Today I begin reading my newest side-along reading book, as I alluded to yesterday. Chapter 1 of Morris Kline's Mathematics and the Physical World is called "The Why and Wherefore." In this chapter Kline asks and answers the age-old question, why should anyone study mathematics?
Each chapter of Kline begins with a quote. Here is the Chapter 1 quote:
Mathematics is the gate and key of the sciences.... Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of this world. And what is worse, men who are thus ignorant are unable to perceive their own ignorance and so do not seek a remedy. -- Roger Bacon (13th century English philosopher)
Kline then proceeds:
"Perhaps the most unfortunate fact about mathematics is that it requires us to reason, whereas most human beings are not convinced that reasoning is worthwhile."
And this quote applies directly to our classes: Perhaps the most unfortunate fact about math class is that it requires us to reason, whereas most students are not convinced that reasoning is worthwhile. So I am reading Kline as my side-along reading book because much of what Kline writes about math applies to the math classes that we teach.
So many students wonder, "When will we ever use math?" Kline provides his answer to this question:
"The primary motivation for the development of mathematics proper and the primary reason for the great importance of this subject is its value in the study of nature" -- that is, science.
This is the underlying theme of the first chapter and ultimately the entire book. Math matters when, and only when, it can be applied to science.
Throughout this chapter, Kline provides examples of problems that are solved using math. We've discussed some of these problems in earlier posts. For example, Kline mentions how selling apples at two for a nickel and oranges at three for a nickel is not equivalent to seeing fruits at five for a dime -- this goes back to the idea of harmonic mean, mentioned in David Kung's lectures. And he also mentions that a highway surrounding the earth at a height of one foot is only 2pi feet longer than the circumference of the earth. This is related to the question the traditionalists were complaining about earlier for being wordy and giving distracting answer choices, like "Einstein" instead of "a person."
Yes, of course I'm mentioning traditionalists in this post. I could label every single post during my reading of the book with "traditionalists," but instead I'll use the "Morris Kline" label. Recall that Kline was the father of the progressive math reform movement. I'll still use the "traditionalists" label during this reading if I go into a more detailed contrast of Kline vs. traditionalism.
But here's one key difference between traditionalism and Kline's ideas: Traditionalists say that because math applies to the sciences, students should master math before they even begin advanced studies in the sciences. And of course, mastering math, to the traditionalists, entails putting in hours and hours of practice on individual problem sets. Only after consistently getting right answers to these math problems may students even open an advanced science book.
Kline's ideas, meanwhile, lead to the opposite conclusion. By being exposed to the sciences, students can begin to appreciate why math is important. I've seen some people say in the comment sections of Common Core articles that they didn't truly learn math until they took advanced science. This is because during math class, they didn't put in the time needed to master it because they found the subject to be unnecessary -- but once they took science, they were then motivated to devote time to studying math. Indeed, a natural consequence of Kline's ideas is that there should be no math class -- all math should be studied as part of science class, when students will be motivated to learn it.
I've mentioned the Physics First movement before. This is the idea that the first science class a student should see as a high school freshman is Physics, not the more typical Biology. By teaching Physics First, students may be more motivated to learn Algebra in their math classes. Of course, traditionalists oppose this idea. They argue that since much of Physics depends on math, Algebra II, Pre-Calculus, and possibly even Calculus should be completed before they even start Physics.
We can tell easily that Kline's book was written more than half a century ago:
"To confine our activity entirely to physical investigations or observations may lead to getting lost in a jungle of physical facts...or it may mean the at present impossible task of getting to the moon."
What should we about pure math that has no scientific application? Kline writes:
"Indeed, the mathematician who advances concepts that have no physically real or intuitive origins is almost surely talking nonsense.... From dust thou art to dust returneth may perhaps not be spoken of the soul but it is well spoken of earthborn mathematics."
Today I also begin reviewing for the PARCC and other Common Core tests. I chose today because in my district, today is when the official SBAC testing window for juniors opens. Even though the window is open, I suspect that the high schools will wait until after the AP to give the SBAC. This suits me just fine -- with the last day of school in the district earlier than last year and the AP at the same time as last year, the SBAC is indeed given closer to the end of the year, which I prefer.
And I timed it here on the blog so that I read the first chapter of Kline's book and cover the first question on the PARCC Released Test Questions on the same day. Recall that these won't be the same as the questions I covered last year. And again, I give practice PARCC questions, even though I live in California (an SBAC state) because the PARCC questions are easier to access. Here is the link to this year's Released Test Questions:
Question 1 of the PARCC Practice Test is on the area of a circle:
1. The circle has a radius of 12 units. Shade an area of 24pi square units.
This question is straightforward. We know that for a circle, A = pi r^2 -- indeed, Kline tells us this in Chapter 1 of his book:
"If the area of a circle could be shown by reasoning to be pi times the square of its radius, then the area of any circular piece of land should also be pi times the square of its radius."
