@WBahn - what do you think of this?

 

@WBahn - what do you think of this?

I haven't had time to watch the video yet. But I did time how long it took me to do it using the concepts I learned in junior high. It took all of 32 seconds.

Namely, I noted that 12 = 3*4 or 2*6 and 6 = 2*3 (and, of course, we have the possibility that 12 and/or 6 is used directly since N = N*1).

Then I noted that since the constant term is positive, we either have the form (x-a)(x-b) or (x+a)(x+b). Since the linear term is positive, it has to be the latter form.

From there it was a simple matter of looking for a combination that works out, which it does for 17 = 8+9 = 4*2 + 3*3, yielding

(4x+3)((3x+2)

Of course, it took me far, far longer to write out how I did it than to just do it.

I'll take a look at the video when I get a chance. There are, of course, many tricks and techniques that can be used to do most things. I like more generalized techniques as opposed to ones that are very narrow in their applicability (and not that the technique I used assumes integer coefficients of the factors, so it's not as general purpose as, say, the quadratic equation. Where I really have a problem is when a technique is presented without noting its assumptions or limitations, or -- and far, far worse -- when a technique is presented without any explanation or justification of why it works. I absolutely hate magical methods that students are expected to just memorize and regurgitate without any comprehension of what it is based on. There is virtually zero learning and understanding when that is done.

I should have an opportunity to watch the video later this evening and I'll post me thoughts.

 

Okay, I just watched it and it was a prime example of what I hate about what passes for "math education" these days.

Absolutely zero explanation for where this magical technique comes from or why it works. Students are just expected to memorize and regurgitate without any comprehension at all.

In addition, his approach is misleading and reinforces the kind of sloppiness that bites students right and left.

He has an expression.

12x^2 + 17x + 6

Then, on the next line, he magically has

x^2 + 17x + 72

This implies that these two expressions are equivalent, which further implies that you can always just divide the quadratic term by its coefficient and multiply the constant term by that same coefficient without changing anything. Neither of these are at all the case, but since that is what people have just seen him show them what to do, what other lesson are they expected to walk away with? Then he has other lines later that are, again, not equivalent to what is above them.

Is it any wonder why kids find "math" so hard when they are presented with a bunch of magical methods that just, somehow, work if only they mindlessly follow the mystical incantations and ignore all of the glaring inconsistencies along the way?

Frankly, I wouldn't be surprised if the guy presenting this method was unable to explain why it works or prove its validity.

 

Okay, I just watched it and it was a prime example of what I hate about what passes for "math education" these days.

Absolutely zero explanation for where this magical technique comes from or why it works. Students are just expected to memorize and regurgitate without any comprehension at all.

In addition, his approach is misleading and reinforces the kind of sloppiness that bites students right and left.

He has an expression.

12x^2 + 17x + 6

Then, on the next line, he magically has

x^2 + 17x + 72

This implies that these two expressions are equivalent, which further implies that you can always just divide the quadratic term by its coefficient and multiply the constant term by that same coefficient without changing anything. Neither of these are at all the case, but since that is what people have just seen him show them what to do, what other lesson are they expected to walk away with? Then he has other lines later that are, again, not equivalent to what is above them.

Is it any wonder why kids find "math" so hard when they are presented with a bunch of magical methods that just, somehow, work if only they mindlessly follow the mystical incantations and ignore all of the glaring inconsistencies along the way?

Frankly, I wouldn't be surprised if the guy presenting this method was unable to explain why it works or prove its validity.

I had the same reaction.

My first thoughts were, "but, why?", and, "what are the limitations?"

I suspect, though I haven't worked it out, that this method only works for quadratics with integer roots.

That's 1st year algebra. What next?

 

It does work for all quadratics. \( ax^2+bx+c \) and \( x^2+bx+ac \) have the same discriminant, and the roots are simply scaled by a.

It's a stupid recipe to teach, setting it as a proof in a more advanced paper would have more merit.

But I really don't see advanced techniques for manually factoring quadratics as a core 21st century skill.

 

But I really don't see advanced techniques for manually factoring quadratics as a core 21st century skill.

The most consequential invention of my entire career was the result of such a solution.

The notebook page with the handwritten solution is hanging framed on my office wall.

