Math meltdown

Summer is the time for desperate students to try to catch up in math, writes Patrick Welsh, a teacher and USA Today columnist. He’s seen the gap between standards and performance at T.C. Williams High School in Alexandria, Virginia, where he teaches English. Students are pushed to tackle high-level classes when they haven’t mastered the fundamentals.

The result of these grand plans, says T.C. math teacher Gary Thomas, is that “we are ending up with kids in upper-level math courses who do not know how to add, subtract or divide unless they use a calculator and who are lost when it comes to fractions.” As a consequence, when he gets kids in algebra II and trigonometry, says Thomas, he is constantly having to backtrack to teach skills students should have learned years ago.

One reason for the teacher frustration is that the state’s math gurus have de-emphasized memorization in favor of “conceptual thinking.” The same philosophy has crept into English classes, where “creativity” has been elevated over knowledge of grammar, and into history classes, where knowing historical trends — “the big picture” — has replaced knowing dates of important events. The result is seniors who are not just incapable of multiplication, but also unable to identify the verb in a sentence or come within 100 years of placing the Civil War.

I’ve never believed that people are better able to understand concepts if they know no facts. Perhaps the ignorant are more creative — but not in a good way.

About Joanne

Comments

  1. It is impossible to think about history if you don’t know the facts of history. Knowing the “big picture” without relevant facts is like growing a tree without leaves; knowing facts alone is like growing a tree tree with leaves alone. The former is lifeless, and the latter is meaningless. History requires both.

  2. According to Lisa VanDamme, learning ‘how to think’ without knowing what to think about is a huge mistake that divorces thought from reality.

    One can learn the facts and learn thinking skills just by learning how those facts were discovered! Just think of it–you can learn about electrons and learn the great thought process that lead to their discovery. Facts and process all at once. There doesn’t need to be this ‘either/or’ idiocy.

  3. Margo/Mom says:

    It would seem to me that we are mixing standards and curriculum. Standards rightly focus on the “big picture,” while curriculum takes this down to the level of specific fact. As an example, the state standard might specify that students understand some key elements of “the novel,” plot, characters, etc–and may add something about literature from different eras. The curriculum, developed at the local level, would be much more prescriptive about exactly which novels, at what time. We should not aspire to state standards that prescribe reading lists.

    I’m not certain that I understand the value of being able to do calculations without a calculator. I personally balked at memorization (early introduction of new math), but my understanding of concepts has been of enormous assistance in getting right answers (buying a dozen items at $1.99 is the same as 12 X $2.00 less .12; I have always had a mental block about 7 X 8, but I do know that 7 squared is 49, plus 7 is 56–because adding a number to nine always results in a first digit that is one less). But the point is that if the details (in any content area) are necessary to reach the big picture specified in the standard, then schools should be building curriculum that teaches this.

    Instead, I see too much dim understanding and helping students to eliminate the two least likely answers and working on problems “in the same format” as that which is likely to be on the test. These short-cuts don’t contribute to education–and the long run they don’t contribute much to being able to do well on tests.

    What might help, is for the gurus at the high school levels to be coordinating with the gurus at the elementary level so that everyone understands how their piece contributes to the “big picture.”

  4. Independent George says:

    Margo/Mom – the benefit comes when you have to manipulate an equation to find its roots (or, in later years, to manipulate an equation into a form that allows you to manipulate another equation). There are no technological shortcuts to that. The calculator is a great tool for people who don’t need calculators; for others, it’s a crutch that inhibits them from ever understanding the ‘big picture’. A student that hasn’t mastered the fundamentals won’t even know how to rearrange an equation into a form that lets him use the calculator.

  5. It’s greatly helpful to have a good feel for numbers and be able to do mental arithmetic. I once worked with a CEO who had a beautiful grasp of numbers. You could tell him “BigCo is looking to buy 10,000 widgets, but only if we can give them a price of $42 per unit.”

    He would run the deal in his head: “Well, let’s see, the cost of goods on those is $36, but we get a 10% discount on the subassembly if we order more than $5,000, so that knocks it down to $32. Shipping will be about $1000, so, that’s what… $330,000 dollars, and that’s gross margins of … hmmm… not quite 20%, so, yeah, that’s a good deal. Take it downstairs to accounting and have them run the numbers for real, but tell the customer we can probably do it.”

    All of this would happen far faster than with a calculator or spreadsheet and he was almost always within 10%. It was a huge advantage to him when negotiating, he always knew things (because he could calculate them on the fly) that the other guy did not.

    I’ve also worked with programmers who could do hex arithmetic in their heads; enabling them to spot certain kinds of errors more easily.

