## The NYT Asks The Wrong Question About Algebra

The interesting question isn’t, “Should every high school graduate in the US have to take Algebra?” Our world is increasingly automated and programmed and if you want any kind of active participation in that world, you’re going to need to understand variable representation and manipulation. That’s Algebra. Without it, you’ll still be able to clothe and feed yourself, but that’s a pretty low bar for an education. The more interesting question is, “How should we define Algebra in 2012 and how should we teach it?” Those questions don’t even seem to be on Hacker’s radar.

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I certainly think every student should have algebraic experience and fluency in some sense, but we definitely need to reconsider the idea that working through systems of 3 linear inequalities is an essential component of a mathematical education.

It occurs to me more and more that programming and science are the best places to utilize and manipulate algebraic expression, so it should be reconsidered, how algebra should be learned and experienced by our students

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### Theron Hitchman

July 31, 2012 - 9:08 am -

No, instead the op-ed was whiny and defeatist.

At one point early on, i thought the piece might go somewhere useful and interesting. The writer raised the issue of examining what the goal of the mathematics curriculum should be, but then let that slide with the general impression that the answer is “not much.”

I like when educational issues get a discussion in a national forum, but i dislike when the starting points are so poorly argued.

2. #### Paul Salomon

July 31, 2012 - 9:32 am -

Well said, Dan!

I certainly think every student should have algebraic experience and fluency in some sense, but we definitely need to reconsider the idea that working through systems of 3 linear inequalities is an essential component of a mathematical education.

It occurs to me more and more that programming and science are the best places to utilize and manipulate algebraic expression, so it should be reconsidered, how algebra should be learned and experienced by our students.

3. #### Mr. K

July 31, 2012 - 10:09 am -

> How should we define Algebra

Don’t we do that by getting together a committee of people who represent a bunch of different (and minor) interests and having them make us a bullet point list?

Less cynically, is it algebra that we need to teach? or quantitative thinking (as mentioned briefly in the article)? or “abstraction”?

Even rephrasing the question as you did still leaves the responses headed for a tar pit that I don’t think will ever be resolved in the national dialogue.

4. #### peterb

July 31, 2012 - 10:59 am -

My co-blogger wrote a rant about this very topic, called “Math Is Hard, Let’s Just Quit”: http://tleaves.com/node/12

I think that there ARE some issues with the way algebra is taught in today’s schools; specifically, the major textbooks in use in middle and high schools are utterly atrocious (in my opinion): boring, not engaging, and guaranteed to kill any love of the subject in most thinking people.

Meanwhile, someone has made an “Angry Birds”-like game called Dragon Box that sneakily teaches algebra. It’s fun and brilliant, and everyone should try it. Because that, my friends, is how we should be introducing younger people to this topic.

Link to article about DragonBox; itself contains links to iOS, Android, and Mac stores: http://www.wired.com/geekdad/2012/06/dragonbox/

July 31, 2012 - 11:07 am -

Does Hacker realized how easily these arguments apply to other subjects? Virtually everything taught beyond sixth or seventh grade is “useless” by his definition and should not be required. The logical conclusion of his line of reasoning is that we should award diplomas after eighth grade or even sooner…

Along with algebra, all works of fiction surely must go — perhaps to be replaced by reading newspapers. Then who really needs to know about thermodynamics or how the digestive system works or — intuitive physics served us well for millions of years and even toddlers know how to void themselves. No science then! History must go as well, to be replaced with a primer in voting so we can understand butterfly ballots along with the CPI (on Hacker’s numeracy wishlist). With foreign languages, the case is the most clear cut — out with them all!

Perhaps Hacker spurs us to reconsider the purpose and goal of education generally and mathematics instruction. More likely, though, we merely waste our time engaging his arguments.

6. #### Robert Berkman

July 31, 2012 - 11:56 am -

I wrote about a similar point yesterday at my blog, bltm.com/blog. My issue is that we need to teach algebra as a verb, not as a noun, which are two very different things.

7. #### Jon

July 31, 2012 - 12:57 pm -

I don’t think you read to the bottom of the article. What the author calls for is a change in how we approach mathematics. I think that’s the point of this site, isn’t it?

8. #### Yaacov Iland

July 31, 2012 - 1:25 pm -

“Our world is increasingly automated and programmed and if you want any kind of active participation in that world, you’re going to need to understand variable representation and manipulation. That’s Algebra.”

The word manipulation is at the heart of this issue. If by manipulation you mean that I should be able to mentally rearrange the speed equation to figure out time and distance, then it’s true that I’ll manipulate variables a lot as an active human being in the modern world. But if by manipulation you mean factoring quadratics, completing the square, the quadratic formula or trigonometric functions, which are the core of the grade 10 course here in Ontario, then I totally disagree that active participation in our world requires those skills.

