Alright, so here’s the background. My mom is working on a doctorate, that I believe will be in Math education. Or at least taking classes, and doing a lot of research on math learning. She is by no means, a published expert, but she does have a lot of information to bring to the table. I wanted to talk to her about the Common Core Math Meme that floated around Facebook, and how some friends reacted to it. In talking, she shared with me a paper she wrote on this very topic. I would love to just copy and paste the whole paper, but I don’t think that would be beneficial to me posting a blog about my thoughts. I’m going to copy and paste some parts of it, and discuss, and I hope that I can get a discussion going like I had on Facebook.
First of all, I want to clear up a little misconception that this meme presented. “In fact, the Common Core State Standards for Mathematics calls for students to understand “the standard algorithm” for mathematical computations (Fuscon & Beckman, 2012-2013). This presumes that all educators know what “the” standard algorithms are.” Common Core does not dictate what methods or algorithms teachers should use. Common Core is a set of standards, what the meme was showing, was curriculum, not Common Core. My mom described the algorithm used in the meme as something called “mental math”. I’m going to leave this here, and hope that I come back around at some point to explain what mental math is, but if I don’t someone in the comments please call it out.
I want to now post a little story about me, that was written in this paper.
My daughter’s fourth grade teacher enjoyed teaching mathematics and used
manipulatives extensively as an introduction to new concepts. Students worked in small groups
to discuss problems before the teacher ever demonstrated a mathematical algorithm. When the
teacher was confident that students had built conceptual understanding, she assigned practice
problems for homework. So, both the teacher and my daughter were puzzled when Jamie,
normally a high-achieving math student, missed every single problem on a multi-digit
Error analysis (Ashlock, 2010) helped the teacher and me identify Jamie’s essential
misunderstanding of the multiplication algorithm she was using: she had transferred the concept
in the addition algorithm of “carrying the one (ten)” to her multiplication process. When the
product of two numbers was greater than nine, Jamie always “carried the one” to the next place
in the multiplicand, even when the product of the multiplier and multiplicand were greater than
19. In other words, when Jamie multiplied 18 x 5, she started with 5×8=40, wrote the 0 in the
ones place and placed a “1”, rather than “4,” in the tens place as a placeholder. Then when she
multiplied 5×10 and added in the “carried” number, she added one ten to get “60” rather than 4
tens to get “90”.
To her credit, the teacher reflected on how her instructional practice might have
contributed to Jamie’s misconception about the multiplication procedure. She realized that
during class, in each problem she had modeled, with and without manipulatives, all the interim
products happened to be less than 19. The teacher had not modeled what to do when the number
was greater than 19, and Jamie’s misconception showed up among other students as well. I
admired the teachers’ professionalism in analyzing what may have contributed to the problem
and recognized how easily, despite a teacher’s best intentions, students can build conceptual
Reprogramming Jamie’s understanding of multiplication took intense work. The more
Jamie and I worked together, the clearer it became that Jamie lacked essential number sense.
As a second grader, she had internalized procedural knowledge of the addition and subtraction
algorithms without essential conceptual knowledge of the reasons for ‘carrying’ and ‘borrowing.’
She did not know that she did not know: she followed procedures that gave her accurate answers
and A’s on the report card. The incorrect transfer of the addition procedure to multiplication
illuminated her misconceptions, but had her teacher and I not used error analysis, even that may
have gone unrecognized. That incident in fourth grade precipitated a gradual slide in Jamie’s
mathematical confidence that Jamie overcame only as an adult when she decided to become an
engineer. That’s a long-term impact for a simple misconception about a computational algorithm.
At the time, I wondered how Jamie could have avoided the conceptual confusion about carrying
and borrowing numbers.
The thing about math, is that there is no one right answer. In the United States, the “standard” algorithms align vertically in columns (McIntosh, et al., 1992; Stanic & McKillip, 1989); are solved primarily right-handed (from right to left) except for division (Pearson, 1986; Van de Walle, n.d.); include the concepts of carrying and borrowing (Hatano, Amaiwa & Inagaki, 1996; Kamii & Dominick, 1998); and reduce calculations to a series of operations on single-digit numbers (Carpenter et al., 1998; Fuson, 2003). Other countries, do not use the same algorithms. Along those same lines, some of these alternative methods being taught that are described as the “new way” are not actually new at all. The lattice method was used in India, many many years ago, and was brought to light in the US in the early 1900s. This Common Core meme that I keep bringing up, actually uses a method that has been around for decades. This is not new a new way of doing math.
I want to get back to the point that I brought up on Facebook. Many of the people that commented talked about concern that this “new method” would make learning math more complicated, and take people away from pursuing STEM careers. I disagree. I firmly believe that anyone who was taught the “standard method” cannot see any other method as acceptable, because we were taught in such a way that there is only one right answer. There is not one right answer when it comes to solving math problems. There are many methods for getting to the answer.
I have a question for you the reader. When you’re trying to figure out how much money you have, and how much an item costs, chances are, you do not pull out a piece of paper and do the “standard algorithm”. If I were to have all readers comment with how they do that, there would probably be a variety of answers. And no one way is right. No one is going to return a check if you don’t calculate the amount using the correct method. So why should we be forcing students to believe that there’s only one right way? The “standard algorithm” may be the most efficient, but with the example that my mom described, it actually caused some rather detrimental effects to my long term mastery of math.
So in the end, here’s my point. Just like with education as a whole, there are many different ways to get to an answer. What works for one person, does not work for another. I want to get away from the idea that there is one true way for how to get an answer in math, and accept that there are many ways to get to that answer. As a student pursuing engineering, I thrived on the idea of partial credit. In most cases, if you got the right answer, or had a logical method but the wrong answer, you got some credit for the work. Why can’t we do that in grade school too?