Why math tutors prosper

Many elementary students never learn basic math facts,  writes Lynne Diligent on Dilemmas of an ExPat Tutor.  They end up in remedial math classes in college. She advocates drill on math facts, more homework and no calculators till 11th grade.

I no longer teach Grade 3; I am now a private tutor. Unfortunately, I am now running across a number of 14-year-olds who are using calculators to add 5 + 3, or 7 + 6, or 9 + 2.

 Diligent also calls for requiring students to learn concepts before moving on, instead of  “spiraling” through the same things year after year.   

And she believes teachers should “instruct and explain, and then follow up with practice to master the skills,” rather than putting students in groups and telling them to figure out problems on their own. But group work is great for math tutors, she writes.

 

About Joanne

Comments

  1. I would say trend 3 mentioned in the article is going on in elementary schools to some extent now if they have a constructivist curriculum like Investigations/TERC.

  2. Thanks for linking to my article, Joanne.

  3. “One or two examples of a certain type of problem are just not sufficient.”

    This is so variable from child to child. 30 problems is not always better than 3. In fact, 3 really challenging word problems would much better for a bright student than 30 easy-peasy equations.

    The problem is that we’re wedded to a “one size fits all” instructional approach that bores bright students with “busywork” while at the same time may not provide enough practice for slower students.

    We’ve got the technology available to individualize math practice. Why don’t more schools adopt it?

  4. “I then spent 30 minutes of my teaching time DAILY, going over these homework problems.

    Causing bright students who got it the first time to become resentful of the slower kids in the class.

    As an adult, I realize these feelings weren’t fair because it wasn’t my classmates’ fault that I was placed into a class that had a too-slow pace and not challenging enough material. But that’s how I felt at the time.

  5. On the other hand, CW, if the students are all matched for ability and readiness, you do still need to go over homework in order to get the full benefit of it.

  6. SuperSub says:

    I guess you could say I was a bright student, and looking back on my education, while I disliked the 30 minutes of HW review and was bored out of my mind, it did serve two purposes.
    One, although I grasped the material after only a few problems, doing (and going over) 10-20 of them taught me one big lesson – Math is 80% process, and 20% focus. Without it I would likely be a lot more careless with my calculations nowadays.
    Two, everyone needs to learn to be bored. Really. Removing boredom simply furthers the aggravation that individuals feel when they have to be bored later in life. Heck, there were points during my wedding ceremony (a long Catholic mass) that I was bored and my mind wandered.

  7. SuperSub says:

    Agreed with the spiraling criticism, though, I’ve tutored enough victims of that scam philosophy.

  8. I’m with CW on the issue of the one-size-fits-all model in the public schools; it’s inherently unfair and discriminatory, not to mention ineffective and inefficient. (do public schools have any awareness of efficiency?) It’s one thing to spend 30 minutes going over homework that was appropriate for the child’s level and entirely another to spend 30 minutes going over homework that was inappropriately easy for some and inappropriately difficult for others, while assuming (hoping?) that it was appropriate for some. That’s the fundamental problem with heterogeneous grouping; it ignores the educational needs of real, individual kids.

  9. My son is a math tutor at his high school and he has had more than a few kids (not special ed) come in not knowing how to add single digits without a calculator.

  10. Roger Sweeny says:

    I’ve had lots of 9th graders who need a calculator to multiply a number by ten.

  11. On the other hand, CW, if the students are all matched for ability and readiness, you do still need to go over homework in order to get the full benefit of it.

    It is worth going over problems with a student that he/she has missed. Again, this is something better done by a computer since each child is likely to have missed different problems. And if the student got 100% correct, then there is little gained by reviewing the answers. If I have calculated the proper answer, what more is there to say about it?

  12. Lynne Diligent is right that many elementary students do not learn basic facts and I believes she’s right that learning basic arithmetic is greatly improved with flash card support at home.

    My guess is that assigning 25 problem to do for homework is better than assigning only 12.

    As for her criticism of spiraling, I’m not so sure. I’ve seen alternatives that backfired.

    Postman derided “the measles theory” of education. If you had it once, you can’t have it again.

    I only taught math for a few years and but I’d have to say that the most common, serious obstacle to learning was student laziness. Those who fell behind weren’t suffering from math anxiety or had a learning style I wasn’t accommodating. The problem was they didn’t do their homework. Because it was too hard? No. Because they didn’t feel like it.

    As for saying the use of calculators impedes the memorization of math facts, it’s just no so. That might be somewhat counter-intuitive because you can imagine that a student never memorized the fact that 3 add 5 equal 8 because a calculator was always there to remember for him. But in the real word, it’s just not so. That’s what research will tell you and over 90% of any full time middle school math teacher.

    There are a number of obstacles to memorizing math facts, but the use of calculators isn’t one of them.

    That has been so widely proven to be the case that when I encounter an anti-calculator point of view, it’s like hearing somebody alarmed about public water fluoridation.

    Allow me to summarize a bit.

    Knowing basic math facts is a good thing.
    There are a number of reasons why some students are slow to mastering basic math facts.
    Parents can help their children memorize math fact and need to help their children learn them if/when they’ve fallen behind.
    I don not claim to be a math instruction expert.
    Calculators do not harm learning not do they delay the mastery of basic facts.
    Having an anti-calculator point of view isn’t supported by research or most experienced math teachers.
    I’m not sure what is the root of the anti-calculator point of view. Maybe it’s the fluoridation in our water?

  13. “My son knows children who can’t add single digits without a calculator.”

    I’m sure that’s true.

    But that doesn’t mean the use of a calculator caused the problem.

    It’s like saying crutches cause people to have broken legs.

  14. “Having an anti-calculator point of view isn’t supported by research.

    While correlation doesn’t equal causation, frequent calculator use in math class is negatively associated with test scores.

    “Nine-year-olds who reported that they used calculators in class every day had the lowest NAEP scores of any response category, while students using calculators only once or twice per month had the highest scores.

    A similar pattern is evident on the TIMSS. Frequent calculator use is negatively correlated with math achievement in several countries. A vast majority of students in the highest-scoring nations (Japan, Singapore, Korea) report that they never use calculators in math class.”

    Source is here.

  15. While correlation doesn’t equal causation,

    Then why mention it?

    Achievement in math is not linked to hard work or lack of calculators. It’s linked to cognitive ability.

  16. Arithmetic can be handled by special ed kids with low cognitive ability. My other son who is borderline IQ doesn’t need a calculator to add. He can also subract, multiply and do long division thanks to his mother.

  17. IQ sets a maximum limit on achievement, but effort matters too. Given the same level of effort, the smarter kid will outscore the dumber one on a math test every time. But a smart-but-lazy kid could wind up with a lower score than an average IQ kid who works his/her tail off studying for it.

    It’s like athletics- talent only takes you so far.

  18. But a smart-but-lazy kid could wind up with a lower score than an average IQ kid who works his/her tail off studying for it.

    First, what’s more likely true is that the genuinely average IQ kid could, with lots and lots of work, equal–not exceed–what the genuinely smart kid does with no work at all. But so what? Why should some kids have to work and work and work, while others learn it effortlessly? That alone indicates that kids aren’t being given the same challenges at school.

    But more importantly, who cares? You really think that average IQ kids not learning math is the problem being discussed here? Seriously? Why the absurd hypothetical?

    I’m talking about kidss with IQs of 85 being given algebra in 8th grade. You’re talking about a kid with a 125 IQ beating out a kid of 135–such is the woeful ignorance that the upper/middle class have about what actual average IQs are.

  19. I graduated in 1981, the first affordable scientific calculators were coming on the market in approximately 1979. I got my TI-55 and while it helped in subjects like Algebra II/Trig, it did not replace the concept of basic skills (Add, Subtract, Multiply, Divide, etc).

    My old Algebra I teacher told us once ‘you guys and gals have no problems doing algebra, you just can’t add, subtract, multiply, and divide.

  20. Crimson Wife, the source you cite is from an organization of parents who strongly favor “back to basics.”

    That’s a source, but it’s a source of opinion, not research.