So the area of the entire circle must be 144pi square units. We need to shade in 24pi/144pi = 1/6 of the circle. But students may have trouble with the computer interface for shading in the circle:
Divide the circle into the correct number of sections by selecting the "More" button. If you divide the circle into too many sections, use the "Fewer" button. Then, select the number of sections to represent the answer.
So students are to divide the circle into six sections, and select one of them to represent 1/6. Besides trouble with the computer, another common student error is to confuse A = pi r^2 with C = 2pi r. We notice that the circumference of the circle is 24pi units -- and they are asked to shade in the exact same number of square units as the circumference of the circle in linear units. So I can easily imagine a student trying to shade in the entire circle.
Believe it or not, this type of question doesn't appear in the U of Chicago text! Lesson 8-9 of the text covers only the area of a whole circle. The previous lesson does indeed cover both the circumference of an entire circle as well as the length of an arc of the circle. Yet Lesson 8-9 fails to cover the area of the part of a circle known as a sector.
Other texts do cover the areas of both whole circles and sectors -- indeed, the lesson on circle area is often called "Areas of Circles and Sectors." Sector area is, admittedly, a glaring omission from the U of Chicago text. This is exactly the reason that I go over the entire PARCC practice test at the end of the year -- I use this time to make sure that everything that appears on the Common Core tests is covered at some point on the blog. I want to fulfill my duty, as the author of a Common Core Geometry blog, to mention all the topics that may appear on the Common Core Geometry tests.
But the issues with entering the answer into the computer are serious. Some people believe that tests shouldn't be given on the computer -- instead, they should be pencil-or-paper only. And they'll point to questions like today's as evidence for their anti-computer position.
I've mentioned before that I like the idea behind the SBAC -- as a computer-adaptive test, students can take tests tailored to their strengths and weaknesses. But this question comes from PARCC -- and unlike SBAC, PARCC is not computer-adaptive. Furthermore, an advantage of computerized tests is that the results could be given quickly -- but we know that the results of the PARCC and SBAC tests still take months to be released. So the putative advantages of having a computerized test are nullified, so they certainly don't outweigh the possible student confusion when entering the answers.
I've said it before, and I'll say it again -- the computerized test should be set up so that every response is trivial to answer. This includes multiple choice as well as free response questions where the answer is a whole number. Integers are OK as long as the students know how to enter negative numbers, as are rational numbers if the fraction bar or decimal point are easy to find. Every question should be rewritten so that either a letter or a number is a sufficient response.
So a much better question to ask is:
The circle has a radius of 12 units. What is the area of 1/6 of the circle?
Find the area of a sector with a central angle of 60 degrees and a radius of 12.
In either case, a diagram showing the sector is shown. The answer of such a question would be 24pi square units -- and then we'd still have the problem of having to enter "pi" in the answer "24pi." Here we have several options:
-- Fill in the blank: ________pi square units. This is my preferred solution.
-- Round off to the nearest square unit (75) or nearest tenth (75.4). This works better if the question is in the calculator section. (Unlike last year's test, this year's doesn't specify calculator use.)
-- Set the question up so that the answer is a whole number, such as:
Find the area of a sector with a central angle of 60 degrees and a radius of 12/sqrt(pi).
Then the answer would be just 24 square units, rather than 24pi. Theoni Pappas, on her Mathematical Calendar 2016, often uses this trick to make her answer be a whole number (the date). Last Sunday's question the area of an equilateral triangle circumscribing a circle of radius sqrt(17/(3sqrt(3))), with the answer turning out to be 17 square units. As we can see, it's difficult to make the area of a sector or equilateral triangle come out as a whole number -- in both cases, not only do we need irrational square roots, but the radicands are themselves irrational! These will be unwieldy for the students to manipulate just to get a whole number answer.
Of course, there's one more way to get the area of the sector to be a whole number, without any irrationals appearing at all:
Find the area of a sector with a central angle of 1 radian and a radius of 12.
The answer here is 72 square units. The problem, of course, is that radian measure isn't taught to Geometry students, but only in Trigonometry and above. Ironically, the Common Core Standard that I quote at the end of this post actually mentions radians -- but recall that the high school standards aren't divided into courses (that is, the content of a single standard is actually taught across several different courses). Radians don't actually appear on the PARCC Geometry test.
Again, I point out that simply making the answers easy to enter isn't enough. I'd make it so that all the tests are computer-adaptive, with the test results given instantly as soon as the test is over. Any question that can't be scored by a computer in real time (for example, Performance Tasks on the SBAC) isn't worth asking.
PARCC Practice EOY Question 1
U of Chicago Correspondence: Lesson 8-9, The Area of a Circle
Key Theorem: Circle Area Formula
The area A of a circle with radius r is A = pi r^2.
Common Core Standard:
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Commentary: Lesson 8-9 of the U of Chicago text only covers the area of entire circles, rather than sectors. To benefit those following this blog and the U of Chicago text, today's worksheet includes some extra sector area problems.