Friday, October 24, 2014

The Common Core Debate, Grade 3 Continued: Proposed Standards

Today was my second day as a substitute teacher. I ended up subbing for an English teacher at a continuation school. But I do have something to say that's related to math. For the classes were held in the library, and while the students were watching a video, I found on the shelf an old favorite book on traditionalist math from the 1980's -- Quick Arithmetic (2nd ed.) by Carman and Carman. I always get a laugh at the Peanuts and other comics interspersed throughout the text.

Now that I've stated that I oppose the Common Core Standards for grades K-3, the next question is, what would I replace them with? For this, let's focus on the third grade, since if we can fix the third grade -- and that's both the standards and the standardized tests like PARCC/SBAC based on them -- then that will automatically fix all the grades below that grade.

The goal is to make the standards more traditionalist in the lower grades. Now, if we were to ask any traditionalist what a third grader should learn in math, the answer is obvious -- the multiplication table.

But hold on a minute. The Common Core already seems to have a standard for third grade multiplication. Here is the relevant standard:

Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

So why would a traditionalist object to this standard? Well, one problem is that the third grade standards fall into five domains -- Operations & Algebraic Thinking, Number & Operations in Base Ten, Number & Operations -- Fractions, Measurement & Data, and Geometry. And this above standard is just the seventh standard within one of the domains -- that is, it's relatively buried within.

But a traditionalist whose third grade child mastered four of the five domains, yet didn't know the multiplication tables, would consider the year a failure and seek to blame the teacher, the school, and the Common Core Standards. By contrast, a traditionalist whose third grader knew very little about four of the five domains, yet can correctly answer what six times seven is and all the other single-digit multiplication problems in one second or less, would consider it a success. And a traditionalist who decides to homeschool a third grader in order to avoid the Common Core would begin by teaching that child the multiplication tables.

And so our first step is to give this standard priority over all other standards:

New Third Grade Standard #1:
By the end of Grade 3, know from memory all products of two one-digit numbers.

Notice that I dropped the first part about division. Division is an admirable goal, but until the multiplication table has been memorized, there can be no division.

But it's not enough just to write this standard. In order to emphasize its importance, we must make sure that it has a prominent place in on the PARCC/SBAC.

Recall that both PARCC and Smarter Balanced are computer-based exams. So there should be no problem with imposing a time limit for each question. So we can have the PARCC and SBAC begin by asking the student to answer a simple multiplication problem, such as "6 x 7 =," and imposing a time limit.

How much time should we give the students? We want to make sure that it's impossible to get the right answer without having memorized the table. Earlier, I stated that the student should be able to answer 42 in one second or less. But I wouldn't want to give a one-second time limit -- I myself had already memorized the entire multiplication table by the end of kindergarten, yet I'd be so nervous with such a short time limit that I'd press the wrong keys on the keyboard. So let's give a more reasonable time limit, say ten seconds.

The first 100 problems on the PARCC/SBAC exams can be the 100 single-digit multiplication facts. This will take no more than 1000 seconds, or almost 17 minutes, to complete, leaving time for the other domains.

And now here's where things really get interesting. If the student doesn't earn a satisfactory score on these 100 questions -- if we want to be tough, we can set 90 to be the passing score -- then the test immediately ends and a failing score is reported for the student, without any question from any other domain being asked of the student.

This would immediately drive home the importance of memorizing the times tables to schools that would be judged by their PARCC/SBAC scores. The way that I have proposed structuring the tests, nothing else at all matters in third grade math more than memorizing the multiplication tables. Therefore, third grade teachers should spend however many weeks it takes for the students to know their times tables -- whether it be a trimester, a semester, or even the entire year up to the window for the tests.

Now let's move on to our next traditionalist standard:

New Third Grade Standard #2:
Fluently add and subtract multi-digit whole numbers using the standard algorithm.

This standard is certainly traditionalist, since it emphasizes use of the standard algorithm for addition and subtraction. The problem is that it doesn't appear in the current Common Core third grade standards -- instead, it appears in the fourth grade standards. There is a gap between when the students learn how to add and subtract and when they are to learn the standard algorithm -- and many schools fill in this gap with progressive algorithms instead. We fill in the gap by dropping this fourth grade standard to third grade to avoid this temptation -- and even that may be too late, since nonstandard algorithms might be taught even in second grade. But for now I keep the standard under third grade, since that is the year that I'm emphasizing right now.

Notice that there's nothing wrong with nonstandard algorithms per se -- after all, even Carman's text, the one I mentioned at the start of this post, mentions nonstandard algorithms in order to motivate the use of the standard algorithm as a kind of shortcut. The problem is when the standard algorithm doesn't appear until a year or two later, as in the current Common Core standards.

Here is the nonstandard algorithm mentioned by Carman:

+  95

That is, it adds up one column at a time. But as I mentioned earlier, the standard algorithm is introduced quickly after this nonstandard algorithm is given.

But a progressive algorithm might look like this: to add 132 + 95, we perform an easier addition, 132 + 100, to obtain 232. Then, since we added 5 too much, we must subtract that 5 to obtain 227 as the answer. I admit that often times, I perform mental math in my head using this method, often called the "Plus-Minus Method", but the problem is that the Plus-Minus method is too confusing for a student in the third grade to perform and understand. This is why the emphasis should be on the standard algorithm.

I see nothing wrong with the rest of the third grade standards in Common Core except that they take time away from the most important two. I decided to list only the standards that are the most important for students to learn. The first two standards listed above should form the bulk of the third-grade math course, and so all current standards that take time away from these should be dropped.

Here are the remaining standards that I've deemed the most important:

New Third Grade Standard #3:
Use place value understanding to round whole numbers to the nearest 10 or 100.

Traditionalists have no problems with students learning to round -- the problem is when students only add or subtract the rounded numbers and never learn how to add the unrounded numbers.

New Third Grade Standard #4:
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

New Third Grade Standard #5:
Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

New Third Grade Standard #6:
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

New Third Grade Standard #7:
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Notice that the current third grade standards include fractions. Using the number line to motivate fractions is a strong traditionalist method. I specifically mention Standard #7 in reaction to a story I once read where McDonalds customers didn't know that a Third Pounder would be larger than a Quarter Pounder -- because they couldn't compare two fractions with the same numerator, 1/3 and 1/4.

New Third Grade Standard #8:
Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

The other main traditionalist method of teaching fractions is by dividing areas, so this standards follows naturally from the fraction standards. As much as I like geometry, the other current third grade standard on quadrilaterals is considered too distracting to be included in a traditionalist curriculum.

New Third Grade Standard #9:
Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

New Third Grade Standard #10:
Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

These standards follow naturally from the others.

My goal is to change attitudes about math. Currently, people who are great at math, say algebra and above, are derided as nerds. Instead, I propose a word to describe those who aren't proficient at basic math, such as the math included in the above standards. Until I come up with a better word, I will spell the word nerd backwards, and call such an adult a dren.

Thus concludes this post. My next post will be on Monday, when I will proceed with Day 52 of the geometry course.

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