    The research on the subject is overwhelming.

  21. Well, we keep trying to reform what works in education from 30-40-50 years ago into stuff which doesn’t work today in our classrooms…Look at the lattice method to try to multiply numbers (I saw a youtube on it)…kid spend 9 mins to try to multiply two numbers and doesn’t succeed.

    Using the old fashioned method, the kid solved it inside of a minute…go figure…

  22. Another thing which works really well…School House Rock…

    Naughty Number 9 was my favorite…

    Also liked the 4 legged zoo…

  23. I recommend reading Keith Devlin’s “The Math Instinct.” He makes an interesting argument, which he supports with research, that the math people learn in schools is virtually never used outside of schools, except by a very small percentage of people.

    First, he shares research which shows that people have a high rate of accuracy when solving mathematical problems using self-made strategies in such contexts as the supermarket, etc… He then points out that the longer people have been out of school, the MORE successful their strategies are. In other words, remembering the school math strategies for solving problems is a hindrance when trying to solve real life math problems. Further, he points out that only a tiny percentage of people are able to use the school math strategies (which are highly efficient in many ways) to solve problems.

    The problem, according to him, is one of an inability to transfer knowledge gained in one domain, and in one style of learning, to another domain in our lives. In other words, knowledge gained about algorithms in schools, regardless of the curriculum used, is too far removed from the actual applications of the math.

    So it seems to me that this means that it doesn’t matter if the kids memorize some algorithms as a kid, that this should not be the primary purpose of mathematics education.

    Another interesting point to make is that mathematicians, engineers, and other professionals who use mathematics are often not the best at arithmetic, but excel in problem solving and applying math they’ve learned to different contexts. They become a profession that uses mathematics because they are able to use it creatively.

    I’m really sick to death of everyone worrying about whether or not kids know how to multiply 6 by 4 and worried that no one seems to be concerned about whether or not kids know WHY we would want to multiply 6 by 4, and HOW this is useful in their actual lives.

    If you don’t think that we have a serious problem with numeracy (the ability to think mathematically and apply mathematics to different contexts), see this map of numeracy levels (including ALL adults aged 16 and older, so like all of you who learned mathematics the old way): http://www.ccl-cca.ca/cclflash/numeracy/map_canada_e.html

    It paints a pretty scary picture of the problems in numeracy across Canada, in one of the best education systems in the world.

    Maybe if we spent a lot more time doing engaging mathematics and applying what the kids learn in context, we might actually have a generation of people who USE mathematics, rather than a generation that complains about how awful math was when they grew up, and how horrible they were at it, but then asks their kids to do the same thing they did.

    Some people actually LIKE mathematics, believe it or not.

  24. Robert): “Having an anti-calculator point of view isn’t supported by research or most experienced math teachers.
    The research on the subject is overwhelming.
    Cite? What research?
    I’m pretty experienced and I oppose calculators before Calc II.

  25. Sean Mays says:

    FWIW: I wrote a position paper against calculators and computers for my math methods class for the ed masters. Much of the research supporting calculators was done by think tanks supported TI, with former TI execs in senior positions. Hardly what I’d call an arms length relationship.

    It’s illustrative, though hardly conclusive, that many nations beating the US on TIMMS and PISA use calculators far less.

  26. Well, I’d say prior to 1980 or so, it was pretty rare to find calculators that commonplace, let alone ones which could assist in doing higher math as well. I guess those of us who went to school before then just got forced to do it the hard way…

  27. Thank you for this article! I remember being so happy that my children’s school did not allow calculators…. now I see why.

  28. My kids have been exposed to the Investigations/Terc curriculum and Everyday Math.

    My own opinion regarding the use of calculators, is that if you are using some of the suggested alternative strategies, at a certain point with larger numbers these strategies become to cumbersome and at that point they switch to calculators.

  29. I’d like to thank everyone for so many thoughtful comments on Part 1 of this subject.

    Here is Lynne Diligent’s Part 2 discussing the current problems in math education:

    http://expattutor.wordpress.com/2011/07/16/why-so-many-elementary-students-arent-mastering-basic-math-facts-part-2-of-2/

    –Lynne Diligent

  30. I’d like to respond to the comments above that I taught in a private American school overseas where we only had one class in each grade, combining all levels of ability.

    When going over homework, I gave the answers to each problem (which students corrected themselves). Then for the six problems in each row, I asked by a show of hands how many students missed each particular problem and chose the two to work on the board which were missed by the greatest number of students. We did this for each row of problems. I felt this was a good middle strategy to prevent too much boredom, yet all students seemed to think was most fair for most of the class.

    In my class, I did succeed in getting all my children to do their homework. I used a system I observed in a Grade 3 class in Colorado. First thing was to check who did their homework. Those who did their homework (right or wrong) got a BIG “A+” on their paper (because having done it, they were ready to learn from their mistakes in the daily the class review) and those who did not do their homework, or who only did part of it got a BIG “F” on their paper, as well as a “Homework Alert” sent home to parents which had to be signed every night. I did not actually count the A’s and F’s in the grades for the report cards, but it sure did get everyone doing their homework, and everyone learning. (And yes, I did explain the system to parents at the beginning of the year–and yes, most were supportive.)

    I’d also like to clarify that I am not anti-technology, nor anti-calculator. Yes, there could be some reasons why a few students have trouble learning times tables, such as learning disabilities. However, by having us work on it daily IN THE CLASS for 2+ months, for about 20 minutes a day, and by asking for student accountablility (in front of peers) it gave students the message that IT IS IMPORTANT TO TRY.

    In my school, we were all required to take touch typing. I hated it at the time, learning on blank-key typewriters, yet that has been the number one skill I have used in every facet of my professional life since (and I also use times tables and estimating, measuring and conversions every day of my life as a mother and housewife, as a shopper, as a human being). My point is that schools need to put TIME in the math curriculum to PRACTICE these skills. Not all parents are able to pick up the burden of daily flash-card practice at home, even though it works. However, PRACTICE DOES NOT HAVE TO BE BORING! It can be fun! Any boring subject can be made into a fun game that students love by any dedicated parent or teacher, and this was the way I did it.

    I’d like to thank all the commenters above for having taken the time to read and and consider these issues, and to leave such thoughtful comments.

    –Lynne Diligent
    Dilemmas of an Expat Tutor
    expattutor.wordpress.com

  31. Vince Lynch says:

    Re: Calculators

    I teach in a one year, post-high school prep school. The following are some things I’ve noticed that I think are the result of allowing students to use calculators too early and two often.

    I see many students who can’t multiply without a calculator because they never learned the multiplication tables. If you can’t multiply, you can’t divide. If you can’t divide, you can’t factor. If you can’t factor, … I suspect if they didn’t have a calculator available, they would have been much more likely to have learned the multiplication tables and the curriculum would have supported it.

    Even more students can’t handle fractions. Because most calculators will perform operations with fractions, the students apparently never spent the time (or possibly had the curriculum time) required to learn how to find common denominators, etc. While calculators work fine when the denominators are numbers, they don’t work when the denominators contain variables. The result is that students can’t perform operations with rational expressions because they never really learned how to perform the operations with rational numbers..

    Many students seem to have little feel for numbers – 50 is the same as 5000. I know that somewhere along the way, they are exposed to estimating. However, if you can’t multiply, estimating doesn’t seem to take.

  32. j.d. salinger says:

    Crimson Wife, the source you cite is from an organization of parents who strongly favor “back to basics.”

    Robert, the source she cites is from a report by the Brown Center for American Education and it was based on data not opinion. What source were you looking at?

  33. Lynne, wanna entertain third graders, get Multiplication Rock (in the Schoolhouse Rock series), they taught an entire generation of kids math, history, government, english, etc…

  34. CW nailed the issue on the head. The problem isn’t that we need more X or less Y, it’s that education needs to be tailored to the individual. All of these discussions about method always seem to assume that the 1 teacher to N students model is necessary. Online education breaks that requirement and allows individualised lessons at the pace of each student, using whatever methods work best for them.