I’m willing to bet good money that if you asked a random sample of adults when the last time they used the quadratic formula, factoring, completing the square or trigonometric functions, over 90% would say high school.
I took a look at the curriculum for grade 10 English and stopped reading after the first ten expectations because I’d used every one of them in the last two weeks. I’m using several of them just writing this comment. Then I checked our History curriculum and discovered that I make frequent use of those as well. I think there’s a pretty good case that the current curriculum in math is less useful to most people than the curriculum in other high school subjects.

And yet I find that my students are terrible at the math skills that we are called upon to use regularly. Most of them can’t make sense of statistics. They can’t decide what is significant, they have very little sense of scale, they are poor estimators. I do some work on improving these things, but I could do a lot more if it was more prominent in the curriculum.

Possibly this is just an indication that the Ontario curriculum is bad. If this is the case, please let me know!

9. #### mr bombastic

July 31, 2012 - 2:02 pm -

Most people have a very limited ability to think algebraically, so it really is irrelevant how useful most people find algebra. It would be like going to a town that is mostly illiterate and asking people how often they use the reading skills that they don’t have.

Instead the question should be whether those that have algebraic reasoning ability benefit from them. If so, is the benefit limited to the work place in technical fields, or do they benefit in every day situations as well.

I rarely write and solve an equation, but I believe I use algebraic thinking in other ways quite a bit – almost every time I make a spreadsheet, when making finincial plans or decisions, etc.

10. #### Dave

July 31, 2012 - 5:32 pm -

I agree with much of Hacker’s OpEd as I interpreted his premise to be we are wasting precious time, effort, resources, and potential forcing ALL students to learn something they may not yet be prepared to learn for a variety of reasons. I base my interpretation primarily on the following excerpts from Hacker’s OpEd. Emphasis added in ALL CAPS.

I also discuss why I agree with Hacker at: http://mathequality.wordpress.com/2012/07/30/is-algebra-necessary/

Excerpt #1: Making mathematics MANDATORY prevents us from discovering and developing young talent. In the interest of maintaining rigor, we’re actually DEPLETING our pool of brainpower. I say this as a writer and social scientist whose work relies heavily on the use of numbers. My aim is not to spare students from a difficult subject, but to call attention to the real problems we are causing by MISDIRECTING precious resources.

Excerpt #2: Mathematics, both pure and applied, is INTEGRAL to our civilization, whether the realm is aesthetic or electronic. But for MOST adults, it is more feared or revered than understood. It’s clear that REQUIRING algebra for EVERYONE has NOT INCREASED our appreciation of a calling someone once called “the poetry of the universe.”

Excerpt #3: Instead of investing so much of our academic energy in a subject that BLOCKS further attainment for much of our population, I propose that we start thinking about ALTERNATIVES. Thus mathematics teachers at every level could create EXCITING courses in what I call “citizen statistics.” This would not be a backdoor version of algebra, as in the Advanced Placement syllabus. Nor would it focus on equations used by scholars when they write for one another. Instead, it would FAMILIARIZE students with the kinds of NUMBERS [& OPERATIONS] that describe and delineate our personal and public lives.

Excerpt #4: This need not involve dumbing down. Researching the reliability of numbers can be as DEMANDING as geometry. More and more colleges are requiring courses in “QUANTITATIVE REASONING.” In fact, we should be starting that in KINDERGARTEN.

Excerpt #5: I hope that mathematics departments can also CREATE courses in the history and philosophy of their discipline, as well as its APPLICATIONS in early cultures. Why not mathematics in art and music – even poetry – along with its role in assorted sciences? The aim would be to treat mathematics as a liberal art, making it as ACCESSIBLE and welcoming as sculpture or ballet. If we RETHINK how the discipline is conceived, word will get around and math enrollments are bound to rise. It can only help. Of the 1.7 million bachelor’s degrees awarded in 2010, only 15,396 – less than 1 percent – were in mathematics.

Excerpt #6: I’ve… [snip] …been impressed by conscientious teaching and dutiful students. I’ll grant that with an outpouring of resources, we could reclaim many dropouts and help them get through quadratic equations. But that would MISUSE teaching TALENT and student EFFORT. It would be far better to REDUCE, not expand, the mathematics we ask young people to imbibe. (That said, I do not advocate vocational tracks for students considered, almost always unfairly, as less studious.)

Excerpt #7: Think of math as a huge boulder we make EVERYONE pull, without assessing what all this pain ACHIEVES. So WHY require it, without ALTERNATIVES or exceptions?

11. #### Matthew

July 31, 2012 - 10:26 pm -

That article was one of the most ridiculous ones I’ve read yet. Now nothing against those who have failed a class, but I can’t help but wonder if Hacker was one of those who failed algebra. It sounds to me like he had a bad experience with algebra and now has a goal of making sure nobody has to experience that awful torture.

