*ascending*on Descending Drive.

But today marks another post in my "How to Fix Common Core" series. The previous post of this series marks its climax, since then I discussed how I would fix the Common Core tests. But in this post, I want to discuss some more concerns of the traditionalists who oppose the Core -- and what better way to do that than to look at the traditionalist websites directly.

Today I discuss two posts from the traditionalist blogger Dr. Katharine Beals. Two commenters on these posts are Dr. Barry Garelick and SteveH. So therefore in these two posts, I address three of the traditionalists listed in my last post.

Beals begins by discussing the fourth grade Common Core standards. In particular, she provides three questions from the fourth grade SBAC:

http://oilf.blogspot.com/2015/07/math-problems-of-week-common-core_10.html

Some of the comments were about the fact that the SBAC is a computer adaptive test. The commenter Auntie Ann writes:

What a computer input nightmare! Those problems would take about 3 seconds with a paper and pencil, but at least 30 on the computer--with a far higher risk of input error.

and then an anonymous commenter adds:

My daughter (5th grader) came out of the SBAC tests - both math and English - feeling uncertain about how she did. Questions as easy as these wouldn't have thrown her (though she did complain about the interface for entering fractions).

The tests are adaptive, so kids out there are getting questions with different difficulty levels. I'm not a fan of the adaptive test - it seems like the adaptive nature of the SBAC test will add another layer of opaqueness to the testing results.

Let me address the second comment first. I prefer adaptive tests because many don't like the idea of the tests being "one-size-fits-all." One great way to make the tests

*not*"one-size-fits-all" is for the computers to ask different students different questions. And which students should receive the more difficult questions? Of course, the more difficult questions should go to the smarter students -- that is, the ones who answered previous questions correct. Therefore, the tests should be adaptive.

But if the tests must be taken on computer, I do agree with making the answers easy to enter. The next anonymous comment tells us how we should do this:

re Auntie Ann - in my child's school the kids were finding it very difficult to answer these questions where you have to click on the answer (rather than just use the numbers on the keyboard).

OK, then, let's officially include that as a proposal:

**Suggestion #6: The computer tests should allow students to enter numbers on the keyboard, including the use of Num Lock and the slash key to the right of Num Lock to input fractions.**

**But still, the current SBAC doesn't take advantage of the full power of a computer-adaptive test. To see this, let's look at the concerns of the traditionalist commenters. We begin with SteveH:**

Fourth grade. And still the results will be bad for many schools. It reminds me of what our town has done with NCLB. The test took the raw percent correct score and converted it into a "proficiency index" with a low cutoff point well below 70% correct. Then they converted it to the percent of students who get over that low cutoff. We're looking better now. Next, our town looked at how we compared to others in our small state. We're fourth in the state! Send out the press releases about quality education.

But the following anonymous commenter disagrees with SteveH:

Steve, I want to believe you really are what you claim to be. But every once in a while you post things that just don't make sense. The raw percent correct does nothing to show whether a cutoff point for a particular test is high or low. The math involved is not particularly complicated, and I would expect anyone who had an education in a technical field to understand this.

Let's figure out what's going on here. Recall that even though the SBAC assigns four different scores, 1, 2, 3, and 4, only two scores actually matter -- proficient (3 and 4) and deficient (1 and 2). SteveH doesn't like the possibility that someone can earn less than 70% and still be labeled "proficient" -- after all, on most grade scales 70% is a C, and surely one should have to earn at least a C in order to be "proficient" at a subject.

In fact, later in the thread, the other traditionalist, Garelick, gives a much lower percentage than 70% for the proficiency cutoff:

And Steve didn't say it does. He is saying that states, school districts, etc., set cut-points for "basic", "proficient", "advanced proficient" and so forth. The cut-point for "proficient" may be well below 70% correct--say 40%. If 80% of the students taking the test scored better than 40%, the what gets posted publically is that 80% of the students in a school district are at the "proficient" level. What that statistic does not tell you is what the average raw score is. If 80% of the students averaged 45%, that tells a different story than saying that 805 of the students are proficient.

Percentages like 40% and 45% are F grades on most grading scales. Clearly, Garelick and SteveH are concerned that students can earn F grades on the test and still be labeled "proficient."