    And the discussion of the calculator is silly. Some students would be much more benefitted from using the time spent usually studying three or four digit addition instead investigating other, non numerically oriented math like logic, proof, etc. The algorithm for digit addition makes more sense with algebra anyways.

  35. Too many don’t even understand basic math well enough to use a calculator. In the past few years, I have encountered retail clerks/cashiers unable to: (1) calculate 5% sales tax on a $10 purchase, (2) calculate a 15% tip for a $40 service and (3) calculate correct change for $3.02 given for a $2.82 purchase – WITH CALCULATORS! They had no clue how to format/input the information they had in order to get the answer they needed. Even when I walked them through it, they obviously didn’t understand – and they were all in their 30s or older. I’m not any kind of math whiz, but I do those kinds of calculations in my head; obviously a foreign concept to the above people. I was also assured by deli clerks at two different stores that they were unable to weigh 1/3 or 2/3 pound on their digital scales. I was assured that they could only weigh in increments of 1/4, 1/2, 3/4 and whole pounds

  36. Ze'ev Wurman says:

    Regarding Cal’s argument that math achievement is not linked to hard work but to cognitive ability, this is no different for math than to any other subject. In other words, *everything* we do, excluding functions of the autonomous nervous system like breathing, is “linked to cognitive ability.” Such bland statements contribute nothing to the discussion.

    The more meaningful question is whether typical K-12 mathematics, as taught in typical classrooms across the land, requires particularly advanced cognitive ability. Is it beyond the grasp — with only a reasonable effort — of a large group of school kids? Fortunately, there is an answer to this question, and it clearly indicates that the overwhelming majority of children can reasonably easily learn what we teach in our K-12 schools, given competent teachers and effective teaching methods.

    But first, let’s consider the question itself. Does anyone believe that a broad and old system like education would willfully impose on itself goals than are inherently beyond a large fraction of its clients? Is the system suicidal? Doesn’t it know, if not through research then from experience, what can be reasonably expected to be learned by children? Do we teach in this country content that is clearly above and beyond what other countries teach? Anyone that believes that is simply out of touch with reality.

    That leads to the obvious answer. Yet there is also data supporting it. In the 1990s UCSMP (yes, that purveyor of the deeply flawed Everyday Mathematics) translated the excellent Japanese math curriculum edited by Kunihiko Kodaira. In is preface it tells the story:

    The Japanese school system consists of six-year primary school, a three-year lower secondary school, and a three year upper secondary school. The first nine grades are compulsory, and enrollment is now 99.99%. According to 1990 statistics, 95.1% of age-group children are enrolled in upper secondary school, and the dropout rate is 2.2%. [...]

    Japanese Grade 7 Mathematics (New Mathematics 1) explores integers, positive and negative numbers, letters and expressions, equations, functions and proportions, plane figures, and figures in space. Chapter headings in Japanese Grade 8 Mathematics include calculating expressions, inequalities, systems of equations, linear functions, parallel lines and congruent figures, parallelograms, similar figures, and organizing data. Japanese Grade 9 Mathematics covers square roots, polynomials, quadratic equations, functions, circles, figures and measurement, and probability and statistics. The material in these three grades (lower secondary school) is compulsory for all students.

    The material described is essentially all of U.S. algebra 1 and geometry curriculum. In other words, here is a country where 99.99% take algebra 1 and geometry by the end of 9th grade. And they are generally successful, as over 95% continue to (then) non-compulsory upper secondary.

    TIMSS also provides some insights. Students in leading countries like Singapore or Korea have their 25th percentile achievement at a level close to our 75th percentile. Unless someone believes that Chinese, Koreans, or Malays, are genetically superior to us, the cause must be in how we teach our students, both in school and outside it.

    The answer to the original question then is that anyone with an IQ within 1-1.5 standard deviation of the average — probably about 80 and above — can master our general K-12 curriculum. Some with more effort, some with less, but none needs unreasonable efforts. Except when they are incompetently taught. Very much like anyone, except a handful of disabled children, can run a 100 yards race. No special “sporting ability” required.

  37. Mr. Salinger, the “source” is from the Society for Quality Education which says up front that it opposes “progressive education.” I think the National Council of Teachers of Math is a far better source.

    Malcom, if you are an experienced math teacher, I think you are the rare exception.

    Vince says, “I see many students who can’t multiply without a calculator because they never learned the multiplication tables.”

    Once again, there’s a failure of logic. Though what Vince says is true, it doesn’t mean the use of calculators prevented the learning of multiplication tables. Again, though one often sees a guy with a broken leg using crutches, that doesn’t mean the crutches broke the leg.

    Lynne, it was very good to read your post. I’ve tried a strikingly similar technique with homework but I’m afraid I’m had less success. I think I need to alert parents as frequently as you do.

    The drudgery of memorizing arithmetic facts is a necessary evil which will prove to have life-long, practical benefits, but it shouldn’t serve as a gate stopper for advancement into higher level math. If you count on your fingers, you can still learn the distributive property.

    Again, I think every child should know arithmetic and know it well and when children don’t learn it, there needs to be aggressive, effective remediation.

    But keeping calculators out of classes that are a step beyond basic arithmetic doesn’t force remediation. Instead, it thwarts forward development.

    I once wrote a math grant. All sixth graders were to be tested in addition, subtraction, multiplication and division. Those who hadn’t mastered all their math facts had a Saturday class taught by college students, 8 AM to noon. They were to go every Saturday until they could prove mastery.

    This way, students could stay in their regular math classes yet would learn to master the arithmetic they had failed to memorize in elementary school.

    Unfortunately, the grant wasn’t funded.

  38. Mom Of 4, I’ve also seen the deli clerk who couldn’t figure out how to enter 2/3rds of a pound on a deli scale (.66) and not being able to figure out what 15% of $40 is, that’s grade school math (at least it was in my day) without using a calculator (which wasn’t allowed in grade school in the early 70′s).

    Robert, the calculator is a tool, if you don’t understand the basic principles behind math, all the tools in the world won’t help you solve a math problem (that’s a fact).

    Want to know what employers are looking for in prospective employees?

    Critical thinking and problem solving skills (math/philosophy)
    Excellent communication skills (English)
    Self Starter/motivated
    Able to work with minimal or no supervision
    Time management skills
    Interpersonal skills.

    A former supervisor once told me, he never liked to micromanage people, but he had to do it on occasion due to the fact that some people simply cannot function without supervision at all times.

    Math (and some other subjects) takes discipline and practice. Giving kids access to a calculator in the 3rd grade is not going to help their math skills what so ever. Which Lynne is making a pretty hefty chunk of change trying to correct.

  39. Mr. Salinger, the “source” is from the Society for Quality Education which says up front that it opposes “progressive education.”

    The article was from there, but the source of the research cited was from the Brookings Institution- a think tank with a reputation for being on the liberal end of the political spectrum.

  40. Bill, you write:

    “the calculator is a tool, if you don’t understand the basic principles behind math, all the tools in the world won’t help you solve a math problem.”

    I agree.

    And the same can be said for memorization.

    If you don’t understand the basic principles behind math, lightening speed recall of the multiplication tables won’t help you solve a math problem.

    Crimson Wife, the research that was referred to was embedded in a page with a lot of spin, some subtle, some not so subtle, composed by opinionated parents, not objective educators. The writing was intentionally misleading. The National Council of Teachers of Mathematics, practically all departments of education in respected universities and over 90% of middle school teachers who teach math day in and day out have an opinion about the use of calculators that are far different from the organization you linked to.

    It’s difficult to discuss the use of calculators because for some it’s an emotional issue (for reasons that escape me) and no amount of authentic evidence will have any impact.

    Really, whenever I bring up the topic to my colleagues who teach math, they look at me, both liberal and conservative teachers, like I’ve just asked them if they think the world is round.

  41. this is no different for math than to any other subject. In other words, *everything* we do, excluding functions of the autonomous nervous system like breathing, is “linked to cognitive ability.”

    To put it mildly, Duh.

    In context:

    Assertion: math achievement is aided by practice. Assertion: math achievement is hurt by calculators. Me: Math achievement is determined by cognitive ability.

