[Fake World] Two Different Definitions Of Relevance

Wolfram uses the word “relevant” to define the word “relevant.” This highlights (again) the contested and circular nature of terms like “real” and “relevant,” terms which everyone says and everyone else nods along to, but which are in reality so soft they fall apart to the touch.

Meanwhile, here’s Ben Blum-Smith, three years ago on this blog, offering a much more useful definition of “relevance”:

The real test of whether a math problem is “relevant” is not “do you use this in ‘real life’,” whatever that means, but “do you want to solve it?” It’s not that you want to solve it because it’s relevant; wanting to solve it is what it means to be relevant.

These two different definitions of “relevance” and “real world” lead to two very design questions for your math class.

One: “How do I make [some difficult math concept] relevant and real world?”

The terms here are contested and circular and for many math concepts the question is impossible to answer.

Two: “How do I make [some difficult math concept] something students want to solve?”

This question is often very hard to answer and the answers are often highly contingent on the culture you’ve developed in your classroom.

But it has the advantage of not being impossible.

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 completely agree with the last statement that the answer to the question of what students will want to solve depends on the class culture you have developed. (And class culture depends on both you and the students.) There isn’t a one-size-fits-all answer in teaching because every classroom is different. I have a hard time keeping my classes at the same pace b/c they’re different and are motivated to do different things.
    If you successful in building a class culture that values curiosity and the coolness/beauty of math, the possibilities are much wider!
    Interesting post, thanks.

  2. I really like Ben’s definition, because it leaves room for the curiosity, beauty, and fantasy aspects of math that attract me. It reminds me of Lockhart’s video for Measurement:

    “That’s amazing! That’s completely unexpected. I would have expected: You make some crazy blob and connect the middles, it’s gonna be another crazy blob. But it isn’t – it’s always a slanted box, beautifully parallel. WHY is it that?!”

    That’s the sort of wonderment that entices students to think about math.

  3. Rather than “solve” – I like to think:

    How do I make [ANY math concept] something students want to (and see how) to use outside of the [math] classroom.

  4. @Denise, thanks for the Lockhart video. I hadn’t seen it yet.


    Rather than “solve” — I like to think:

    How do I make [ANY math concept] something students want to (and see how) to use outside of the [math] classroom.

    Can you define “use outside of the [math] classroom” a little more? Are there mathematical concepts that don’t have a use there? What do you do with those?

  5. I have an arsenal to back me up now when people of all ages asked me, “When will I ever use this in real life?” In my classroom, I’m quite frank with them right from the get go and tell them they will probably never use it. At this point, (and as you’ve mentioned in a previous post) I’ve lost them, not matter what explanation I give them. I’ve taken several routes in answering this question, but it’s pretty much useless once the question has been ask. My goal is to prevent the question from even happening by creating a task that students WANT to solve.
    I am grateful to have read this and several post to help me realize the difference between relevant/real world tasks and perplexing tasks. But, this leads me to another problem, how do I rewire my brain to think creatively enough to make tasks irresistible!? Any ideas on how to create an irresistible task on Mean Value Theorem?

  6. Re: Design Question 2 – Student motivation and classroom culture.

    This reminds me of another distinction you made about “two very different schools of thought.” On the one hand seeing “content and management (as) functionally the same,” and on the other hand developing them seperatley. (https://blog.mrmeyer.com/?cat=49)

    I just got back from a visit to the Ron Clark Academy*, so maybe this is more vivid for me right now, but I think we could place tremendous emphasis on motivation and culture without thinking twice about content. I mean, the ‘Standard Model’ of teaching is incredibly robust in this aspect with recomendations like “Have a clear system of classroom procedures (for everything!)” and “Kids will do amazing things for candy.”

    So having a bunch of songs, dances, buttons, and trampolines can be a great way to bring some enegry into your room, but there’s a stark contrast here when Mr. Lanier decides to Ditch the Stickers because although ” It still seems good to me to recognize one’s success and progress—and stickers aren’t an awful way to do this, but I think there are better ways. The best, of course, is when what one has done or gained or accomplished is the reward.” (http://lanier180.wordpress.com/2012/09/05/ditching-stickers/)

    Likewise, there’s a difference between having a long list of clear expectations, values, rules, and procedures (or whatever) and Mrs. Nguyen saying “Maybe I’ll go over our classroom rules and procedures on Day 2. It should take all of 10 minutes to go over them anyway. (If you need more than 10 minutes, then I think you have too many!).” Because Day 1 has been all booked up for a “few years” now with “doing math.” (http://fawnnguyen.com/2012/07/29/20120728.aspx)

    *The Ron Clark Academy is a tremendous school. Although their trademark methods were definitly in-your-face energetic magic tricks (non-content), they were also very clear about the necesity of rigorous content standards. From interacting with their students, I’d say they are succesfully cultivating a genuine interest in academics… as well as dancing, singing, and general purpose social skills.

