Is Algebra Necessary?

[JVN]

Yes, it is necessary. It is my opinion that everyone should have some basic understanding of the math and physics that defines the world around them. It is my hope that natural curiosity would drive people to seek out this understanding. Algebra is of course (imho) a big part of this understanding.

Building along the same line that Karthik started: I recommend everyone read "A Mathematician's Lament" by Paul Lockhart (sometimes just referred to as Lockhart's Lament).

http://www.maa.org/devlin/LockhartsLament.pdf

-John

[artdutra04]

Originally Posted by Michael Hill

I could go the rest of my life and have European History not be relevant at all, but I still took it. If someone is really contemplating taking Algebra out of the curriculum, the bar is being set far too low.

There is a huge difference between mindless memorization of dates, names and learning about the public opinions, socio-economic conditions, and other factors that lead to things like World War II happening.

History (like other subjects) should be about teaching people how to analyze events and conditions to identify patterns and draw conclusions. A lot of people were taught that communism is bad, but few can identify the factors that lead to communism starting in the first place.

If more people were taught these critical thinking skills, then they would be better equipped to make decisions in every day life.

[Siri]

Originally Posted by Lil' Lavery

Because both of those skills apply to your ability to function as a member of society and citizen. Algebra, when viewed as a discrete skill, does not seem to be something that a typical citizen must master.

I don't mean to be antagonistic, but how discrete is discrete? Analyzing the three caskets in The Merchant of Venice? Writing a 5-paragraph character sketch? Do I really need to know the Treaty of Utrecht to understand the importance of its modern ramifications? ...By the way, I don't think I was taught the latter any more than the applications of a character sketch. Is "teaching" algebra, the long, confusing, repetitive jumble of letters that has no impact on my future really that different from teaching, say, Wuthering Heights (the long, confusing, repetitive jumble of letters that has no impact on my future*)? [EDIT: What Arthur said.]

This article seems to set up a false dilemma. Algebra's not the problem, the algebra curriculum is. Basic algebraic concepts are critical to quantitative literacy in society, as the article itself explains. Karthik nailed it, but I wanted to point out an additional line that struck me as...bizarre.

I fully concur that high-tech knowledge is needed to sustain an advanced industrial economy. But we’re deluding ourselves if we believe the solution is largely academic.

What does "academic" mean? Webster's first** definition is relating to a school/higher education. Under that definition--why not? Especially when the preceding example is about a community college. Shouldn't academics be giving us the knowledge we need to sustain our society?


*I actually didn't mind this book.
**Webster's seventh is "having no practical or useful significance". Seems like a good example of the root problem.

[Akash Rastogi]

Originally Posted by artdutra04

If more people were taught these critical thinking skills, then they would be better equipped to make decisions in every day life.

I really loved history in high school as well as biology, math (geometry, algebra, calc), and especially physics all because it was taught by pretty great teachers. It taught me to break down a problem, analyze root causes, and evaluate what to do next or interpret data (even in our AP history classes). AP bio was awesome when the teacher taught us how to diagnose simple diseases or predict outcomes when, let's say, a few proteins are missing. AP history classes I had really really made me learn how to analyze a situation and find a root cause/event. Heck, even AP English classes I had made me analyze (maybe over analyze) why a wall in a character's room was yellow. I loved it all though.

Maybe more kids in high school need to take AP classes or non-AP classes need to emulate the type of thinking and skills AP classes/instructors teach? These classes were more fun than my regular/honors classes.

Art's got it right, so does Karthik. You need to be taught well and without just creating a memorized pattern for a student to follow in order to solve a problem. (obviously this isn't always true in math/physics/chem/bio)

/end babbling +$0.02

[ManicMechanic]

We provide "crutches" for the lack of algebra, when knowing it would save paper and money. Tax booklets have 40+ pages of tables with each income level in $50 increments printed with the corresponding tax. If the average American could do the simple piecewise function calculation:

T(x) = 0.15x for x<= 40,000 (or whatever the cutoff income is for the 15% rate)
or = 6000 + 0.28(x - 40000) for x > 40,000 (assuming 28% tax on excess earnings

it would take only an inch of paper on one page, not 40+ pages.

