both words are too easy to mix up unless they are repeated many, many times separately first and much later together to compare and contrast.
I teach algebra. Believe it or not, students still mix the two up! But they’re such useful concepts to put polynomial addition and multiplication into context with an area model. I love to use the x variable to talk about pools or windows that are proportional but different sizes. Thanks for your insights!
Thanks Anne. I am less concerned about when they are taught this than I am that they come from an authentic context and are derived from there. Lots to think about.
What if??? What if we taught and assessed problems like those about area, perimeter, circumference, and volume with the real-world words (such as how many tiles to cover, how much fencing, how many gallons to fill) then taught students the vocabulary words as a “Did You Know?”. These “Did You Knows?” could be seen as math history in a lesson. Students would not be assessed on the terms, but their interest may be piqued in a natural way and they could choose or not choose to use the historical math term during discourse about such problems.
This is definitely something worth exploring. Personally, I’d prefer giving them the context in as an authentic form as possible. Sometimes when we’ve translated to a word problem, a lot of the authenticity goes away.
Wondering if the every day use of the word ‘area’ is one of the points of confusion. If one asks an elementary grade student to show the area where they play basketball, will the student walk around the edge of the court or will they traverse every inch of the space inside of the lines? I am thinking that some kids will walk along the edge of the court to show the area in which they play basketball. So having students name their actions might be best to do first and then add the precise mathematical language. I think for ELLs this might also be very helpful.
Once we move to teaching area with the more abstract array, I find that students do not connect the numbers along the edges of the array, what we call dimensions, as representing an actual measurement. There is some disconnect going from a real rectangle (whatever it is, garden, playground, classroom, I-phone carton) and scaling it down to its representation on a piece of paper. Dilation is often taught in 8th grade math and yet we are asking much younger students to understand that the array is a replica of something they can see, touch, walk around/on. To me, this is another huge disconnect for students.
Estimating to me is the most important skill we want students to gain. Do the students even have a sense of the magnitude of a square foot, square meter? We can put the problem into context but the units of measurement might still be very abstract for them. Without practice about size, being able to see something and estimate its magnitude is another thing that might help students with the concepts of area and perimeter. Students should be able to estimate the distance from one point to another and then move to square units. If anyone has connections with a flooring company, square foot tiles would be perfect for use in a classroom. When I was in a room with the tiled floor, was great, but there is nothing like holding that tile and working with it.
Thanks Marsha. It’s certainly important to build estimation skills. I think that these questions and wonderings are worth exploring. What I’ve come to realize is that students can find the answers to questions with context, but with abstract labels, it can tend to fall apart.
Going back to the original basketball court, you could just modify it to say that the coach wants the students to do sprints around the court but doesn’t know how far that would be.
Yes, perhaps you could. My broader point is that we need context and hopefully authentic content. I don’t often see people running around the outside of a court, and even if I did, I imagine it wouldn’t be perfect rectangles.
I totally agree with the concept of providing fake contexts. Additionally, I think presenting a rectangle with two side measurements is tapping into student context of the array area model that students are being exposed to in 2-4 grades. Introducing multiplication as groups and number of groups has students thinking about how much area is being taken up inside the rectangle.
A much clearer context if you want to use a basket ball court would be to say, you need to put bleachers all around a rectangular basket ball court with sides measuring 20′ x 45′. To cover all sides of the basketball court with bleachers, how many total feet of bleachers are needed?
I think “clearer context” is relative. Are bleachers really measured in linear feet? Could the fact that we’re measuring perimeter with three dimensional shapes cause other problems? What happens when students want to know how many people will sit in the bleachers, which would matter in real life?
Again, rather than trying to create a context to meet a problem, I prefer find one that already exists in real life.
My theory about why students struggle with area and perimeter is that the concept of a square unit is a completely abstract concept. I’ve found that students are much more likely to substitute perimeter for area than area for perimeter, and I think this is because they are familiar with instruments that measure the length. Students have seen and interacted with rulers, yardsticks and tape measures, but we have no similar physical square foot or square yard that we thump down on the ground and count to figure our how much carpeting or tile we need. When students reach back into their minds to figure our how one would measure the size of a room, the only thing they can dredge up is that a tape measure might be useful, so it must be the distance around.
I teach that peRIMeter can be remembered by thinking of the RIM of a basketball net.
Robert Kaplinsky, I see this happening also with addition and subtraction. All of your comments regarding perimeter and area apply here as well,
As another way to add context, I feel it is valuable to teach vocabulary deeply. For example, the ‘peri-‘ of perimeter comes from Greek and means ‘about or around.’ (Most students will know that a periscope helps sailors in submarines look around.) ‘Meter’ is also of Greek origin and means ‘measure.’ (This is an easy one for students to grasp as we use a meter stick to measure.) Once students know the parts, they never forget the meaning of perimeter.
Students would benefit from exposure to the vocabulary prior to being formally taught, so they are not grappling with new vocabulary, conceptual understanding, and operations simultaneously. Asking students to color the whole “area” of the figure/picture/shape in [red] during art projects, having students sit on the “perimeter” of the circle, etc. No in-depth explanation in warranted; modeling and repeated exposure is adequate for foundational understanding.
♥️ Tricia’s drawer analogy to give an everyday example of the perimeter (trace drawer when open), area (bottom of drawer), & volume (how much you can put in the drawer).
My students enjoy math songs to help them remember concepts. Our favorite for remembering Area vs. Perimeter is the Musical Cats-inspired animation by Bizilions “Area Around The Perimeter” https://www.youtube.com/watch?v=qU8aWpRd6Qw
We don’t even have to change the context.
The basketball team has to run the perimeter of the court. How far do they run in one lap? Or
The basketball coach told the player to run around the edge of the court. How far does the player run?
I LOVE the drawer analogy. I’ve used the fence a situation, but then realized that a fence would only be three sides.
I