
(One Of Many Reasons) Why Students Hate Algebra - gaika
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pmiller2
I taught high school level algebra for two years, so I have a moderate
familiarity with this subject.

I, too, dislike these types of problems, and I refer to them as "math book"
problems, because that's the only place you'll ever see them. I mean, if Luigi
can paint 1/2 of a room in 3 hours and Mario can paint 1/4 of a room in 2
hours, the way you find out in the real world how fast they can paint the room
together is to throw them in the freaking room with two sets of painting
equipment and let them go to it. If it's not going fast enough for you, then
you go find Mario's friend Bowser and hire him to help them.

But, I don't think lack of realism is the worst thing about these types of
problems. I think lack of fun is. I have a graduate degree in mathematics, and
I frequently work on problems that have little connection to anything (so far
as I know) in the real world, just because it's fun for me to do so. To me,
the real problem with high school algebra textbooks is they're so goddamn
boring -- and I'm saying this as someone who really, really _likes_ math.

If I had my way, there wouldn't be such a thing as "algebra 1", "algebra 2,"
"trigonometry," etc. as high school math courses. I'd teach "Year 1 math,"
"year 2 math," and so on, and put the focus on problem solving rather than any
one particular corner of math. If the problems are interesting, the students
will do the work and love it. I've seen it happen. I can't imagine that such a
problem-solving based course would prepare students for either college or the
real world any worse than what we're currently doing.

~~~
tokenadult
_I don't think lack of realism is the worst thing about these types of
problems. I think lack of fun is._

I can recommend an algebra textbook with fun. And it was written by a
mathematician with rather better credentials (theorems that bear his name in
higher mathematics) than most algebra textbook authors. The book is Algebra by
Israel Gelfand and Alexander Shen

<http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773>

I discovered the Gelfand-Shen textbook through Professor Richard Askey's
review of that book.

[http://www.aft.org/publications/american_educator/fall99/ame...](http://www.aft.org/publications/american_educator/fall99/amed1.pdf)

Askey's review is actually a review of Ma Liping's book Knowing and Teaching
Elementary Mathematics that quotes a passage of Gelfand's book in a sidebar. I
later saw a glowing description of Gelfand's books in the same series in a
bibliography by a U of Chicago mathematics student.

<http://www.ocf.berkeley.edu/~abhishek/chicmath.htm>

I use the Gelfand textbook to teach supplemental math lessons for gifted
elementary-age students. They LOVE the Gelfand problems. They haven't even
gotten to the really funny parts of the book yet, which such sections as "How
to Confuse Students on an Exam." Gelfand's book(s) exemplify what you are
looking for in math textbooks. Of course they are not used in very many public
schools in the United States, nor in very many remedial college classes.

~~~
pmiller2
Thank you so much for introducing me to this textbook. I've only read a little
sample (what Amazon will let me see when I "look inside"), but I can see it's
an improvement over at least 95% of the books I've seen.

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lmkg
I am reminded of a talk I attended in college about (elementary-level) math
education. One of the first intro problems to estimation was something like
27+75, and the presenter showed us a (real) dialogue where the student gave
the exact answer, and the teacher had to "correct" them to the estimate, and
then proceeded to talk about how this wasn't exactly correct. The teacher's
guiding the student through the "proper" reasoning was convoluted, and not
effective teaching. The presenter concluded that if you teach estimation, you
should use an example like 209385324579+394875293745, where estimation is
actually faster and easier than exact arithmetic.

Now, what real-life example demands systems of equations? I suspect none,
until you get into industrial applications. Therefore, maybe you should ask
what the kid wants to be when he grows up, and create an example in that
setting.

~~~
lliiffee
> Now, what real-life example demands systems of equations?

Finding the cheapest phone card to use is a good example.

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bitwize
As soon as I read that problem my brain went on instinct and quickly
calculated you need 3 vans and 2 cars. And somehow I'd skipped over the part
that implies you need the number of vehicles to total five.

But I guess I'm just wired like that. And one thing I'm getting out of this
how-do-we-teach-math kerfuffle is, people who are "just wired like that" think
differently from the people who most need to learn math. Paul Lockhart would
advocate dropping the pretense of real-world relevance altogether: "Suppose I
were thinking of groups of five and seven things (as I very often am). How
would I divide 31 things into groups of five and seven with none left over?
It's the _beauty_ of solutions to problems like this, which exist purely of
and in the imagination, that we are denying today's children!"