Lack of knowledge of compound interest is a great contributor to the mortgage crisis. A friend from China (an English major) told me "Most of us have our apartments (the equivalent of houses) fully paid off because we know the effects of the formula A = Pe^rt." We can blame our legislators and bankers, but each person is responsible for managing his/her own finances and making informed/prudent choices is something that people in many other countries seem do better than we do. In the county where I live, over 1/3 of homes owned in 2006 have foreclosed, and over 50% of the remaining currently owned homes are "underwater" (with less equity than debt). I can't help thinking that while a knowledge of algebra might not be "necessary", it could potentially save a lot of grief.

Originally Posted by JVN

Building along the same line that Karthik started: I recommend everyone read "A Mathematician's Lament" by Paul Lockhart (sometimes just referred to as Lockhart's Lament).

http://www.maa.org/devlin/LockhartsLament.pdf

-John

As a jr. college math instructor, I have to admit that I'm not crazy about the appeal to math as an art, with self-discovery as a necessity.

In robotics, not many students can invent the (mecanum) wheel, but many more are capable of copying and using the designs of others. Strict copying (similar to memorization and application of formulas) allows students to practice until understanding kicks in, and allows them to reach much higher levels (and generate more enthusiasm) than self-discovery would in the same timeframe. Shortly after the Vex platform was released, I watched students spend months building robots less functional than the Squarebot. That's where self-discovery without copying gets you (especially if you have solid but not stellar talents), and I would never want to return to that.

Exceptionally talented people seem to pick up the concepts and use them for their own purpose, no matter how badly a subject is taught. They're also the ones who see patterns and beauty in discovery. However, I have a child who struggles with math, and for her, "drill and kill" math has been very effective. Like washing dishes, she may find it boring, but with hard work, she can do it,and there's pleasure in doing something well, even if you didn't discover it yourself.

Motivating students to do the boring work helps, and connecting math to previously discovered concrete examples (rather than to what they can imagine/discover) seems to work well with my lower level students. For example, whenever students ask "Why do we have to know about singularities?", the Tacoma-Narrows Bridge, coupled with a discussion of resonance caused by undefined frequencies gets their attention. I don't expect them to discover the formulas on their own. And I don't see being on a time-table to teach a broad-based curriculum as necessarily bad.

[ratdude747]

Originally Posted by Karthik

Is Algebra Necessary? If it's taught poorly, as it is by most teachers, and beaten down into a series of repetitive drills that rely solely on short term memory, then no it's not necessary. However, if it's taught properly, and used as a vehicle towards higher level problem solving and analysis, then yes, it's absolutely necessary. The skills learned when algebra is taught properly come into play every day. One of my best math teachers always used to say "I'm not teaching you math, I'm teaching you how to think. Hopefully that'll be useful to you someday..."

Originally Posted by Lil' Lavery

The problem is not Algebra. Failure to pass Algebra, poorly taught or otherwise, is the symptom of a far greater cultural problem.

Both quoted for truth. Sad but absolutely true.

[KrazyCarl92]

I will preface this by acknowledging that I am heavily biased on this topic, but I see this bias as justified.

I think the writier here falls into the trap of being a sheep (if you saw Karthik's presentation from St. Louis then you know what I'm talking about). His best evidence in the entire essay are rates of failure and dropout and educators quoted as saying algebra is the problem. Where the mistake comes in is immediately assuming that taking algebra out of the equation solves the problem.

This would be akin to if you asked someone from a robotics team why they were losing 90% of their matches and they said to you "because of our ball collector". Is it then the right move to take away your ball collector entirely? Will that make you win? Certainly not, and likewise removing algebra will not better educate our society! As a student, I would be insulted that people consider lowering standards because people are supposedly not capable. Maybe I'm failing to see something because math has always interested me and come easy to me, but I strongly feel close to 100% of people are inherently capable of passing basic algebra.

