2010-10-19-Access to Math and Science
Seminars@Hadley
Access to Mathematics & Science for Blind Students
Presented by
Dr. Cary Supalo
Mick Isaacson
Moderated by
Billy Brookshire
October 21, 2010
Billy Brookshire
Ladies and gentlemen, it’s time to begin our show. We’d like to welcome you to Seminars@Hadley. Today we’re going to be talking about access to math and science for blind students. My name is Billy Brookshire; I’ll be moderating. Your presenters are Mick Issacson and Cary Supalo. These guys have just made incredible headway in trying to make access to math and speech for blind students a lot more accessible. We thank them for that. And I know you’re going to enjoy their presentation today.
With that in mind, without further talk let’s turn it over to today’s first presenter, Mr. Mick Isaacson. Mick take it away.
Mick Isaacson
My name is Mick Isaacson; I’m going to talk to you about ambiguity and speaking mathematics. Ambiguity and speaking mathematics can be problematic for students with print disabilities. When we say “print disabilities” we’re referring to students who are blind, students with low vision and also a group of students who have a certain learning disability that inhibits processing of printed material.
What do we mean by ambiguity and spoken mathematics? I’m going to take a closer look at this. This is actually an area that I’ve done some research on a couple of years back. There seems to be unawareness in the main stream education system, particularly mathematics, teachers of mathematics. As to the fact that there is ambiguity in mathematics. Let’s first take a look at what we’re referring to when we say ambiguity in mathematics.
I would like you all to listen to this utterance; “The square root of A plus B plus C.” Think about what that is, how would you write that equation out? That utterance, the square root of A plus B plus C actually has multiple interpretations. In this particular case there are three of them. We could interpret that as the radical sign with A plus B plus C all contained with the radical sign. Or it could be interpreted as the radical sign with A plus B contained within the radical sign and the plus C outside of the radical sign. And the third interpretation could be the radical sign with just the A contained within the radical sign and plus B plus C contained outside the radical sign.
There are three interpretations of that relatively simple utterance. Let’s take a look at one more example just to get the point across. I think that you probably get it by now but listen to this utterance and think about how you would write this down; “E super script M plus J plus K.” Now that simple utterance, E super script M plus J plus K it also has three possible interpretations.
The first interpretation could be E with M plus J plus K in a super script. It could also be E with M plus J in a super script and plus K returning to the baseline. It could also be E with only M in the super script and then returning to baseline with plus J plus K continuing in the equation but not in the super script, in the baseline. There is another example of ambiguity in spoken mathematics.
Many of you are familiar with Abraham Nemeth. He’s a retired blind professor of mathematics. And when he was studying to become a professor of mathematics, to get doctoral degree and his masters degree; he found that ambiguity in speaking mathematics inhibited his learning during college and graduate studies. So he developed a set of rules for how to speak mathematics non-ambiguously.
Let’s take a look at how we would apply these rules to some of these previous examples. Let’s go back to the example of root A plus B plus C, the one with the radical sign. Let’s consider the example of the radical sign with A plus B plus C contained within the radical sign. Using the Nemeth rules you would speak that non-ambiguously as “start root A plus B plus C end root.” So you would demarcate the beginning and the end of the radical sign and what’s contained within it.
Let’s take a look at the example of the radical sign with A plus B contained within the radical sign and the plus C outside of the radical sign. Using the Nemeth rules that would be spoken as “start root A plus B end root plus C.” And let’s take a look at the third interpretation in which the radical sign contains only the A and the plus B and plus C are contained outside of the radical sign. That particular expression would be spoken non-ambiguously as “start root A end root plus B plus C.” I think you’re starting to see a pattern here. Basically the Nemeth rules are demarcating the beginning and the end of an instance in where ambiguity could occur. In this case it’s centered around the radical sign.
Just to give you a more thorough set of examples, let’s take a look at the example of the E with M plus J contained as a super script. Using the Nemeth rules we would say “E super script M plus J plus K” now you would think “Oh, we need to say the end of what’s in the super script.” But in that particular case because it’s the end of the expression, it’s implicit that you don’t have to say the end of it. I think you’ll get the point when we come to the next example here.
The expression now is E with M plus J as a super script and the plus K in the baseline. It’s returning to baseline. In that particular example we would say “E super script M plus J baseline plus K.” And as you can see it’s demarcating again, the beginning and the end of the super script, which is the case where the ambiguity arises. So it’s demarcating again the beginning and the end of an instance of ambiguity. And of course the example with just M in the super script would be spoken as “E super script M baseline plus J plus K.”
Since Nemeth developed those rules he used during his college years those are pretty informal. He developed them and he used them himself. About ten years ago, maybe not quite ten years ago, a company called GLC worked with Nemeth to refine those rules that he used. And they subsequently have become Math Speaks. Those rules were subsequently developed and they were tested by some researchers at Purdue. Myself being the lead investigator on that particular study.