    Nothing in there to make math different from other academic subjects. It’s just that we’re talking about math.

    The more meaningful question is whether typical K-12 mathematics, as taught in typical classrooms across the land, requires particularly advanced cognitive ability. Is it beyond the grasp — with only a reasonable effort — of a large group of school kids?…The answer to the original question then is that anyone with an IQ within 1-1.5 standard deviation of the average — probably about 80 and above — can master our general K-12 curriculum.

    You are, to put it politely, delusional.

    K-12 is an awfully big category for math. So let’s define it as 9-12 math, which is currently defined as geometry, second year algebra, trigonometry, a few odds and ends, and calculus.

    And yes, a complete grasp of this math is beyond the effort of a large group of school kids.

    I would hypothesize that an even basic understanding of the math subjects listed above is, without question, beyond the ability of anyone with an IQ below 100. That defines well over half of African Americans and Hispanics, and roughly half of all whites and Asians.

    Moving down to K-8, there’s no question that low ability students do better with repetition and drills in arithmetic, whereas high ability kids don’t require it. Algebra, ostensibly an 8th grade subject, is bound to the same IQ requirements I mentioned above.

    A person with an IQ of 80 is fully able to participate in society, but largely incapable of abstract concepts. High school academics is loaded to the teeth with abstractions.

    Does anyone believe that a broad and old system like education would willfully impose on itself goals than are inherently beyond a large fraction of its clients? Is the system suicidal? Doesn’t it know, if not through research then from experience, what can be reasonably expected to be learned by children? Do we teach in this country content that is clearly above and beyond what other countries teach?

    To answer the questions in order: yes, yes, yes, and no, the other countries aren’t suicidal and so they don’t impose these goals on all children. They sort and track. And, in some cases, the countries have populations that are entirely white and Asian, and thus aren’t bound by the same lower level constraints that we are–and yet, they still track.

    Are you really that naive? Our country refuses to even explore the possibility that the achievement gap is cognitively based, for obvious and understandable reasons. And in fact, the achievement gap is almost certainly based in large part on cognitive differences (and I have no idea what the cause is, so don’t pretend I do).

    Yes, we are suicidal when we declare these goals achievable by all. We never used to. We used to accept that only smart kids, regardless of color, went on to study advanced topics. Then we realized, to our horror, that “smart” wasn’t equally distributed.

    This is obvious. Are you really that clueless that you don’t understand the distortions the achievement gap forces upon us? Or is it that you just don’t understand what it means to have an IQ of 80?

  42. {Cal wrote]… the other countries aren’t suicidal and so they don’t impose these goals on all children. They sort and track. And, in some cases, the countries have populations that are entirely white and Asian, and thus aren’t bound by the same lower level constraints that we are–and yet, they still track.
    This is absolutely false. Neither Japan nor Korea track students in grades 1-9. In Japan, promotion to the next grade is not dependent on achievement; they are socially promoted. In Japan, there is no gifted education.

  43. Here’s the reference for no tracking in Japan.

    http://www.ed.gov/pubs/JapanCaseStudy/chapter3.html

  44. “practically all departments of education in respected universities.”

    Why don’t you try asking the math, engineering, and hard sciences departments whether or not they support the extensive use of calculators in elementary & middle school math classes? I’d be willing to bet that STEM professors are not nearly so enthusiastic about calculators as the ed school profs are.

    I don’t have a problem allowing occasional calculator use AFTER students have mastered the pencil & paper algorithms. A calculator is a useful time-saver once students have had enough practice to realize when an input error gives them a garbage answer.

  45. In regard to the cognitive ability of the Japanese, they Japanese have every bit as much variability in ability as we do in the U.S. As kids progress though the compulsory grades (1-9), the slower kids get further and further behind, but they still learn vastly more than if they had been tracked or ability grouped. (This isn’t done within classes, either. As groups, neither Chinese nor Japanese have been shown to be inherently more intelligent than Americans. This was shown by Stevenson et.al in
    Stevenson. H. W, & Lee, S. Y. (1990). Contexts of achievement: A study of American, Chinese, and Japanese children. Monographs of the Society for Research in Child Development, 55(1-2), p. 4

  46. Ze'ev Wurman says:

    Quoting Cal: “To put it mildly, Duh.”

    Exactly my point when referring to your post.

    More to the point, when I wrote about typical K-12 content, in math I referred essentially to about Algebra 2. Neither calculus, nor pre-calc, are typical or expected for the majority of high school kids. Check the data — less than a third of graduates take those. Similarly, any AP or true honor course is by definition not typical.

    Even more to the point, I appreciate your answers but nothing stands behind them except your own interpretation of reality. Further, you never bothered to explain the Japanese data or the TIMSS results I quoted, in view of your beliefs.

    Finally, your sweeping over-generalizations border on the irresponsible. While those countries may “sort and track,” TIMSS samples across all types of schools and hence comparing our 75th percentile to their 25th is entirely appropriate. Similarly, throwing generalizations like “white” or “Asian” indicates gross ignorance. Neither is a homogenous category and the fact that in Singapore, which is at least as diverse as we are, the Malay minority achieves much higher than we do is highly significant. The Malays were once considered in similar terms you refer to our Black and Hispanic minorities.

    Again, nobody argues IQ plays no role. I am arguing that what we require in our typical K-12 education does not require IQ above at least one standard deviation below the average.

  47. Crimson Wife writes:

    “Why don’t you try asking the math, engineering, and hard sciences departments whether or not they support the extensive use of calculators in elementary & middle school math classes? I’d be willing to bet that STEM professors are not nearly so enthusiastic about calculators as the ed school profs are.”

    I would imagine that professors of math and engineering would preface any opinion being saying it’s out of their area of expertise.

    Well, we might be in agreement on many issues regarding calculators. It’s a broad topic.

    When I was a teacher and I assigned for homework a problem of multiplying a two digit number with a three digit number, I wouldn’t accept any answers where the student didn’t show his work. Getting the answer with a calculator in this case doesn’t improve learning. “I didn’t show my work because I did it my head.” I got that excuse and rejected it. Nope. In a case like this, a calculator does the process that the student is supposed to learn. A calculator would still be good to use, but only to check the answer, not to do the problem. If and when calculators are used to multiply when the objective is to learn multiplication then yes, they impede learning.

    But let’s say there is a different problem:

    Fred, John and Alex are roommates. They all agree to split food expenses three ways even though Alex has an eating disorder and over-consumes products with fat, sugar, and red dye #16. In the kitchen is a hollowed out Buddha that Fred made for his mother in ceramics class for Mother’s Day. But since his mother isn’t what you’d call a universalist, it’s remained in Fred’s possession ever since he saw it on a card table in front of their house during a Saturday yard sale. The asking price was $5.00. Every time somebody buys food for the house with his own money, he puts the receipt with his name on it in the Buddha. Every time somebody has some extra change to contribute to the food fund, they put it in the Buddha along with a note that includes their name, the date and the amount. Every three months they like to even out the account so nobody has contributed more than anybody else. This means that sometimes John will give Alex a few bucks or Fred will give Alex one amount and John another. It all depends on the receipts and the contributions. In March, April and May of this year, John bought food totaling $200. Alex’s total was $175. Fred’s total was $185. John put in $15.50. Alex put in nothing and Fred put in $75.00. At the end of May, the guys empty the Buddha of the receipts, the notes and the currency. The question, is, how do they divide up the currency and then who needs to give whom what in order that everybody has contributed equally for those three months?

    In solving this problem, the use of a calculator would do no harm.

  48. This is absolutely false. Neither Japan nor Korea track students in grades 1-9. In Japan, promotion to the next grade is not dependent on achievement; they are socially promoted. In Japan, there is no gifted education.

    It’s absolutely false that Japan and Korea are Asian? I was assuming that tracking begins in high school.

    I am arguing that what we require in our typical K-12 education does not require IQ above at least one standard deviation below the average.

