Wednesday, November 27, 2013


Cultural Capital at Home

We’ve hit the age that every parent dreads: the age at which your kid’s math homework involves actually re-learning algebra.

I knew this day would come, but nothing really prepared me for it.

The Boy is smart, and wants to be an engineer when he grows up, so he’s in advanced math.  And he’s mostly fine with it.

But this week he brought home...exponents.


So far, it isn’t too bad.  Experience tells me, though,  that it won’t be long before he gets to fractional and/or negative exponents, neither of which I really understood the last time I saw them, which was sometime during the Reagan administration.  

They teach math differently now than they used to.  Back in the day, I learned to “carry” and “borrow,” like God intended.  Now, apparently, they “regroup.”  It’s actually a smarter approach, once you figure out what the hell it is, but I admit being brought up short the first time I saw it.  Still, though, arithmetic is arithmetic.  Now we’re getting to the more abstract stuff.

My own history with math was odd.  I cruised along quite comfortably until I hit geometry.  I hit geometry in the same sense in which a watermelon dropped from a skyscraper hits the sidewalk.  It wasn’t pretty.  “Prove this is a rhombus.”  I’m sorry, what?  I’m hoping that proofs have gone the way of borrowing and carrying, but I suspect not.  

I know, as a card-carrying college administrator, that I’m supposed to condemn math phobia wherever I see it.  And there are fine and good reasons for that.  I made it through my mandatory stats classes for poli sci in both undergrad and grad school, so it’s not like the sight of numbers gives me the vapors.  But as a human being, rather than just a placeholder for an office, I have to admit a pretty glaring blind spot where “geometry” is supposed to be.  I’m not proud of that, but it is what it is.  

(On those 8th grade “vocational aptitude tests” they used to give, I always crashed and burned on “spatial reasoning.”  I’ve long considered the ability to do complicated jigsaw puzzles a sort of magic.  And let’s just say I give my phone’s GPS a pretty good workout.)

This is where the whole “first-generation” thing really hits home.

The Boy has parents who went through this stuff, and who can at least try to help him when he gets stuck.  We can help him make the maddeningly difficult leap from “I know that” to “you mean, I use that here?”  When he gets discouraged or confused, we’re there for him.  I even took a crack, this week, at explaining that a period inside quotation marks at the end of a sentence does double duty.  He was skeptical, but went with it. Nobody said English grammar makes sense.

This is how cultural capital gets transmitted.  Well-meaning people, doing right by the people in their lives, reproduce privilege.  The kids who don’t have parents at home who can help them through the rough patches in their homework are likelier to get lower grades and draw negative conclusions about what they’re doing.  The kids who get help when they need it are likelier to make it through.  Neither is a certainty, but the probabilities are real.

I don’t see withholding help as any sort of solution.  Parents are supposed to help their kids.  Wasting TB’s talent because others don’t have educated parents wouldn’t help anybody.  That’s not the point.  I didn’t go into education for a living only to withhold education from my own kids.  

As he gets older, I’ll have to add some other lessons.  Those will involve trying to figure out what happens when some people have the wind at their back, and others have the wind in their face.  And what it means to try to live an ethical life in a stratified country.

We’ll give thanks this week for the very real blessings we have.  As they get older, I’ll try to get them to ask just how those blessings got there in the first place.

Tom Leyrer answered your "regroup" question back in 1965. This lipdub is awesome but there are others.

If TB has mathematical skills, he may survive and thrive in a discovery approach to math, but that might not be the case for others in the same curriculum. I believe that many things attributed to high-stakes testing in math are actually due to methods that do not teach effective algorithms in the early grades. For example, you can't do synthetic division if you never saw the standard long division algorithm and the guessing approach to fractions is even worse. (It is the math version of "look say" reading instruction.)

Along the way, proofs appear to have vanished from our state's schools, which is really bad for the future engineers. They never learned to present their work in a logical order, so they have to learn it in calculus and physics while also learning other things.