But here's what I say: 40% doesn't always indicate

*failure*. After all, a baseball player with a .400 batting average would be one of the best hitters of all time, and even a .400 on-base percentage would make a player one of the best hitters of the season. Returning to more mathematical examples, a college student getting 40% of the questions right on a test on the PUTNAM test that I mentioned last December or on the category theory that I discussed throughout July, would be

*impressive*.

Of course, this isn't a category theory test, but the fourth grade SBAC. SteveH seems to imply that if the three questions posted by Beals are representative of what appears on the SBAC, it's not unreasonable to expect students to get 70% of them right in order to be "proficient."

On my proposed test, students have to gain 100 points per year to be considered average for their grade level -- that is, our fourth grader who starts at a score of 400 should reach 500 to be considered the average entering fifth grader. And so we ask, what percent of the questions would students have to get right in order to gain those 100 points?

Recall that on my test, students can earn one or more points for correct answers depending on the difficulty of the problem, while student lose one point for incorrect answers. But notice that even if we knew that every single question is worth one point, that doesn't tell us how many questions a student needs to get right. A student could gain 100 points by getting 100 questions correct, or by getting 200 right and 100 wrong, or even 1,100 right and 1,000 wrong.

On this basis alone, we could make the percentage correct arbitrarily close to 50% if the students answer enough questions correct. Of course, in reality students can't answer that many questions, because there's a 30-minute time limit.

Still, we can

*calculate*how many questions there would need to be. If students gain one point for every right answer and lose a point for every wrong answer, and we want the students to earn 100 points and get 70% right to be considered proficient, then we can figure it out using -- after all, this is a math blog --

*algebra*. We can let

*x*= the number right and

*y*= the number wrong. The equations to be set up are:

*x*-

*y*= 100

*x*= 0.70(

*x*+

*y*) --> which can be written as 3

*x*= 7

*y*

*The solution to this system is*

*x*= 175,

*y*= 75. So we can have the student answer 250 questions and answer 70%, or 175, correctly to earn proficient. Of course, 250 questions may sound like a lot, but if these are simple one-digit multiplications, as may appear on the third-grade test, then 250 questions is about eight per minute -- and we

*want*the students to answer those questions quickly.

Of course, for fourth grade, we want there to be longer questions. We may decide that on average, each questions should be worth two points. If we want two-point questions, then we can solve the following system:

2

*x*-

*y*= 100

3

*x*= 7

*y*

*The exact solutions are not integers -- they are approximately*

*x*= 63,

*y*= 27. (We see that this is only an approximation as this gives us 99 points, not 100.) There's a total of 90 questions in 30 minutes, so that works out to be about three questions per minute. Questions that take longer than 20 seconds to answer would need to be worth more points -- these can be balanced out with enough one-point questions to ensure that on average, about 70% right is needed to earn a proficient score.

Now let's get to the other Beals post that I wanted to discuss. We move from fourth grade to algebra, and Beals is herself referring to an article published in Ed Week. Here are links to both the original Ed Week article and the Beals post:

http://www.edweek.org/ew/articles/2015/06/03/common-core-algebra-seen-as-tougher.html

http://oilf.blogspot.com/2015/07/going-deep-with-common-core-algebra.html

We notice that the first commenter on the Ed Week article is none other than Dr. Ze'ev Wurman -- one of the other traditionalists I mentioned in my last post. Now you can see why I included that list of traditionalists -- the same mathematicians regularly post comments on articles that pertain to Common Core Math.

The article discusses how many districts -- in particular, San Francisco -- use the Common Core Standards to discourage eighth grade Algebra I. The justification is that Algebra I contains topics formerly included in Algebra II -- such as exponential functions -- that may be too difficult for eighth graders, and so the district recommends that students wait until ninth grade to take Algebra I.

We already know that this is unacceptable to the traditionalists, because only by taking Algebra I in eighth grade can students hope to reach Calculus by senior year. In his comment, Wurman takes this even a step further -- about the inclusion of exponential functions in Algebra I, he writes:

In terms of "difficulty," Common Core Functional Algebra I deals early with more complex functions such as exponential or piece-wise linear functions because students are not expected to be able to handle them beyond knowing what they are and feeding them to a machine. Consequently, this allows proponents to claim that it deals with "more advanced" concepts. Traditional Algebra I develops technical skills that on one hand support the sciences (e.g., balancing chemical equations, solving mixture problems, solving gas laws problems, solving distance and velocity problems, etc.) and on the other hand are a solid basis for further study of Algebra 2 and analytical geometry and trig. Consequently, it develops more slowly and more coherently.

