[PS] Metal Lids

Foundations and Pre-Calculus – Mathematics 10, Pearson.


John Scammell [twitter, blog] via e-mail:

I suppose they think it is contextually brilliant because they supported it with a pretty picture. My two questions are, “Who cares?” and “Why don’t they just count if they care so much?”


Great problem. I wonder if “Who cares?” is a good lamp for guiding our curriculum design, though. There will always be students who don’t care, because engagement is relative. The part that strikes me about the problem is that you could replace “cones” with any random unit, or even gibberish, and it wouldn’t diminish the relative engagement of the problem one bit, from student to student, because there isn’t anything inherent to those cones that leads to that system of equations.

At this point, I have a lot of submissions I can’t (yet) post because I can’t personally prosecute the charge of pseudocontext. You need to convince me. I’m relying on you all to make the case for or against pseudocontext in your e-mails and in the comments. And definitely check out Ben Blum-Smith’s recent description of the term.


Talise folded 545 metal lids to make cones for jingle dresses for herself and her younger sister. Her dress had 185 more cones than her sister’s dress. How many cones are on each dress?


  1. Scan an example of pseudocontext.
  2. Email it to dan@mrmeyer.com
  3. List the textbook title, edition, and publisher.
  4. Give me your interpretation of the term “pseudocontext.”
  5. Let me know if you’d like credit (name, blog or twitter) or if you’d prefer anonymity.
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. I’m amused that they’re made out of “metal lids” instead of specifying that jingle dress cones are traditionally made out of tobacco lids. That tidbit doesn’t change the problem, but it does make the context feel that much more pseudo-fied.

  2. Dan-
    I am a big fan of your work. I teach history and my belief is that relevance and engagement are 2 cornerstones to quality instruction and quality learning. I read your “WCYDWT” and it is fascinating to the point where it makes me want to be a math teacher!

    At any rate, as an avid reader, my thoughts always come back to, why aren’t more sports integrated math instruction? See: ESPN commercial…


    As a 30 yr old male in more fantasy sports leagues than I would like to admit, I see a world of potential math instruction that could be both relevant and highly engaging.


  3. @Doug: I’m both a math teacher and a sports fan, and I too see lots of opportunities in sports for exploring math. (One of my favorites: a major leaguer can have a batting average of .333 on the first day of the season; but if his batting average is .334, then it must be June at the earliest.)

    I believe, unfortunately, that there’s a huge downside to sports-related problems: people who aren’t sports fans are extremely confused and put off by everything that we take for granted. If you want a taste of what this feels like (and you’re American like me), go read the recap of a cricket match. Imagine being presented with a math problem using all of the nouns that cricket fans toss around so casually.

    In addition, I hope that one day we will live in a world where girls are just as likely to be interested in and excited by sports as boys. But that’s not the case today. So the differential between how well sports math problems would engage sports-savvy students and non-sports-savvy students would also exacerbate the existing obstacles that girls face when learning math in the US.

    For these reasons, I’ve resolved not to use sports problems in my math classes.

  4. What I find most ridiculous about this problem is the grade level. 10th grade?!? This should be a 3rd or 4th grade problem. I guess I have been spoiled by teaching Singapore Math?

  5. Cathy Campbell

    October 2, 2010 - 3:41 pm -

    Yes, this is a simple question and not very interesting. I think the publisher used this context to support aboriginal students as they would be familiar with dancing and this type of costume. I believe it is a First Nations girl dancing and her dress is decorated with the metal lids. In our Western Canadian Program of Studies we are trying to infuse the First Nations, Metis and Inuit perspective more often.

  6. What bothers me about this, is this is how textbooks publishers, teachers… seem to think is an authentic way to address the struggles of First Nations students (and other ethnic groups) with mathematics. It is important that students be able to “see” themselves in the resources and in the classroom, but the assumption is that this picture, with the highly culturally-stripped question, is somehow worth a check mark in that column. There is great significance in the dance, the dress, and even what the jingles might be made of, yet none of that is mentioned. Instead, all that’s done with it is to make an excuse for doing some Western mathematics.