~~~
pmiller2
But, 3 vans and 2 cars isn't the only valid solution, unless you assume every
vehicle has to be full. Four vans and 1 car works, and a previous poster gave
the example of 5 vans, which also works. The fact that there are multiple
valid solutions when you remove the constraint that each vehicle should be as
full as possible makes me think this is a lousy example for illustrating
systems of equations. It is, however, a great problem for teaching
combinatorics, since, for each solution, you can count the number of ways to
fill the vehicles, and thus the total number of distinct ways to transport the
31 people.

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astine
Might this be titled as: "Why Students Hate _Word Problems_?"

The OP's argument is that common word problems in high-school textbooks are
very contrived and that it problems made to be less contrived and more like
what a student might expect to experience they might have more success in
keeping their interest.

I'm not so sure of this conclusion. Making examples that aren't contrived but
still illustrative and simple enough to retain their didactic value is _very_
difficult, especially if you are attempting to make it relatable to teenagers.

~~~
ekiru
However, making contrived examples, even if illustrative, does make students
think math is pointless and detestable. Many high school students, in my
experience, really do develop a belief that they dislike math because they are
led to believe that math is only good for things as boring, useless, and
contrived as solving a linear system to figure out how many cars and vans to
bring.

In this case there are even many other simpler and faster solutions compared
to the proposed one. The obvious solution if the problem actually occurred in
reality, barring other unmentioned limitations, is to take five vans. In many
math courses, if a student given that problem during the lesson on linear
systems were to suggest taking five vans, the teacher would mark the answer
wrong. Seeking the simplest solution involving the fewest and simplest
steps(and thus fewest opportunities to make a mistake) is a virtue in
mathematics. Is not a shorter and simpler proof of the same theorem usually
the most well-respected amongst mathematicians? Likewise should our
mathematics education teach us to seek the shortest and simplest route to a
correct solution.

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nickpleis
I hated math. Absolutely loathed it. I had a terrible algebra teacher (8th
grade) that left me a gap I never truly closed until college.

However, the 'lights' finally came on in my senior year of high school. I took
an experimental trigonometry course, instead of calculus. This course focused
on the application of trigonometry to real life problems.

It was by far the best class I've ever had, and it opened me up to
understanding the application of ALL of the math that I had been taking for 12
years. It was truly seminal to me. That I had gone 11 years without ever being
taught how to actually apply those principles is, in retrospect, mind
boggling.

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ElliotH
I'm still in education, doing my A levels - and I experience questions of a
fixed form all the time.

Now this seems quite good in principle. In lessons you learn about the same
ladder against the same wall or whatever, with very slightly changed values
each time. Then you do the past papers - and it has the same problems - so you
end up with a nice qualification at the end.

The problem with this is what I experienced today. The examiners changed the
exam - not massively - just enough so that the standard solutions people had
learned became worthless.

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teeja
This is actually a useful word problem, they only left out one part that would
make it more practical: cars cost $50 to rent, vans cost $100 to rent. Now
you've got a reason to prefer cars over vans.

I (and in my experience most students feel the same) always liked learning
better when it was about concrete problems and not just an abstract operation.
If I'd had to learn programming without real-world applications, I wouldn't
have bothered.

I explain this by pointing out that physics has theoreticians and
experimentalists. Seems to me that's about mental style. Also, it's easier to
communicate an idea when you've got a physical metaphor.

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ams6110
It's funny, algebra is when I first started liking math. I hated arithmetic
and got poor grades because for some reason I could not memorize addition and
multiplication facts very well so I was slow on the tests.

With algebra I started to see that I could actually do something useful with
math. It became more about thinking than memorization. Yeah a lot of the
problems were contrived but at least it was more interesting than long
division.

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jrockway
I always liked math for the sake of math. Math that has "real-world"
applications is boring.

(Of course, Calculus, number theory, and differential equations have lots of
real world applications, but the "real world" that these apply to is not
"daily life", and is interesting as a result.)

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lucifer
Why on earth would we want to "waste" the time of teachers and students on
such un-"realisitic" problems? Its not like they would have to solve the usary
equations in the fine print in a bank contract in the future ... and if we
ever need people who have encountered systems of two (yikes!) equations, I'm
sure we can outsource it to India and China and elsewhere.