The writer needs to take a closer look at what the root cause of the issue is. Is it, as he suggests, a structural problem where we are trying to educate people in a subject that they are incapable of learning and it is a waste of their time and effort? Is it ineffective teaching in mathematics (and furthermore, is this a developmental problem early on or a problem isolated to MS, HS, and early college?)? Is poor parenting the culprit? There are too many parents I have seen who are willing to blame anything but themselves or their child for failure; the problem MUST be that someone else messed something up. Is the problem with the students? Are they actually incapable or are they just unmotivated and don't want to work for their success? In my opinion, every case is different and all of these problems contribute to each failure in some way, except for the first problem which this writer suggests.

Originally Posted by Akash Rastogi

I really loved history in high school as well as biology, math (geometry, algebra, calc), and especially physics all because it was taught by pretty great teachers. It taught me to break down a problem, analyze root causes, and evaluate what to do next or interpret data (even in our AP history classes). AP bio was awesome when the teacher taught us how to diagnose simple diseases or predict outcomes when, let's say, a few proteins are missing. AP history classes I had really really made me learn how to analyze a situation and find a root cause/event. Heck, even AP English classes I had made me analyze (maybe over analyze) why a wall in a character's room was yellow. I loved it all though.

Maybe more kids in high school need to take AP classes or non-AP classes need to emulate the type of thinking and skills AP classes/instructors teach? These classes were more fun than my regular/honors classes.

Well said. I loved the way my AP classes always challenged us to understand the "why"s and "how"s behind history, science, and mathematics. I really appreciated the teachers who went the extra mile to get us to THINK and ASK QUESTIONS. It's really given me the skills I've needed now in college, skills which many of my classmates are still struggling to learn because the expectations were never that high for them.

I still keep in touch with my favorite teachers and get together from time to time. This past week I met with my AP biology teacher and when I asked about how the next school year looked, she expressed frustration that the college board was removing the most challenging parts of the AP bio and AP US history curriculums. Specifically they are removing cellular respiration and the cell cycle from the cirriculum. The goal is to get more people to take AP courses, which I agree is a good objective, but to do so by dumbing down the expectations is ruining what made AP classes worthwhile and challenging for many of us in the first place!

"Far better it is to dare mighty things, to win glorious triumphs even though checkered by failure, than to rank with those poor spirits who neither enjoy nor suffer much because they live in the gray twilight that knows neither victory nor defeat."
-Theodore Roosevelt

[Chris Hibner]

I was thinking about this very issue about a week ago. I saw some sad statistic about high school math skills and it got me thinking about what the problem is.

One thing that I've thought about recently is: are the right things being tought in elementary school? It seems that the cirriculum is very antiquated.

One example: long division. Is this really a necessary skill for people in modern times? (See sidebar below) Part of the problem with teaching long division (in my opinion) is the law of primacy kicks in and screws people up once they get to fractions. You'd be shocked at how many high school students (as well as people beyond high school) that I talk to that still don't understand that a fraction means division. After elementary school, the concept of ratio, proportion, and fractions account for 99%+ of all division applications, yet people are screwed by primacy by learning long division first. Just watch the youtube video of the woman trying to explain miles per hour as a great example.

It seems that a lot of the curriculum is antiquated and the early part of math education could be replaced to better prepare students for algebra and problem solving. I'm not yet sure what the solution is.


Sidebar: There is only one application in the modern world for long division that I am aware of: checksum computation (as a part of polynomial division). In this case, shouldn't it be tought in classes about checksums or in an algebra class? Once you get to that age and level of math, you can learn long division in a matter of minutes. Is it really necessary that elementary students spend cumulative months on learning it? It seems the time would be better spent elsewhere. Not to mention that primacy would then ingrain in people's heads that division = fraction and long division is a special case (instead of the other way around).

[Retired Starman]

There are many interesting facets to this problem.

