That's an interesting result that I hadn't really realized.

Just one example of algebra at work.

Few people have the faintest idea what their effective tax rate it and even less of an idea how to figure it out.

In round numbers the median income in the U.S. is about $50,000. Someone single making that amount and not claiming anything other than the standard deduction and exemption will have a taxable income of $39,600, which places them in the 25% marginal tax bracket. However, their federal income tax is $5,639, for an effective tax rate of just 11.3%. The ubiquitous family of four is in the 15% bracket but their tax liability is $2,233 for an effective tax rate of just 4.5%.

This is leveraged even further in retirement. If your income is purely from retirement savings and Social Security, then you will no longer be paying the FICA tax, so now your gross income only needs to be $46,175 to be comparable. A person that makes this level of income (i.e., about median) their entire working life will have a SS income of about $20,000/yr, so they need to draw the remaining $26,175 from their IRA to cover this. The single person is in the 15% bracket but will only pay $1,900 in taxes, or 7.3% of their distribution. But arguable a more apples to apples way of viewing it is that they are paying that $1900 to keep a $50,000 lifestyle, or 3.8%. Which way makes more sense depends on exactly what you are looking at.

But the family has an even better situation. Assuming they are now a family of two (the kids aren't living in the basement) and one spouse claims half the other spouse's benefit, then they only need to withdraw $16,175 and that isn't even close to their combined deduction/exemption of $20,850. So they don't owe ANY taxes on their traditional IRA savings, at EITHER end. But they would have paid 15% up front on a Roth.

That's not to say that a Traditional always beats a Roth -- it doesn't, but it does most of the time for most people. Drawing too much from a Traditional can make some of your Social Security benefits taxable, for instance. Also, in early years when you aren't making much, a Roth can make a lot of sense. My daughter is working for me and 90% of her wages are going into a Roth. Since she has NO payroll tax or income tax at her earnings level, all of those funds will never be taxed (assuming they don't change the law, which I fully expect they will at some point).

The best strategy is to have a well-crafted blend that reflects all of your income sources in retirement and how they interact from a tax standpoint so that you can withdraw from Traditional accounts up to the point of being taxed (including SS benefits) and then draw from Roth above that. Even that isn't the truly optimal line, but it is in the ballpark.

 

Fair enough.
I was referring to intermediate level college algebra, which the article mentioned at the start of this thread, but if we are basically talking high school level (or less), that's a different story.
I certainly think that anyone with a high school education should have passed a high school basic level algebra course.
But if they don't have that, then they should indeed have to gain that level of proficiency to get a college degree.

I was just questioning the need for everyone having to pass a higher level course.

What IS "intermediate level college algebra"?

It is REMEDIAL middle school and high school algebra taught to college students. That's it. The name "college" algebra is there purely to avoid hurting little Johnny's self-esteem.

Think about it. Someone that is mathematically properly prepared for a technical major probably took the normal algebra sequence in, roughly, 7th through 9th grade then went on to geometry and trigonometry and very possibly calculus their final year in high school. In college, they never take any "college algebra" of any kind, which pretty much rules out the possibility that any form of "college algebra" involves non-high school algebra, otherwise they would need to take it to fill in the gap.

 

I remember the days of college preparatory courses in high school. That didn't impede the rest of the student body from having algebra II in high school.

Me, too. Most schools (if the district had enough students) had a reasonably well balance blend of college-prep, arts, and vocational offerings.

Maybe it's time to do a survey of the higher paid positions to see how much higher math they use in the course of a year. Surely they must understand what's being produced.

I'm not so sure. I think many of them will underestimate how much they use just because they use it, or the principles behind it, so naturally they don't even realize they are doing it.

 

Example of a need for algebra skill in everyday life please.

My family farm was surveyed by a licensed surveyor with decades of experience (business name = Houchens). The polygon did not close by 10 feet!
Without algebra, that mistake would still be showing up in land sales now, 30 years later.
(My math also added 8650 square feet to my list of possessions.)

Funny story goes with that. The front desk clerk in the survey office asked me if I added up the length of the lines. I replied, "First, I changed your Polar coordinate system to a Cartesian coordinate system, then I use the sine and cosine to find the y and x components...
I looked up and she was gone.
Just that fast she realized I was no country bumpkin and she ran...not walked...ran to get the Main Man who happened to be a lady. Betty Houchens.
I gave the lady my math and 10 minutes later I said, "Bingo. She's running this on a computer and she's running it twice or she'd be done by now."
Mission accomplished: Algebra.

