Knowing Math Facts

Jo Boaler's writing kept popping up on my RSS feeds and on twitter this summer (I read her book too). Her writing about math education in the US has me thinking a lot about fact fluency and memorization. Boaler, if you do not know, is one of the most vocal advocates against memorization when learning math (see this article for a good sense of her argument or this paper for a more detailed analysis). She argues that using timed tests and traditional flash cards to get students to memorize their facts is increasing student anxiety around mathematics and causing students to believe that mathematics is about memorizing rules.

As a former 5th grade and now 6th and 7th grade teacher, I spend a lot of time thinking about multiplication facts and their importance in math education. But, this is one topic where I just cannot seem to decide where my beliefs lie. So, I am going to spend this post sorting through some feelings and see where I come out on the other end.

Background

Before I start talking about what I am thinking now, I want to share a bit about my past practices.[17]Sharing info about your practice -- especially when you are not necessarily confident that it was best-practice is always nerve-wracking, but I welcome any honest feedback, suggestions, and constructive-criticism. I taught fifth grade the last five years, and students entered my class with a wide range of knowledge of their multiplication facts. Some had them memorized completely, some could reason to find unknown ones, and others simply did not know them and had no strategies for solving them.

In my first year of teaching fifth graders, I quickly found that there was a positive correlation between students who REALLY did not know their facts and those that struggled later on in the year.[18]More on what it means to "know" one's facts later. In case you are wondering, this correlation was not caused by a use of timed tests or other tasks that rewarded students for working quickly later in the year. With the exception of multiplication facts, I do not time assessments and have tried to remove all biases that reward fast workers.[19]I give lots of wait time on questions, allow students to work at different paces, don't reward students who finish first, etc. It seemed to me, rather, that students with weak knowledge of their multiplication facts struggled to follow the ideas of other students in discussions and to make sense of new ideas. An obvious example is that students with weak multiplication facts had a difficult time factoring numbers even when given unlimited time and understanding-based strategies.

To help students who entered my course with a weaker knowledge of their facts, I began spending class time helping students better learn their facts. About twice per week, we began class by completing a 60-fact sheet. I STRONGLY emphasized that the goal of the activity was improvement. I had students correct their own sheets and then share their times with me.[20]We settled on a 10 second penalty for incorrect facts to encourage accuracy. And, I was careful to get excited for student improvement and not for fast times. We did this about 12 times in each section, though some students did extras outside of class. Here is what the average of each section's times looked like over time:

Vertical-axis: Average seconds taken. Horizontal-axis: Number of times attempted

Average seconds taken to complete 60 single-digit multiplication facts by class section over time

Using minimal class time, I saw an average improvement of about one second per fact. Last year, as I moved to standards-based grading, I made one of my standards "complete 60 single-digit multiplication facts in under three minutes" which I think most will agree is much slower than many teachers expect from their fifth grade students. To meet the standard, each student only had to break that barrier once, and they were allowed as many attempts as they needed. I was especially impressed by one student who completed about two sheets per day for a whole week until the student was able to meet the standard.

This practice definitely seemed to help students throughout the year, though I have no evidence other than anecdotal observations to back this up. Moreover, I could be biased in thinking that this practice is more helpful than it really was. Maybe, for example, I credit this practice for things that were actually unrelated.

My concerns with this practice is that, when push comes to shove, I am grading a timed test. I think I am doing so in the healthiest way possible, but that does not change the fact that speed determines some aspect of success (albeit a very minor part) in my math course. I feel my time expectation is very reasonable and allows students to use strategies other than memorization to answer questions, but I still have concerns about the effects of this assessment practice on students.

Alright, with that out of the way, let me share some of my current thinking on student learning of math facts.

Knowing Vs. Memorizing

The first idea I want to get into is semantic: the distinction between memorizing and knowing. I think that most folks use these terms synonymously, but Boaler does not. This quotation highlights the difference as she sees it (emphasis mine):

I asked Boaler if rote memorization might be a beneficial supplement to a rich mathematics curriculum that emphasizes creative problem solving. Just the way that the fast repetition of scales is useful for a Juilliard musician, for example, or vocabulary drilling is useful for a foreign language student. But Boaler says that “mathematical ideas” are different, and stands by her position that times tables are unnecessary. “I never memorized my times tables as a child because I grew up in a progressive era in the U.K.,” Boaler said. “It’s never held me back.”

