The Common Core State Standards, as you can tell from the contrasting pieces I published this week (here and here), are nothing if not controversial. Until now much criticism has been leveled at the English Language Arts Standards, but here’s a somewhat scary, detailed look at the math standards. It was published on the blog CCSSI Mathematics.
The author of the blog writes anonymously. He is a former high school math teacher and now an attorney in Manhattan. Knowing the times tables helped him pass the bar exam, he said, and he has won cases by knowing math better than his adversaries. A math major in college, he still occasionally teaches part-time to test out some of his evolving theories of math education.
Here’s the blog post, and if you want to follow along with specific items from the standards, you can find them here:
The results were 83%, 64% and 52% correct, respectively. Why was performance on the last task, arguably the easiest of the three, the worst?
Answer: Probably because it was the only one in which calculators were not allowed.
Without obsessing on the calculator issue, the authors of this blog [CCSSI Mathematics] believe in learning basics, not as rote exercises, but for the purpose of being able to apply them in the furtherance of solving more advanced mathematics problems and equally important, to bring those basics to the abstract level.
We don’t know if the poor results on the third question were due to lack of concept (not knowing it’s multiplication?), weak (multi-digit multiplication) skills, or calculation error.
For the moment, we’ll look at the third factor, because of the other seemingly endless debate (in addition to calculators): times tables memorization. We’re going to try to end this silly debate once and for all (ha!). There is sufficient rationale for knowing one’s times tables: even at the highest levels of mathematics, there comes a time that two numbers must be multiplied. But beyond its obvious utility, there is greater depth to the times tables than just memorization. There are important patterns inherent in the times tables that can lead to a deeper understanding of the nature of numbers and ultimately, lead to more profound problem solving and abstract abilities. The times tables can play a central role in fostering some fairly advanced mathematical thinking, even in elementary school — but more on that later.
The times tables fit as a subtopic in the more general topic of multiplication, the other steps that lead to multiplication facility being how the concept is introduced, multi-digit multiplication skills, and how multiplication is used in problem solving.
Once again, there are problems with the Common Core starting at the very beginning of its treatment of a topic: how it introduces the concept. CCSSI begins a piecemeal approach to multiplication in 2.OA.4:
“Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.”
Without saying as much, this is repeated addition. Certainly that’s how one begins, but then what? Following CCSSI, students will spend an entire summer after second grade thinking that 4 + 4 + 4 + 4 + 4 = 20, and so what? It is both ridiculous and demeaning to leave students hanging with repeated addition as the way to total an array.
Instead, finding the total number of objects in an array by repeated addition should culminate in an introduction to multiplication as a shorthand notation. There’s no rhyme or reason for separating these two concepts.
In other words, repeated addition should be seen, but quickly recognized as an inadequate solution to a common situation, and therefore, should be rejected soon after its appearance. Repeated addition should certainly never be tested.
Following Common Core, it is not until some months later that students reach 3.OA.1:
“Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.’’
How many months later are students introduced to multiplication proper? According to CCSSI, which explicitly avoids mandating the order in which topics are presented, 2.OA.4 could come in the middle of Grade 2 and 3.OA.1 a full year later. Again, any time gap is ridiculous.
Repeated addition as a concept certainly belongs in Grade 2, but multiplication?
If Grade 1 focuses on addition and subtraction, and the first half of Grade 2 is devoted to carrying and borrowing, then in the latter part of Grade 2, students should arrive at the sum of arrays (repeated addition) and an immediate introduction to multiplication as a concept and its notation. Calculating the times tables (yes, the times tables should be calculated as an activity, one row at a time) allows students to make a valuable connection to counting by twos, threes, etc., the latter skill an explicit requirement which has been left out of CCSSI for some reason. (See 1.OA.5, 2.OA.3 and 2.NBT.2.)
Teachers can post a large blank times table chart on the wall (seemingly mysterious at first), with 0-9 on the top and side, and fill it in slowly as they are calculated. We’ve seen numerous useless math visual aids plastered around elementary classrooms, but we think an evolving chart would be both motivational and intriguing.
After the times tables have been filled in manually (which justifies each value), it’s time to memorize them. 3.OA.7 states:
“By the end of Grade 3, know from memory all products of two one-digit numbers.’’
Yes and no.
The key to making good use of the times tables and recognizing their critically important patterns is to memorize them first. The problem with 3.OA.7 is that too many multiplication things are happening in the Grade 3 curriculum for students to lack fluency in the times tables as a prerequisite.
How are students supposed to become proficient at division (3.OA.4, 3.OA.6, looking for the correct quotient) and find equivalent fractions (3.NF.3) during Grade 3 if they can’t already rapidly multiply number pairs? We won’t dissect each standard, but 3.OA.2, 3.OA.3, 3.OA.5, 3.OA.7, 3.OA.8, 3.NBT.3, 3.MD.2, and 3.MD.7 all involve the use of multiplication. Students who can recall the times tables fluently will excel, those who cannot multiply will lag.
Memorizing the times tables over the course of the year is too late; students won't make the necessary connections.
Parents: Even if your child’s school is following Common Core, reject CCSSI’s approach. Buy a set of flash cards and drill the times tables into your child’s head over the summer, before she begins the third grade.
NAEP has already highlighted that American students can barely multiply, even with a calculator. We need to make it a national priority that every student entering the third grade should understand the concept and know their times tables.
In contrast, by introducing repeated addition in Grade 2 and making knowing the times tables the final goal of Grade 3, CCSSI has dragged out multiplication unnecessarily, spoonfeeding the concept and focusing on basic skills instead of developing higher order thinking.
Consider the following “new style’’ sample question released by PARCC, one of the two major assessment consortia, that conforms with 3.OA.7, fluency of multiplication within 100:
It is a dreary future we have to look forward to.
To conclude this section, we look at 3.OA.8, which captures the spirit of CCSSI’s low-level aspirations for our students:
“Solve two-step word problems using the four operations.”
Certainly, the newly learned operations of multiplication (and division) can be applied to solve word problems. (“8 children attend a birthday party. Each child gets 3 cookies at the party and 2 cookies to take home. How many cookies are needed?”)
And where are these kinds of problems ultimately leading us? The answer lies in 5.NBT.5:
“Fluently multiply multi-digit whole numbers using the standard algorithm.”
So the ultimate goal is to be fluent at basic skills and get those NAEP questions correct (now by fifth grade instead of fourth).
Is that all the learning we can elicit?
This is only part of the post. The rest is here.
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