<p>I am NOT a math whiz, have never been, would never be.</p>
<p>You’d might be surprised at how many mathematicians don’t do well at basic arithmetic. Its not a ton, but, for folks with PhD’s in the subject…</p>
<p>I agree, typically, there are a lot of mathematicians in academia that usually make tons of simple errors (not adding properly, forgetting a constant, etc) all the time. Calculation ability isn’t necessarily correlated with mathematical ability.</p>
<p>A lot of students who excelled at arithmetics struggle with algebra.</p>
<p>For the vast majority of people, statistics and logic would be much more helpful than calculus and trig.</p>
<p>My theory is that if someone TRULY understands arithmetics, he should easily learn algebra. If he TRULY understands algebra, he should easily learn pre-calc and introductory calculus.</p>
<p>Geometry is a a different matter.</p>
<p>bomgeedad:</p>
<p>A central argument of Liping Ma is that teaching arithmetics is harder than teaching algebra. The reason is that students and teachers need to TRULY understand, rather than just memorize formulas.</p>
<p>A personal experience with my daughter supports the above posts. She always felt that she was stupid in math–struggled with algebra and precalc. Though geometry(in HS) and statistics in college came easy to her–must be a different brain function?</p>
<p>marite, could you elaborate on this central argument of Liping Ma? I don’t understand what is supposed to be different about algebra so that it can be taught as mindless formalism but arithmetic can’t.</p>
<p>Liping Ma’s argument is not that algebra can be taught as mindless formalism. I didn’t say that, did I? If you think this is what I was suggesting, it is because you think that arithmetics is being taught as mindless formalism. And that is precisely Liping Ma’s argument: it should not be.
Remember that a key factor for undersanding algebra is understanding fractions.
Well, Liping Ma supplies several examples of American teachers not understanding how to teach fractions–perhaps they don’t really understand fractions to begin with.
She feels that many teachers blow off basic math; they teach “math facts” but not what lies behind these “math facts.” Teaching iarithmetics well requires what she calls Profound Understanding of Fundamental Mathematics–which many lack. If teachers do achieve that PUFM, they will find teaching algebra easier and students will learn it more easily. Again, not through the application of cookbook formulas but through really learning what is involved. Her work has won the endorsement of advocates on both sides of the math wars.
Here is a link to one review of the book:</p>
<p><a href=“http://www.ams.org/notices/199908/rev-howe.pdf[/url]”>http://www.ams.org/notices/199908/rev-howe.pdf</a></p>
<p>Here’s another review of Ma’s book: </p>
<p><a href=“http://www.aft.org/pubs-reports/american_educator/fall99/amed1.pdf[/url]”>http://www.aft.org/pubs-reports/american_educator/fall99/amed1.pdf</a></p>
<p>Thanks, Tokenadult. I was looking for that review but could not locate it.</p>
<p>Roger Howe and Richard Askey are on either side of the math wars and both praise Liping Ma.</p>
<p>The link to Richard Askey’s review of the Ma book shows the review with a two page piece about understanding why multiplying two negative numbers results in a positive number. When my kids were in elementary school and had trouble with this concept I found a slim little book at the library called “Realm of Algebra” by Isaac Asimov that explained the concept quite nicely and simply by emphasizing that if you think of the negative sign as meaning “the opposite of” all the mystery goes away.</p>
<p>Sadly, the Asimov book has been out of print for some time, although I see you can get used copies on amazon.com in the 50 to 70 dollar range. (yes, $60 for a 150 page paperback!)</p>
<p>For one of my kids I used the analogy of jumping backwards on the number line to explain the concept of multiplying by a negative number. i.e. if the first number is negative you face the negative side of the number line. Multiply by a positive number you jump the direction you are facing and end up with a negative number. Multiply by a negative number and you jump backwards.</p>
<p>I am familiar with (several reviews of) Liping Ma’s book and some of her online articles, but have not read the actual book. What I have not seen or noticed in her work is a claim that there is some significant difference between algebra and arithmetic that makes the latter harder to teach.</p>
<p>I may have been carried away in my comparison of arithmetics and algebra, using a quote I read somewhere–and which I cannot trace back. In the meantime, however, I found an interesting article. The use of Mathland and Math Trailblazers may be intentional. But both texts have been much criticized by math traditionalists.</p>
<p><a href=“http://www.math.uic.edu/~jbaldwin/pub/kessel1.html[/url]”>http://www.math.uic.edu/~jbaldwin/pub/kessel1.html</a></p>
<p>I think true understanding of arithmetics would make learning algebra easier. My reasoning is that the positional numeral system can be thought of as a special case of a polynomial where x=10. What you learnt in long addition can be applied to polynomial additions. In a sense it is even easier because there is no carry. Same for long subtraction, multiplication and division for the counter part in polynomial. This is especially clear if you just use the coefficients in your polynomial add/subtract/multiply/divide.</p>
<p>Factoring is more difficult in algebra than arithmetics, but once you master it, you can now use your knowledge of fraction operations in the rational expressions operations.</p>
<p>Therefore it is very important to have a solid foundation in arithmetics.</p>
<p>Not quite from the horse’s mouth:</p>
<p>
</p>
<p><a href=“http://www.maa.org/editorial/knot/AlgebraCritique.html[/url]”>http://www.maa.org/editorial/knot/AlgebraCritique.html</a></p>
<p>“For the vast majority of people, statistics and logic would be much more helpful than calculus and trig.”</p>
<p>I agree. I think that statistics is unfortunately trivialized, when it really has much more application to modern life than people realize (economics, demographics, psychology & psychometrics, urban planning, environmental planning, medicine, much more). As to logic, there is a serious lack in modern education. The fact that many so-called math “whizzes” have not been trained in this discipline is more than evident on CC. When I was in high school during the Pleistocene Era, we all had to take a separate course called Logic. Of course this was a traditional Jesuit high school, so it shouldn’t have been that surprising. I have found it extremely helpful & applicable to many areas of my life.</p>
<p>For the vast majority of people, statistics and logic would be much more helpful than calculus and trig.</p>
<p>Maybe…there’s actually a good bit fo calc/measure theory involved in stats.</p>
<p>re: marite,</p>
<p>It is obvious that thoroughly mastering earlier material (arithmetic) makes it easier to learn later material (algebra). Also obvious that both subjects are taught 10 times better in China than the US. The actual question we were discussing was whether Dr Ma or anybody else claims that algebra is intrinsically easier to teach than arithmetic, which would be a surprising and non-obvious statement.</p>