As for TB's question about exponents, there are lots of good videos produced by college math faculty for developmental classes. It is not an uncommon problem! Ask your math faculty if they have specific recommendations.
Geez Matt, you just described my math experience to a T. After a terrible year in 9th grade geometry, I decided that math was not for me and did all I could to avoid it through high school and college. That didn't change until I took the required/dreaded stats sequence in my first year of grad school and discovered how much fun it was (for real, and nobody was more surprised than me).

For what it is worth, I don't think proofs dominate high school geometry classes anymore. Or at least, they didn't when my son took geometry on the math fast-track a few years ago. It was much more applied than what I remember being taught.

Of course you're right about the importance of family background and attitude, but I think that is less straightforward than you are suggesting. In the mostly blue-collar community where we lived for much of my children's elementary & middle school experiences, many of the kids whose parents worked in manufacturing (mostly auto industry) and construction had great math and spatial reasoning skills. My sons' friends were always building stuff at home with their dads/brothers/uncles where math was involved on an applied level. Their parents might not have gone to college, but they tended to be hands-on, practical people with good work ethics and strong puzzle-solving skills.

When it comes to learning math, I think those baseline attitudes can take a person a long way - maybe further than having a parent with an advanced degree and an engrained math phobia.

I think the cultural capital comes into play more when kids are trying to navigate the complexities of choosing colleges, applying, identifying scholarship opportunities, and then dealing with the particular bureaucracies and red tape that go along with college life. Then, having family who know how the system works and who can point a kid in the right direction is a major advantage.
Oh - and check out Khan Academy. Math, explained well, online and free.
"I’m hoping that proofs have gone the way of borrowing and carrying, but I suspect not."

Argh no! To do this is to reduce math to mere computation, which is certainly a necessary skill but can be pointless and boring. Paul Lockhart, in his essay "A Mathematician's Lament", describes that as simply having music students playing scales up and down again without getting into any real music. Heck, as a physicist, I need to be able to do computation. But while technique practice is extremely useful, that's not music. Learning how to do mathematical proofs is a very good exercise in logical thinking, showing how you can support an argument from one step to another and hopefully coming up with creative ways to do so.

I'm sure in your work you've ended up thinking "Well, there may be a case to say that X is due to Y. Now just for the sake of argument, what if Y wasn't happening? Would we still see X?" This is the sort of thing you do in math proofs all the time!

WITH THAT SAID, I'd readily agree that many (perhaps even most) primary-and-secondary school math teachers really don't grok on how to explain much of this, and instead bring out the more horrendous flow-charts for how proofs work. Knowing about parallel lines is a bit boring until you start playing around with them on spheres and hyperbolic spaces, at which point you start to see how how they can be more interesting.

As for the discussion of cultural capital--it's a problem.
Math is just language, really. Negative exponents don't "really" intrinsically have a meaning anymore than words do. Someone assigned them a (somewhat arbitrary) meaning. In this case someone who noticed the pattern that 2^3 * 2^2 = 2^5 (the exponents add) and realized that you'd be able to write 2^3 * 2^(-2)= 2^1, if you chose to define 2^(-2) as 1/2^2. That's neat - keeps the pattern going. But it was still a kind of arbitrary choice for how to define that symbol.

Same for fractional exponents. 2^(1/2) * 2^(1/2) = 2^(1) if you choose to define 2^(1/2) = sqrt(2), which is a pretty good reason to define that symbol with that meaning.

Has nothing to do with multiplying a number times itself a negative or fractional number times - don't even think about that. It seems like it makes no damn sense because it really makes no sense. It's just a symbol which is assigned a meaning, mostly out of convenience.

