D: True or False: 6+8=3+11
I've become painfully aware recently how sloppy communication can be. I am surprised how many times I have to reply to an email asking for clarification because of unclear writing or simply not taking the time to think things through. And that's with other scientists.
The GK-12 program I run at the University of Nebraska places graduate students in upper elementary, middle and high schools to work with teachers on improving math and science education for their students. We spend an entire day of the week-long orientation discussing communication. I break it down roughly into "scientific" communication and "normal person" communication. That's probably not a fair breakdown, but we really have to emphasize to the students that, although it is perfectly OK to reply to a scientist's idea with "here's why that won't work", it's a death knell for the relationship if you do that with a teacher (or, often, a spouse).
At the end of the year, one of my students made an observation I wholeheartedly endorse. "I like scientific communication better," she said, "It's just faster." And clearer, I would add.
Although Jennifer and I come from very different disciplinary backgrounds, I think one of the reasons we've hit it off is that we share the trait of wanting to use words properly. Jennifer recognizes that scientists and mathematicians use words and symbols to convey very specific meanings. If I use the word "velocity", she's likely to ask if there's a reason I didn't say "speed". (Speed is a scalar, velocity is a vector. Sometimes it makes a difference, sometimes not.)
Nowhere, perhaps, is the specificity of symbols more rigorous than in mathematics. My mother was a graduate student in math, then economics, while I was in elementary and middle school. I remember seeing her scribblings filling up scads of yellow legal pads and asking her once "when do I get to learn this language?" And math is definitely its own language. One of the biggest problems teaching (or communicating) science and math is that sometimes words mean different things in the discipline than they do everyday life.
But the equal sign should be an easy one, right? It means, well, equal.
Apparently, American students have a much less clear idea of the meaning of the equal sign than their Chinese, Korean and Turkish compatriots. A study by Capraro, et al in Psychological Reports (106(1), 49-53 (2010)), which draws on their previous data in Li, et al. (Cognition and Instruction, 26, 195-217 (2008) compares 6th grade students from different countries. Both papers originate from the research group of Mary Margaret Capraro and Robert M. Capraro at Texas A&M University. Incidentally, "et al." translates literally in Texan to "and them".
The results from the first two questions I posed from their study (6+9=__+4 and __+8=12+5) were surprising/appalling. Only 28.6% of American students got these questions right. The Chinese and Korean rates were in the 90+% range and the Turkish rates were 61% and 79% respectively.
As is often the case, we learn more by looking at the wrong answers than the right ones. The first two problems were designated "Type A" and "Type B" - similar, except the missing numbers are on opposite sides of the equal sign. The third problem - the one that lent itself to the title of this blog - is classified as "Type C" and provides a slightly different probe of the understanding (or misunderstanding).
For the "Type C" problem __+3=5+7=__, American students got the first blank right 23.8% of the time, while the rates for other students were 98.6% (Chinese), 86.5% (Korean) and 60.2% (Turkish). Interestingly, the correct rates for the second blank were much more comparable: 86.7% (American), 97.9% (Chinese), 93.3% (Korean) and 86.0% (Turkish).
What this strange disparity between the first and second blanks tells us, the authors argue, is that American students disproportionately don't understand the equals sign. The most common wrong answer for the first blank was "2". It is true that 2+3 = 5, but 2+3 definitely doesn't equal 5+7. Almost 90% of the students recognized that "5+7=12", but a significant number of those got the first blank wrong. This is apparently a common misconception among American students that isn't seen nearly as much in student from the other countries studied: the belief that the answer is the number immediately following the equal sign.
In their previous study, the authors used the True/False question "6+8=3+11?" to test understanding of the reflexive property of the equal sign. Reflexive, which I had to look up, means a=a. The popular phrase "it is what it is" embodies the mathematical philosophy of reflexivity. The educators doing the study, though, realized that students with the misconception I mentioned above - the answer is the number immediately after the equals sign - would get this question wrong for that reason and not because they don't understand that a=a.
In their new study, they replaced that question with "160=___", and expected the blank to be filled in with "160". But a number of students put an operation in that blank, like 80*2 or 40+120. Those answers are not wrong, but (strictly speaking), but they indicate that those students look at the equal sign as indicating that a mathematical operation is required and not solely as a representation of equality.
I've always thought of the equal sign as the pivot on a see saw. Whatever is on the left has to balance with whatever is on the right. If I fill in the blank with a 6, there's only 9 on the left and 12 on the right, so the see saw isn't balanced. OK, I can't draw a picture for the multiple equals signs on one line, but you get the idea. Who ever thought that something as seemingly simple as 'equal' could be so complicated?
At this point, you might be thinking that this seems like a bunch of quibbling over a precise definition of interest only to the highly mathematical. One of the original motivations for this study was the low performance of American students relative to their international counterparts on standardized tests like the TIMSS and the PISA tests. There apparently isn't much explicit attention to the equal sign in middle school curricula and, the authors (along with other math educators) believe that not understanding the equal sign puts students at a distinct disadvantage when it comes time to learn algebra. If you put 'x' in place of the blank, you realize you're actually doing algebra answering the questions I posed at the start of the blog.
These misconceptions carry over to physics. For example, consider the force on an object falling: F = mg. The force gravity exerts on a ball falling through the air is equal to the product of its mass times the acceleration due to gravity (in the absence of air resistance - sorry, I felt compelled as a professor to add that. I couldn't help myself.)
Students identify gravity as a force, but a significant number of them also identify a force "F" in the above equation as distinct from the force of gravity. The problem gets worse when there are multiple terms on the right-hand side of the equation due to multiple forces.
It is amazing that something so seemingly fundamental can so impact a student's education. One of the (many) reasons I am looking forward to Jennifer's book is that I speak fluent calculus. Imagine trying to explain to someone how to walk. That's what me teaching calculus is like. One of the best reasons for using peer teaching (students teaching each other) is that they explain things in ways I wouldn't have thought to use. Listening to them explaining how they understand an idea helps me realize how I can explain it better. It's research like this that reminds me that sometimes the better part of teaching is listening.
ADDED 8/13/10: An interesting study notes the need for better prepared mathematics teachers, as well as a significantly strengthened math curriculum. Jennifer and I have been talking a lot here recently about stereotypes. Although we've focused on those in the media, this article, by researchers at the University of Chicago in PNAS, suggests a scary chain: Female first and second grade teachers who are anxious about math pass that anxiety along to their female students. More female students are likely to agree with the suggestion that boys are better than girls at math after being exposed to this anxiety, and the female students who did agree with this stereotype performed worse in math as the year went on. Great article and PNAS makes the full text publicly available.
AND: A Christopher E. Granade speaks on the topic of 'equal' - a very nice post focusing on the importance of relationships and how that is really at the base of math and science.