    A person who is familiar and easy with numbers will use numbers to their advantage where a person not familiar and easy with numbers won’t even bother with a calculator.

    Do you see lots of people with calculators in the grocery store? No, but there are some of us there who are calculating anyway…

  6. Basal Readers in the 1970’s were considered a bad idea.

    Whole Language in the 1980’s was considered a bad idea.

    Skill memorization has never produced a strong reader, writer or math mathematician.

    Strong thinking skills and concept development is essential but also not enough.

    My point, there is no one way to do anything in schools. In ELA we need technical skill as well as creative skills. In mathematics we need to know facts and have conceptual understanding. My point again, there is no one way.

    We should not do what we always do in education; throw the baby out with the bath water. To ignore the importance of big picture, creative, and conceptual understanding would be a move in the wrong direction again. The best teachers in the best schools blend concepts and fact memorization. They blend creative thinking with technical knowledge. They teach big picture and important events. It is all important.

  7. Regarding “how to think” and “what to think about,” see my post thinking and memorizing.

  8. Margo/Mom says:

    “A student that hasn’t mastered the fundamentals won’t even know how to rearrange an equation into a form that lets him use the calculator.”

    Neither will a student who “learned” mathematics by memorizing the addition, subtraction, multiplication and division facts.

  9. Barry Garelick says:

    Neither will a student who “learned” mathematics by memorizing the addition, subtraction, multiplication and division facts.

    Traditional math isn’t taught solely by memorization. Memorization is something that is necessary in order to implement concepts that are taught and learned. I didn’t learn math soley by memorizing the facts, but such memorization was surely a large part of it. And the fact that you can get to 8 x 7 by knowing another fact is testimony to that.

    Independent George’s said it very well. There are many things that you must know to automaticity.

  10. Margo/Mom says:

    “And the fact that you can get to 8 x 7 by knowing another fact is testimony to that.”

    But only because such an emphasis was put on understanding what multiplication is, what it means to multiply 8 X 7. I remember at the same time I was learning these concepts, I friend from another district told me quite confidently that multiplication went up to 9 X 9. When I expressed surprise that she thought this, she rethought and admitted it went up to 12 X 12, because she had seen this in her older brother’s text book.

  11. >> I’m not certain that I understand the value of being
    >> able to do calculations without a calculator.

    facts and a feel for numbers are critical for assessing if one hit the correct calculator button, much less being able to troubleshoot work in steps. I hate hearing someone quoting a answer from a calculator that is clearly wrong

  12. dcowart

    “Skill memorization has never produced a strong reader, writer or math mathematician. ”

    Carl Friedrich Gauss is considered by many to be the greatest mathematician who ever lived. According to 1 biography I read he had memorized the table of common logarithms. He enjoyed mental calculations. He calculated the orbit of the minor planet Ceres.

  13. I got it wrong. Gauss did not memorize logarthms. He calculated them in his head as he needed them to compute the orbit of Ceres.

  14. Independent George says:

    In mathematics we need to know facts and have conceptual understanding. My point again, there is no one way.

    Nobody disputes this. The point I’m making is conceptual understanding doesn’t happen in isolation; you need a solid grounding in the basics before you can start filling in the connections.

    For example, there are dozens of known proofs of the pythagorean theorem. All of the proofs are grounded in more fundamental ideas that have to be mastered in order to understand the larger proof. Most of them require a minimum prerequisite knowledge of geometry – for example, the definition & properties of line segements, complimentary/supplementary angles, congruence/reflections/rotations, properties of polygons, etc. Each of these, in turn, requires a base level of knowledge of its own to be understood.

    That’s the nature – and, in my opinion, the beauty – of math; each lesson necessarily builds upon previous lessons, in new, unexpected, and often unintuitive ways. This also means that deficiencies in earlier lessons necessarily precludes the ability to learn later lessons.

    Things like the order of operations, the properties of real numbers, the axioms of arithmetic, and, yes, the multiplication table are considered fundamental because they continue to be used at all levels of math, and need to be learned to the point of automaticity.

  15. I think math should be taught (as with many things) by a combination of application and practice. How can we calculate the area of this field? OK, now that’s we’ve found it was 3.5 acres, what skills did we use to reach that result? OK, now we’re going to practice those skills so that they become automatic.

    The application answers the tired old “why do I have to learn all this?” question (tired, but fair). The practice hones the skill until it’s automatic, so that when you move on to the next level of difficulty, you’re only adding the new stuff, not still sweating the old stuff.

    If you’re teaching math (at least up through algebra) and can’t come up with a plausible application of any given skill, then you might ought to think about teaching something else.