    If we really want to be respectful of other cultures and their mathematics, it’s about time that we started letting the people who are from those cultures have a real say in our mathematics classrooms. Mathematics is an incredibly culturally biased subject, and it’s about time that we started addressing it in that way. This is not to say that students don’t learn Western mathematics, as it is such a dominant player in our Western society; however, it’s time to learn that it’s not the only way to think and do mathematics.

    This is actually one of the reasons that often like the things I find on this blog — because they’ve been stripped to the bones — here’s an idea, a picture, a video, a general question — now you go out and investigate it in your terms, through your understandings, through your ways of knowing (cultural and other), and then lets bring those ideas together.

  7. A majority of my students are American Indian. Many of them do travel to pow-wows and participate – some dance jingle.

    I know them well enough to guess what their reaction to this would be. They are culturally savvy enough to see when someone is “culture dropping.” Believe me, I tried it a few times my first couple of years. Can you say lead balloon?

    My guess is that this would actually be more of a distraction for my students than anything. In fact, some might find it mildly offensive in that a textbook has taken something culturally significant and distilled it down to “who has more cones?”

    Also, I believe the number of cones has significance for some people. A quick check of Wikipedia (http://en.wikipedia.org/wiki/Jingle_dress#Contemporary_Design) shows that childrens’ jingle dresses typically have 100 – 140 cones, so the numbers used in the problem might be a distraction as well.

  8. @Gale #6: can you elaborate on what you mean by “Mathematics is an incredibly culturally biased subject”? Are you refering to the (pseudo)examples we use to teach math? Or are you talking about the actual content of math courses?

  9. I’m taking a class on research issues in mathematics education. Last week we had to read an article on cultural bias in math classes. Bishop, A.J. “Mathematics Education in Its Cultural Context”, Educational Studies in Mathematics 19 (1988) 179-91.

    The publishers here have done exactly what you assert. They have taken a pretty picture, forced a context on it to turn it into a math question, and then patted themselves on the back for not only making a math question, but fulfilling the requirement to include FNMI perspective in their book.

  10. While no doubt there is great cultural significance in the dance, the dress, and what the jingles might be made of, none of that is relevant to the math problem. Math does not depend on cultural context; that is what makes it so powerful. “Western” (modern) mathematics may not the only way to think and do mathematics, but it is the only one that is of any importance to contemporary technological society.

  11. @Greg – I don’t think sports as a basis for math problems is an issue of whether girls are interested in sports (I don’t think you do, either, since you mostly refer to sports-savvy and not-sports-savvy rather than boys and girls).

    I think the issue is that if you use sports as a basis incorrectly, you are requiring knowledge of a system of rules that students might not know AND are sometimes not giving them time to find out. On a homework problem, it’s just frustrating, as the student has to find someone to ask how many points a touchdown is worth. On a timed test, they might not have time to ask, or they might not be allowed.

    The pseudo-content questions seem to avoid this problem, but I feel like WCYDWT embraces it somewhat. It makes sense, based on the environment when the questions are asked: [PS] seems intended for independent work, and WCYDWT encourages questions to be asked.

  12. I’m going to agree with Diane and Tyler, the fact that this 3rd grade level problem is in a 10th grade Pre-Calculus speaks volumes to the expectations of students.

  13. Can someone explain to me how this is third grade? You can guess and check this out pretty easily but the fastest way to solve it (unless I’m mistaken) is with a system of equations.

  14. There are a total of 545 lids. The larger dress has 185 more, so imagine setting those aside:
    545 – 185 = 360 lids left.
    Those 360 lids are evenly split between the dresses:
    360 ÷ 2 = 180.
    So there are 180 lids on the smaller dress and 180 + 185 = 365 on the larger.

    This type of reasoning is very common in Singapore Math word problems. See examples in my Word Problems from Literature series. The problems are often modeled with bars (a sort of visual algebra), to make the number relationships easy to see.

    total = 545

  15. I should also point out that this method is taught not just as a trick to solve this sort of problem, but as one element among many in developing number sense and mental math skills. Numbers are manipulated in many ways, taken apart and put back together, worked with and played with. It makes a great foundation for algebra.