So we see that according to Wurman, the claim that Algebra I is too difficult for eighth graders because it teaches exponential functions falls apart.

Moreover, many traditionalists argue that the Common Core Standards don't even expect juniors to finish a true Algebra II course, but only expect "pseudo-Algebra II." Beals tells us what she expects to see in a true Algebra II course:

Here are some examples of what they are missing, and will miss--throughout their years of Common Core-inspired high school math:

--simultaneous equations involving more than two variables.

--simultaneous quadratic equations.

--equations involving abstract quadratic patterns.

--real-world situations that take some real mathematical thinking to model mathematically.

In the original Beals post from which I am quoting, she provides a link to an old Wentworth Algebra text from about 100 years ago. I have nothing against old texts -- after all, we've been discussing the old Legendre Geometry text from about 200 years ago. Of course, the feeling I get from the traditionalists is that no text in the last 50 years or so covers high school math adequately. (One argument they give is that ever since photographs have been inexpensive to print, the texts contain too many photos and not enough math.)

Let's look at each of these missing topics in more detail. I agree that I always think about systems of equations in more than two variables -- or to be precise, in

*three*variables, as all of the Wentworth examples in the link are in three variables -- as being Algebra II. Indeed, whenever I tutor an Algebra II student, I always tell him that systems in three variables are the first real Algebra II topic -- all the previous topics in the text are actually a review of Algebra I. The last topic listed there, "situations that take some real mathematical thinking to model," linked to a Wentworth problem that also lends itself to a three-variable system.

I wasn't quite sure what Beals meant by "simultaneous quadratic equations," at least not until I clicked on the Wentworth link. The first problem given there is:

"The difference of two numbers is 6, and their product exceeds their sum by 39. Find the numbers."

That is, this is a word problem that can be modeled with a system of equations, and when it is reduced to one variable, the resulting equation is quadratic. OK, I rarely see this type of problem in any Algebra II text, whether pre- or post-Common Core.

"Equations involving abstract quadratic patterns" was another mystery to me, until once again, I had to click on the Wentworth link. Actually, I have seen this type of problem in Algebra II texts before -- there they went by the name "quadratic in form." The first example is

*x*^4 -5

*x*^2 + 4 = 0 -- Beals explains how one has to make the substitution

*y*=

*x*^2 to obtain the equation

*y*^2 - 5

*y*+ 4 = 0, which is a quadratic equation that the students already know how to solve. Recall the old joke I mentioned in a prior post about mathematicians who seek to reduce problems to those already solved -- that is definitely the key to solving equations that are quadratic in form.

I've tutored Algebra II students to solve equations that are quadratic in form before. I suppose that my experience tutoring them is not that different from that of the following commenter:

Unfortunately, a lot of students have trouble with these kinds of exercises. The ones I run across tutoring get stumped with problems that look like #1, which is really rather easy if you know your stuff.

Notice that this commenter goes by the username "Niels Henrik Abel." That name should look familiar to readers of this blog, as it refers to one of the two 19th-century mathematicians who independently proved that the quintic is the "equation that couldn't be solved."

We look at the other problems, which get increasingly complex. The last problem on the page, converted into ASCII, is, 2

*x*^2 - sqrt(

*x*^2 - 2

*x*- 3) = 4

*x*+ 9. Beals explains that a clever substitution, namely

*y*= sqrt(

*x*^2 - 2

*x*- 3), converts this into 2

*y*^2 -

*y*- 3 = 0. (Beals makes a typo here -- the middle term should be negative, not positive.)

We notice that the December post in which Beals posted these problems is titled "Puzzle Math, II: abstract pattern recognition." She writes:

And how many self-proclaimed math experts realize, in the spirit of Puzzle Math, just how much fun it can be to unearth or sculpt out the underlying patterns and use them to disentangle what at first glance seems hopelessly disordered, unaesthetic, inelegant, and, in short, unpatterned?