  16. John Dewey says:

    But only because such an emphasis was put on understanding what multiplication is, what it means to multiply 8 X 7.

    Yes, and the fact that someone taught it poorly so that your friend thought multiplication goes up to 9 x 9 does not imply that memorizing math facts is unnecessary. Writers of textbooks in the 40’s, 50’s and 60’s were saying exactly what you were, but still expected students to memorize math facts. That some teachers taught math poorly is not the author’s fault nor the fault of those who claim that students need to master such facts.

  17. Miller Smith says:

    One must memorize that the symbol 2 means two of anything. If you don’t memorize that simple little fact you can not move one iota into anything more difficult.

  18. Mrs. Davis says:

    Sorry I was busy. Thanks for covering for me.

  19. Parent2 says:

    Mastery of the basic vocabulary of an area of study is necessary, but not sufficient, for further study. Thus, students must learn basic arithmetic in order to grasp fully more advanced mathematical concepts, and they must learn grammar in order to parse sentences, paragraphs, and compositions. Well educated parents seem to make an effort to remedy such lacks in their children’s education. The pity is that other children are left to the mercies of school administrations.

    The present credit crisis gripping our country illustrates a lack of mathematical understanding on the part of much of the populace. I read pundits blaming debtors for greed. Perhaps they’re right, but having read about the state of American education, I think it is at least debatable that many debtors had no idea how quickly an increase in an interest rate would add up. I think the NAEP math scores agree with this conjecture.

  20. I’m still amazed–although given the last 30 years in American schools, I should not be–when anyone argues against learning facts in any subject, especially mathematics. Give me a college student who can do what what the CEO in Rob’s post can do with numbers and I am a very happy professor, parent, employer, etc. Personally, I can do in my head exactly what Rob’s CEO can do with numbers. I haven’t had to use algebra, geometry, or trigonometry since I graduated from high school in 1969. Yet, I publish research papers with complex statistical analyses. Yes, yes, I know that complex math such as algebra is used in these analyses. But, the primary information that I need to know is the numbers. As long as I understand what the numbers mean, I don’t have to use or understand complex mathematics.

  21. The most effective math programs are those that take a balanced approach, teaching both the underlying concepts and also the traditional algorithms. The traditional math I grew up on was great at teaching me how to calculate the correct answer, but I never really understood why the formulas worked. The fuzzy new math programs such as the infamous “Everyday Mathematics” one that don’t ever teach kids the efficient algorithms are a disaster. What’s needed is something in between these two extremes.

    There are programs out there like that such as Singapore and the one that we use in my family’s homeschool called Right Start- I only wish that more government-run schools adopted them.

  22. I’m very irritated at the idea that adults feel free to just “opt out” of math. I’ve been meaning to write about it for a long time now, so I finally did:

    http://roborant.info/main.do?entry=1409

  23. I happened across this article when it was published and read it to the group of high school math teachers at a workshop I was presenting. We all agreed that the article was right on the mark! We as math teachers struggle to teacher kids upper level mathematics because they haven’t mastered their basic facts! They are so tied to their calculator that they cannot do simple calculations without it.

    And to those of you who don’t see why people need to know basic facts – have you ever been to a store or fast food restaurant and had the person behind the counter unable to count change because the register isn’t working? Why is it socially acceptable to be math illiterate but not reading illiterate?

    You may not have used you high school algebra or geometry since high school, but you have used the thinking skills you acquired in those high school math class. In high school, most kids do not know what career they will go into, but whatever career it is, they will have to know HOW TO THINK LOGICALLY. Math is the universal language that all kids can speak to learn logical thinking and reasoning. When my students ask when they will use a specific skill outside of my math class, this is what I tell them. They are doing brain calisthenics in my class, stretching their “logic muscles.”

    Think of how hard it would be to read the newspaper if you didn’t know the alphabet, or even a few letters. Kids who don’t know their math facts are at the same disadvantage in their high school math class.

  24. Think of how hard it would be to read the newspaper if you didn’t know the alphabet, or even a few letters. Kids who don’t know their math facts are at the same disadvantage in their high school math class.

    I think that the closer analogy is “how hard it would be to read the newspaper if you had to sound out every word, every time you encounter it.” Kids who just figured out 8+5 via calculator turn around and do the same thing for 5+8 two calculations later.

    One of my kids had a requirement to provide a calculator, any calculator, for 5th grade math. When I asked the teacher if they had any discussion about order of operations on cheap dollar-store calculators versus scientific calculators, the teacher wasn’t aware that something like 3+4*5 had more than one answer. The kids are trained to trust the calculators without having a clue because the math apple doesn’t fall far from the teacher tree unless parents are very, very vigilant.

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