How to Fix Common Core: The Middle Grades, 6 and 7

When I think about the opposition to Common Core math, I ask myself the question, which grade level is the least objectionable grade? We know that Core opponents dislike the lowest grades because they feel that math should be more traditionalist, with a stronger emphasis on the rote memorization of basic math facts. On the other end, the problem with the highest grades is that the Core doesn't recommend Calculus in senior year, and so the grades leading up to senior year don't necessarily prepare students for Calculus.

So we conclude that the least objectionable grades are the middle school grades. Sixth and seventh grades are the farthest away from the lack of rote math facts in the earliest grades and the lack of Calculus in senior year.

But even in these grades, the Common Core math standards aren't perfect. Sixth grade is the last grade that mentions use of the standard algorithm to perform arithmetic:

Compute fluently with multi-digit numbers and find common factors and multiples.

CCSS.MATH.CONTENT.6.NS.B.2
Fluently divide multi-digit numbers using the standard algorithm.

CCSS.MATH.CONTENT.6.NS.B.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

The problem is not that these require the standard algorithm -- which is what traditionalists want -- but that any standard mentioning the standard algorithm has a corresponding standard one grade lower where students are expected to learn the same operation using a nonstandard algorithm, along with requirements to explain how they performed the operation. So we see that the sixth grade standards above correspond to the following fifth grade standards:

CCSS.MATH.CONTENT.5.NBT.B.6
Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

CCSS.MATH.CONTENT.5.NBT.B.7
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

So a traditionalist would prefer that the two sixth grade standards 6.NS.B.2 and 6.NS.B.3 be dropped down to fifth grade, and the two fifth grade standards 5.NBT.B.6 and 5.NBT.B.7 be removed entirely from the Core.

The problem with the seventh grade standards is that even as early as the seventh grade we must make sure that students are heading for the door leading to senior year Calculus. To reach senior year Calculus, we must have Algebra I in eighth grade, which means that the seventh grade must have a full Pre-Algebra course. The Core delays some Pre-Algebra standards until eighth grade since the standards are intended to prepare for ninth grade Algebra I.

I've mentioned earlier that I don't see much difference between Common Core eighth grade math and the Integrated Math I course. So we could keep the Core seventh and eighth grade standards, then place freshmen directly into the Integrated Math II course -- this will then put them back on pace to reach Calculus by senior year.

Still, the old planned California adjustment to the Common Core moved some of the Core eighth grade standards down into seventh grade Pre-Algebra. As I mentioned in last week's "How to Fix Common Core" post, the California plan was doomed because it required eighth graders to take a full Algebra I course plus additional geometry topics so that they could pass the SBAC, and that amalgamated course was deemed too much for eighth graders.

Here are some of the eighth grade standards California proposed moving down into seventh grade:

Know that there are numbers that are not rational, and approximate them by rational numbers.

CCSS.MATH.CONTENT.8.NS.A.1
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

CCSS.MATH.CONTENT.8.NS.A.2
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

So far, this post mentions pushing some sixth grade standards to fifth grade and some eighth grade standards to seventh grade. So we might as well push some seventh grade standards to sixth grade -- and this are some of the standards pushed down to sixth grade in the California proposal:

Apply and extend previous understandings of operations with fractions.

CCSS.MATH.CONTENT.7.NS.A.1
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

CCSS.MATH.CONTENT.7.NS.A.1.A
Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

CCSS.MATH.CONTENT.7.NS.A.1.B
Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

CCSS.MATH.CONTENT.7.NS.A.1.C
Understand subtraction of rational numbers as adding the additive inverse, p - q = p+ (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

CCSS.MATH.CONTENT.7.NS.A.1.D
Apply properties of operations as strategies to add and subtract rational numbers.

As it turns out, the Common Core 7 standard in which pi is taught for the first time is also moved down to sixth grade under the 2010 California proposal. So the first Pi Day lesson would be encountered in 6th grade.

CCSS.MATH.CONTENT.7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Since I agree wholeheartedly with the desire for traditionalist math in the earliest grades, and partially with the idea of allowing as many seniors as possible to take Calculus, I can go along with moving most of the standards listed here one grade level down. The only move I am a little concerned with are pushing the seventh grade standards involving rational numbers (with the emphasis on integers) down into sixth grade. I subbed in some sixth grade classes last week and integers is definitely a topic on which many of the sixth graders struggled. Of course, if integers were a sixth grade topic, the teachers wouldn't wait until May or June to introduce integers, which may mean that sixth graders will have more time to understand them. Notice that the California proposal only moves the addition and subtraction of integers into sixth grade -- multiplication and division of integers remain in 7th grade.

So let's make it official:

Suggestion #3: Move several of the Common Core Standards down a grade level, as follows:
-- Any standard mentioning "the standard algorithm" should be moved down, with the corresponding lower grade standard mentioning "strategies based on place value" dropped completely from the standards.
-- The middle school standards that are moved down in the 2010 California proposal to prepare students for eighth grade Algebra I should be so moved.

Recall that while I support the proposed sixth and seventh grade California courses, I don't support the proposed eighth grade California course. I formerly recommended Singapore's New Elementary Math 2 for eighth graders, but this is too similar to the proposed California course. Instead for eighth graders, I prefer Saxon's Algebra 1/2, any known published Common Core 8 or Integrated Math 1 course, my proposed integrated course posted earlier on the blog, or else simply a traditionalist Algebra I course with no geometry thrown in.