Anyway, that's what I tell myself to make me feel better about it. And I'm and physicist. :-)

Sadly, proofs in geometry have gone the way of the dodo. Proofs were the only instruction that most students got in logic and reasoning, and while the actual proofs themselves aren't especially useful to most people, the skills they taught are extremely valuable. Schools talk a lot about critical thinking skills, but by eliminating proofs have actually stopped teaching them.
Funny because it's true.

Although I'm growing tired of the "privilege" cliche, which adds little to the discourse.

Very sharp insights by Anonymous at 5:37.

And I always smile when I see "grok" in a comment.
True, Anonymous @ 5:37. My building trades relatives have a much more powerful, much more fluid understanding of math than my liberal arts relatives. Let's leave out STEM people, because they have special talent and daily experience with upper-level math, regardless of where they started out on the economic scale.

Interestingly, it's been demonstrated that the GI Bill took thousands of farm boys coming back from WWII and turned them quite easily into engineers, because they already had intuitive grasp of many of the principles of physics and chemistry.
I'm amused that most commenters are replying to the math part of the post because what really struck me was the last lines, "As he gets older, I’ll have to add some other lessons. Those will involve trying to figure out what happens when some people have the wind at their back, and others have the wind in their face. And what it means to try to live an ethical life in a stratified country." These are things I think about all the time, as a parent and as a professor. Thanks for highlighting them in the higher ed blogosphere this week.
The math is the part Dean Dad expressed confusion about. He's got the sociology covered.
You should have The Boy check out Art of Problem Solving ( They have everything from message boards where kids can geek out about math to high quality online classes that can complement a dreary in-school class.
I think the cultural capital comes into play more when kids are trying to navigate the complexities of choosing colleges, applying, identifying scholarship opportunities, and then dealing with the particular bureaucracies and red tape that go along with college life. Then, having family who know how the system works and who can point a kid in the right direction is a major advantage.
For what it's worth, you're kind of solving the sociology problem by educating your son on how to work the system. Most of my friends in school were first gen immigrants and I acted as sherpa for them when we were applying to college. If your son befriends the right people, he could easily “spread the wealth” sociologically speaking. My friends and I all got in somewhere and all of us graduated. Students who can help themselves into and through college do much better than those who require parental assistance – and teenagers rarely listen to their parents anyway.

As for geometry, teach your daughter or son to piece and quilt or do simple woodworking and they will learn all they need to know. Repetition is an underappreciated aspect of learning math. Asian parents get this with their endless tutors and Kumon. Crafts work to teach math because having mastered a single lesson they force you to repeat it over and over (this is perhaps more true in quilting) which ingrains it in a way a few repetitions never would. I suspect my friends in the liberal arts forgot their math skills because they were weak to begin with and then never reinforced through work.

The best lesson you can teach your kids is to persist even when something is difficult. Showing them that you still have a good attitude towards a subject despite struggling with it is a more important lesson than the ability to solve a quadratic equation. Just keep being encouraging and honest about your own experience. Your son and daughter will learn what they need to know about the subject matter and about having compassion for themselves and for others who are struggling.

I have yet to see a well-written, comprehensible K-12 math textbook. When it comes to assisting my daughters with their math homework, I turn to the internet to relearn math skills that have atrophied in the, ahem, many decades since I last studied them.

Fortunately, Autistic Youngest appears to have an amazingly skilful math teacher this year for Advanced Functions. I've only had to sit down with her twice and assist in the problem-solving process. Her teacher has armed her with helpful skills and guides that supplement that still-subpar textbook.
Your story makes me so sad. :( I spend a LOT of time with my grad students getting rid of their math phobia and giving them back the skills they should have learned K-12.

Lots of great comments above on the math so I won't add... though I do have a favorite geometry textbook-- Geometry for Challenge and Enjoyment.
(Though I worry that link might be one that's violating copyright, as it appears to be a free download of a pdf of the book.) I learned from it, taught my sister out of it, and bought a copy (on an amazon sale) so I'm ready when DS hits Geometry, just in case it goes out of print. Sometimes in faculty meetings I'll prove that two triangles are congruent just for fun. And it wasn't just me-- everybody in my class from Brilliant Freshman to Struggling Senior loved the class. Most likely your son will not have proof-based geometry though which will also be sad, because so much of the logic is similar to what's needed for computer programming.
That looks like a link that breaks copyright. Probably not a thing for academics to encourage…
Speaking of proofs in Geometry, does anybody even know about Wff 'N Proof any more?

