1) Give students problems that actually matter to them.
I briefly mentioned this idea in a prior blog post ("My Vision for the Future of Secondary Mathematics Education"), and this is a driving point throughout the texts of Paul Lockhart (which I also talked about in that same prior blog post). If mathematics is an art (as Lockhart argues), then it must be understood and practiced as an art. It is enough for mathematics to be done for its own sake, for its own inherent beauty and pleasure.
This doesn't mean that there are no benefits to it. Far from it! Are there benefits to the arts of music, of painting, of poetry? You know that there is plentiful research and there are countless more writings on the benefits of the arts for the holistic development and thriving of our students and our society. As students are assisted to find problems that actually matter to them (which you can do when you know them well) and given space to work on them at their own pace, they are naturally and implicitly developing their skills in investigation, critical thinking, and problem-solving. It is OK for students (in fact, it is OK for people, of ALL ages) to work on problems that they just find fun, without any identifiable applications. (Interestingly enough, historically, the applications are actually usually discovered later!) The political pressures might tell you it's not OK: we're wasting time, kids aren't learning or growing fast enough. But how much of what students have learned in math-class-as-it's been-done can we say are contributing factors to their growth as human beings 10 or 20 years later, and how much of it just felt like hoops to jump through and motions to go through to graduate high school and move on with their lives? Real, lasting growth takes time. And so does real mathematics. Our teachers needs to believe this so that our students can believe this. It must be built into the fabric of the classroom culture.
I strive to model continuing to work on new mathematics for fun. I've been thinking lately about the famously unsolved (unproven) Collatz Conjecture. If you're not familiar with the Collatz Conjecture, here is a great video tutorial on it (by Numberphile) that you should stop and watch at least the first few minutes of: https://www.youtube.com/watch?v=5mFpVDpKX70. "Erdos actually said this is a problem for which mathematics is perhaps not ready. Turns out all this fuss is about a problem that any fourth grader can understand." This is an utterly fascinating problem! Are there any practical applications to this? Not that mankind is currently aware of. But people keep working on it. Is it because they're hoping to find a practical application of it? Doubtful. They're just working on it because it has sparked their curiosity and their wonder, and in this way has made itself matter to them. This is just one of many unsolved, mysterious, "pure" mathematics problems that have captivated people just by what it is, by its inherent being.
2) Give students problems that they will actually have to solve outside of math class immediately or in their foreseeable futures.
Sometimes (though I would argue that it's not as often as you might think) what makes a problem "matter" to a student is its authentic applicability to him/her. That's been somewhat recognized for years, and so this is a repeated outcry we in mathematics education have all heard: give students "real-world" math problems! But unfortunately, the good intent there has often led to the creation of problems that, while they may include real-world elements, are contrived and impractical -- they're not problems that any sane person would actually encounter in the "real world". Here are two examples:
Example #1:
Maria is two years older than twice her age seven years ago. How old is Maria?
In what situation would you ever know that someone is two years older than twice her age seven years ago and not know her actual age!?
Example #2:
Summer Nail Salon is having a special this month on services. Over the weekend, they performed 33 manicures and 40 pedicures, bringing in a total of $1509 in receipts. So far this week, they have administered 13 manicures and 43 pedicures, with receipts totaling $1330. How much does the salon charge for each service?
In what situation would you ever know all the information given in this problem and not know the how much was charged for each pedicure and each manicure!?
Again, the authors of these problems have thrown "real-world" elements (age, manicures, pedicures) together into a fake situation to lure students into using a mathematical idea (e.g. systems of equations) in an impractical way. And, intrinsically, students know this, even when adults deceive themselves of it.
Now, does that mean that there is no value to students solving problems like these? Not necessarily! Refer back to my first wider-reaching thought/solution: "Give students problems that actually matter to them." What matters to students is highly subjective, relative, variable (which, again, is why teachers need to know their students well). If a student finds these problems fun and/or interesting and want to do them, let them do that! And they will garner the aforementioned benefits of developing their mathematical thinking. Just don't pretend that this is a practical application of mathematics. That will only send students down a line of thinking that mathematics is anything but practical.
If students want practical mathematics, then instead of these contrived "real world" problems, we must give students authentic problems that are applicable to students immediately or in the foreseeable future -- that is, the students must be able to foresee and believe that they will have to apply this in the future. When asked, students, especially older students, should be able to state with clarity and conviction the applications of their mathematics. The students need to believe that they will encounter problems like these outside of math class. Students knowing why what they're doing matters is the hinge factor for engagement and learning.
Dan Meyer has written a handful of insightful blog posts over the years about "real-world math" (http://blog.mrmeyer.com/category/fake-world-math/), so I'll close by quoting him on "relevant" math problems:
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.