    Fine. Show me studies demonstrating that students with an 80 IQ are capable of mastering Algebra, let alone Algebra II. Of course, we don’t have any at all. Which means that your argument, like mine, is based purely on “your own interpretation of reality”. Except mine is much closer to reality, since the test scores show something much closer to my assertions than yours, including higher test scores for whites and Asians, even accounting for poverty.

    As for the Japanese studies, not everyone passes, right? And they have far fewer people with IQs below 90 than the US does. So I’m not sure what you think Japan’s experience refutes.

  49. “I would imagine that professors of math and engineering would preface any opinion being saying it’s out of their area of expertise.”

    I’ve heard several college professors who teach math, science, or engineering complain about the inadequate preparation of many of their students in recent years compared to in past decades. So yeah, I think their views are relevant.

  50. Ze'ev Wurman says:

    Cal,

    Please apply your famous math, reading, and analytical skills.

    Kodaira writes: The first nine grades are compulsory, and enrollment is now 99.99%. According to 1990 statistics, 95.1% of age-group children are enrolled in upper secondary school, and the dropout rate is 2.2%.

    95.1% corresponds to 1.66 standard deviation below average, or IQ of about 75 and up. All those were deemed capable to continue to the non-mandatory upper secondary, after taking Algebra 1 and Geometry by grade 9. Factoring in the 2.2% dropouts brings us to everyone above 1.47 standard deviation below average, or IQ of about 78 and above, successfully completing Japanese upper secondary schools.

    Estimates of “national” IQ are rather questionable and highly imprecise, and vary between plus(!) 7 to minus 7 points between United states and Japan. Even accepting the extreme disadvantage for the US, this would adjust the numbers above to 82 and 85 respectively.

    Your reality seems to reside entirely within your own mind.

  51. Crimson Wife,

    You write,

    “I’ve heard several college professors who teach math, science, or engineering complain about the inadequate preparation of many of their students in recent years compared to in past decades.”

    So have I, but none of them pointed to the use of calculators as a cause.

  52. Cal wrote: “Me: Math achievement is determined by cognitive ability.”

    Cal I must disagree with you here. While this is true to SOME extent, there are a whole group of people who are “math avoidant” because they just don’t “get” math. But I know from personal experience that this can change.

    In fact, I used to be one of those people. When I had to start teaching elementary math, I felt terrified. But after three years, I became good at it. I saw that my way of THINKING had changed, in exactly the same way someone who can’t draw naturally can learn very quickly to draw well, given a good teacher. The process for math and art are exactly the same (I went from being hopeless at both to being reasonably good at both, even though I never had either as a natural skill to start out with the way some people do). In both cases, one has to be taught how to see and think in a different manner. But a good teacher can help a student to see or think in the new way (however, it’s much quicker to learn to see and draw than it is to change one’s thinking to understand math).

    One other problem I didn’t even mention in my original post was that I estimate that at least 50% of elementary teachers chose that level because they are math avoidant. So this is one reason why you have elementary teachers often skipping the teaching of story problems. I certainly was never taught how to do them. But once I became good at math, I always taught every story problem in the book (and went over them for homework).

    I actually think teachers who formerly had problems themselves as students are sometimes much better math teachers than those who came by their math skills naturally. Those with natural ability have much more difficulty in understanding why the students “just can’t get it” than do those who have overcome their math difficulties as children and endured the same frustration and lack of clarity.

    –Lynne Diligent
    Dilemmas of an Expat Tutor
    expattutor.wordpress.com

  53. Aaah, I see Cal has started down the path of racial superiority. Nice work, and I thought this might be a fruitful conversation.

    Cal, the research about the difference in IQ between races was debunked because the authors never bothered to adjust for:

    1. The effect of not having enough food.
    2. The effect of not getting enough sleep.
    3. The effect of poor ventilated tenement buildings with lead lined paint.
    4. … and many other things.

  54. I’m truly on the fence.

    It is definitely plausible that a deli clerk wouldn’t know how to convert anything other than 1/4, 1/2, or 3/4 into a fraction.

    On the other hand, I think it’s equally plausible that a clerk who makes dozens and dozens of measurements a day would obviously know how close 2/3 is to 3/4, and maybe saying “the scale doesn’t allow that” would be their way of having a bit of fun with a fussy customer.

    But in either case, I’d just tell the clerk the exact measurement that I wanted, giving the clerk the information he needs/ending the charade. Disaster–paying for an extra .08/lb of pimento loaf I don’t want or need!–averted.

  55. Lynne, I’m glad you’ve contributed further here on this website

    Good food for thought.

    Thank you.

  56. Cranberry says:

    http://magazine.jhu.edu/2010/12/back-to-basics-for-the-“division-clueless”/

    “As another experiment, Wilson gave a short test of basic math skills at the start of his Calculus III class in 2007. The results predicted how students later fared on the final exam. Those who could use pencil and paper to do basic multiplication and long division at the beginning of the semester scored better on the final Calc III material. His most startling finding was that 33 out of 236 advanced students didn’t even know how to begin a long division problem.”

  57. Roger Sweeny says:

    If and when calculators are used to multiply when the objective is to learn multiplication then yes, they impede learning.

    Which is why it hurts me when one of my 9th grade science students needs a calculator to multiply a number by 10. They obviously didn’t learn place value, and are not going to have much of a “feel” for quantities.

  58. Roger,

    I grew up on Multiplication Rock…I watched ‘My Hero Zero’ yesterday, and as we both know, adding a zero after any number multiplies it by 10, adding 2 zeroes multiplies by 100, and adding 3 zeroes multiplies by 1000…

    Why more elementary schools don’t use Multiplication rock to deal with math is beyond me…

    Place value is very important in IT and in programming where you might need to turn single bits on in a byte or word, but it all comes back to basics, without understanding place value, it’s just another hurdle for students to overcome.

  59. Cranberry, there is the issue right here. How does a student get into Calc III without knowing long division, multiplication, and basic math facts?

    Here is a very scary scenario…Assume all the technology in the world failed tomorrow, what would society need to do in order to carry out basic tasks?

  60. Cal ” Me: Math achievement is determined by cognitive ability.”

    Please calibrate this.

    Cal: “So let’s define it as 9-12 math, which is currently defined as geometry, second year algebra, trigonometry, a few odds and ends, and calculus. ” …

    “I would hypothesize that an even basic understanding of the math subjects listed above is, without question, beyond the ability of anyone with an IQ below 100. That defines well over half of African Americans and Hispanics, and roughly half of all whites and Asians.”

    Hypothesize? You mean guess.

    “Moving down to K-8, there’s no question that low ability students do better with repetition and drills in arithmetic, whereas high ability kids don’t require it. Algebra, ostensibly an 8th grade subject, is bound to the same IQ requirements I mentioned above.”

    “high ability kids don’t require it”

    You’re guessing again.

    “bound to have the same IQ requirements..”

    ditto.

    “Except mine is much closer to reality, since the test scores show something much closer to my assertions than yours, including higher test scores for whites and Asians, even accounting for poverty.”

    Test scores show what, exactly? How would schools use this information?

    “Our country refuses to even explore the possibility that the achievement gap is cognitively based, for obvious and understandable reasons.”

    Here it is. This is so predictable. There are no other reasons for an achievement gap?

    You know, it’s possible to not put all of your eggs into an acheivement gap/IQ basket. So how, exactly, does a school use IQ information? Do they have a policy that says that it is OK if half of the African Americans and Hispanics don’t get to algebra in 8th grade? Where is the rest of the calibration of IQ and content? What if these kids don’t know what 6*7 is in fifth grade? Are they the ones with an IQ of 85, or is it some other problem? How do you know when schools are doing a good job or a bad job?

    It’s possible to expect schools to separate kids by ability or willingness to learn without resorting to an IQ or race argument, unless that is your real agenda.

  61. Roger Sweeny says:

    SteveH,

    Alas, most schools try to “separate kids by ability or willingness to learn” as little as possible. A major reason is that the advanced groups will have more than their equal share of “whites” and “Asians” and less than their equal share of “blacks” and “hispanics” and educators very very much don’t want that to happen.