 

It does work for all quadratics. \( ax^2+bx+c \) and \( x^2+bx+ac \) have the same discriminant, and the roots are simply scaled by a.

This broke it wide open for me. Thanks.

Given:

\(
ax^2 \; + \; bx \; + \; c = 0
\)

The roots are, via the quadratic formula (which students SHOULD be required to derive, such as by completing the square):

\(
\{ x_1, x_2 \} \; = \; \frac{-b \; \pm \; \sqrt{b^2 \; - \; 4ac}}{2a}
\)

The factored expression is then

\(
\left( x \; - \; x_1 \right) \left( x \; - \; x_2 \right) \; = \; 0
\)

So let's form a second equation (noting that this equation is NOT equivalent to the original equation):

\(
x^2 \; + \; bx \; + \; ac = 0
\)

The roots of this equation are

\(
\{ x_3, x_4 \} \; = \; \frac{-b \; \pm \; \sqrt{b^2 \; - \; 4ac}}{2}
\)

From here, it follows directly that

\(
\begin{align}
\{ x_3, x_4 \} \; &= \; \frac{-b \; \pm \; \sqrt{b^2 \; - \; 4ac}}{2} \\
&= \; \frac{a}{a} \cdot \frac{-b \; \pm \; \sqrt{b^2 \; - \; 4ac}}{2} \\
&= \; a \cdot \frac{-b \; \pm \; \sqrt{b^2 \; - \; 4ac}}{2a} \\
&= \; a \cdot \{ x_1,x_2 \} \\
\end{align}
\)

Which gives us the final scaling result.

\(
\therefore \; \{ x_1,x_2 \} \; = \; \{ \frac{x_3}{a}, \frac{x_4}{a} \}
\)

Anyone that has reached the point where they understand where the quadratic equation comes from should be able to follow this development (or even one that is more handwavy but has the essentials).

As to it's practical utility, I can see it being useful for students working algebra problems and perhaps later higher-level problems, but I don't see it having much practical utility in the real world. If the quadratic expression can't be readily factored without using this trick, it probably can't be factored that much more easily using it to make the effort of committing the technique to memory and you will likely be much better served just using the quadratic equation to find the roots. That has enough utility in the real world that you should be able to internalize it pretty thoroughly. I have used it many times over the years and haven't had to look it up for well over three decades. In fact, the last time that I couldn't confidently recall it I didn't have any reference handy (this was before the Internet went mainstream) and so I derived it on the fly by completing the square. Out of curiosity, since I'm pretty sure that that was the last time I ever completed the square, I was curious if I could still do it without referring to anything, and I was. Not because I had memorized the technique -- I hadn't, which became obvious very quickly -- but because I understand the fundamental concepts the underlie it well enough that I could figure out what to do based on what the objective was.

 

I agree, he sheds no light on the "why" but does at least begin by saying its a "trick". Actually most of the guy's videos are decent and do explain the reasoning much better, this is a good one:

 

I agree, he sheds no light on the "why" but does at least begin by saying its a "trick". Actually most of the guy's videos are decent and do explain the reasoning much better, this is a good one:

A few thoughts come immediately to mind when I see the title and the problem as shown on the title screen:

First, the overwhelming majority of college majors are non-STEM and I wouldn't expect someone getting an art history degree to be able to even begin tackling a problem like this. Even most STEM majors never touch on geometry or trig. So saying that less than 1% can solve it isn't saying a whole lot. The statistic that would be meaningful would be what fraction of students that SHOULD be able to solve it can't.

Second, what is the basis for the claim that less than 1% can solve it? Is it actually based on at least some data, or is it just another exaggerated and completely made up claim to act as click bait?

Third, what are the constraints of the solution? Is the student expected to be able to arrive at an exact solution using pen and paper? That seems unlikely, but it's possible that the specific radii used were chosen because it is a special case that actually reduces to something that can be solved that way. Or can they use techniques that are generic but yield arbitrary numerical values for areas and angles and then can use a calculator to crank the numbers (or, can I through them a set of trig and log tables and let them use those instead)?

Fourth, are they allowed to look up specifics on things that they haven't seen since they took trig in high school. Or do they have to solve it using only what they can remember and/or derive on the fly?