I briefly mentioned this idea in a prior blog post ("My Vision for the Future of Secondary Mathematics Education"), and this is a driving point throughout the texts of Paul Lockhart (which I also talked about in that same prior blog post). If mathematics is an art (as Lockhart argues), then it must be understood and practiced as an art. It is enough for mathematics to be done for its own sake, for its own inherent beauty and pleasure.
This doesn't mean that there are no benefits to it. Far from it! Are there benefits to the arts of music, of painting, of poetry? You know that there is plentiful research and there are countless more writings on the benefits of the arts for the holistic development and thriving of our students and our society. As students are assisted to find problems that actually matter to them (which you can do when you know them well) and given space to work on them at their own pace, they are naturally and implicitly developing their skills in investigation, critical thinking, and problem-solving. It is OK for students (in fact, it is OK for people, of ALL ages) to work on problems that they just find fun, without any identifiable applications. (Interestingly enough, historically, the applications are actually usually discovered later!) The political pressures might tell you it's not OK: we're wasting time, kids aren't learning or growing fast enough. But how much of what students have learned in math-class-as-it's been-done can we say are contributing factors to their growth as human beings 10 or 20 years later, and how much of it just felt like hoops to jump through and motions to go through to graduate high school and move on with their lives? Real, lasting growth takes time. And so does real mathematics. Our teachers needs to believe this so that our students can believe this. It must be built into the fabric of the classroom culture.
I strive to model continuing to work on new mathematics for fun. I've been thinking lately about the famously unsolved (unproven) Collatz Conjecture. If you're not familiar with the Collatz Conjecture, here is a great video tutorial on it (by Numberphile) that you should stop and watch at least the first few minutes of: https://www.youtube.com/watch?v=5mFpVDpKX70. "Erdos actually said this is a problem for which mathematics is perhaps not ready. Turns out all this fuss is about a problem that any fourth grader can understand." This is an utterly fascinating problem! Are there any practical applications to this? Not that mankind is currently aware of. But people keep working on it. Is it because they're hoping to find a practical application of it? Doubtful. They're just working on it because it has sparked their curiosity and their wonder, and in this way has made itself matter to them. This is just one of many unsolved, mysterious, "pure" mathematics problems that have captivated people just by what it is, by its inherent being.
2) Give students problems that they will actually have to solve outside of math class immediately or in their foreseeable futures.
Sometimes (though I would argue that it's not as often as you might think) what makes a problem "matter" to a student is its authentic applicability to him/her. That's been somewhat recognized for years, and so this is a repeated outcry we in mathematics education have all heard: give students "real-world" math problems! But unfortunately, the good intent there has often led to the creation of problems that, while they may include real-world elements, are contrived and impractical -- they're not problems that any sane person would actually encounter in the "real world". Here are two examples:
Example #1:
Maria is two years older than twice her age seven years ago. How old is Maria?
In what situation would you ever know that someone is two years older than twice her age seven years ago and not know her actual age!?
Example #2:
Summer Nail Salon is having a special this month on services. Over the weekend, they performed 33 manicures and 40 pedicures, bringing in a total of $1509 in receipts. So far this week, they have administered 13 manicures and 43 pedicures, with receipts totaling $1330. How much does the salon charge for each service?
In what situation would you ever know all the information given in this problem and not know the how much was charged for each pedicure and each manicure!?
Again, the authors of these problems have thrown "real-world" elements (age, manicures, pedicures) together into a fake situation to lure students into using a mathematical idea (e.g. systems of equations) in an impractical way. And, intrinsically, students know this, even when adults deceive themselves of it.
Now, does that mean that there is no value to students solving problems like these? Not necessarily! Refer back to my first wider-reaching thought/solution: "Give students problems that actually matter to them." What matters to students is highly subjective, relative, variable (which, again, is why teachers need to know their students well). If a student finds these problems fun and/or interesting and want to do them, let them do that! And they will garner the aforementioned benefits of developing their mathematical thinking. Just don't pretend that this is a practical application of mathematics. That will only send students down a line of thinking that mathematics is anything but practical.
If students want practical mathematics, then instead of these contrived "real world" problems, we must give students authentic problems that are applicable to students immediately or in the foreseeable future -- that is, the students must be able to foresee and believe that they will have to apply this in the future. When asked, students, especially older students, should be able to state with clarity and conviction the applications of their mathematics. The students need to believe that they will encounter problems like these outside of math class. Students knowing why what they're doing matters is the hinge factor for engagement and learning.
Dan Meyer has written a handful of insightful blog posts over the years about "real-world math" (http://blog.mrmeyer.com/category/fake-world-math/), so I'll close by quoting him on "relevant" math problems:
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.
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