1. Our educational system was never designed to do what we expect of it now. In the early days of public education, the goal was to each basic reading and arithmetic skills to as many people as possible to make them adequate workers. At that time, it was expected than about 20% would complete elementary and secondary school with the rest dropping out to join the work force. Only about 2% were ever expected to finish four years of college, mostly to become ministers, lawyers, and doctors. We have far exceeded these expectations with about 80% graduating high school and 20% finishing a Bachelors degree. Algebra has always been part of the high school and college curriculum, but we have cut out Greek and Latin, which were once considered necessary. It appears that Algebra has passed the test of time as a valuable subject for all students.

2. When you look closely at nearly every subject, the argument can be made that we really don't need to teach it because it is difficult, boring, and never needed in life. History? Just a bunch of facts about dead guys. Philosophy? Well, my ideas are just as good as anything the philosophers of the ages have said. Literature? Hay, I read what I want. Music Appreciation? If I want to listen to music, I'll use my iPod; besides, most of what they talk about is stuff written by more dead guys. So with the possible exception of Reading, there really isn't much that can't be thrown out.

But I'm of the opinion that a core curriculum, taught to everyone, is a necessary part of our cultural identity--a common knowledge base we all draw on, sometimes subconsciously, in our everyday activities. Look at the cartoons in the New Yorker magazine, for instance. Nearly all require a vast common cultural knowledge base in order to understand. Likewise with things as mundane as T.V. commercials. Take away to core cultural knowledge and you become unable to function effectively in society.

3. Algebra, like all technical subjects has moved more and more from the practical to the theoretical in the past two decades. Personally, I don't like this move since it takes the recognition of the reason for learning away. If I have MY problem to solve, I'm much more interested in learning the theory. Otherwise, the phrase, "Why do we have to learn this, I'm never going to use it," becomes very true. But it is also true that if you don't learn it, you for sure will never use it. Math teachers at one of the school where I have mentored robotics say that the robotics students pick up math concepts quicker than their cohorts who are not in robotics because the robotics students have real world experience to tie the new math concepts to. Ratios for instance, confuse the normal students, while the robotics students see the concept as the theoretical construct they have used with gears and chain sprockets. We need more practical applications to drive the teaching of the theory. If people saw how algebra could help them, they world be more motivated to learn it.

4. Algebra is taught badly in many if not most schools. I've had friends say they were good at language, but not at Algebra. In reality, both math and language are functions of the left side of the brain, which is rational, sequential, linear, and organized. Algebra at its most basic level is the language of mathematics. When people tell me they were good at language and not algebra, I tell them that Algebra is a foreign language with facts and rules that they never learned. To illustrate this, I'll usually say something like this, "I'm thinking of a number. If you multiply my number by three and add four, you get 31. What number am I thinking of?" Most immediately reply with the correct answer, yet freeze up if I present them with the same statement in equation form (3x +4 = 31). Somehow a teacher never conveyed to them the shorthand language nature of Algebra. At the heart of all algebraic statements is the equal sign which should always be understood as "is the same as". If this isn't taught first, people will have trouble with Algebra for ever.

5. In actuality, most students start to fall behind in math with the introduction of fractions, not the introduction of Algebra. By the time many students get to Algebra, they are already hopelessly lost.

6. Should we keep Algebra? Probably so. It is and always has been part of what we consider knowledge common to all educated citizens. Sure, it's a gatekeeper to that Bachelors degree, but I really don't have much of a problem with that. If you want to be an Art Historian but can't pass Algebra does that mean you won't have a degree? Maybe. Does it keep you from taking art history classes to your heart's content? Probably not. Does it mean you won't be a fully educated person in society? Yep, afraid so. Would it motivate you to work harder in Algebra? Probably so, if you want that degree so you can be recognized as an educated citizen.

Well, enough of my ramblings. You might not agree with anything I've put on the table, but I've been thinking about this for at least 45 years and am now entering my curmudgeonly years. I vote to keep Algebra.