I was trimming trees in Kentucky. One limb refused to dislodge, even with two men tugging on it. I knew the Law Of Tangents and took that limb out single handed without climbing the tree.
I just had the guys in the top tie a rope to the limb and I worked from there.

I needed to lift a Ford Aerostar off its engine (because the engine comes out the bottom).
I built a huge (wooden) triangle shaped frame which would be strong enough to lift a car.
How do you think I figured the length of the legs required to get the car high enough?
Algebra.

My business partner wanted a ramp to get his motorcycle in the back yard.
How do you figure cutting the lumber for the right slope?
Algebra.

A platform is needed to hold an air conditioner to the side of a building.
Same problem as the motorcycle ramp.

Building my saw horses.
Installing solar panels.
Cost for BTUs of fuel oil vs natural gas vs electricity.

Half of these can be done without algebra, but algebra sure makes it easier.
I had to take Algebra I, twice, before the light bulb in my head came on, but I'm sure glad I did.

 

What IS "intermediate level college algebra"?

It is REMEDIAL middle school and high school algebra taught to college students

If that's true then I had a misunderstanding of what constitutes intermediate level college algebra.

 

If that's true then I had a misunderstanding of what constitutes intermediate level college algebra.

So what was your understanding of what intermediate level college algebra was?

For a very long time whenever the opportunity arose I would ask just what "college algebra" was and could never get a clear answer. But one day I was talking to a math professor and I asked him. He didn't know either (since he had never taught it). So we went to the Math Department Office and looked in the textbooks in their library and both realized almost immediately that it was basically 8th/9th grade algebra with perhaps a bit of trig thrown in depending on the text.

The course names vary quite a bit. In high school the content is usually called either Algebra II or Pre-calc. At the college level, sometimes their is only one course and it may be called College Algebra or Pre-calc or sometimes something else, such as Algebra for College Students. Some places have two courses and tack the "intermediate" onto the second course but may also call it "pre-calc". At UCCS they use the titles "College Algebra" for the first and "Elementary Functions for Calculus" for the second. In technical majors neither counts toward graduation because they are considered remedial courses.

 

Perhaps you can, but I don't think the average person can do exponential functions in his head, even if he knows algebra.

True. I am rather good with algebra and I must use a calculator to do calculations for money across time.

 

I think many of them will underestimate how much they use just because they use it, or the principles behind it, so naturally they don't even realize they are doing it.

I think this is very true; many, perhaps most, of us in electronics aren't a whole lot more aware of how much we rely on what we learned in Algebra I & II, than fish are aware of the water they swim in. I know I'm not, unless I take the trouble to stop and think about it.

I sometimes wonder: how much of some people's difficulty in mastering basic DC circuit analysis is actually caused by a failure to master basic Algebra? Some examples:

"I have a 5 volt power supply and an LED whose nominal forward voltage drop is 1.5 volts, and I want 15 mA to flow through it. What series resistor should I use?"

"I need to make a voltage divider to get 3.3 volts from a 12 volt supply. The top resistor is 10 kΩ. What resistance should I use for the bottom resistor?"

"I have a 5 volt DC motor and a 12 volt power supply. I made a voltage divider to power the motor, from a 68 kΩ resistor and a 47 kΩ resistor, but when I connect the motor it doesn't run and the voltage drops down to only a few millivolts. What am I doing wrong?"

"I have a sensor that outputs zero to 10 volts full scale; over the measurand range I'm interested in, its output goes from 1.9 volts to 8.1 volts. I need to change that output range to zero to 3.3 volts to drive my PIC's ADC. I know there's a level shifting and scaling circuit to do this, using three resistors and a negative reference voltage supply; but how do I tell what resistor values to use?"​

It's possible that in all four cases, the person is missing that most basic of all circuit analysis principles: that the sum of all currents entering and leaving any node in a circuit is zero. That's certainly the case with some of the people who show up here with simple questions like the above; they just aren't aware of the principle.

But I suspect that for others, the stumbling block is not the principle itself but the horrifying (to them, anyway) realization that in order to deploy that principle in answering their question, they are going to have to write equations and solve them. The first three questions each involve writing one equation and solving for one unknown; the last question requires solving two simultaneous equations in two unknowns. OMG, it's Algebra! Brain freeze ensues.

I wonder how often that is the case? Rather frequently, I suspect.

 

Our University changed in regard to the necessity of Math for a degree, not to take away anything from what has been commented on but rather point out if my degree is not in the STEM fields then what purpose does it serve than to make money for the University. Apparently those individuals who had Mommy do their homework yet couldn't pass a basic knowledge driven algorithm based program don't need a degree and should just be the ignorant hair stylist, bus boy, dishwasher etc. and they have no place in that field of work.