Signed,
a 17-y/o taking multivariable calculus in college this fall.

12. #### Otto

August 1, 2012 - 3:25 am -

The reasoning is quite specious and the overall tone of the article is that of an aloof academic with little knowledge of the “real world.” We shouldn’t be surprised, it is the Times.

13. #### Dan L.

August 1, 2012 - 4:00 am -

@Paul who said “It occurs to me more and more that programming and science are the best places to utilize and manipulate algebraic expression, so it should be reconsidered, how algebra should be learned and experienced by our students.”

Having a practical problem to solve is a hallmark of engineering. I can’t think of a better context for using mathematics. It would seem that getting lost in teaching the tools, techniques, and proofs for their own sake is a (if not THE) divergent point.

14. #### Stephanie

August 1, 2012 - 5:47 am -

Dan – your question “what is algebra?” is the great crux of any article talking about math. You see, I ask this question of teachers across the country all the time and there is never a unified understanding. People think algebra is the content that is covered in any HS math class that isn’t Geometry. I would be interested to hear what Hacker’s definition of Algebra is. I don’t think Algebra is defined by all that stuff, but rather should be thought about for what the word means.

The free online dictionary states, “1. A branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set.” Algebra is when math moves from concrete representations such as counting to abstract understandings of relationships.

Thinking if it this way… is there no reason why anybody would need to study algebra, or to be able to define a representation abstractly rather than with concrete things? Are we back in the caveman days?

Dave writes in his argument AGAINST Algebra for all: “Thus mathematics teachers at every level could create EXCITING courses in what I call “citizen statistics.” This would not be a backdoor version of algebra, as in the Advanced Placement syllabus. Nor would it focus on equations used by scholars when they write for one another. Instead, it would FAMILIARIZE students with the kinds of NUMBERS [& OPERATIONS] that describe and delineate our personal and public lives”, clearly showing that his interpretation of Algebra is what happens in today’s classrooms. Instead, his description here really talks about what Algebra IS (and he doesn’t even know it).

I think ALL students need to understand algebra. They need experiences with all kinds of representations and to understand math in an abstract way. I also believe that we need to get creative in doing that, which is why I LOVE this blog. But perhaps first we need to make sure that when people talk about Algebra they really know what it is.

15. #### Belinda Thompson

August 1, 2012 - 7:01 am -

A secondary math education professor said to me that algebra is difficult for students because so much of the content is new. She gave a percent that I forget now, but it was high and based on some textbook analysis. My response was that there’s no way that could be the case because students in Algebra 1 deal largely with relationships between operations and equality, topics they’ve studied since they entered school. None of those ideas change. Not to pass the problem down, but if we decide what algebra instruction should entail, we might also discuss what entails appropriate preparation for the study of algebra.

16. #### blaw0013

August 1, 2012 - 7:20 am -

@ddmeyer – you have a NorCal neighbor who might disagree with you on the current and future role of some “need” for variable representation and manipulation. @worrydream http://bit.ly/Qqhjp8 What do you mean by these two words?

17. #### Santosh

August 1, 2012 - 7:32 am -

I am with Stephanie about the public perception of ‘Algebra.’

My schooling was outside the US, and I used to find it so confusing when people in the school system would refer exclusively to PreAlgebra, Algebra I and Algebra II. ‘Algebra’ in the common parlance is obviously the name of specific courses, not the name of a set of ideas. In order to draw the distinction, I find myself using ‘algebraic thinking’ more and more.

As math teachers, we are obviously failing in communicating our role when the educated public cannot describe the subject we teach half-accurately. People have a good idea of what they learn/learned in History, American History, World History, etc. Same with Physics, Newtonian Physics, Quantum Physics, Nuclear Physics, etc.

I hated history because it seemed to be a boring sequence of dates and conquests. I suppose many people hate algebra for the same reason – a boring sequence of operations and transformations.

What are the big ideas that tie the post-elementary math curriculum together?

18. #### Lorraine

August 1, 2012 - 9:05 am -

One reason to study Algebra is its useful connection to Symbolic Logic.

The ability to reason is clearly important in all walks of life and in multiple activities (e.g., analyzing newspaper articles, understanding debates).

Some knowledge of Symbolic Logic can be really helpful in sorting out types of arguments (e.g., transitivity – if A > B and B > C then A > C, true/false arguments and so on), so that you can make better decisions.

Symbolic Logic builds on Boolean Algebra. Learning Algebra in high school was – at least for me — important preparation for learning Symbolic Logic in university, which has been useful preparation for debating, analysis of arguments, and logical reasoning.

The problem, as I see it, is that too FEW individuals learn Symbolic Logic, NOT that too MANY learn Algebra.