Dr. Bob

Chairman's Award is not about building the robot. Every team builds a robot.

[JuliaGreen]

To graduate with a Michigan high school diploma, each student must have 4 credits of math, which includes:

1 full credit of Algebra
1 full credit of Geometry
1 full credit of Algebra II
and a math course (can be applied math, financial literacy, accounting, computer science, business related, etc.) as a 12th grader

Michigan defines a full credit as a learning experience which is a school year. It also allows for the math to be delivered in a Career Tech Education environment. Algebra II can be sequenced to be covered in two years, instead of 1.

Yes, this is tough - but it does not have to be delivered at "sit and get" in columns and rows of desks - but don't get me started on that!

Julia

[BigJ]

In my mind the difference between history/language and algebra (BOTH are important and should be taught, no matter how much I hate writing) is that algebra (and higher mathematics) literally defines the universe we live in and is the definition of many problems we come across in daily life. History and language are facets of our culture.

[Lil' Lavery]

Originally Posted by Michael Hill

I work for the U.S. Government in an area that deals quite a bit with foreign politics. I'd say I'd pay more than enough attention to both national and international politics.

Then I'm struggling to see how you could possibly think European history is not important. See artdutra04's post.

Originally Posted by Michael Hill

The article is a cry for help because teachers fail to find a way to show how mathematics has practical real-life use, which is tragic because it closes the door to many opportunities for students who have unrealized passion for a technological area. A student may not grasp Algebra completely, but they may later find love for geometry or calculus. I had a similar issue with linear algebra in college. I excelled in differential equations, but some of the concepts in linear algebra can be abstract. But it was just the opposite situation for other students. Some found linear algebra to be a breeze, but struggled with calculus.

This is a direct quote from the article:


He wants to take it out to make high school easier. That's the direction he wants to go. It's all about making graduation rates higher. But graduation rates mean absolutely nothing if you learned absolutely nothing.

You're taking an out of context quote from the article and attempting to use it to state that's what the article's premise is. The article is not arguing to make high school easier, but rather to change what we're teaching to be more applicable to everyday life. A large portion of the third page discusses possible alternatives to algebra to help make it more engaging and applicable to students with varying passions.

Let me make this clear, I don't agree with the article. I don't want algebra removed from our essential curriculum. I just don't want the article's viewpoints being misrepresented and the debate coming down to gut reactions rather than informed conversation.

Originally Posted by Siri

I don't mean to be antagonistic, but how discrete is discrete? Analyzing the three caskets in The Merchant of Venice? Writing a 5-paragraph character sketch? Do I really need to know the Treaty of Utrecht to understand the importance of its modern ramifications? ...By the way, I don't think I was taught the latter any more than the applications of a character sketch. Is "teaching" algebra, the long, confusing, repetitive jumble of letters that has no impact on my future really that different from teaching, say, Wuthering Heights (the long, confusing, repetitive jumble of letters that has no impact on my future*)? [EDIT: What Arthur said.]

I don't disagree with the general premise laid out by you, Arthur, Karthik, and others about the ability to reason and critical thinking skills that are cultivated by algebra. In fact, I strongly agree. But what I meant by "discrete skill" was that algebra as a tool isn't particularly useful to a large portion of the population (though ManicMechanic makes a strong counterpoint to this). You don't need algebra to balance your checkbook or figure out your monthly finances. Most people are probably never going to balance an equation outside of an academic setting.

The problem, in my eyes, is not algebra or even how algebra is being taught. The critical reasoning and logic skills that are important behind algebra and that students should take away from algebra (even if they don't grasp algebra as a tool) don't start in high school. The development of these skills starts much earlier in life. By the time a student is in algebra is often far too late in their development to try and fix the issue.

[JackN]

I just graduated from Michigan State University with a degree in Journalism. In school/the start of my professional career, there was no need to really utilize algebra or calculus. I took no math credits while at MSU because I had credit from AP classes, so I don't have a very good idea of what math classes are like in the college level.