This would be my approach:

Problem 1: Find the area of the space that is outside any of the circles but inside the triangle.
Solution 1: Find the area of the triangle (Problem 2) and subtract the area of each of the pie-shaped segments of the circles (Problem 3) that are within it.

Problem 2: Find the area of a triangle given arbitrary-length sides.
Solution 2: There is a well-known formula for this (it start's with an 'H', Hipparcus? Heron? I don't recall), but I don't have it memorized and doubt I can reconstruct it, at least not in a reasonable amount of time. But I can probably find the area using a more brute force approach in, say, ten minutes or so.

Problem 3: Find the area of a pie-shaped segment of a circle.
Solution 3: Find the included angle of the arc (Problem 4) and use that and the radius to determine the area.

Problem 4: Find the included angle of the arc.
Solution 4: This is also the angle of the triangle vertex at the center of that circle, which can be found using the Law of Cosines. While I can always recall the Law of Sines, I would need to look up the Law of Cosines. This is something, however, that I'm quite confident that I can derive on the fly pretty quickly using analytic geometry,

So now the question becomes whether this constitutes an acceptable "solution" to the problem without actually carrying it out? It is an algorithm that will lead to the answer and is decomposed into steps that can be executed once two pieces are either looked up or derived.

 

https://arstechnica.com/tech-policy...threats-to-profession-as-industry-encroaches/

The Leiden Declaration, which has already drawn hundreds of signatories, warns that recent AI developments are threatening “characteristic values” of mathematical research, “often in ways that disproportionately affect students and early-career mathematicians, and hence the long term future of the discipline.”
First, it points out how AI models can “produce plausible but unreliable (or even incorrect) arguments which are difficult to distinguish from correct mathematical proofs.” Such developments put reviewers under increasing pressure and are “jeopardizing our ability to implement traditional standards for the correctness, transparency, and independent verifiability of proof,” the declaration warns.
“Inaccurate AI-generated drafts are cheap to produce, and there is a risk of cluttering the literature with claimed results that are simply wrong,” said Leslie Ann Goldberg, head of computer science at the University of Oxford, in a statement. “Once that happens, the errors are likely to propagate as new results are built on faulty foundations.”

Second, the declaration highlights how “models trained on published works frequently return outputs that do not properly cite the human works they synthesize,” while also pointing out that many current AI models were trained on data obtained through “exploiting licenses and access arrangements” or “simply violating copyright protections.”

Worried about losing their monopoly of imaginary numbers?

 

Evening All. I agree we do have a major math problem going on. But I can only speak for the situation in my country. I do not see why high school children are given the option to do math literacy instead of pure math. What I do know is certain skills will be lost along the way, or are we moving towards having robots as our engineers and humans just being the general labor. I hope that changes. On a personal note, I'm 44 years old and going back to university next year to study Electrical Engineering, anyone have suggestions on what textbook I can work through in the meantime to get my math up to scratch, I did any math studying 20 odd years ago last

 

I don't know to what degree this is connected to the math crisis, but I think it is reflective of some of the shifts that may be contributing.

I've been working with a few students as a tutor for a while now and have noticed that they will often have a problem on an assignment that is labeled as a "Case Study". In my day, a case study was virtually always drawn from some real world situation. If it was in a textbook or handout, it was often just a narrative of what happened and what was done and what the consequences were told from an educational perspective. If it was part of an assignment, then the student was usually tasked to solve some problem that was clearly tied to the situation, though it was often streamlined and simplified to match the level the student was at or what the topic was that we being taught.

But all of the "case studies" (in assignments) I have seen lately are clearly contrived narratives with little, if any, connection to a plausible real world situation and the problem that the student is asked to solve has zero connection to the narrative and there is no need for the student to actually read or understand anything about the narrative to solve the problem. It is merely fluff that is thrown out to make it appear like some kind of applied problem.