Now here's the problem I have with these questions: once again I'm thinking not from the perspective of the A student who delights in spending time solving these "puzzles." Rather, I'm thinking from the perspective of the D or F student, the poor sophomore (i.e., the grade level at which the traditionalists want students to take Algebra II) whose parents tell him that he can't join any school sports, get his drivers license, or work at a part-time job until he raises that math grade. The students works hard until he finally learns how to solve quadratic equations, and then he's greeted with this page of 20 Wentworth equations that are quadratic in form, with all those cubes and fourth powers and square roots everywhere.

(The fact that this was posted on Saturday, December 6th is ironic, considering that the first Saturday in December is, if you recall, the day of the PUTNAM exam. To our hypothetical student, this assignment is as hard as the PUTNAM -- and college students

*choose*to take the PUTNAM. He has to do this assignment for a

*grade*!)

Surely this student wouldn't think of this as a fun "puzzle" to solve. If, say, a jigsaw puzzle is too difficult to solve, he can just put all the pieces back in the box and do something else. No jigsaw puzzle stands between him and the ability to play sports, drive, or earn money after school the way that these 20 problems would. In short, I'm thinking about the student who sees math as a

*barrier*that

*closes*doors, not an

*opportunity*that

*opens*doors (to some STEM job in which he has zero interest).

I know that when I tutor students on systems in three variables, many of them complain that it takes too long to solve them. They often feel that if it takes more than a minute to solve a problem, that's too long and the problem isn't worth solving. Let's look at Question 20 above -- where we left off at the equation 2

*y*^2 -

*y*- 3 = 0 -- and finish solving the problem, as Beals suggests, by factoring:

2

*y*^2 +

*y*- 3 = 0

(2

*y*- 3)(

*y*+ 1) = 0

2

*y*- 3 = 0 or

*y*+ 1 = 0

*y*= 3/2 or

*y*= -1

This gives us two values for

*y*, but we are supposed to find

*x*, not

*y*. Recall that we made the substitution

*y*= sqrt(

*x*^2 - 2

*x*- 3), and we try plugging in the first value of

*y*:

-1 = sqrt(

*x*^2 - 2

*x*- 3)

Notice that the radical symbol ("sqrt" in ASCII) refers to the

*principal*square root, and the negative number -1 cannot be the principal square root of anything. So we only have to consider

*y*= 3/2. At this point, our student may be glad that he only has to deal with one value of

*y*, but upset that it involves a fraction:

3/2 = sqrt(

*x*^2 - 2

*x*- 3)

9/4 =

*x*^2 - 2

*x*- 3

9 = 4

*x*^2 - 8

*x*- 12

4

*x*^2 - 8

*x*- 21 = 0

(2

*x*+ 3)(2

*x*- 7) = 0

2

*x*+ 3 = 0 or 2

*x*- 7 = 0

*x*= -3/2 or

*x*= 7/2

I noticed that in all of the Wentworth problems involving radicals, the equation in

*y*always has two solutions -- one positive and one negative. Had both values of

*y*been positive (or zero), we would have had to solve

*three*quadratic equations to solve

*one*problem -- one equation in

*y*and

*both*of the equations for

*x*-- and of course our student would complain that he has to solve three equations just to finish one question. As it stands, he will complain that he has to solve

*two*equations just to finish each question.

And that's assuming that he doesn't make a mistake. We saw Beals

*herself*make a sign error when solving this problem. I'm not blaming her for making that mistake since, after all, anyone can make a sign error (probably the most common error in any algebra class). But that's just it --

*anyone*can make that mistake, including the students. Imagine how our hypothetical student would feel if he made the same mistake. He probably wouldn't notice it at first, since all that sign error would do is switch the sign of the answers (

*y*= -3/2 or +1 instead of +3/2 or -1). But if we try to solve for

*x*using the value of

*y*= 1, the resulting quadratic is not factorable. The student will now feel frustrated since it could take him a long time to find the sign error.

Let me tell you something. I enjoy solving math problems, and if I were a high school student assigned to solve these problems, I might delight in finding the clever substitution that would solve these problems --

*at first*. But if I were given a problem like Question 20 --

*especially after having spent considerable time on 19 other problems*-- even

*I*would be fed up with this assignment. And if I made the same sign error that Beals made -- even

*I*would consider it tough to find and fix. All I want at that point is just to be

*done*with this assignment so I can move on to something else -- which could be homework for another class.