    This doesn’t mean that members of the “over-represented” group are smarter. There are lots of cultural reasons for such a result. But everyone in the ed business knows it will happen. And they desperately don’t want to look like bad people. So we pretend that putting wide ranges of ability, preparation, and interest in the same class will work out fine. The alternative is to look like a racist. That is a big big big no-no.

  62. Roger, the fact that you have students that need calculators to multiply doesn’t mean that their use of calculators prevented them from learning to multiply.

    You’re not the only one here who leaps over logic to land on that conclusion.

  63. How do you explain away the NAEP and TIMSS results that students use calculators frequently in class do much worse on math tests than students who rarely use them? I find it hard to believe that the difference is test scores is 100% due to innate differences in ability and 0% due to certain curricula downplaying mastery of pencil & paper calculations in favor of using calculators.

  64. Robert, I agree with your correlation/causation point, but I’d like to know why you think ES-MS and lower HS kids should use calculators? What is the advantage of doing so? Because the SAT now uses calculators (does the PSAT?), delaying usage until pre-calc/junior year is a non-starter, but what is the reason they should use them prior to HS? By that, I mean why is it beneficial to kids, not to the adults in the system.

    I think Lynne has a point about the math aversion of ES teachers; I’ve heard many admit it freely. Would they admit they couldn’t read well or didn’t like reading? I also think her comment that teachers who had to work hard to learn math might have particularly good insights about common student problems and how to avoid/remediate them is a good one. Of course, I think it’s also likely that kids who “get it” quickly and easily benefit from a teacher with similar background, particularly at HS levels.

  65. Math aversion (avoidance) is a main reason why so many elementary school are so ill equipped to teach math (and perhaps other subjects). I hear so many people say ‘they suck at math’, but I wonder, were any of them really taught math correctly?

    Sigh

  66. Roger,

    When I attended public schools, grouping students by ability was quite common (1970′s), and in many cases, it was done for a reason so that the students and the teacher could cover the material at (roughly) the same pace.

    Now we have eliminated grouping and lumped the students together. What does the student who is extremely bright get asked to do, help the student who isn’t grasping the material (which if done once in a while isn’t a problem), but if done constantly, cheats the bright student out of an education as well as the student who can’t grasp the material.

  67. There are no other reasons for an achievement gap?

    Around the edges, sure. But as the bulk of the gap? It’s the most logical explanation.

    The obvious way to determine the relationship of cognitive ability to academic achievement would be to conduct a whole bunch of studies. No one dares. I understand why.

    Do they have a policy that says that it is OK if half of the African Americans and Hispanics don’t get to algebra in 8th grade?

    No. They have a policy saying that kids who want to take algebra in 8th grade should take algebra in 8th grade. They also use other predictive tools (math scores, algebra readiness tests) to encourage those kids who appear to have the skills to move to algebra but are taking easier courses.

    They make sure that all math courses are educational and develop the students as thinkers, regardless of whether it’s arithmetic or algebra.

    Where is the rest of the calibration of IQ and content?

    There was lots of research in the 50s, 60s, and 70s on algebra predictors. IQ is only part of it, but of course, there’s little evidence that low IQ kids do well in math. However, not all high IQ kids are good in math. However, I think we could probably improve on that these days. We just stopped researching any relationship between cognitive ability and academic achievement, because we didn’t like the answers.

    What if these kids don’t know what 6*7 is in fifth grade? Are they the ones with an IQ of 85, or is it some other problem?

    First off, speaking as a math teacher, I have noticed a huge correlation between kids who are utterly clueless as to their multiplication tables and their cognitive ability. However, I think this is an area that needs research, as I’ve also noticed more than a few kids who know their mult tables cold but can’t grasp algebra in the slightest.

    But we won’t know these answers until we start to examine these issues with cognitive ability–of which IQ is just one major, but not exclusive, part.

    How do you know when schools are doing a good job or a bad job?

    This is really funny, because right now, we are declaring that schools have done a bad job without any reference at all to cognitive ability. Low test scores? Must be bad teachers, bad curriculum, poverty. It couldn’t possibly have anything at all to do with the fact that far more kids in that school’s population don’t have the same mental abilities. Nope. Look somewhere else. So you’re apparently fine with ignoring cognitive ability to determine whether schools are doing a good or bad job, but factoring it in? Oh, my lordie, what shall we do?

    The obvious place to start would be to factor in IQ and other cognitive tests into the school’s profile. So if a school has terrible test scores but an average IQ of 87, that suggests there might be huge challenges to improving learning outcomes.

    To show how completely ignorant we are, we don’t even know this. You ask how, in this mythical future, we could tell good schools from bad. But right now, we have no idea what the average IQ of a school population is and whether or not there are schools with low average IQs but better outcomes than other schools with the same IQ profile.

    Seems a pretty key piece of info, to me.

    Naturally, the reason we avoid it is because it’s politically and ideologically very disturbing.

    “Why are so few blacks and Hispanics signing up for algebra?”

    “Well, we’ve run comparisons and, controlling for cognitive factors (IQ being just one), there is no racial disparity in algebra starts. Pass rates, test scores, achievement levels: no racial disparity.”

    “But why are so few blacks and Hispanics signing up for algebra?”

    Yeah. That’s not a fun conversation. And I get it. I really do. I understand the challenges. I’m just fed up with the pretense. We don’t ignore it because it doesn’t matter, but because we’re terribly afraid it does.

    However, we don’t really know. It seems completely obvious that cognitive ability plays a part in this, we have no idea what comes next. Why not see if there’s a way to teach algebra better to kids with lower cognitive abilities? Why not see if we can do a better job teaching algebra to kids with high IQs but low algebra proficiency scores? What teachers are better working with low ability kids, which are better with average kids, which do best with really smart kids? Small class sizes seemed to work really well with adolescent black boys–was that actually a proxy for IQ? Would it be better to put kids with lower IQs in small classes–does that help with outcomes?

    Ignoring cognitive ability, most of these studies are just a joke. Factor it back in, and we might make more progress.

    And of course, maybe it turns out that there’s no relationship at all. Whoohoo! Unlikely, but until we check it out, there’s no way to know.

  68. Roger Sweeny says:

    Roger, the fact that you have students that need calculators to multiply doesn’t mean that their use of calculators prevented them from learning to multiply.

    I never said it did. I’m sorry I gave you the impression that I was jumping from one to the other.

    However, thinking about it now, one thing calculator use does do is that it allows students to get the right answer when they don’t understand why, and can make it look like they do understand when they don’t. I assume that ninth graders are supposed to understand place value. Many of mine don’t–but they can successfully multiply a number by ten using a calculator, and they have wound up in ninth grade.

  69. momof4,

    There are a number of reasons why I believe it’s a good idea for students to use calculators in math class.

    Starting with upper elementary, math problems are not about how to do arithmetic, but when to do it.

    Take for example the math question of how many chairs are in the auditorium that has 23 rows of 15 chairs each. The math concept is that multiplication is called for. If a student knows that, he can get the correct answer quickly with a calculator. Without a calculator, it wastes a lot of time that could be spent on further learning. Also, without a calculator, a careless error could be made which would invalidate the proof that he understands the problem and knows how to solve it.

    Higher level math should not be weighed down by working out tedious arithmetic. Calculators are great tools for speed and accuracy.

    Knowing the times tables and recalling them with speed and accuracy is not math. It’s memorization. Very helpful memorization and an essential foundation for learning math, but it’s not math.

    There is the belief that when students are required to work out their arithmetic by hand, it gives them increased practice and strengthens their mastery. That is certainly a logical argument to not use calculators. But when I’m trying to teach that “of” means multiply, I don’t want that point derailed by additional skill requirements and the tedium of using them.

    For example, if I’m giving a spelling test and I mark a word wrong because of poor but readable penmanship, that defeats the purpose of the test, even though both spelling and penmanship are important.

    We all agree that calculators are good tools. My car is a good tool, too, and that’s how I get to work. Oh, I suppose I could walk, but that would take more time and tire me out. And I don’t think driving will make me forget how to walk.

  70. Ah, Roger.

    Your comment that questions the use of calculators is the first one that makes some sense.