  7. I was at the Computer-Based Math Conference, and while I saw and heard plenty of things that I strongly agreed with and strongly disagreed with – much more than is usual for me at conferences – I have to point out that there’s not nearly as much difference between Wolfram’s (perhaps unfortunately worded) message and Blum-Smith’s as this post would imply.

    CBM showed us a very brief glimpse at a curriculum in progress that included questions that would meet both definitions of relevance. Examples: “By how many steps are [two people] separated by on Facebook?” “How much can you compress a photo before you notice the difference?” “Am I normal?” They were organized in eight large topics, including “Knowing where you are in space,” “Math in the natural world,” and “Winning” that would definitely fall into the “do I want to solve it” category for most of the high schoolers I know.

  8. Michael:

    I was at the Computer-Based Math Conference, and while I saw and heard plenty of things that I strongly agreed with and strongly disagreed with — much more than is usual for me at conferences — I have to point out that there’s not nearly as much difference between Wolfram’s (perhaps unfortunately worded) message and Blum-Smith’s as this post would imply.

    Indeed, there is an enormous overlap between “math that kids want to solve” and “math that is ‘relevant’ or ‘real world’.” (Whatever those terms mean.) But there is also a large territory of math that isn’t relevant for a job and won’t be applied to the world outside the classroom but that kids can still want to solve.

    In a follow-up, Wolfram didn’t admit the value of that territory, which is his loss and ours.

  9. But there is also a large territory of math that isn’t relevant for a job and won’t be applied to the world outside the classroom but that kids can still want to solve. In a follow-up, Wolfram didn’t admit the value of that territory, which is his loss and ours.

    Indeed, and bingo – and probably the main thing that I strongly disagreed with. There wouldn’t be any computer-based anything if Euler hadn’t invented graph theory just for fun, hundreds of years before it had any practical relevance.

  10. Conrad Wolfram:

    …for “relevant to” I mean (hope I said!) “actually used today in”.

    Even if you believe that the main purpose of mathematics education is to prepare students with skills that are relevant to a job or the “real-world,” isn’s it unwise to prepare them for what is “relevant” today instead of helping them to be able to adapt to what may be “relevant” in the future?

  11. Real world and relevance may be taking away from understanding Wolfram’s point. And in this one tweet exchange the thrust of the idea has been undersold. Computing has changed what mathematics is. The importance of the “real world” comment here is only to underscore how the essence of the subject is shifting. Other disciplines are not changing fundamentally in the same way because of technology. The problems we choose, whether they are “real world” in the sense of what you will need in a high paying job today, or “relevant,” as in what do students want to solve or what will engage them, need to include the concepts that have been created since computing technology became ubiquitous. The role of teachers building classroom culture and engaging students was at the heart of the conference.

  12. I gave my students a post-test extra credit sheet full of problems like this one [http://goo.gl/E5Ao1f] the other day.

    They ate it up. And not a single student asked me when they were going to have to use the situation in real life. That’s a good thing, because I would not have had a good answer for them.

  13. Over the past few years, I’ve been coming to understand that there are huge differences between and among the many different forms of the fundamental student question about the “relevance” of mathematics. Right now, I understand these questions to fall into two main categories.

    The first is the kind of question that asks, How is this new/strange/perplexing idea/concept/skill connected to other things/ideas/skills/concepts in our world? These forms of the question express a kind of openness to perplexity and to student agency and to the possibilities that math can help us to unlock.

    However there are also forms of this question that are not genuine questions at all but rather blind, mute, and highly emotional defense mechanisms – rhetorical forms of a question that seeks not to actually *ask* a question about mathematical relevance but rather to express a stance of resistance against engagement in the classroom community. In this regard, they are a kind of “acting out” against the didactic contract – an attempt to use willfulness as their own last, best defense against attempts to guide them to owning their own agency. For whatever reason, rightly or wrongly, any such effort seems to feel to these students like a threat to their very subjectivity.

    I’m not trying to defend or attack this second form of questions and questioners – I’m extremely interested in coming to understand what is going on with these students because it seems to me that their relationship with the learning process has become so poisoned they feel compelled to defend themselves against it.

    So I’m curious to hear what others may have to say about this phenomenon.

    For whatever reason, there still seem to be a lot of students (meaning > 0) who feel the need to resist what others of us experience as math’s “call to adventure.” I think that cultivating perplexity and a healthy classroom culture of inquiry and a spirit of delight in discovery can help to erode this resistance among a great number of students. But there remain some students who continue to see this as a form of entrapment, and I have no clue how to help move them through this self-defeating defense system.

    Thanks for another great post+conversation about all this.

    – Elizabeth (@cheesemonkeysf)

  14. @Elizabeth,

    It seemed like an important realization at the time: when kids ask “when will I ever use this in real life?” they don’t necessarily mean the question literally. What they mean by that question and what we can do about it is, as you point out, a fairly open question, though.

  15. I had to give a speech on this recently at my school in front of a lot of parents – and I got in a bit of trouble for my answer.

