Teaching Math With "Fidelity"

I recently saw a discussion among parents on the internet which included a note that one parent received from his son’s teacher:

“Do not teach your child the ‘standard algorithm’ for computations until he or she has learned it in school.” 

Having seen similar notes publicized on the internet, I wondered what other warnings/notices parents received from otherwise well-meaning teachers. To find out, I wrote a post on the “Science of Math” Facebook Group page, quoting the above note and asking for other examples people might have seen or received.

Members of the “Science of Math” Facebook group are teachers as well as non-teacher parents, though there are also education consultants and instructional coaches. While the purpose of the group is to foster greater awareness and understanding of teaching practices backed by solid research, there are many differences of opinion on what constitute “evidence” and effective practices. 

What I received in reply to my post were not more examples of teachers’ notes but rather a flurry of comments, mostly from teachers and most of them angry. Some accused me of throwing teachers “under the bus”, others stated that there was indeed a problem in teaching the standard algorithm too early. A small portion of the commenters, however, felt that teaching the standard algorithm should not be delayed but should be taught first before introducing alternative place-value strategies. Below is a sample of some of the comments I received.

Comments against teaching the standard algorithm first:

“Parents SHOULDN'T teach the algorithm before teachers have had a chance to teach the concepts.”

“There is a progression that needs to happen. Memorizing a formula without the mental support is more of the old way of doing math.”

“Over-reliance on the standard algorithm is no good. It leads to a lack of place value understanding and a lack of understanding with regard to properties of operations.”

“Research shows students are less willing to learn other methods after learning the algorithm.”

Comments in support of teaching the standard algorithm first:

“Some kids need to practice procedures before they really understand the concept.”

“The solution I’ve found is to teach the algorithm and the other strategies concurrently. Show them that the algorithm is merely a shortened representation of the other methods.”

The comments in favor of teaching the standard algorithm first are echoed by cognitive scientists and researchers in education. In an interview with Anna Stokke on her “Chalk and Talk” podcast, Dr. Ben Solomon, a psychology professor and a researcher in math assessment and intervention, states:

“We always should be teaching students the most efficient way to solve a problem first. That helps them develop the confidence that they can calculate that answer, and can also do it fluently, almost effortlessly. Once they reach that point, then if you want to teach how additional strategies demonstrate how the algorithm can work in different perspectives, that’s fine. But always use the standard algorithm as the foundation; it has been shown to be the most efficient means to solve a problem.”

The comments against teaching the standard algorithm first, derive primarily from a paper written by Constance Kamii and Ann Dominick. Kamii claims that learning the standard algorithm at an early age eclipses any understanding of how and why it works and concludes that it has a “harmful effect” on young children. This paper has been criticized for its flawed experimental design; but it has continued to survive the criticism and has lived a long life.

The Common Core Math Standards (CCMS) have contributed to the longevity of Kamii’s and Dominick’s paper, by requiring students to use “place-value” strategies for adding and subtracting multi-digit numbers through the third grade. The standard algorithm for addition and subtraction does not appear in the CCMS until fourth grade, a placement that has resulted in a widespread interpretation that the standard algorithm is to be delayed until that time. The result is that students in first through third grades have been required to draw pictures and use convoluted and inefficient strategies for adding and subtracting multi-digit numbers. By the time students reach fourth grade, the standard algorithm appears as just one more way to add and subtract – one more side dish in a never-ending meal in which the main dish remains hidden. It has generally resulted in more confusion over what strategy to use, rather than conferring any “deeper understanding”. 

The interpretation of CCMS that delays the teaching of the standard algorithm until fourth grade is, in fact, not correct. Nothing in CCMS prohibits teaching the standard algorithm earlier than fourth grade as verified in writing by two of the authors of the standards themselves: Jason Zimba and William McCallum. Jason Zimba wrote an article to this effect, stating that the standard algorithm should be taught in first grade and in subsequent grades thereafter, bringing in other place-value strategies concurrently.

This disclosure has not been well publicized. No mention of the standard algorithm being taught earlier than fourth grade appears on the Common Core website. Nor is it discussed in professional development seminars or in guidance given to teachers on implementing the CCMS. Most importantly, textbook companies have not disclosed this; the standard algorithm does not appear in most textbooks until fourth grade. Younger teachers who don’t know better adhere to the textbooks; older ones may diverge but are often told by the school to teach “with fidelity” to the textbook. 

What I found especially interesting were comments expressing disappointment in the Facebook group itself. Specifically, some teachers were perplexed to see comments supporting the teaching of the standard algorithm first, stating that they thought the “Science of Math” group was supposed to be about science.  There was a distinct guardedness amongst teachers who had been taught and continued to believe that traditional means of teaching math are ineffective. These teachers believed that the opinions that went against their beliefs were “anti-science”.

What I was seeing first hand was a phenomenon that has occurred since Emily Hanford’s “Sold a Story” podcasts that exposed the ineffective methods used to teach reading in schools. This series has attracted much attention and proponents of these methods have come under fire and are being discredited.  The result of the series has 1) created a fear among math reformers that the same thing will happen in math education, and 2) emboldened some math teachers to take a stand against the prevalent methods and ideologies of math education. There has been an increase in teachers who are aware of the cognitive science behind teaching as well as teachers who are willing to speak out against ineffective practices.

This growing awareness is the subject of an excellent article, by journalist Holly Korbey. She describes this growing cadre of teachers who “didn’t learn about the scientific evidence behind teaching and learning in their teacher training [but] from their own curiosity, and sometimes from other teachers on social media.” She calls these teachers “insurgents” who once entering their classrooms are confronted with ideas not supported by research or scientific evidence. In interviewing teachers for her article, Korbey states that such teachers “often walk a fine line: they want other teachers to know about learning research, but don’t want to get tuned out.”

The arguments I saw on the “Science of Math” Facebook group were indicative of what she describes. What was encouraging to me was that these “insurgents” were willing to speak out and found a forum in which to do so. I am hoping that there will be more teachers willing to not only speak out on social media but to teach math using methods backed by cognitive science, “with fidelity”.