    Thanks for bringing that up.

  71. As for ability grouping in math and spiraling, these are interesting questions.

    It makes sense to me to group by ability and require mastery before moving to the next step.

    And I would probably firmly believe it if not for the fact that the math teachers I know and respect don’t share that opinion.

    Armchair theories about teaching math are easy to formulate. Too easy.

  72. In theory, calculators would be used as Robert describes: to save time when the paper-and-pencil method would take too long, in situations where the student fully understands the paper and pencil process and the arithmentic facts that are used to carry it out. In reality, as many are pointing out, the use of the calculator allows students to bypass the arithmetic facts (whether in their head or as used in carrying out multi-step arithmetic operations), to be unaware of how place valuw works, to be unable to maniupulate fractions, and so on. And it allows them to avoid having to develop number sense that is needed in many math situations before you can even begin to choose an approporate operation.

  73. Mark Roulo says:

    It makes sense to me to group by ability and require mastery before moving to the next step.

    And I would probably firmly believe it if not for the fact that the math teachers I know and respect don’t share that opinion.

    I am guessing that there is *SOME* point at which these teachers would be in favor of ability grouping. For example, would they favor teaching differential equations in the same class they they were teaching single digit arithmetic? Or no?

  74. As groups, neither Chinese nor Japanese have been shown to be inherently more intelligent than Americans.

    The average IQ of Japan is estimated to be 105.This puts almost half the Japanese population above the 106 threshold believed to be essential for certain important occupations; the USA caucasian population’s smart fraction is 34%.

    95.1% corresponds to 1.66 standard deviation below average, or IQ of about 75 and up.

    Only for a mean of 100 and standard deviation of 15.  For e.g. a mean of 105 and a standard deviation of 13, more than 95% would have an IQ of 83 or more.

  75. Mark, yes.

    These math teachers believe in grouping by grade level and teaching classes with a broad range of ability and base-line knowledge. Though if a student is significantly above or below grade level, then yes, they are want that student to be in another class.

    These are teachers in the middle grades.

  76. All four of my kids attended the same school between ES and HS. For the older two, it was a 7-8 JHS and, for the younger two, it was a 6-7-8 MS, despite overwhelming opposition from the community (apparently County rules required a vote, but the Board didn’t have to pay attention to the result). When my younger kids arrived, the MS had lost the academic climate of the JHS and instead focused on feelings, so I can believe MS teachers believe in the diversity model (or at least feel it necessary to pretend to).

    However, I have not heard a convincing argument why 10-15 minutes of the teacher’s time, in a heterogeneous class, is as effective as a whole period of the teacher’s time, in a homogeneous class. If teachers are as important as they would have the public believe, why would students not benefit from more of their time?

  77. “Take for example the math question of how many chairs are in the auditorium that has 23 rows of 15 chairs each. The math concept is that multiplication is called for. If a student knows that, he can get the correct answer quickly with a calculator. Without a calculator, it wastes a lot of time that could be spent on further learning. Also, without a calculator, a careless error could be made which would invalidate the proof that he understands the problem and knows how to solve it.”

    I just asked my DD who will be starting pre-algebra as soon as she finishes her current math book to solve that problem and timed her with a stopwatch. She calculated the correct answer via pencil and paper in 19.9 seconds.

    Now if you had said how many seats are their total in a baseball stadium with 23 sections each consisting of 85 rows of 15 seats per row and 17 sections each consisting of 36 rows of 7 seats per row, now *THAT* I could see using a calculator to speed up solving the problem. But I’d still want to see the equation (23 x 85 x 15) + (17 x 36 x 7) written out on the paper to make sure that she knows what she would do to solve it by hand.

  78. Momof4,

    When I attended 6th grade it was part of our district’s forced busing plan, where we would head over to the predominantly black community (this was back in 1974-75 when I was in 6th grade). I got moved from one 6th grade center (yeah, that’s what our district called ‘em) to another and back again within a 3 month span, and did it ever mess up my education.

    Our Junior Highs (middle schools today) would be grades 7-8 or 7-9 depending on the school size and capacity, but for the most part, we didn’t have the diversity stuff (remember this was the mid to late 70′s) and the focus was pretty much academics, except for the students who wanted to get into trouble on a constant basis (they got shipped off to what we called ‘opportunity school’ which would be known as an ‘alternative learning environment).

    The issue of the math concept present of an auditorium with 23 rows of 15 chairs each. This is standard multiplication, and hardly requires a calculator, one can figure this out in their head by doing 20 x 15 + 3 x 15 = 345 chairs. Now, if I’m converting hexadecimal values such as 0xBEEF to a decimal value, while I know how to do the conversion with pencil and paper, this would call for the use of a calculator.

    Big difference here is that while 23 rows of 15 chairs is routine math, the second problem requires more calculation and knowledge of converting between base 16 and base 10.

  79. palisadesk says:

    “Take for example the math question of how many chairs are in the auditorium that has 23 rows of 15 chairs each. The math concept is that multiplication is called for. If a student knows that, he can get the correct answer quickly with a calculator. Without a calculator, it wastes a lot of time that could be spent on further learning.

    Heck, you not only don’t need a calculator, you don’t need a pencil and paper either. Multiplying 23 by 15 is just multiplying by 10 and adding on half the product. Average (even slow) students can get the hang of multiplying and dividing by 10 quite well, in their heads. 23 x 10 is 230, add half of that — half of two hundred is 100, so 330, plus half of 30, which is 15, for a total of 345.

    Operations with 10, and doubling or halving are fairly easily taught to most kids. Many aren’t good at mental math generally, but can deal with doubling, halving and operations with 10 (or 100). A good trick I treach kids to calculate tips and various real-life percentages is what I learned as the “One-Percent Solution.” You need 12% of, say, $54.60? Well, 1% would be 54.6 (if you’re figuring tips, round to 55 cents). 10 percent would be $5.46, add 2 more one-percents: $1.10, so $6.65. Of course if you need to calculate to three decimal places you probably do need a pencil and paper.

    For everyday purposes, this works great and lots of middle school kids can do it easily with a little practice.

  80. I agree with this, basic computational concepts should be mastered (so called tricks or shortcuts) so that students can make progress even when they don’t have access to a calculator.

    I once took a class in bioinformatics where the instructor covered basic statistics (mean, mode, variance, std. deviation, sample, etc), but for the exam, no calculators were permitted (all the formulas and information was on the professor’s website, and it was accurate).

    I had a raw score of 95 (the class average was 69), and out of the 28 or so students, the 5 persons with the highest scores were all over the age of 40, with the exception of one electrical engineering major. The rest of the students were under 30, with the bulk being in their early 20′s.

    The professor was extremely disappointed, as we had covered the material in class, the formulas, etc. With the exception of using square root, and perhaps some exponentiation, there was nothing harder on the exam than add, subtract, multiply, divide, percentages, and fractions (I suspect the poor scores actually happened because many of the students simply did not know the basics of math).

  81. Crimson Wife,

    You write:

    “But I’d still want to see the equation (23 x 85 x 15) + (17 x 36 x 7) written out on the paper to make sure that she knows what she would do to solve it by hand.”

    We agree!

    I think in my first week as a math teacher I learned that “showing your work” had to be an ironclad requirement.

  82. When I mentioned diversity in my last comment, I wasn’t referring to racial/ethnic categories but to heterogeneous classrooms in terms of ability and/or preparation.

  83. And I would probably firmly believe it if not for the fact that the math teachers I know and respect don’t share that opinion.

    Then you don’t know all that many math teachers. Surveys show consistently that math teachers are far more likely to oppose heterogeneous classrooms–something like 80% or more.

    Research unequivocally shows that ability grouping leads to better results, something prominently observed in the California math teaching standards handbook.

    Moreover, if they are truly big on teaching heterogeneous, they are not merely supportive of it in middle school, but all the way through.

    So the math teachers you know and respect are a) outliers, b) ignorant of research, and c) inconsistent with all the other idiots who support heterogeneous classrooms. Awfully convenient math teachers for an online cite.