The human brain is forgetful by nature, she argues, and what she wants is students to develop the number sense to calculate 7 x 8 quickly even when their brains can’t recall the math fact instantly. (For example, you might remember that 7 x 7 is 49 and then add 7 to that to arrive at 56). Students who learned primarily through rote might freeze during an inevitable moment of forgetfulness, and be unable to think through the problem and come to an answer efficiently.[21]From this article which I linked to a the beginning of the post.

I am completely fine with any student knowing 7 x 8 by working off of 7 x 7 = 49. To me this constitutes knowing, and I think that is reflected in the fact that I am willing to give students three seconds to calculate an average fact.[22]For most students, 7 x 8 takes longer than the average fact. Using this strategy probably does not constitute having 7 x 8 memorized, but I do not see memorization as necessary when efficiency can be achieved just through knowing. It is likely that repeated use of a strategy like this would result in memorizing the answer, but I would actually prefer my students go through that process rather than just memorize without the underlying number sense.

Quickly or Efficiently

One word sticks out for me in that quotation: "quickly." It is not enough for a student to know a fact through number sense, they need to be able to know it quickly. It is as if their strategy for knowing needs to work well enough to simulate memorization. I think this is the wrong way of thinking about this. I suggest a better way is to think about efficiency.

I want my students to know their multiplication facts well-enough to output answers at an efficient rate for their grade level. For my fifth grade students at my school, that averaged to about 3 seconds per fact or faster. My experience suggests that students who were less efficient than that struggled to process, understand, and learn future ideas like factoring and fractions. Boaler's paper seems to counter with the following statement:

(A New York curriculum) links the memorization of number facts to students’ understanding of more complex functions, which is not supported by research evidence. What research tells us is that students understand more complex functions when they have num- ber (sic) sense and deep understanding of numerical principles, not blind memorization or fast recall (Boaler, 2009).

Here, though, I think we are again looking at the difference between memorizing and knowing. Memorizing math facts is not going to help students understand ideas like factoring and common denominators, but knowing them is likely a prerequisite. Boaler says that students with strong number sense learn later ideas more easily but students with strong number sense are able to know multiplication facts more efficiently. I want my students to know their multiplication facts because of strong number sense but efficiency matters. In the example of 7 x 8, using 7 x 7 + 7 requires number sense. But, so does skip-counting by 7s or 8s. Neither method is memorization, but one is certainly more efficient than the other and that efficiency matters.

Is it worth the cost to assess efficiency?

One thing I know that Boaler and I would agree on is how to promote that efficiency. The answer is not to spend more time memorizing facts but to build number sense.[23]See update at bottom of post for a study that supports this idea We should help student learn, for example, that one can use 2 x 8 to figure out 4 x 8 and not have them just make a new set of traditional flash cards for each set of facts. Deepening number sense and understanding is a meaningful way of improving efficiency while memorization is not.

If you think that efficiency is valuable -- and I do with limits -- then it is important to assess students on this efficiency. I cannot think of a way to do this without using some form of timed test. But, as Boaler points out, timed tests come with lots of negative consequences like unproductive beliefs about mathematics and decreased student-confidence. The questions is whether the benefits of promoting efficiency outweigh the costs of timed tests, but the answer to this may vary by each student, by school, or by grade. So...sigh...I just do not know if it is worth it.

Conclusions

After all of that, I still don't have a side of the fence to confidently sit on. If I was teaching fifth graders again this year, I would still use timed assessments to measure students' efficiency with their facts but I would not feel great about it. When it comes down to it, though, I do think efficiency is important. It is not the only goal -- or even the most important -- but it does matter. There must be a way to promote and measure efficiency without irrevocably harming students self-conceptions.

One thing I know I would improve is how I encouraged students to become more efficient. In the past, I just sort of threw the students to the wolves and told them to "practice" or "study" their facts. I know their 4th grade teacher was great about using number sense to learn multiplication facts, so I just leaned on her work. But, this probably resulted in lots of attempts at memorization. Instead I would use class time on number talks and strategies that support knowing facts through understanding.

I am interested to hear where you all stand. Do you think there is any place for timed assessments in math class? If not, then how do you promote efficiency?

Update on 2/4/16: @DavidWees, a must-follow on twitter, shared this article on how to help students develop fact fluency -- memorization versus teaching strategies. It's a little dense, but the introduction, discussion, and conclusion get the point across clearly even if the minutia of experimental design and p-values are not your things.[24]Are there really people out there who DON'T love experimental design and p-values?