Advice on the Teaching of Standard Algorithms Before Common Core Says it is Safe to Do So Dept.

EdReports.org is an organization that rates textbooks/curricula with respect to how well they align with the Common Core standards. There are no ratings on the effectiveness of a curriculum or textbook–just whether it adheres/aligns to the standards.

They published a guideline for how to use EdReports’ reviews of texts.  Of interest is that under “Focus” for K-8, the key criterion to be assessed via their gradated ratings is: “Major work of the grade and no concepts assessed before appropriate grade level”

What captures my attention about this is the “no concepts assessed before appropriate grade level”.  Sounds similar to “no wine before its time” but it has more sinister implications in my opinion.

In my investigations and writing about Common Core standards, I have heard from both Jason Zimba and Bill McCallum, the two lead writers of the math standards. They have assured me that a standard that appears in a particular grade level may be taught in earlier grades. Jason Zimba also wrote an article to that effect.  So for example, the standard algorithm for multidigit addition and subtraction appears in the fourth grade standards. This does not prohibit the teaching of the standard algorithm in, say, first or second grade. A logical take-away from this would be that students need not be saddled, therefore, with inefficient “strategies” for multidigit addition and subtraction that entail drawing pictures or extended methods that have been known to confuse rather than enlighten.

Nevertheless, the interpretation of the “focus” of Common Core is used as a means to judge whether a textbook is aligned.  The term “focus” is discussed on the Common Core web site in a description of instructional “shifts” expected from implementation of CC. The instructional shifts do not appear as a standard anywhere within the Common Core content or the 8 Standards of Math Practice.

This means that if a publisher wishes their math book to sell, they will make sure that their assessment packages (otherwise known as tests and quizzes) do not include standards for later grades than for which the test/quiz is intended.  The result is that students will be tested on the inefficient convoluted and pictorial methods that are used in lieu of the standard algorithms.

While there are teachers who can overlook this, and accept a student’s use of a standard algorithm, there may be teachers new to the profession who adhere to the pre-packaged assessments. Furthermore, such practices may be reinforced by Professional Development vendors who specialize in Common Core.

The Common Core website insists that pedagogy is not dictated by Common Core:

Teachers know best about what works in the classroom. That is why these standards establish what students need to learn, but do not dictate how teachers should teach. Instead, schools and teachers decide how best to help students reach the standards.

In view of all this, my advice is to allow students to use the standard algorithms on a test even if it appears in a later grade. As far as alternate strategies, be aware questions on state assessments may include them. While scores on the state tests are not used in figuring the grade a student receives in a class, they may be used to qualify students for gifted and talented programs.

Bottom line advice: Teach the alternate strategies. Just don’t obsess over them.

 

Out on Good Behavior, Dept.

I am currently writing a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder”  When the series is complete, it  will be published in book form by John Catt Educational, Ltd.”

The chapters are being published in serial form at the Truth in American Education website.  If you are curious, the first seven chapters are available for your reading pleasure and can be found here:

Chapter 1 , Chapter 2 , Chapter 3 , Chapter 4 , Chapter 5 , Chapter 6 and Chapter 7

Rich Problems, Dept.

If you hang around long enough in the world of math education you’ll hear people refer to “rich problems”. What exactly are rich problems?

 

One definition is: “A problem that has multiple entry points and has various levels of cognitive demands. Every student can be successful on at least part of it.”

 

My definition differs a bit: “One-off, open-ended, ill-posed problems that supposedly lead students to apply/transfer prior knowledge to new or novel problems that don’t generalize.” 

 

For example: “What are the dimensions of a rectangle with a perimeter of 24 units?”

 

 A student who may know how to find the perimeter of a rectangle but cannot provide some of the infinitely many possibilities is viewed as not having “deep understanding”.

Rather, the student’s understanding is viewed as “inauthentic” and “algorithmic” because the practice, repetition and imitation of procedures is merely “imitation of thinking”.

 

I teach 8th grade algebra and use a 1962 textbook by Dolciani. Here’s a problem from that book that I gave to my students:

 

“If a+1 = b, then which is true? a>b, a<2b, or a<b?”

 

This is my idea of a “rich problem” but don’t tell anyone. I might get fired.