For those interested, there is a trick to this. You successively add the first several items until the amount first exceeds 43.94. This occurs after summing up the first twelve items, with the sum being 46.81. The items are over the limit by 2.87.
You wish to decrease your spending by $2.87, so you check if there are any items you have not chosen that cost exactly $2.87 less than an item you have chosen. A smart fifth-grader would quickly find that the car ($5.18) can be replaced by Jacks ($2.31). This concludes the problem-solving process.
This question was intended for fourth- and fifth-graders (https://www.insidemathematics.org/sites/default/files/materi...). Put yourself in the shoe of a smart fifth-grader who knows nothing about NP-completeness or knapsack problems. This problem is given to you, and you know it as solvable. This problem would look intimidating, but a good student would look for patterns, and the best way to start looking would be to sum the first few items (who knows, maybe the first items would sum up to exactly $43.94). After summing up to 46.81, a good student would "naturally" stop adding new items and ask himself "what can be done now?" All that he needs is now a mental "click".
Essentially, this problem tests students' intuition. It tests whether students can remain unintimidated in the face of difficult-looking questions. They need to start adding, and they need to have that one insight of replacing one of the items. Insights like these are extremely common and rewarding for young math learners.
I won't be surprised if many fifth-graders can solve the problem in five minutes.
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?
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.
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.
..or down the columns instead of across the rows! It is indeed likely that the kids who got it (as claimed by the child) probably lucked into it in the afore-described way. But don’t you think that thinking about it first makes more sense than “just start summing”? And if you think about it, and know what an NP hard problem is (as this kid, shockingly! apparently did), you quickly realize that it’s impossible. Then, you need to have the meta-reflective insight that no one would give a third grader an NP hard problem, and that there must be a trick, so then, and only then, do you start to look for patterns, .. and then the bell rings and class is over and you spent the whole time meta-reflecting and looking for a trick in just one stupid problem. Sucks to be smart!
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.
"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.
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.
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.
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.
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.
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.
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.
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.
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.
Some of this language: “a smart fifth grader would”, “a good student would”, stuck out to me. I have experienced people talking like this at work and I think it can be harmful. I watched a group humiliate one person by saying “how could you get a job here and not know that?” I also had a coworker once offer to help me solve a problem, during which they said something like “any competent engineer should be able to …” I had only been there a couple of weeks and it really undermined my confidence.
I don’t think recognizing a trick problem is necessarily about being good or smart. Is the author of the cited blog post dumber than a fifth grader?
I really did appreciate the post, but I also felt weird about that language. Not offended or anything, but it stood out.
Not to detract from the nice explanation in that post, but I do think "smart" can be a bit of annoying term. It sort of implies an innate property. I'm pretty confident anyone can become really good at anything, given enough motivation and time. I'd prefer the term "experienced" or something to that effect.
> I'm pretty confident anyone can become really good at anything, given enough motivation and time.
And I'm pretty confident that some people can become really good at said thing faster, due to some form of innate talent. And as we all have finite time, that's a pretty important trait.
People, and their lives, are extremely high dimensionality objects, and most of those dimensions are essentially continuous. However, in terms of cognition, we are all quite close to one another along any given dimension. If we weren't, you'd really notice, like someone on the far end of "The Spectrum". However, small deltas in a high dimensionality space add up to a large vector length. Still, these are never so important as context: Where you grew up, whether you fell in love with someone early, whether you had access to the best schools for your target, etc. are much more important factors than the combinations of people's intrinsic features.
> I don’t think recognizing a trick problem is necessarily about being good or smart.
It goes a bit beyond that; this problem solving algorithm is pretty clearly unworkable. Even here where it can come to an answer in reasonable time it is questionable.
Any student that is happy to employ this algorithm is either extremely smart to the point where they can process a lot of arithmetic very quickly, or not very smart and setting themselves up for failure later on in life when the first guess at an answer doesn't work.
Ordinary garden-variety smart (as in, likely to be good at maths but not Euler or Gauss) should fail to answer this question on the basis that there is no technique beyond brute force, and brute force is impractical without a computer.
The author has the “problem” of knowing too much. I’ve seen this myself where presented with a Singapore Math problem solvable easily by a 3rd grader I reach for more complex algebra because “I know that’s what the problem really is” when there are much easier ways to solve that kind of problem visually.
I think this approach is a bit backwards. The world needs more competence, and people should be encouraged to admit their gaps in knowledge and learn from others.
While that may be true, I consider it to be a lousy way of teaching and an even worse measure of mathematical competence, regardless of age.
I was fairly adept at math in fourth and fifth grade—well ahead of my peers. While I didn’t know terms like NP-Complete and NP-Hard, I would have immediately recognized that such a problem didn’t have a straightforward, timely solution for someone doing it by hand. I would have considered it a trap—in the sense that it’s a time sink—and moved onto the next problem.
