For some reason, I’ve been on the email list of an outfit called “Conceptua Math” for the longest time. I think their products are terrible and I’m always amused by the amount of hyperbole they put into their different press releases (“New Breakthrough in Adding Fractions!”, “Our Latest Innovation: Fact Fluency!”) Like any person with a few minutes to kill here and there, I like to check up on what they’re doing, if only to roll my eyes and count the number of days until someone finally pulls the plug on this enterprise.
Today’s email featured a video about Conceptua Math’s latest innovation on something entitled “Fact Fluency.” I don’t know if CM purchased the licensing rights to this particular phrase, so taking credit for putting these two words together in this combination seems odd, to say the least. I got about 90 seconds into the video before I clicked off and decided to write this post (and all my legions of followers know how often I blog….)
So let’s start at the beginning:
YO, PEOPLE AT “CONCEPTUA MATH” – NOT ALL MULTIPLICATION FACTS ARE EQUAL!
I knew this program was going to be a stinker the minute I saw the multiplication table they used to illustrate the skills that would be “taught” in this module, which included all the “facts” from 1 – 10. The only kudos I can give the folks over at CM is that they had enough sense not to include the “0” facts, as if kids need to be “taught” and “drilled” on those.
So let’s begin at the beginning: there are two different types of multiplication facts: conceptual facts and pair-associated facts. Conceptual facts do not need to be taught individually, because they make use of the principles of mathematics in order to generalize them to any number presented. The best examples of conceptual facts in multiplication are the 1s and 10s. In multiplication, 1 is known as the “identity element,” as any number multiplied by it is left unchanged. Knowing that 1 is the identity for multiplication means that the principle can be applied over and over again, without regard to the other factor being used. The remedy for students who don’t know their ones facts is not to drill them over and over again, but to have students develop an understanding of the uniqueness of 1 in multiplication and then explain how it can be applied and extended. Yet, in the video demonstration of this fact fluency module, we see questions like “1 x 4 =” Please, give me a break! Better yet, give me 1 x 345,580 of them! (or don’t you know the 345,580 tables yet?)
Another example of a conceptual fact is the 10, which is easily generalizable by students who are exposed to 4 or 5 examples. However, the difference here is that we often see students “overgeneralize” this concept by assuming that every number multiplied by 10 just gets a zero added to the end. But this only works for whole numbers, which is why we see many students who insist that 1.2 x 10 = 1.20. What they failed to understand (probably because they used the “adaptive computing” model at Conceptua math) is that multiplying by 10 has the effect of changing the place value of each digit in the number – that is, each digit moves one place to the left, which is directly connected to the fact that we use a base 10 system of counting. Thus, 5 x 10 is not about skip counting by 5’s ten times, but just moving the digit 5 from the ones to the tens place. If students understood this in 3rd grade, there would be many fewer of them insisting in 6th grade that 4.5 x 10 = 4.50.
There are so many more ways that Conceptua Math offends me (including the fact that they continually refer to research, yet their entire program seems predicated on doing the exact opposite of what actually works), but what can you do? They’re in the education bidness, where every product is a breakthrough or innovation, and I’m in the bidness of actual teaching and learning. I’m looking forward to my next Conceptua Math newsletter, because there’s nothing I love better than “innovations” and “breakthroughs.”