In one of my entry level Journalism classes I was able to earn extra credit because I understood how a logarithmic scale worked and could explain it in class (we were discussing the Richter Scale). I have a friend who is currently working as a page and graphic designer for a major magazine who took several tries to pass college algebra.

I live with one student who is getting a degree in engineering and another getting a degree in horticulture. About six months ago we had a discussion about the Republican primary process, neither one of them could name two people who were running for the nomination (this took place when Herman Cain, Ron Paul, Mitt Romney, Michele Bachmann, Rick Santorum, John Huntsman, and Rick Perry were all still in the race). I found this just as frightening as people not able to do algebra.

Every single person needs a basic understanding of political science, math, biology, physics, and writing to be able to function in society. Having baselines for understanding across all fields can improve our communication and can make it easier for us to relate to each other. The US is become more sharply divided along economic, educational, religious and political lines maybe figuring out a common level of education could help us solve the problem.

[Michael Hill]

Originally Posted by Lil' Lavery

Then I'm struggling to see how you could possibly think European history is not important. See artdutra04's post.

Because not once has anyone likened anything in a briefing to the Habsburg rein of Austria. Also, I used that in the sense that it has no real-life use outside of high school for most people.

Originally Posted by Lil' Lavery

You're taking an out of context quote from the article and attempting to use it to state that's what the article's premise is. The article is not arguing to make high school easier, but rather to change what we're teaching to be more applicable to everyday life. A large portion of the third page discusses possible alternatives to algebra to help make it more engaging and applicable to students with varying passions.

I realize that isn't the full premise of the article (though a generous portion of the article is precisely that, complaining that math is too hard). However, his whole argument is based on the fact that students aren't doing well in Math. If students were able to grasp algebra easily, this article would have never been written in the first place. The author is a prime example of someone who advocates the "dumbing down" of America just like No Child Left Behind. Algebra is something that many middle schoolers (including myself) took. It's not too much to ask of someone in high school to be able to take it either. High school today is no longer about educating students, but giving them a degree to get them out of the door....a McDiploma.

[Siri]

Originally Posted by Lil' Lavery

I don't disagree with the general premise laid out by you, Arthur, Karthik, and others about the ability to reason and critical thinking skills that are cultivated by algebra. In fact, I strongly agree. But what I meant by "discrete skill" was that algebra as a tool isn't particularly useful to a large portion of the population (though ManicMechanic makes a strong counterpoint to this). You don't need algebra to balance your checkbook or figure out your monthly finances. Most people are probably never going to balance an equation outside of an academic setting.

I agree with you. My question is, how often are the discrete skills we teach useful to a large portion of the population? Reading (at tax form and newspaper level) and addition (arithmetic), sure. But manual matrix row reduction and analyzing causes of corruption in the Janissary corps? The school algebra curriculum is made up of a lot of discrete sub-skills that themselves bare no real impact on daily life. The same way basically every other subject is. I can certainly expound on uses for and abilities cultivated by these*, but to argue that algebraic computations are unique in their uselessness seems inaccurate to me.


This is not to say it mightn't warrant additional attention, if only due to its relatively spectacular corruption from critical thinking skills and/or its supreme difficulty for many students. I'm not sure that this is actually true, but if it is, algebra reform (or scrapping and re-doing, as Lockhart would prefer) may well deserve priority*. It's the argument that the subject itself is any more or less valuable--in discrete skill form or otherwise--than any other school subject which I dispute.


*EDIT: In that way, I partially disagree with Michael Hill. I, too, see a level of disingenuousness in focusing on algebra as "not necessary" while really arguing in large part that it's hard. These are two entirely different issues. However, if one were looking for a place to start productive reform, one might reasonably start with the subject of most difficulty to the most students. (I haven't perused the statistics in the Hacker article nor read contrary arguments, so I haven't yet judged if algebra is actually that subject.) I'd venture that this is Hacker's actual argument here--perhaps without knowing it.