Here's another thing that the traditionalists also forget. At this age, teenagers often ask questions in math classes such as "Why do I have to learn this?" or "When will I ever need to learn this?" And while one might be able to give a real-life justification of a problem requiring a system of equations (even one with three variables), one would be hard-pressed to find any real-life problem requiring an equation quadratic in form with radicals. So now our student is spending a long time, making sign errors left and right, all to solve an equation that he will never need to use in real life.

I suppose the only reasonable answer to "When will I ever need to know this?" is Calculus -- the clever substitutions used to reduce the problem to a quadratic are similar to the

*u*-substitutions that

are often used for integration. (Indeed, I'm surprised that Algebra II teachers don't use the variable

*u*, rather than

*y*, for these substitutions to get the students thinking ahead to Calculus.) But I doubt that our student will accept "Calculus" as the answer to his question -- all this long work in a hard math class to prepare him for even longer work in an even harder math class.

Now I keep referring to this

*hypothetical*struggling student -- but keep in mind that according to Beals, an

*actual*student completed these Wentworth problems:

I’ve already blogged about the abstract patterns in math that are vanishing from today’s algebra classes. But my daughter just completed a particularly elaborate set of problems in her 1920s algebra book (Wentworth's "New School Algebra") for which abstract pattern recognition is absolutely essential:

Recall that Beals is a homeschooling parent -- her daughter, an eighth grader, is already working on Algebra II problems. The key word is that the daughter

*completed*these problems -- the young teen presumably didn't complain about how hard the problems are, how long they take to solve, how easy it is to make a mistake, or how useless they are in real life. That's lucky -- we know that in an actual school classroom, it's almost inevitable that these complaints will arise.

Now let's return to the subject of this post -- how to fix Common Core. Should the sort of question that Beals likes appear on the Common Core tests? On one hand, I suppose that a multi-step problem similar to those appearing in Wentworth could appear on the computer-adaptive test. Such a problem would certainly be worth more than one or two points. Our conversion rate calculated above is that it should be worth about six points per minute (since it takes about as long as three two-pointers) that such a problem would take. So if you believe that such a problem takes, say, five minutes to solve, then it should be worth 30 points. This makes sense -- the student will have displayed many important Algebra II skills (substituting, simplifying, factoring, solving) in doing this one problem. Missing the problem will still cost only one point (especially if all the student did was take a few seconds to guess), but one may wish to award partial credit for certain wrong answers. For example, in the above problem the right answers are

*x*= -3/2 or 7/2. Wrong answers for partial credit include giving just one of these two answers, giving the wrong signs (3/2 or -7/2), giving

*y*instead of

*x*(3/2 or -1), including extraneous answers because one tried to solve the case where

*y*= -1, and so on.

But there's a problem here. Our proposed computer-adaptive tests only go up to Pre-Algebra because we wish to remain neutral regarding the traditionalist vs. integrated pathway. Indeed, we proposed the ACT as the test for students who completed Algebra II to take. For all classes beyond Pre-Algebra, we can only give standards, not specific test questions -- then the test writers would based the tests on the standards (as ACT is said to be aligned to the Common Core Standards).

Many traditionalists prefer homeschooling because they say that they can teach their children at the level that they wish without having to worry about the other students. Recall that Beals's daughter is an eighth grader in Algebra II. Some traditionalists say that if only the school would place, say, their eighth-grade child in a full Algebra II class, they wouldn't need to homeschool -- the eighth-grader would easily get an A in the Algebra II class.

But it's tough for public schools to do this -- especially since eighth graders go to middle school, a separate campus from most of the Algebra II student. Let's say that my proposed test and its scoring system show that the eighth grader is so far above grade level that she belongs in Algebra II. But most Algebra II students are in high school -- on a different campus. Even the smallest classes at a public school have about 20 students, so unless we can find 19 other eighth graders who are able to take Algebra II on the middle school campus, the cost of having a class there would be unjustified. Of course, some charter schools in certain states, such as BASIS (mentioned on my blog in April), have no problem with letting eighth graders take Algebra II.

Meanwhile, in Washington State, there is a debate concerning the use of the junior SBAC exam as an exit exam. The original proposal was that students had to earn at least a score of 3 on each section -- ELA and math -- to earn a diploma. Now it's possible that scores in the higher 2-range will allow the student to graduate.