If it later turned out that the solution to the problem required that I not actually understand it (from my perspective at the time), I would grow frustrated. I was the sort of asshole fifth grader who would not hesitate to chastise a teacher for giving us what I believed to be unrealistic, dumbed-down math problems. When in life am I going to encounter a knapsack problem that just happens to be quickly solvable that way? The chances of that are slim; the technique I’ve now learned has no application in the real world.
Take factoring, for example. I would regularly get frustrated with factoring assignments: how do you expect me to factor a collection of polynomials on a timed test if you cannot describe a deterministic algorithm for doing so? “If you’re having trouble, stay after school and I’ll help you.” After school, the teacher would helpfully demonstrate factoring several polynomials. I’d ask how the teacher knew how to guess the particular numbers they chose. What if the polynomial can’t be factored, and I waste time trying to find a solution? What if the numbers are large? The teacher would inevitably grow frustrated: “I won’t give you problems like that. Stop worrying about it.”
I’ll be the first to point out that I was never particularly nice to teachers who my judgmental 12-year-old brain considered incompetent, but I have little sympathy for this particular scenario. I wanted to understand the problems; I didn’t want canned solutions. My teachers knew I was perfectly capable of factoring the numbers with which we were presented, so they would get frustrated: why is an intelligent student stuck on a problem they’ve already solved?
Eventually, I found a book on prime numbers at a used book store and got the answers I needed.
Wow. Factoring quadratics were a similar pivotal moment in my maths education. Teacher couldn't explain how to do it in a way that fitted my brain. My confidence took a hit as did my love for the subject.
You forgot the other half of the question - is your solution the only possible answer? - at which point you can really only evaluate all possible combinations. I agree it's one of several ways to test for a 'clever' student, though.
> at which point you can really only evaluate all possible combinations
You only need one counterexample, and in this case you can easily find it. E.g., you can note that the duckie in GP's solution can be swapped for the pinwheel+whistle without changing the solution's total cost, ergo the solution isn't unique.
I think a lot of people are looking at this like a coding interview, where it's assumed that your algorithm must work for all possible inputs, rather than just the inputs stated.
It’s easy to understand how a student at this level would approach it and give an answer.
Everybody’s problem here, mine included, is that this is a toxic way to teach math. One which has no use in math, which does not teach how to properly approach a problem and which in fact teaches and artificially rewards a bad approach: “just try it! It’ll work because all problems, even ridiculously difficult ones, are artificially customised to be easy for you to solve!”.
It’s bad. It’s awful. It’s not good. It’s everything math education shouldn’t be: boring, artificial, useless, constrained and a lie.
I don’t want children to solve this problem. I want children who can like math, to like and learn math. This is one of many drops in the jar of bullshit that will eventually overflow and turn them off.
No you don't. You only need to find a combination of toys in your answer that sum to another toy to easily see its not the only possible answer. If your solution involves the duckie for example that can be replaced by buying both the pinwheel and whistle. Or just find another combination of toys that works.
No rational human being would see this problem and just start adding the first few items. A smart student would skip the problem. It is counter-intuitive that just adding the first few items would be correct (why would they make it so obvious). If anything I would start with the last few items. Problems like this are the entire problem with math education in the U.S.
> It tests whether students can remain unintimidated in the face of difficult-looking questions.
That's a very good goal. But for this problem, your success depends on whether you chose the same strategy that the teacher cherry-picked prices for - or just got lucky that another strategy happened to work.
"Intuition" implies that the student somehow understands the structure of the problem, but the prices in the problem could be adjusted ever so slightly in such a way that the strategy you outline would lead nowhere at all.
The only possible intuition to be found here is realizing that the only way to solve the problem is to throw shit on the wall until something sticks.
> I won't be surprised if many fifth-graders can solve the problem in five minutes.
A lot of fifth-graders could win on a scratch-off ticket in five minutes, too.
You wish to decrease your spending by $2.87, so you check if there are any items you have not chosen that cost exactly $2.87 less than an item you have chosen. A smart fifth-grader would quickly find that the car ($5.18) can be replaced by Jacks ($2.31). This concludes the problem-solving process.
This question was intended for fourth- and fifth-graders (https://www.insidemathematics.org/sites/default/files/materi...). Put yourself in the shoe of a smart fifth-grader who knows nothing about NP-completeness or knapsack problems. This problem is given to you, and you know it as solvable. This problem would look intimidating, but a good student would look for patterns, and the best way to start looking would be to sum the first few items (who knows, maybe the first items would sum up to exactly $43.94). After summing up to 46.81, a good student would "naturally" stop adding new items and ask himself "what can be done now?" All that he needs is now a mental "click".
Essentially, this problem tests students' intuition. It tests whether students can remain unintimidated in the face of difficult-looking questions. They need to start adding, and they need to have that one insight of replacing one of the items. Insights like these are extremely common and rewarding for young math learners.
I won't be surprised if many fifth-graders can solve the problem in five minutes.