  16. Here’s a problem from the Grade 3 Primary Mathematics materials:

    Adam and Carly have 135 baseball cards altogether. If Adam has 41 more baseball cards than Carly, how many baseball cards does Adam have?

    The fastest way to solve it is by understanding the relationships, which the model method helps students see.

    FYI, the math books used in Singapore include National Education in the Teacher’s Resource Pack. In the 3B Teacher’s book for Shaping Maths, the unit on Time is introduced using pictures of festivals and celebrations: Diwali, New Year, Christmas and Hanukkah. Here’s the note in the teacher materials from that page:

    “Though we have many different races and religious groups in Singapore, we all still pursue the same dream – The Singapore Dream. NE message: We must preserve racial and religious harmony

    You don’t see that in the U.S. version of the curriculum.

  17. I think there’s a mistake in Denise’s bar diagram. The top right bar should be labelled 185, not 180.

    But as Denise’s comment suggests, the only math skills you need to solve this problem are counting and pairing. You can do it on a tabletop with a jar of pennies: Take out 545 pennies. Separate them into two groups to represent the lids on the two dresses. Take 185 pennies out of the larger group. Equalize the two groups by pairing to correct for any error in your original separation. Then put the pennies you took out earlier back into the group representing the larger dress.

    You now have two groups of pennies which contain the same number of pennies as the number of lids on the smaller and larger dresses, respectively. If you count the two groups of pennies, you’ll find that the smaller one has 180 pennies and the larger one has 365 pennies. That tells you how many lids are on each of the two dresses.

    So you’ve solved the problem. And you didn’t need to subtract or divide. You didn’t need know how to use numerals, much less equations. Counting and pairing sufficed. (Of course, in the real world, with the two dresses at hand, one could solve the problem even more simply just by counting the lids themselves.)

    So if the purpose of the problem is to get the answer, or to learn how to solve problems like it using manipulables, or to develop general number sense and mental math skills, then it might be considered a third grade problem. But does the fact that it can be solved in this way mean that the problem can’t (or shouldn’t) be used in higher grades to teach more advanced math skills, such as how to manipulate equations?

  18. Denise,

    Thanks for that explanation. I played around with a Singapore bar model a bit last year, and was able to do systems questions like the one you described above. As long as one of the variables was the same in each equation, I could model it. Can you tell me if it is possible to model and solve using bars a system like:

    2x + 5y = 20
    3x + 2y = 19

  19. @Zeno
    Thanks for the correction! I was trying to do too many things at one time yesterday and didn’t edit as well as I should. As for the question of whether such a problem could be used to teach systems of equations — yes, in algebra 1 it might. But did you notice that this book *claims* to be pre-calculus?

    I suppose you could use bars for that system, but I don’t know why you would want to. Algebra is much easier when the coefficients are not related at all.

    The bar model can easily be used for systems of two variables where one of them drops out. This, for example, might be a 4th or 5th grade problem:

    2x + 5y = 20
    2x + 2y = 19

    Systems with three variables are also used, but these are very simple. For instance, here is one from my Hobbit Math article:

    x + y + z = 123
    x = y + 15
    z = y + 3

    And in 5th-6th grade, just before switching to algebra, there are problems where the student needs a multiple of one of the equations, such as:

    2x + 5y = 20
    x + 3y = 11

    Of course, these aren’t given as variables, but as word problems. Usually they have to do with purchasing an assortment of items, or fixing a variety of foods for a party, or collecting and trading stickers, etc. — so that the students have something concrete to imagine.

  20. Dan, can you point to any examples of word problems where the context is *not* pseudocontext (i.e. steering clear of this bizarre need to be culturally sensitive, environmentally aware & politically correct)? Perhaps if we all carry around a library of “appropriate contexts” that work for different classes of math problems then we’ll be able to rewrite the obnoxious ones when we see them.