The exam tests up to Algebra II, so this makes Algebra II a

*de facto*graduation requirement. Then again, according to the traditionalists this is only

*pseudo*-Algebra II. If genuine Algebra II denotes problems at the level of the Wentworth text, then

*no*-- that level of math absolutely should

*not*be a requirement for a diploma. Students who can't solve equations that are quadratic in form with radicals should not be

*blocked*from having a diploma -- the only real door to a job (even a non-STEM job) and a middle-class lifestyle.

This is the problem with the traditionalist mindset -- the view that math is only an opportunity to open doors and never a barrier that closes doors. In Beals's most recent post, she writes:

What's challenging and interesting about negative numbers, for example, isn't that they represent numbers less than 0 or numbers on a particular side of 0 on the number line, or that they have such concrete instantiations as distances below sea level or temperatures below freezing. What's challenging about negative numbers is grasping that a negative times a negative is a positive and correctly distributing and multiplying out the negative numbers in a complex expression. A class that spends two weeks on ways on which negative numbers correspond to distances relative to sea level is wasting precious time and making students think that negative numbers are boring.

Now suppose during the lesson on negative numbers, a student who sees math as a barrier asks, "When will I ever need to know this in the real world?" Seeing that negative numbers correspond to distances below sea level in the real world sounds like a reasonable answer to that question to me. If something is

*challenging*-- as Beals admits negative numbers are -- the first response from the many students who see math as a barrier is

*not*that it is "interesting," but that they shouldn't have to learn it unless it will be relevant to them in the real world.

The first commenter, who goes by the name lgm, wrote the following:

I had to explain why a negative times a negative is a positive to my child. His teacher left the definition out. Working more problem sets wouldnt have helped as it would have just resulted in slightly more rote memorization of cases. He needed good teaching to know it and use it down the road. The answer for me is thus a decent textbook, to make up for being stuck in a class of uninterested students who just want a pass; or grouping by instructional need, which would be a politically incorrect honors program. Having neither from the public school, we bought a used Dolciani and the child learned. Khan takes too much time...reading and discussion is faster.

We recognize the "uninterested students who just want a pass" as the students who feel that math is a barrier to their future success. "A used Dolciani" clearly refers to the "decent textbook" -- most likely one from decades before the Common Core -- that the author desires for the child. A quick Google search reveals that this is indeed the case (but Dolciani isn't quite as far back as Wentworth). And "Khan" of course is the Khan Academy -- the most well-known online math tutor.

But then we see the phrase "politically incorrect honors program." I am unsure what lgm means here, but it probably refers to an honors program where entry correlates with demographics -- most likely either gender or race.

I've discussed gender several times here on the blog, including the efforts of several writers (such as Dr. Danica McKellar) to make sure that girls learn math as well as the boys do. Judging by the later comments, lgm could very well be referring to gender. But it's also possible that lgm refers to race -- recall that many schools abandoned tracking because it led to segregation. We see that lgm's son would do well on the higher track, but schools don't provide a higher track because of the tendency to place students of other races on the lower track.

At this point I'm afraid I have to get into politics. As I've said before, I try to avoid politics whenever I can, but the problem is that Common Core is inherently a political issue. Still, I try to stick to the politics of, if not Common Core itself, at least education. Readers who prefer to avoid politics completely can just ignore the rest of this post.

A few years ago, I read about someone who was placed on the lower track due to his race, despite being more than academically capable for the higher track. Now here's the political part -- the person I am describing here just happens to be Dr. Ben Carson -- a declared presidential candidate who even participated in last night's debate. Indeed, I am a little reluctant to tell this story due to the author's ties to politics. But I admit that I read this story years before he considered running for president. If I'm going to post anything that seems political, I'd much rather post it now during summer vacation than on a school day when I should be posting mathematics only.

And that leads to another problem with this story -- I can't remember in which of his books does Carson tell this story. It's another case where I read something years before I post it on the blog, and then can't find the exact source when I decide to post it. I keep scanning his books to find the story, but to no avail. Here's all that I remember -- both the future doctor and his brother were placed on the lower track when entering the high school despite each graduating at the top of his respective class at his (racially mixed) middle school. Their mother had to complain to the school before they were finally placed on the higher track.

