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That is... absurd. What an absolute waste of time, setting arbitrary patterns in interesting math questions- do we seriously want students, upon seeing a tough

That is... absurd. What an absolute waste of time, setting arbitrary patterns in interesting math questions- do we seriously want students, upon seeing a tough problem, to "just start adding"? What rationale could anyone possibly have to teach this mindless nonsense to children, unless it was to prepare them for other standardized tests later in life, which are unanimously the product of mathematically impotent failed bureaucrats?

ivalm 5y ago
Oftentimes problems are hard, being able to come up with a useful heuristic (in this case greedy + fixer) is probably the most valuable thing one can do. Most math problems in real life aren’t solvable, you have to learn to approximate/bootstrap your way through.
- quacked 5y ago
  Which I agree is an extremely useful skill to have, but I wouldn't call this problem "bootstrapping", I'd call it "a trick question". If the question had asked "provide a general solution to get you pretty close to the answer", I'd retract my criticism.
  - noduerme 5y ago
    Write an essay on what the logic was to get you there - sure! I wish math classes were always taught that way!
phoenixreader 5y ago
I can only venture a guess. Many interesting mathematics results come from people simply observing patterns when doing adding, subtracting, etc. Many insights can come when people are not afraid of getting hands dirty. It's about getting rid of a fear of computation and developing a system for exploring unknown problems. For example, consider this problem for geometric series: 1 + 2 + 4 + 8 + 16 + 32 + ... + 4096 = ?. This problem may look completely intractible and tedious without a calculator. But once you start adding: 1+2=3, 1+2+4=7, 1+2+4+8=15, 1+2+4+8=31, ..., even a child can spot the pattern. Now that they are amazed, they will be motivated to learn more about the cause of this pattern, etc, etc.
- quacked 5y ago
  And I in turn can only venture a guess that you have entirely forgotten the brain-liquefying tedium of modern mathematics education, as shown by the fact that you immediately revert to actually interesting patterns that a properly-educated mathematician might think about. The dollars-and-cents addition problem mapped onto the "fun" toy store narrative is not mathematically interesting until assigned the type of analytical firepower outlined in OP's blog post. It is a deliberately obtuse trick question posed as a learning tool; children labelled "gifted" will figure it out and feel the dull satisfaction of killing another "problem", and those without that benefit will simply experience more fear and boredom.
  - supportlocal4h 5y ago
    The word "label" is very important here. "Smart" children are identified early. After that they are smart because everyone knows they are smart. It is very hard for a person who knows they are not smart to convince a society who knows they are not smart. Dumb people are sometimes lucky and get something right. That doesn't make them smart. Smart people sometimes make mistakes. That doesn't re-label them as dumb. The important thing is the label you start with. Smart people will recognize the grammatical "mistake" in the previous sentence and understand that this post came from a dumb person.
- young_unixer 5y ago
  But in the case of OP's problem, starting to sum from the first item is an arbitrary choice that "just happens" to need only one change for the sum to be exact. Where "just happens" means that it was artificially inserted by whoever wrote the problem. Would the problem work if the student started summing backwards from the last item? How many items would the student have to swap in that case?. For how many random combinations of items would the problem "work" (i.e. require exactly one change?) On the other hand, in your example you're not just trying random things until something works. Noticing that 1 + 2 = 3, etc is more analytical than adding members at random and hoping that the person who created the problem thought of the same random pattern you thought about.
  - 5y ago
karxxm 5y ago