    In Sweden, the idea of trying to make math more “real world” is actually written into the school law and high schools have been criticized for not doing enough in this regard. In junior high (which I currently teach) there is less pressure since we are required to give them a more general background, but in Sweden already in high school students have to pick a “program”, which is more or less a major and the classes they take are tailed to that program, though there are some general ed classes (like math).

    For example, in high school students might opt into the “vehicle” major, where there learn all about cars with the idea that most of them will end up being truck drivers or mechanics or something along those lines, and not go to university. They still have to take math the first year, but it should have a lot of car problems in it according to the law.

    Anyway, at my junior high parent meeting, the issue about “when will we use this” was on the table.

    I first asked how many of the parents had heard of Mozart. Nearly all raise their hand. I asked how many of them worked with classical music in their jobs and none raised their hands (with kids I have asked the same except with them I ask how many of them listen to classical music in their free time).

    Same for Picasso and painting.

    Then I asked how many had heard of Gauss, and only 6 out of 200 raised their hands (more than I expected actually). I explained who he was and asked why they thought this was okay from a cultural point of view, especially considering he probably has had a lot more direct impact on their daily lives and our society than the other two.

    I made comparisons to art and history and literature – things society agrees that in some way enrich our lives and are part of this bigger concept called “generally educated” (not a great translation of the Swedish word I use here, but close), even if we don’t work with them in our jobs, even if we don’t “use them in the real world”.

    I pointed out that studies have recently shown that literature gives doctors a better bedside manner and increases empathy in most people. The point being that sometimes the benefits are not obvious. I talked about how I thought math could enrich a persons life and view of the world even if they didn’t use it in their job.

    A lot of them liked my speech, but some didn’t like the idea that I wasn’t pushing the practical “calculate your taxes” reason for learning math. My administration wasn’t thrilled with me either.

    One parent said they got chills (the positive kind) from my speech, another parent came up and grilled me about the use of percent in “real life” and how dare I not talk to the students about that.

    It is a very sensitive topic here.

  16. I agree with the idea that the “When will we ever use this?” question is often utilized by students who are bored or unengaged from the learning task. However, I’d like to throw another conjecture into the conversation: this is cultural and students are scared to death of mathematics and not confident in their abilities to practice mathematics. Students use this question in a contentious manner because they are struggling to make conceptual connections and move beyond the concreteness that has been their math world into the abstractness which math evolves. Normally, I would say that when individuals are struggling with something, they inherently want to get better at it through practice or hard work. But in the case of mathematics, it has become sort of a “badge of honor” to not be “good” at math. Our culture has formed this idea that mathematics is something only “smart” people get, and the rest of us just need the basics to get by in our daily lives. (I’m not here to debate what has caused this cultural mindset, but maybe the “old-school/back to basics/[insert jagon here]” way of teaching mathematics was not so great after all.) I’ve sat through numerous parent-teacher conferences in my 10 years of teaching, and every year, multiple times, I have parents tell me that they “just don’t get math” so they pass their students to their spouses, older siblings, etc. This mindset is often verbalized in front of their student, further perpetuating the idea that it’s ok not to work at getting better with their proficiency in mathematics. I’m not an expert in Maslow’s hierarchy of needs, but I do know that students hold their parents as the most important provider of their needs. So if it’s ok that their parents aren’t good at math, and they don’t see them working to get better at math, than that mindset is ok for them too, regardless of the relationship built within the classroom, the possible connections to future employment, or just the sheer curiosity of a perplexing task, real-world or not.
    * I understand that this does not apply to ALL students, but I wouldn’t hesitate to generalize the ideas to a lot of students.

  17. Using “wanting to solve it” as a definition of math relevance misses the point, completely. So many kids are motivated in school by extrinsic factors — grades, standardized test scores, obsession with college admissions, etc. So if we had a national math curriculum that assessed students on the basis of speed and accuracy of performing low-level math operations on an abacus, a slide rule, or an obsolete adding machine, we’d undoubtedly have lots of students who would want “to solve it.” What would that tell us? Would that give us assurance that we’re teaching kids math that matters? Clearly, no. The harsh reality about K12 math education in the U.S. today is that kids are required to take lots and lots of math, are drilled and assessed on the basis of how many low-level calculations they can do by hand, and yet almost no adult in our country uses any math beyond decimals, fractions, and percentages. Look no further than AP Calculus, which teaches things no one uses in the real world (unless you’re an . . . AP Calculus teacher), but is the “gold standard” for math proficiency at the K12 level.

  18. I’m reasonably certain Blum-Smith meant “want to solve it” in the intrinsic, intellectually-needy sense. In any case, that’s the sense in which it’s used here.

  19. In any event, Dan, it’s hard to understand why what’s relevant in math education should be defined by student perspective on wanting to solve it. I’m completely committed to providing educational opportunities for students to pursue meaningful challenges in school, driven by intrinsic motivation. By why does that drive what’s relevant math experiences? That seems like a complete non sequitir. I think we collectively have an obligation to push educators to engage students in learning experiences that help them develop important skills. At a minimum, I’m hoping we all agree that spending most of high school math education on mastering rapid performance of low-level calculations isn’t meeting any reasonable goal for our students.