  84. Cal said:
    “The obvious place to start would be to factor in IQ and other cognitive tests into the school’s profile. So if a school has terrible test scores but an average IQ of 87, that suggests there might be huge challenges to improving learning outcomes”

    But this still doesn’t provide calibration on an absolute basis. It may be true that no one “dares” to do this research, but how is this a prerequisite for improvement? It may argue against a focus on some sort of achievement gap, but that only goes so far (and you won’t win that argument). Expecting this kind of solution only allows you to blow off steam on your IQ soap box.

    It has, however, been clearly documented that even bright kids are not getting what they need with curricula like TERC and Everyday Math. It’s clear that these curricula do repeated partial learning rather than distributed practice or true spiraling. You don’t need IQ testing to make this case. You can argue for separating kids by results without regard to any IQ test. You can argue for improvement on an absolute scale.

    When I was growing up, kids were held back a grade or forced into summer school if their grades were low. Nowadays, schools just pass kids along until it’s too late to do anything about it, and it’s too late to figure out why. As kids get older, everything looks like it’s a problem with the kids, or parents, or society, or even IQ. If these tough decisions were made each year, more people would pay attention to exactly what the problems are.

  85. Deirdre Mundy says:

    The only thing worse (in my experience) than being tasked with teaching a heterogeneous classroom is being expected to teach “Algebra 2″ to a group of students who should really be in remedial math.

    One should NOT have to spend weeks on Y=MX+B with a group of “Algebra 2″ students. (The class was homogenous, but there was no way to bring them up to speed from “We don’t know fractions” to “We’re ready for Pre-Calc” in a single year! Especially when mixed in with a huge dose of “We know you’re not allowed to fail the whole class and we’re just killing time until graduation.”)

    Argh. There’s a reason why I now only teach math in situations where I can choose the students AND teach them a ability-appropriate class.

    Though their scores DID tend to be lower when they got to use calculators. Without them, they would at least TRY to do the problems. With them, they punched in a bunch of random numbers and operations and wrote down whatever appeared on the screen, and then blamed the calculator for their mistakes.

  86. Richard Aubrey says:

    The assertion is that math tutors are prospering. Presuming that’s true, it’s a matter of conspicuous consumption on the part of status-hungry parents, or the kids need math tutors to learn math.
    Keep arguing. The tutors can use the business.

  87. Michael E. Lopez says:

    DM-

    I can’t tell from what you wrote if it’s the teaching slope-intercept form at that level or its taking weeks that is troubling you. I would have thought one shouldn’t have to spend a single day on slope-intercept form in Algebra II.

  88. Mark Roulo says:

    I would have thought one shouldn’t have to spend a single day on slope-intercept form in Algebra II.

    20-ish years ago I attended University of California (campus to go unnamed … not Cal). I took an undergraduate macro economics course. We spent one full TA session on slope/intercept. I think about 1/2 the class still didn’t get it.

    I no longer assume that students actually learn or retain anything from previous classes :-(

  89. Moebius Stripper (remember her?) kept testing her students on binomial expansions.  The same “students” kept failing with the same errors even after repeated explanations.

    (Are they really students if they don’t bother to study?)

  90. @Richard…I don’t know that I would agree that hiring a tutor is a matter of conspicuous consumption..but who knows. I can say I know many parents that seek out other resources to help their kids with math, whether it be flash cards, workbooks, or even “afterschooling” their child in another math curriculum like Singapore math.

    Tutors are getting more business in school districts where curriculums like Investigations/TERC or Everyday Math is being taught. This isn’t a new phenomenon either.

  91. Cal, you might be right on target when it comes to mixed grouping and spiraling.

    My sampling of math teachers could very well be skewed.

    Ability grouping seems it would be easier to manage and instruct but the teachers I talked to, at the middle school level, pointed to the benefits of heterogeneous grouping. Even though it’s more work for the teacher, they claimed the benefits outweighed the difficulties.

    But again, the sampling wasn’t broad, and it’s not proof. It just made me question my own assumptions.

  92. Ron Johnstone says:

    After finishing 18 holes of golf I began adding up the scores of each member of our group. Nearly instantly I heard a voice over my shoulder, “92, 88, 97, 82″. I continued adding the columns only to confirm that these were in fact the correct scores for each player. The two young members of our group were flabbergasted by this demonstration of quick addition but I was not surprised in the least. There he was again, my father of 85 years grinning from ear to ear with pride in this special ability. While driving home I asked him how he had become so proficient in this area. He told me about how for most of his school career he just did enough to get by, something he was not too proud to admit. He went on to tell me how that changed for him during junior high when his math teacher (Mr. ?) taught his class in a very different way. My dad described the class as “mystery math”, where groups had to get together to solve problems in order to unscramble mysteries using math, he was instantly hooked. He said that virtually every student (including the not so smart ones) had mastered these skill(s) which are clearly just as sharp today as they were over 70 years ago.

  93. I’m assuming this thread is dead, but I thought I’d wrap this up:

    Several of you are calling me a racist. This is precisely what I mean when I say we’re too scared to investigate this; because morons like you declare any discussion of cognitive ability to be de facto racism. I’m not racist. I am not an IQ fundamentalist and pay attention: I don’t think cognitive ability is a racial trait.

    It is merely a fact that cognitive ability correlates (and is probably a lot of the cause of) academic achievement. Ignoring it is stupid, but understandable. The reason we are uncomfortable with cognitive ability as a factor is entirely wrapped up in the fact that cognitive ability as we currently define it is not proportionately distributed in all races and ethnicities.

    That’s a fact. Screaming and whining about my racism is insulting, but ultimately changes nothing. Worse, it only delays the inevitable. If everyone (including you all) is right about cognitive ability, which is why we avoid it, then the decision to ignore it is only harming the very students who need the most help. You can all sit there and scream until the cows come home about how pointless it is, but putting a kid of any color with an IQ of 85 in an algebra or geometry class isn’t going to help him a bit if 85 turns out to be a number highly associated with algebra failure.

    And Linda, not all people with average to higher cognitive ability are good at math. That has nothing to do with the issue, although finding out why would be useful, and one of the things we could investigate if we weren’t avoiding it. But pointing to your experience as a reason why inner city schools are failing is a serious sign of Really Not Getting It.

    But this still doesn’t provide calibration on an absolute basis. It may be true that no one “dares” to do this research, but how is this a prerequisite for improvement?

    The first prerequisite for improvement is to stop declaring we’re failing. Right now, there’s a good possibility that “failure” means “not being able to teach algebra and Hamlet to kids with IQs under 90″. But we’re not even sure it’s possible to teach either to kids with IQs under 90. So before we can improve, we have to define our goals more meaningfully, because “all kids can achieve” ignores the impact of cognitive ability–and research has indisputably showed that cognitive ability has links to academic achievement.

    This is obvious; the only people who ignore it are those who want to scream racism. But I have said nothing about race and any links to cognitive ability–and let me make it clear again: I do not think that cognitive ability is linked to race. Correlation, not cause.

    It may argue against a focus on some sort of achievement gap, but that only goes so far (and you won’t win that argument). Expecting this kind of solution only allows you to blow off steam on your IQ soap box.

    If you focus on the achievement gap and closing the gap is impossible because of the distribution of cognitive ability, then the achievement gap will never be closed. The achievement gap has never, ever been closed, not even in small samples, at the high school level and only at the barrier level in elementary school.

    Given that this will continue, the argument will eventually be won, because your argument–that closing the achievement gap IS possible–is unachievable.

    You can already see this building in the talk of “college isn’t for everyone”. No one wants to mention cognitive ability, so instead they’re just pushing back on college.

    I’d rather not that result, because ultimately it’s an evasion. But from your standpoint, it means the same thing: your side will lose, we will stop preparing everyone for college and accept that not everyone can achieve in the same way. The reason I’d like to avoid just going the “college isn’t for everyone” route is to see if we can improve teaching outcomes by taking low cognitive ability into account with research and experimentation.

    But one way or another, the “everyone can achieve” side is sliding towards the “lose” column. I’d just like to see us be more honest about why.