19. #### Dan Meyer

August 2, 2012 - 7:24 am -

Mr K:

Even rephrasing the question as you did still leaves the responses headed for a tar pit that I don’t think will ever be resolved in the national dialogue.

Who said anything about the entire nation agreeing on a definition of Algebra? I’m proposing we don’t start the conversation with the wrong question.

Brian:

@ddmeyer — you have a NorCal neighbor who might disagree with you on the current and future role of some “need” for variable representation and manipulation. @worrydream

Victor wants students to be able to pose and answer questions in dynamic, visual environments … created by someone who understood variable representation and manipulation. If you understand Algebra, you can create those environments (a la Victor). If you don’t, you only get to pose and answer questions in the environments that people like Victor create.

20. #### M Gilmartin

August 2, 2012 - 2:23 pm -

I agree with you that while algebra and the type of thinking it promotes is important perhaps we need to reexamine how we teach it. After reading the op-ed I came away with a similar, though not as drastic, view as blink in that we could apply this line of thinking to a myriad of other subjects taught in schools, but that does not negate the importance that these subject have in promoting critical thinking and exposing students to different fields of potential interest. Paul Salomon’s comment is intriguing as computers have definitely influence the way people utilize knowledge so it might be time for us to better in corporate them in what we want students to get out of school. Salomon’s comment also made me think about how my brother is still in high school and while he has taken a computer class because of his own personal interest there is no such requirement that all students must take a technology course, with technology especially computers becoming so invasive in our lives is it perhaps time we teach at least some basics to students outside of how to use Word and PowerPoint?

21. #### Scott Portnoff

August 3, 2012 - 9:40 pm -

#17 Belinda Thompson was on the right track. Both Dan Myer and Hacker assume that students arrive in an algebra class with grade-level math skills.

My experience in urban L.A. schools is that at least 2/3 of those students have gaping holes in their math foundations, and have been failing mathematics for years before they reach algebra (with the expected decline in self-confidence and a gradual generalizing of poor self-concept to other academic areas). Ask them to add 1/2 + 1/3 and you’ll get anything but 5/6. If they don’t know fractions, they can’t solve proportions, which are the beginning of one-variable algebra. In fact, anyone who has spent more than 5 minutes teaching high school math and has taken the time to investigate the skills of students who fail (looking at math proficiency scores, Math Diagnostic tests, etc) knows this.

These kids are not stupid, and in fact can fill in the holes pretty quickly now that they’re older, IF remediated. When shown how to solve some fractions-related problem, they go: “Really? But that’s so simple!” My experience is that most such students are dying to experience success in mathematics, and once they do so, are grateful and eager for more – and become much less of discipline problems.
Although I thought the film “Waiting for Superman” concerned itself with all the wrong topics, one of the things they did get right was when they pointed out that the charter schools that remediate students as soon as they get them are models of what can be done in this area.

If you put students with grade-level math skills in an algebra class, most students will succeed. And there are even strategies for boosting the performance numbers (SEE Brad Fulton). Algebra (and in general) mathematics is not the problem.

Rather, the question should be: how do you ensure that K-8 students do not fall behind in basic math foundational skills?

22. #### Belinda Thompson

August 3, 2012 - 10:17 pm -

@Scott #17

To be clear, I’m not arguing for getting students to arrive at the door of Algebra 1 with better basic skills. Rather, I’m arguing that they might arrive with a disposition toward thinking relationally about quantities and being flexible in representing relationships. Really understanding equivalence, as well as the relationships between addition and subtraction, and multiplication and division, would go a long way. My basic contention is that Algebra 1 need not be the beginning of algebraic thinking.

23. #### Mr. K

August 5, 2012 - 9:48 am -

>Who said anything about the entire nation agreeing on a definition of Algebra?

Your two questions were “How do we define Algebra?” and “How should we teach it?”. It seems to me that any discussion about the second is predicated on some agreement on the first, and I’m just pointing out that this isn’t trivial.

24. #### Kelly Holman

August 6, 2012 - 1:12 pm -

I think as a group, we agree with Hacker more than we disagree:

Algebra is often taught poorly, with little connection to other subjects.

People need quantitative reasoning, logic skills, and the ability to apply math skills to real problems.

Most people don’t need all the theory of math.

Almost all I’ve seen of how algebra is taught deserves everything the article says about it, and I would make it optional. Almost all I’ve seen on this blog would provide very good answers to Hacker’s desire to change how math is taught.

25. #### prisnerich

August 15, 2012 - 6:43 am -

Are the kids not learning algebra/math since they will not apply it in real life? Or are people not applying algebra/math in their lives since they didn’t really learn and understand it when they were kids? If it is the second, we should just make an effort to teach algebra better.