Now, suddenly Johnny realizes being a bus boy isn't the way to go and he's going to be crapped on until he gets a degree in something that pays more than 50k a year in some areas and 100k in other area's and decides to go back to school. This time he prepares himself to place in courses required to take upper division degree programs (To avoid additional cost's) yet only to find the Algorithm puts him back down to lower level Math and because it's accepted nation wide that it must therefor be his fate to take a required Math class even though he truly qualifies in Algebra 1 for a C# programming class. But, for me when I asked the Math Department does a Algebra 1 have Trig in it? As I felt lied to.

So, when I said to them your study Guide has nothing about Trig as a preparatory and yet that said Algorithm rates me otherwise, then they say I'm sorry kv it's a national testing program so, it's not our fault. I said, BS (In my mind of course) while still in my mind I thought, rather it's all about money and the sooner you fix it, the sooner you'll get people in here making you more money because you've just removed half the fear of spending more than you earn, thus driving all the Johnny's deeper into debt, as well as putting that slap down both socially and mentally as they fall further into despair.

I have a Job at a University to pay for my classes so for me it wasn't a debt, and so far it's paid me around 25k for the lack thereof in debt to them. My pay increases year by year because of my education. Finally now this year the Academic Elite finally see the bigger picture and put the smack down on the Math Debt. They Switched to a program called ALEX and if a student using the program goes little by little and passes each part of it's system they become qualified at their rate and needs for their degree thus eliminating the power of the Math Department extortion for Math Classes as it only costs a student $15 and their precious time, if not then they better find another degree that fits their skills and abilities.

e.g. go back to a qualified counselor to see where they fall and find whats in their Wheelhouse.


kv

 

So what was your understanding of what intermediate level college algebra was?

Something beyond high school algebra.
To me "intermediate" did not mean "beginning".

 

Example of a need for algebra skill in everyday life please.

At the plant I used to work at, I was a Maintenance Tech under the Maintenance Manager. We had quarterly meetings to discuss overall status of the plant, machine upgrade plans, PM schedules, downtime accruals, etc. In one of the meetings he quoted some astronomical downtime percentage for a machine that I had down for upgrades for just over a week. After the meeting I asked him how he was calculating downtime percentage and he sent me his spreadsheet.

The formula for downtime percentage for every machine was (down days ÷ up days) so in a 30 day month, if a machine was down for 10 days and up for 20 days, it had a downtime of 50%. I pointed out to him that it should be (down days ÷ total days) for a downtime of 33%.

It took him a whole shift of arguing with me, googling, and calling smart people, to finally come around. He said that he had been using the spreadsheet for over a decade and that the spreadsheet was turned over to him from the previous maintenance manager and he had never altered the formula. He said that spreadsheet is sent from him and up the chain, his boss, boss's boss, etc about 4 levels higher, and major business decisions are made based off it. I was astonished that nobody ever caught the discrepancy. Surely in the 15+ years the spreadsheet had been in use, there was a month in which a machine had more down days than up days, and formula would yield >100% down time; no red flag?

This is only one example, but it, among a host of other examples, tells me that we have a widespread math deficiency before you even start talking about algebra. How many people over the years in various levels of leadership in that company didn't have a basic grasp on math? Apparently all of them; dozens of highly paid people.

And when I uncovered the error, guess how many in attendance at the party in my honor? No fanfare for the guy who singlehandedly improved downtime scores for all of Maintenance dept. Across the board on all machines. Someone probably got a bonus or a raise for the drastic and sustained improvement. Probably someone a couple paygrades above me, getting pats on the back on the golf course and thinking the reason is because of the awesome leader his is, inspired his underlings to perform better. Meanwhile I am expected to keep my discovery a secret, lest any stupid people get hurt by the mirror.

 

I've thought about education many times. Lord knows I've been on the receiving end of plenty of it. There are some fundamental challenges for which I have no answers.

One such challenge is that you cannot know in advance what any student will 1) need to know (in the sense of using it on a job or for some important task such as @#12's land purchase), 2) will enjoy and will be a better person for knowing (such as the arts and history), and 3) will never really see again in any form outside of their education. Trade schools – targeted at making their students into productive workers – should focus on #1. Universities need a mix of #1 and #2. Those students might end up anywhere and they need a broad education. Everyone should look for #3 items to prune. I would suggest "women's studies" and the like but I digress.