Sounds like you are a very smart mathematician who knows how to write a 4th grader math question which isn’t just „sum things up“. In fact, this question is more than „just sum things up“, as described pretty well by the person above your comment. I am a scientist. When I see a problem I have never seen before, I start to analyze it using the tools I know. This helps me to learn more about the given problem and supports me finding a „clever“ Solution in the next iteration.
- quacked 5y ago
  This problem must be taken in the full context. This wasn't a free-form "explore the infinite bounds of mathematical expression" lesson, it was part of a worksheet. Worksheets are meant to be completed as proof of mathematical ability, and the relationship between one's ability to complete a worksheet and their intellectual potential is a poisonous lesson to teach to anyone. (Reference the past few pages of A Mathematician's Lament [1], or the whole thing if you have the time). I am not opposing the idea that to do math, you must occasionally "get your hands dirty" as another commenter says; I am opposing the idea that it is useful to set this question in front of third graders in the form "find the solution and move onto another problem". As OP's article proved, there are many, many interesting implications that arise from this problem, and an investigation of those implications requires much more care, passion, and analysis than is implied by the form in which the problem is presented (as a solvable worksheet). For what it's worth: if you want to teach the lesson "sometimes you have to just start adding things up", just write that down in plain English and teach that. Put this question in a giant book of trick problems with an explanation of why each question is a trick, don't put this on a worksheet for of children who are required to attend class and then expect them to pick up on the subtleties of instruction after they write down the answer and get their grade back. I am not a very smart mathematician; I am a middling intellect and have been my entire life. It was not until I was two years out of my engineering degree that I started investigating the world on my own and began to find out the true size and power of mathematics, and now it always angers me when I see the shallow imitations of real pedagogy and instruction passed off on the world's children as "useful math worksheets". My self-esteem and lifetime mathematical achievement were butchered by worksheets just like these. [1] https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament....
marcodiego 5y ago
I agree. GP explanation is no different than spotless guessing. I would probably start summing the most expensive ones in the hope that it would lead to less work to do.
bryanrasmussen 5y ago
>That is... absurd. >unless it was to prepare them for other standardized tests later in life yes, it teaches you how to understand standardized tests and to exploit them.
noduerme 5y ago
Unless standardized tests have changed radically for the better recently, I don't think this question is designed to teach someone how to score well on one. This is a thinking problem that could be approached in a lot of different ways. It's really the opposite of a standardized test problem.
psyklic 5y ago
You can quickly determine that you need _at least_ seven items by summing the largest. It then happens that there are two possible seven-item solutions (468 solutions in total). To figure this out, I did in fact "just start adding." That said, this problem isn't about mindlessly adding. It's about figuring out a way to systematically discover the answer. Most math challenge problems are designed with a method in mind; it's your job to figure that method out, and that's where the fun lies.
kevindong 5y ago
On the other hand, it teaches rapid prototyping and just trying out theories since each theory is fairly cheap to test out. There's a somewhat popular TED talk on how kindergarteners frequently outcompete adults in the marshmallow challenge (tl;dr; build the tallest structure possible using assorted materials but a marshmallow must be on top) because, among other reasons, they're willing to just try out different approaches and fail fast rather than take a long time to think up of an approach that ultimately fails. https://www.tanveernaseer.com/5-lessons-on-fostering-team-success/ https://www.tanveernaseer.com/5-lessons-on-fostering-team-su... https://www.marshmallowchallenge.com https://www.marshmallowchallenge.com

