Can Colorado School Board Members Do Sixth Grade Math?
by Ari Armstrong, December 8, 1999
Below is an article I wrote for the Independence Institute. Following the article is additional commentary which responds to criticism and provides further details and analysis. Another essay on the same subject but using different examples can be found at my home page.
Here's a challenge for Colorado's school board members: resign if you can't solve all the sixth grade mathematics problems you approve for our students. But this challenge is unfair. That's because some of the math problems assigned to Colorado sixth graders are riddled with ambiguity, literally impossible to solve. So a better challenge for educators is to make sure the materials used in schools are coherent and sensible.
Forcing a sixth grader to come up with an answer for a problem that doesn't have any answers is confusing and unfair and it turns students off to math at a young age. The problem of ambiguous, poorly written mathematics material is widespread, ranging from Crawford to Boulder, from the elementary grades through high school. That doesn't mean all the materials used are poorly written, but enough are to warrant concern.
Take, for example, a worksheet given to sixth graders at Deer Creek middle school in Jefferson County last semester. The worksheet is supposed to teach basic probability, but its explanation of probability is incoherent and half of the problems on the sheet are impossible to solve as written. According to this worksheet, "An outcome which can happen has a probability of one." That's just plain silly: the statement implies that anything which can happen necessarily will happen. So it's a sure bet that the next President of the United States will be Donald Trump. And the Broncos are busy planning their January Superbowl victory party.
Fortunately, part of the worksheet makes sense. It defines the "probability of an outcome" as the "number of times [the] outcome could occur" divided by the "number of possible outcomes." This is hardly adequate to explain probability to a beginner, but at least it is accurate. But is a worksheet that's only half right good enough for sixth graders who are new to the subject?
The worksheet contains 20 problems. The last ten problems are impossible to solve. The section reads, "There are 26 students in Mr. Johnson's algebra class. Fourteen are girls," and so forth. Question 11 asks, "What is the probability that a male student will be absent?" Of course, if we assume that exactly one random student will be absent from the class, then we can come up with an answer. But that condition is not specified in the worksheet. The answer to the question as written is impossible to determine, as it depends on how often students are absent, whether boys are absent more often than girls, and so on. A sixth grader, just starting out with these concepts, is not going to be able to learn anything from the worksheet. The probability that a student will end up utterly confounded and frustrated by the exercise approaches 100%.
However, such exercises do leave their impressions. With such hopelessly inane work, many students come to feel that math is too difficult for them and that getting a good grade is all about second-guessing what the teacher's answer key says. Such worksheets are excellent for fostering rote memorization and for preventing students from learning how to think critically for themselves.
And if one worksheet isn't enough to show that there is a real problem here, the textbook used by the same class, entitled Mathematics: Applications and Connections, Course 2, offers another probability question. It says on page 501, "A soft drink machine contains cola, ginger ale, root beer, orange, and diet cola. Without looking, choose a soft drink." We are supposed to find the probability of choosing a cola or diet cola. Even if we make the extra assumption that each button on the machine corresponds to one type of drink, we still don't know if some drinks get more than one button. Many pop machines stock more slots with the more popular variety. Again, the problem is nonsensical and distant from the real world.
Besides including ridiculous problems that have no answers, the textbook also contains politically correct social projects, such as asking people why they quit smoking (page 409). Our kids won't know any math, but at least they won't be as likely to smoke!
Far and away the biggest problem with many modern textbooks is that they make complete hash out of the natural conceptual hierarchy of mathematics. For instance, the Mathematics: Applications and Connections text separates the related concepts of decimals (Chapter 2), fractions (Chapter 5), and percents (Chapter 11) and spreads them out among more advanced subjects such as algebra (Chapter 6) and geometry (Chapter 8). If we wonder why some students have trouble thinking logically, it's partly because they are taught from disintegrated materials.
The "experts" have had their chance to write good course materials for our students. They have failed miserably. Now the responsibility lies with Colorado's teachers, principals, school board members, and especially parents to be more critical of the materials used in our children's learning.
Ari Armstrong is a private tutor and also a Senior Fellow at the the Independence Institute, a think tank in Golden which studies education policy, http://i2i.org.
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Update December 9, 1999
A letter in reply to a critic:
Your contention that the *students* are the ones who are "disabled" is extraordinary and potentially damaging to the students. It is clear that the disability lies with the authors of the materials cited, who are unable to write intelligibly or explain mathematics concepts accurately.
You're giving far too much credit to the authors of these materials. You seem to want to believe that the authors intentionally left it to the students to make additional assumptions. On the contrary, the authors were merely careless (or incompetent), as demonstrated by their ridiculous assertion that "an outcome which can happen has a probability of one." (This relates to the authors of the worksheet; the authors of the textbook are somewhat better but still vague.)
You are correct that an important skill is the ability to check assumptions, determine when too little information is given to solve a problem, and make reasonable additional assumptions that make a problem solvable. For instance, that's what I did with the materials in writing the article.
However, children who do not have the slightest knowledge of a subject can hardly be expected to make additional assumptions for a problem concerning that subject. The probability materials cited were used to (try to) teach the concept to students who had never before in their lives been exposed to the subject. These students didn't even know what probability *is*, much less what additional assumptions might be required to work the problems. You can't run before you can walk, and you can't solve advanced probability questions with insufficient data before you even know what probability means.