Mary (Anonymous@6:07AM) reminds me that the non-intrinsic meaning of exponents gets even more interesting when you move from integer values to fractional values (which is about the time that many student's minds explode) to any real number, and it doesn't stop there. You know you are dealing with mathematics rather than computation when you figure out how to have an imaginary number as an exponent.

Tell TB to google "Euler's Identity". When he gets to logarithms in a few years, he will probably have a calculator that knows that ln(-1) = i*pi if it is put in the right mode, and a teacher who doesn't even know that this can be done. It would be totally mystical if not for the fact that it can be used for computation in electrical engineering and various parts of physics. Ditto for group theory and code breaking (and physics).
The latest comment by CCPhysicist reminds me of a time in high school that DD may recall (perhaps with more phobia). A friend who is now an economist pointed out that his favorite equation is e^(i*pi) + 1 = 0. It combines 5 important number from math into a single true equation.

It turns out that I have ultimately used many of the concepts that arose in math, including geometry. It probably helped that my PhD advisor loved geometry himself. DD, I can provide TB (or the blog) specific examples when the need arises.

It's a relevant captcha code today: 1 connexN
Good points above about the links to the book.

The publisher apparently doesn't want the book to be found on their own site. (ISBN search leads to no results.) It does seem available via Amazon:
Geometry for Enjoyment & Challenge
Saying one is tired of the "privilege" cliche is like saying one is tired of the "gravity" cliche.
Oh just go get lost in a new city on purpose and drive your way out of it without GPS. Your spatial reasoning skills will thank you. And let your son and daughter roam 40 miles or so from home- spatial reasoning is tied to roaming distance, and that's the distance my father went at 12 years old. Totally reasonable, right?
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I would just add that some of us got by without parental help. My mom, a former secretary, was very helpful in teaching me to type and edit my assignments. But she never went to college, I don't think she even took geometry, and wasn't able to assist in most of my classes in high school. My father is an engineer and very bright, but saw homework as our domain. In fact, I'm pretty sure he had no idea what classes I even took.

That said, I survived, and now my husband and I (both faculty at our state university) can help our kids with their homework, although my husband (MBA, PhD who teaches data analytics and advanced stats) doesn't get our 4th grader's fraction homework. Everything comes full circle!
Interesting thing about gravity, it's never going to change.
If a parent with a mathematics background does not "get" their kid's 4th grade fraction homework, look into the "Singapore" math books. I have not seen them, but they are widely used by homeschoolers and parents who gave up trying to fight the latest fad that doesn't teach basic math.

Just tell the kid that these are secret methods that should only be used to check their answer or on standardized tests.
The strongest symptom of privilege is present in anyone who uses the word "privilege" as a pejorative. That only undermines the hard work of developing a middle class.
As I lay here on bedrest waiting for the arrival of my first child, I'm so very thankful that between my husband and I we've got most of the educational areas covered. He's got the science and math. I've got the history and english (and basically anything with words in it).

I've got lots of friends who are teachers and as bad as I was at math in school, I could never pass the way they teach it now. I actually enjoyed geometry and proofs. That and stats were the only parts of math that I actually understood.

But my husband doesn't get why I know how to diagram a sentence, quote the prologue from canterbury tales in middle english from memory or analyze poetry. Or my excessive love for all 4 history channels we get on the TV.

So I think we've got all our bases covered unless we end up with a super athlete.

My point is that it's okay to hate math, be bad at math and generally pick fields were you don't have to do it if you don't like it.
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