ivalm 5y ago

Oftentimes problems are hard, being able to come up with a useful heuristic (in this case greedy + fixer) is probably the most valuable thing one can do. Most math problems in real life aren’t solvable, you have to learn to approximate/bootstrap your way through.

quacked 5y ago
Which I agree is an extremely useful skill to have, but I wouldn't call this problem "bootstrapping", I'd call it "a trick question". If the question had asked "provide a general solution to get you pretty close to the answer", I'd retract my criticism.
- noduerme 5y ago
  Write an essay on what the logic was to get you there - sure! I wish math classes were always taught that way!

phoenixreader 5y ago

I can only venture a guess. Many interesting mathematics results come from people simply observing patterns when doing adding, subtracting, etc. Many insights can come when people are not afraid of getting hands dirty. It's about getting rid of a fear of computation and developing a system for exploring unknown problems. For example, consider this problem for geometric series: 1 + 2 + 4 + 8 + 16 + 32 + ... + 4096 = ?. This problem may look completely intractible and tedious without a calculator. But once you start adding: 1+2=3, 1+2+4=7, 1+2+4+8=15, 1+2+4+8=31, ..., even a child can spot the pattern. Now that they are amazed, they will be motivated to learn more about the cause of this pattern, etc, etc.

quacked 5y ago
And I in turn can only venture a guess that you have entirely forgotten the brain-liquefying tedium of modern mathematics education, as shown by the fact that you immediately revert to actually interesting patterns that a properly-educated mathematician might think about. The dollars-and-cents addition problem mapped onto the "fun" toy store narrative is not mathematically interesting until assigned the type of analytical firepower outlined in OP's blog post. It is a deliberately obtuse trick question posed as a learning tool; children labelled "gifted" will figure it out and feel the dull satisfaction of killing another "problem", and those without that benefit will simply experience more fear and boredom.
- supportlocal4h 5y ago
  The word "label" is very important here. "Smart" children are identified early. After that they are smart because everyone knows they are smart. It is very hard for a person who knows they are not smart to convince a society who knows they are not smart. Dumb people are sometimes lucky and get something right. That doesn't make them smart. Smart people sometimes make mistakes. That doesn't re-label them as dumb. The important thing is the label you start with. Smart people will recognize the grammatical "mistake" in the previous sentence and understand that this post came from a dumb person.
young_unixer 5y ago
But in the case of OP's problem, starting to sum from the first item is an arbitrary choice that "just happens" to need only one change for the sum to be exact. Where "just happens" means that it was artificially inserted by whoever wrote the problem. Would the problem work if the student started summing backwards from the last item? How many items would the student have to swap in that case?. For how many random combinations of items would the problem "work" (i.e. require exactly one change?) On the other hand, in your example you're not just trying random things until something works. Noticing that 1 + 2 = 3, etc is more analytical than adding members at random and hoping that the person who created the problem thought of the same random pattern you thought about.
- 5y ago

karxxm 5y ago

Sounds like you are a very smart mathematician who knows how to write a 4th grader math question which isn’t just „sum things up“. In fact, this question is more than „just sum things up“, as described pretty well by the person above your comment. I am a scientist. When I see a problem I have never seen before, I start to analyze it using the tools I know. This helps me to learn more about the given problem and supports me finding a „clever“ Solution in the next iteration.

quacked 5y ago
This problem must be taken in the full context. This wasn't a free-form "explore the infinite bounds of mathematical expression" lesson, it was part of a worksheet. Worksheets are meant to be completed as proof of mathematical ability, and the relationship between one's ability to complete a worksheet and their intellectual potential is a poisonous lesson to teach to anyone. (Reference the past few pages of A Mathematician's Lament [1], or the whole thing if you have the time). I am not opposing the idea that to do math, you must occasionally "get your hands dirty" as another commenter says; I am opposing the idea that it is useful to set this question in front of third graders in the form "find the solution and move onto another problem". As OP's article proved, there are many, many interesting implications that arise from this problem, and an investigation of those implications requires much more care, passion, and analysis than is implied by the form in which the problem is presented (as a solvable worksheet). For what it's worth: if you want to teach the lesson "sometimes you have to just start adding things up", just write that down in plain English and teach that. Put this question in a giant book of trick problems with an explanation of why each question is a trick, don't put this on a worksheet for of children who are required to attend class and then expect them to pick up on the subtleties of instruction after they write down the answer and get their grade back. I am not a very smart mathematician; I am a middling intellect and have been my entire life. It was not until I was two years out of my engineering degree that I started investigating the world on my own and began to find out the true size and power of mathematics, and now it always angers me when I see the shallow imitations of real pedagogy and instruction passed off on the world's children as "useful math worksheets". My self-esteem and lifetime mathematical achievement were butchered by worksheets just like these. [1] https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament....

marcodiego 5y ago

I agree. GP explanation is no different than spotless guessing. I would probably start summing the most expensive ones in the hope that it would lead to less work to do.

bryanrasmussen 5y ago

>That is... absurd. >unless it was to prepare them for other standardized tests later in life yes, it teaches you how to understand standardized tests and to exploit them.

noduerme 5y ago

Unless standardized tests have changed radically for the better recently, I don't think this question is designed to teach someone how to score well on one. This is a thinking problem that could be approached in a lot of different ways. It's really the opposite of a standardized test problem.

psyklic 5y ago

You can quickly determine that you need _at least_ seven items by summing the largest. It then happens that there are two possible seven-item solutions (468 solutions in total). To figure this out, I did in fact "just start adding." That said, this problem isn't about mindlessly adding. It's about figuring out a way to systematically discover the answer. Most math challenge problems are designed with a method in mind; it's your job to figure that method out, and that's where the fun lies.

kevindong 5y ago

On the other hand, it teaches rapid prototyping and just trying out theories since each theory is fairly cheap to test out. There's a somewhat popular TED talk on how kindergarteners frequently outcompete adults in the marshmallow challenge (tl;dr; build the tallest structure possible using assorted materials but a marshmallow must be on top) because, among other reasons, they're willing to just try out different approaches and fail fast rather than take a long time to think up of an approach that ultimately fails. https://www.tanveernaseer.com/5-lessons-on-fostering-team-success/ https://www.tanveernaseer.com/5-lessons-on-fostering-team-su... https://www.marshmallowchallenge.com https://www.marshmallowchallenge.com