In addition, the mindset of the teacher was NOT, "Answer these problems if possible, or explain why a solution is impossible, and then make additional assumptions if necessary to solve the problems." Rather, the teacher, working from an answer key, simply told the students to come up with the single, correct "answer."
I grant you that a good teacher can make up for poor materials, but no competent teacher would use poor materials in the first place.
But I didn't even cite the worst problems in the article (again, for lack of space). All the problems I cited in the article at least permit additional assumptions that make them solvable. However, such is not the case for the final three questions on the worksheet. To review, the worksheet details:
There are 26 students in Mr. Johnson's algebra class. Fourteen are girls. Nine students have black hair, twelve have brown hair, four have blonde hair, and one has red hair. Fifteen have brown eyes and eleven have blue eyes.
Question 11 asks, "What is the probability that a male student will be absent?" IF we assume that exactly one student will be absent, AND we assume that males and females are absent with the same frequency relative to the sizes of each population, THEN we can say that the probability of a male being absent is the number of males divided by the total number of students, or 12 out of 26.
But consider question 18, which asks, "What is the probability that no student will be absent?" HUH? Obviously, our previous assumption that exactly one student is absent can no longer hold. There's simply no way to answer this question, except to analyze attendance records and figure out the frequency with which students are absent.
Question 19 continues, "What is the probability that all students will be absent?" Which assumption would you consider "reasonable" here? A student could "reasonably" answer, "Based on the assumption that an asteroid will smash into the earth, the answer is 100%." There's simply no way to give any sort of sensible answer to this question. The most we can say, based on historical data, is that the chances of all the students being absent (on any given day, yet another assumption) is very low, though greater than one. There's no way to quantify the answer, though.
The final question asks, "What is the probability that you answered all questions in this lesson correctly? Explain your answer." The only answer possible is, "Based on the reasonable assumption that those attempting to teach probability in the classroom, including the teacher and the authors of the material, are completely incompetent, the probability of a student answering the questions 'correctly' approaches 0."
I simply advised the student I was tutoring in the matter to leave the final three questions blank, as they don't even permit additional assumptions (at least not ones remotely close to the bounds of reason). The teacher wrote at the top of the worksheet, "-3, good job!!" Unfortunately, the same cannot be said of the teacher.
What is the best way to foster the skill of identifying unstated assumptions? I submit that an effective way is NOT to give beginning students ambiguous problems and require them to come up with the single, correct answer. The effective way is to first teach students the underlying principles, and then to openly explain how ambiguous problems can be tentatively resolved by making additional assumptions. (In some cases, students may simply state that a problem does not contain enough information to solve.)
The tendency to label the *students* as "disabled" or "disordered" is just a convenient way to excuse the ineptitude of the teachers and other educators. When a student has trouble learning concepts, the reason couldn't *possibly* be that the teacher is a moron who flunked math in grade school but who now coaches sports while "teaching" on the side. The reason couldn't *possibly* be that the materials are hopelessly confusing. The reason *has* to be that the *students themselves* are "disabled." HOGWASH!
The same goes for so called "attention deficit disorder," a label assigned to children whose natural reaction to their tedious and mind-numbing classes is boredom and restlessness.
To stigmatize a student as "disabled" or "disordered" is traumatizing to the child and it tends to make him or her even more wary of the troublesome subject. Obviously, *some* problems are genuinely physiologically based, such as dyslexia, and such problems require additional work. But if a normal student needs additional help understanding a concept in math, he or she is not "disabled," but simply in need of quality instruction.
You argue that an important skill is the ability to pick up on others' unstated assumptions. This is true. Again, though, this skill can only be developed once a person understands the fundamentals of the issue at hand.
The authors of mathematics materials can skillfully teach students how to look out for "loaded questions," problems which permit no answer without additional qualifying assumptions. However, such authors should be consciously aware of which assumptions have been stated and which have not. Obviously, the authors of the worksheet were aware of no such thing.
The problem with the materials cited is that they unintentionally require students to make assumptions which are NOT reasonable. There's simply no reason why anyone should *assume* that precisely one student will be absent from a classroom on any given day. There's no reason for a student to *assume* a pop machine will display one button for each different type of soda. Such assumptions are unwarranted.
You want to stigmatize those unable to make questionable assumptions as "disabled." But the real intellectual disability lies in making unwarranted assumptions. (As the saying goes, to "assume" is to make an "ass" out of "u" and "me.") Just ask NASA, the scientists of which *assumed* a set of data was already in metrics, even though it was not. The result? NASA crashed a multi-million dollar space-craft.
Or, if he were still alive, we could ask Ismael Mena about what he thinks of assumptions. "But we *assumed* we had the right address," the Denver police lamented after they shot Mena eight times on an unconfirmed drug report.
No, a teacher who urges students to make unwarranted assumptions does them no favors.
Mathematics more than any other subject has the inherent ability to help students learn how to think with precision. Ambiguous materials only confuse students and "teach" them that mathematics is a guessing game in which the answer-key (or some authority figure) is the only ultimate source of truth.