This challenge of not knowing the future is aggravated by several factors. All students are different and will follow different paths in life. Do the dance students need to learn calculus, and vice versa? Maybe, but it's silly to treat them the same. Another aggravating factor is that the things one generation needs to know are obsolete for the next. I had to learn to use a slide rule. Would I waste his time teaching my grandson that? Of course not, except maybe as historical perspective.

Each discipline competes for more mindshare of students, all with very good arguments why their field deserves more. Education is a good thing and more is always better, so it's hard to argue against any of them in isolation. The mathematicians make a good case but so do the music teachers. The problem is prioritization, and this needs to be a deeply personal decision. It's the job of parents, teachers and students, and they're all underprepared for it. Government even less so.

Another challenge is that students tend to be receptive to learning certain ideas at certain times in the development of their brains. We teach history to little kids that couldn't care less because we want to be sure they know some history even if they drop out of high school. That's pretty unlikely these days, and the price we pay for this insurance is wasting their precious time when they might learn math or another foreign language far easier than they can at any other time in their lives. Education consumes a lot of a young person's time, and that time is not used efficiently.

The final challenge I'll mention is one we can actually do something about. The 'progressive' fantasy that education should be the same for everyone and will produce the same positive outcome in everyone is, in a word, absurd. It's like saying that the only thing that kept me from playing in the NBA was more practice and coaching. That's ridiculous. As this thread as shown, how many people will actually benefit from learning calculus? Maybe 5% of those exposed to it, and maybe only 0.5%. (The lower value feels more accurate to me personally. I'm almost the only person I know that can do calculus, present company excluded.) It's not so hard to identify very early who's in that small population. Ask any 7th grader and they'll pick out the 2 or 3 people they know that will likely be the ones. There is no shame in recognizing a student would be better served by a trade-school approach rather than maintaining the illusion that, if we just send them to the right school, they'll become the next Gates or Zuckerberg. That illusion is like spending resources preparing me to be Michael Jordan when it's abundantly clear that isn't going to happen.

It's an old-fashioned idea that we should focus the resources of higher education on those that will likely produce a return on that investment, but we need to get back to that. Don't pretend that we are not different, embrace our differences. Exploit the hell out of them. That's what makes us great, freeing people to pursue their unique potential. Quit sending the top 20% of the bell curve through the same mind-numbing experience of public school. Quit sending the lower third (or half, or whatever portion) of the bell curve to college with the dream that they'll be wealthy executives within a few years of graduation. The reality is they'd be better off learning welding or plumbing or carpentry or whatever thousand other things they might actually enjoy and excel at. Most commerce in this country is small businesses run by hard-working people and I'd bet only a few of them need college graduates to get the work done. Wishing it were different doesn't make it so.

 

how many people properly understand exponential growth? That single curve is a powerful thing, especially of it is talking about your money... compound interest... credit card debt...etc My math profeciency comes in handy every day.

Yes, that type of applied mathematics is necessary and much needed yet I doubt most of us can ever recall those subjects being covered in any math class for more than a day or two when we were back in junior high and even then more than likely the classwork and related study was absurdly lame in regards to real life usages and application. Whereas the largely useless algebra and other higher maths that was drilled into us for weeks or months every semester for years most of us have forgotten 95+% of it simply because we in fact never have need for it ever in real day to day life.

I know I have. I design and build complex electric and electronics plus mechanical stuff all the time yet I never have to use any math beyond the basic level to do it. Sure, it could be done with far more complex math but why? The end answer I need is what I am after , not how unnecessarily complex I can make the work to get it.

 

One from just today: My wife wants to tile our bathroom and the little foyer between it and the two bedrooms on either side of it. How many gallons of tile adhesive does she need? That's a simple algebra problem that a huge fraction of people can't figure out how to solve.

I would just read the label on the bucket being it will clearly state how many square feet of area it will cover then divide the known area by how many buckets rounding up the answer to whatever full number I arrive at.

845 square feet to cover,
One bucket covers ~300 square feet.

845/300 = ~2.82

2.82 rounds up to 3 buckets. No algebra required.

you over complicated a simple job with unnecessary math.

 

You can also use back-of-the-envelope approximations such as the rule of 72.

Unfortunately in reality the rule of 72 does not in fact hold true.

In theory it's fine but in real life until the end goal reached every single variable is just that, a variable. Anyone who has ever taken out a longer term loan and had to renegotiate the terms due to a unforeseeable a the beginning change in life or banking policy knows that.