While I'm sure there are a number of factors at play, even assuming that I'm not imagining a trend where none actually exists, the thought that comes to my mind is that two factors might stand out: First, searching for suitable real-world test cases involves much more effort compared to just making something up, and educators are increasingly less willing to put in the effort to do their jobs at a higher level if a lower bar will be acceptable. Oddly, the ability to search for good real-world case studies is immensely easier that it was in the pre-Internet days, which underscores the relative willingness of people to put in the work. Second, recognizing a good case study and then crafting relevant, but accessible, problems from it is not only more difficult than creating a normal problem wrapping some fluffy words around it, it also requires, to be particularly good, that the person crafting it understand the significance of the real-world problem and the nuances it involves. Increasingly (though this has been the case for a long time), educators today have very little or even no real-world experience. They came straight from academia to the classroom, so they have no real-world appreciation for complex real-world problems.

 

It seems clear to me that social media is a big factor in the decline of standards. Over the past few years I've noticed that the understanding of the word "truth" has changed, no longer grounded in objectivity and rationality, it is now more a reflection of what's necessary to maintain a belief.

In other words, rather than beliefs being based on evidence and reason, the "truth" is now based on one's beliefs - a reversal of the dependency.

One chooses what to believe and then makes up whatever "truths" are needed to "prove" the belief.

Social media has enabled this and sadly (frighteningly) the mass of humanity seem eager to adopt this approach to "knowledge", we see this in politics now as the new normal but it is making inroads to other areas too.

When a student is immersed in this (as the vast majority of young people are) then education loses value, when you live in a world increasingly disinterested in reality, then you too can easily lose interest, preferring instead to pick what you want to believe and grabbing as many "facts" as you can from your favorites sources.

I've always enjoyed science fiction, especially short stories, where some future technological event or discovery causes people to deal with problems that can't happen in real life. In almost all science fiction though there is one constant, the people and their values in the future have generally the same ethics and values as they do in the present.

Unfortunately, as the future unfolds before us, this isn't the case, ethics and values are no longer the basis for one's action, but a means to and end. The young people of today are increasingly living in and shaping this new way of thinking and humanity is setting itself up for a catastrophe unlike anything we've seen before.

 

Over the past few years I've noticed that the understanding of the word "truth" has changed, no longer grounded in objectivity and rationality,

Past few years?

Immanuel Kant started plowing this ground in 1781, laying the foundation for some of history's most destructive philosophy.

That's been further bastardized into today's relativism: "My Truth" vs. "Your Truth", and "Who are you to say what's objectively true?"

2+2 = 5 and all that.

 

I guess the real catastrophe began when human switched from spiritualism to rationalism,

I'd argue: spiritualism (pre-enlightenment) -> partial rationalism (enlightenment) -> relativism/subjectivism (20th century on).

 

This catastrophic world needs to be more Hegel-ized...

Kant -> Hegel -> 100+ million dead in the 20th century....

... -> kids can't do math (to help bring this thread back on topic).

 

As I and many people point out, we don't teach our kids philosophy here, yet they do in say France or Italy etc. I think that's a factor because it exposes kids to the idea of reasoned argument, rational reactions to being disagreed with and so on.

Mathematics overlaps philosophy in some areas and engaging a child's mind is what philosophy is all about and that can't be a bad thing for a kid learning math.

Some of the most well known mathematicians were also philosophers like Wittgenstein, Descartes and Russell.

 

Past few years?

Immanuel Kant started plowing this ground in 1781, laying the foundation for some of history's most destructive philosophy.

That's been further bastardized into today's relativism: "My Truth" vs. "Your Truth", and "Who are you to say what's objectively true?"

2+2 = 5 and all that.

Perhaps but such ideas weren't amplified by the rich and powerful using mass communications technology. That's what's new, social media is used today to popularize quite a lot of baseless beliefs and huge numbers of people go along with it, embrace a claim or belief because it suits them nothing more.

When enough people believe some claim then to all intents and purposes it becomes true, the world acts as if it were true.

 

Perhaps but such ideas weren't amplified by the rich and powerful using mass communications technology.

Yup. In the old days, they just used bullets.

 

Yup. In the old days, they just used bullets.

Well they still do (the global spend on arms is larger than at any time in the past, almost 3 trillion annually), but "they" have always used fear, and today is no different but it is more sinister and has far greater reach than has ever been possible because of social media "platforms" that many young people spend every waking hour fretting over.

It's much easier to control a mass of scared people than a mass of informed, thinking people, you just tell them you will take care of them, you will confront those things that threaten them etc etc.