Back at the end of March, I mentioned how I had purchased an old seventh grade textbook for $2 at a library book sale. It is the McDougal Littell seventh grade text, and once again, because the book is dated 2001, it is based on the old pre-Core California standards. In March, I discussed only Chapters 8 to 10 of this text, since these are the chapters most relevant to this geometry blog. I now post the full table of contents, since today's blog topic isn't geometry but rather middle school math. This is a good example of a text covering the 7th grade standards given by Suggestion #3 above:

1. Operations with Numbers
2. Operations in Algebra
3. Operations with Integers
4. Algebra and Equation Solving
5. Rational Numbers and Percents
6. Operations with Rational Numbers
7. Proportional Reasoning
8. Geometry Concepts
9. Real Numbers and Solving Inequalities
10. Geometry and Measurement
11. Graphing Linear Equations and Inequalities
12. Polynomials

I've mentioned earlier that my favorite number of chapters in a text is ten -- this way the chapters line up with the months. My second favorite number of chapters, though, is twelve -- especially in a middle school where trimesters are common. The number 12 is divisible by two (six chapters per semester), three (four chapters per trimester), four (three chapters per quarter), and six (two chapters per hexter). Each chapter ends up spanning exactly three weeks rather than one month.

In this text, a lesson on pi (which is actually a review from 6th grade, but necessarily to prepare for the formulas for surface area and volume) appears as Section 10.1. On an Early Start calendar, recall that Pi Day occurs near the end of the third quarter. It's therefore conceivable that three quarters of the text (i.e., Chapters 1 to 9) of the text will be covered by then, and so Section 10.1 can be the lesson for Pi Day itself. On a Labor Day Start calendar, one is most likely to have covered only eight chapters by Pi Day rather than nine, so the pi lesson can't be taught on Pi Day unless one either speeds up the pace a little or switches Chapters 9 and 10. Then again, switching Chapters 9 and 10 might not be such a terrible idea after all -- such a switch unites Chapters 8 and 10 (the two geometry chapters)
as well as Chapters 9 and 11 (inequalities appear in both chapters -- 1D in 9, 2D in 11).

But one might want to speed up a little in order to get to Chapter 11 before the PARCC or SBAC -- which leads to my other suggestion for today. I've already mentioned how to improve testing in the eighth and tenth grades to encourage keeping the door open to Calculus -- eliminating the test completely for eighth graders (so that schools can offer Algebra I without being penalized on the PARCC or SBAC test based on Common Core 8) and using the ACT as a tenth grade test (since the ACT contains Algebra II questions).

So now let's discuss testing for sixth and seventh grades. We know that one concern about Common Core testing is that so much time is devoted to reviewing for the test, as well as the test itself.

I believe that this is a major reason for so many parents opting their students out of the Common Core tests. One problem is that each testing session lasts over a hour. This is not so bad in elementary schools where students have a single teacher the entire day. But in middle schools. students can have five or six teachers for 50-55 minutes each day. So one can't have a testing session of an hour or more without having some sort of block schedule. It doesn't matter if teachers say that only a few days in class will be spent reviewing for the test -- all the parents have to see is the note their child takes home announcing a full month of block schedules, with testing being given as the reason for the block scheduling, and they will want to opt the students out of the test.

So my goal is to reduce the amount of testing -- enough so that no block schedule is necessary. In fact, I wonder whether the testing time can be reduced down to just 30 minutes for the math test. Then this would give, say, 10 minutes to set up the computers and 10 minutes to shut them down, all in the course of a single 50-minute class period, with no block schedule needed.

What sort of questions could be given on such a short test? I expect, under our SPUR categories associated with the U of Chicago test, most of the questions to be Skills questions -- with perhaps one or two Uses questions thrown in. Of course, Properties questions can be quick as well -- but of course, many traditionalists oppose such Properties questions. Of course, questions where students have to explain their work would be gone. There would be no more PBA section of the PARCC, or Performance Task section of the SBAC, as these are the most time consuming sections of the tests.

Of course, we could set it up so that each subject takes only 30 minutes to test -- but then we wonder, what about the writing section? I'm not quite sure what to do about a subject like writing. This is a math blog, and so I will focus only on the subject at which I am an expert, namely math. And 30 minutes should be enough to determine whether a student is at, above, or below grade level, if the questions are chosen well enough.

I also would prefer that the test be given closer to the last day of school. Since under my plan, eighth graders don't have to take a math test, we could schedule the math test while the eighth graders are preparing for their promotion activities -- but then again, eighth graders would still have to take an ELA (and possibly a Next Generation Science) test.

If we look back at the McDougal Littell text above, we notice that Chapter 12 is on Polynomials -- a preview of Algebra I (just as sixth graders are now previewing for 7th grade and integers). We expect a seventh grade exam to test material from only Chapters 1 to 11, so it would be reasonable to give the test a little before the last day of school, at around the 11/12 mark of the year. This would put the test the week before Memorial Day weekend at a Labor Day Start school. At an Early Start school, this would be around the last week of April or first week of May.

By the way, I also wrote earlier that a good computer exam should also test material that is above grade level, if the student responses to previous questions warrant it. When I spent one month in an Honors Math 7 classroom that used the McDougal Littell text mentioned above, the pacing guide suggested skipping to Chapter 12 near the middle of the year before resuming the natural order. Such students would therefore be able to answer polynomial questions on the end-of-course test. The tricky thing about seventh grade is that acceleration here is hard to define -- would an accelerated 7th grader be answering Algebra I questions or Integrated Math I questions (since in my plan and many other integrated plans, polynomials don't appear until Integrated Math II, two years after 7th grade).

Of course, there's no excuse not to test accelerated sixth graders on seventh grade material, since 7th grade material is uniform. If integers are covered early enough in the year, there might be some 6th graders who can handle it.

Suggestion #4: Give the standardized test as late in the year as possible -- if not the last week of the year, certainly during the final 3-6 weeks. Let the math section of the test be no longer than thirty minutes on the computer. Allow students who finish the test in less than thirty minutes to spend the remaining time answering questions above grade level.

My next post will be in about a week.