I once took out a $10,000 loan on a 5 year payback plan. knocked it out in less than 3 because unforeseeable variables in my life gave me extra money to work with one year but took a penalty for paying it off too quick.

 

If that were the case, then why has the tuition been going up at two to four times the rate of inflation for nearly four decades?

Greed in the systems. That why. Too Many colleges and universities blow a huge amount of their budgets on stupid crap that has nothing to do with anyone educations or their institutions own long term well being.

Partly because those who now run them never had any practical life application class that taught them things like'
Real life financial management and accountability,
How to run a business at all levels.
Never had a history class that actually taught them about any historical event that will ever relate to their own lives and how to avoid repeating such past screw ups.
Never had a human relations and social sciences class that taught them anything about how real life works.
and many many more skills that they would and do need to do their jobs correctly.

In my Technical Trades education it would be like going to college for a welding degree and spending the whole time on theory, largely irrelevant history regarding how various metals were discovered yet covers nothing about what one will see and work with everyday, math that has no practical application to the work and then ending with never having picked up an actual welding rod and some metal and learned hands on techniques regarding how to actually put some pieces together.

 

Second, something like taking the formula for the area of a circle given the radius and then using that to find the radius needed to have a certain area is NOT arithmetic, it is algebra.

In my book that's geometry to which most of it can still be done with basic arithmetic if you use the right processes.

I use it in fabrication work regarding hydraulics all the time like when figuring out what size of hydraulic cylinder I will need for a given load and to what possible range of rotation around a pivot point I need.

Sure, you can make all of that more complicated to the point of needing algebra but why?

 

"I have a 5 volt power supply and an LED whose nominal forward voltage drop is 1.5 volts, and I want 15 mA to flow through it. What series resistor should I use?"

"I need to make a voltage divider to get 3.3 volts from a 12 volt supply. The top resistor is 10 kΩ. What resistance should I use for the bottom resistor?"

"I have a 5 volt DC motor and a 12 volt power supply. I made a voltage divider to power the motor, from a 68 kΩ resistor and a 47 kΩ resistor, but when I connect the motor it doesn't run and the voltage drops down to only a few millivolts. What am I doing wrong?"

"I have a sensor that outputs zero to 10 volts full scale; over the measurand range I'm interested in, its output goes from 1.9 volts to 8.1 volts. I need to change that output range to zero to 3.3 volts to drive my PIC's ADC. I know there's a level shifting and scaling circuit to do this, using three resistors and a negative reference voltage supply; but how do I tell what resistor values to use?"

1:
5 - 1.5 = 3.5
3.5/.015 = 233 Ohms.

No algebra

2:
12 - 3.3 = 8.7
8.7 / 10,000 = .00087
3.3 / .00087 = 3793 ohms.

No algebra.

3:
No basic understanding of motors and how to test them thus no algebra will fix that.

4:
Use a 1.9 volt zener to cancel out the 1.9 volt bottom end value then use a pair of resistors in series with it to make a voltage divider to drop the remaining voltage down.

All passives and no algebra.


Thats the problems I see equated to wanting to use algebra where practical sense, basic math and experience shows it's not needed.

 

Here's a great idea: let's dumb down our education system even more than it already has been!

Say Goodbye To X+Y: Should Community Colleges Abolish Algebra?

I wonder how many years will go by before we see calls to eliminate addition, subtraction, multiplication and division from the curriculum because they're "too hard"?

The future is here. And it ain't pretty.

Dreadful on to deprave -- but not surprising!

To deny the basal requirement of mathematical appreciation is to deny all that distinguishes our species as creatures of mind!

Of course this is but another step in advancement of the fascist agenda oft promulgated via 'pleas for sensitivity' to the perfidious concept of 'innate skill' (or, CIP, lack thereof)! Clearly --excepting cases of neurological defect/injury-- 'idiocy' is an illusion! - a mere rationalization of abject sloth and/or base cowardice! - A decadence of mind that must under no circumstances be condoned, supported, or abetted in any way - is humanity to emerge as anything more than 'sophisticated primates'...

Best regards
HP

 

Clearly --excepting cases of neurological defect/injury-- 'idiocy' is an illusion! - a mere rationalization of abject sloth and/or base cowardice! - A decadence of mind that must under no circumstances be condoned, supported, or abetted in any way - is humanity to emerge as anything more than 'sophisticated primates'...

Do you ever wonder if, headed in the direction we are, we'll eventually reach a time when one generation passes from the scene and the next generation can't figure out how to keep the lights burning, keep clean water flowing from the taps, keep the petrol flowing from the gas pumps, and keep the roads maintained?

I do.