It's stories such as this that gave tracking a bad name. Many traditionalists state that tracking would be beneficial to the students who are above grade level, but know (as lgm does above) that too many people would track by race. Some argue that dividing students into grade levels is also a sort of "tracking" -- in this case, tracking by age. There appears to be only two choices -- tracking by age, or

*de facto*tracking by

*race*-- and in order to avoid the latter, schools end up choosing the former.

This is why, whenever I state my own "tracking" system (the path plan), I try to include checks into the system in order to allow students of all races -- including potential Dr. Carsons -- to succeed. My path plan doesn't actually divide students into tracks, where students on the lower track are doomed to receive a substandard education. Instead, students are divided into paths that don't line up exactly with the grades, so that an older student who is below grade level may be in the same class as a younger student who is above grade level. This makes it harder to place members of a race on a lower track to give them a substandard education, since they would be in the exact same class as younger members (of other races) who are above grade level.

Of course, members of a race may still find themselves always on paths that correspond to being below grade level. My other check is to have the

*computer*score determine on which path a student should be placed, rather than a human teacher who may (perhaps subconsciously) track by race. No system is perfect, but my hope is that this avoids at least some problems associated with tracking.

I have already mentioned one of the participants of last night's presidential debate. In order to avoid appearing to endorse one candidate over another, I post a transcript of a short part of the debate -- in

particular, the part where Common Core is debated.

Governor Jeb Bush supports the Common Core standards, while Senator Marco Rubio opposes them and prefers state control. I stated my own preference -- I have no problem with national standards as most of the nations that we wish to emulate have such standards. Meanwhile, some countries also have control at the level of (their analog of) states. And so, while 50 sets of standards is a lot, I have no problem with state control either. So I favor neither Bush nor Rubio in this debate.

This transcript is courtesy

*The Washington Post*.

http://www.washingtonpost.com/news/post-politics/wp/2015/08/06/annotated-transcript-the-aug-6-gop-debate/

President Obama's secretary of education, Arnie Duncan, has said that most of the criticism of Common Core is due to a, quote, "fringe group of critics." Do you think that's accurate?

BUSH: No, I don't. And I don't believe the federal government should be involved in the creation of standards directly or indirectly, the creation of curriculum or content. It is clearly a state responsibility.

I'm for higher standards...

(APPLAUSE)

BUSH: ... measured in an intellectually honest way, with abundant school choice, ending social promotion. And I know how to do this because as governor of the state of Florida I created the first statewide voucher program in the country, the second statewide voucher program, in the country and the third statewide voucher program in the country.

And we had rising student achievement across the board, because high standards, robust accountability, ending social promotion in third grade, real school choice across the board, challenging the teachers union and beating them is the way to go.

And Florida's low income kids had the greatest gains inside the country. Our graduation rate improved by 50 percent. That's what I'm for.

BAIER: Senator Rubio, why is Governor Bush wrong on Common Core?

RUBIO: Well, first off, I too believe in curriculum reform. It is critically important in the 21st Century. We do need curriculum reform. And it should happen at the state and local level. That is where educational policy belongs, because if a parent is unhappy with what their child is being taught in school, they can go to that local school board or their state legislature, or their governor and get it changed.

Here's the problem with Common Core. The Department of Education, like every federal agency, will never be satisfied. They will not stop with it being a suggestion. They will turn it into a mandate.

In fact, what they will begin to say to local communities is, you will not get federal money unless do you things the way we want you to do it. And they will use Common Core or any other requirements that exists nationally to force it down the throats of our people in our states.

(APPLAUSE)

BAIER: And do you agree with your old friend?

BUSH: He is definitely my friend. And I think the states ought to create these standards. And if states want to opt out of Common Core, fine. Just make sure your standards are high.

Because today in America, a third of our kids, after we spend more per student than any country in the world other than a couple rounding errors, to be honest with you, 30 percent are college- and/or career-ready.

BUSH: If we are going to compete in this world we're in today, there is no possible way we can do it with lowering expectations and dumbing down everything. Children are going to suffer and families' hearts are going to be broken that their kids won't be able to get a job in the 21st Century.

My next post will be early next week, on spherical geometry. After that, I hope that by the time of my next post in the How to Fix Common Core series, the SBAC scores for California will have been released so that I can analyze those in that post.
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