What we did is we basically tested the Math Speak rules to see how easy they were to learn. Low and behold they were very easy to learn, very intuitive. In a nutshell what we did is we had students at Purdue who were in education, they were going to become teachers. They volunteered to participate in the study. We had them listen to Math Speak rules, a little training session. We gave them about a five minute training session. And we tested them on radical fractions, super scripts and absolute value.
Later we tested them for whether they actually could interpret the expression correctly. In less than five minutes we had students performing at close to 100%. What this shows you is that the Math Speak rules are very easy to learn and to understand.
An overview of ambiguity, we already spoke about that. You need to be aware of the beginning and the end of an instance of ambiguity. And you need to clearly demarcate them. Essentially that is the essence of what the Math Speak rules do. They demarcate the beginning and the end of an instance of ambiguity.
In the first example it was with the radical demarcating the beginning and the end of the radical. Just to let you know, ambiguity is all over the place in the field of mathematics. You can find it with grouping. When we say “grouping” we mean things in parenthesis. You can find it with super scripts. It’s there with logarithms, with subscripts, with radicals, matrices, absolute values and fractions, just to name a few of the instances where you can find it.
This is another piece of research that we did. Math teachers will use math textbooks. If you were to go out and pull a math textbook off the shelf and find an expression. There is a very high probability that expression that you pull out of there will be spoken ambiguously if the teacher were to speak it with non Math Speak rules. Being the researchers that we are, we actually went out and we pulled a bunch of textbooks off the shelf of what was called “The Teacher Resource Center” at Purdue. And we sampled them. We did a stratified sample. We analyzed the expressions we pulled for whether they could be spoken ambiguously.
Of over 1,500 mathematical expressions we found that 72.5% had the potential to be spoken ambiguously. Now, if you’re a math teacher and that’s where you’re pulling this stuff from you’re most likely going to be having a source of ambiguous expressions? We ask questions to math teachers and you say “Can you speak this as if you were going to speak this equation to your class?” We give the teachers expressions that have potential ambiguity and we say “Would you speak this as you normally would?”
We did that and we took those expressions, we took the teachers recording and we went back and we analyzed them for whether they actually were spoken ambiguously. 86% of the recording of teachers speaking mathematical expressions were spoken ambiguously. Now, that’s a lot of potential ambiguity.
If you’re a blind student or a student with a visual impairment, or even a student with a print disability, something that inhibits processing mathematics or printed material, and your only source of information is listening to the teacher. Your primary source of information input. That becomes a problem that teachers are speaking ambiguously. Of course they’re putting it up on the board or on the overhead projector but if you’re blind or you can’t process that information it really is of little benefit to you. Teachers really need to be aware of the fact that they are speaking mathematics ambiguously and that there actually is ambiguity in spoken mathematics.
Just as an aside, in addition to measuring ambiguity in teachers speaking mathematics, we also took 20 of those expressions and we had the teachers read them a second time. And there is a whole lot of inconsistency in how they’re actually spoken. So the same expression would be spoken differently by different teachers. That actually is a similar problem in that if you’re starting to first learn mathematics or if you have a problem processing mathematics it’s very difficult to learn if you’re being presented the information in an inconsistent manner.
Let me give you an example here. Fractions are a very common construct that is taught. When we went back and we analyzed how fractions were spoken there were four different ways and this was out of eight teachers, teachers would say “blank over” so whatever the quantity is over. “Blank divided by. The quantity blank over or all over.” So those were four different ways the same teacher, how they spoken fractions. That can be a very big problem for students. It inhibits learning. Consistency is very important in teaching mathematics. At least in my opinion it is.
Finally, we’re going to wrap up here. After we did that study we did the debriefings and I asked the teachers about their awareness of ambiguity in spoken mathematics. Most of them were not even aware of the fact that there was ambiguity in speaking mathematics. And I also asked the teachers if they were in favor of learning about ambiguity in mathematics during their training to become a teacher; if it would be something that would be beneficial. And they all were in favor of such training.
Summary; textbooks, the content of textbooks has a high potential of having content that will be render ambiguously if speech is used, if everyday speech is used. Teachers tend to use everyday speech when speaking mathematics and hence, communicate ambiguously. Teachers are unaware of or they really don’t know how to deal with ambiguity in spoken mathematics. And they all tend to support training for non-ambiguous communication of mathematics.
Implications, content and unaware teachers; that creates a situation that may inhibit learning mathematics by students with print disabilities. Teacher training programs on how to speak mathematics non-ambiguously would be beneficial. That’s really about all I have to say. My talk has been primarily a talk on increasing awareness among educators, particularly in the area of mathematics, that there is potential ambiguity out there. And it is a problem for many of your students and you ought to take a look at how you are presenting your materials and possibly adapt your presentations to accommodate student who cannot access it visually.
One last thing, I just basically touched on the mathematics, Math Speak. If you’re interested you can go to the kin
www.gh-mathspeak.com//examples/grammar/rules/
you’ll find a more comprehensive listing of the Math Speak grammar rules. Thank you, that’s the end of my talk.
Billy Brookshire
Thank you, Mic. We appreciate that very much. There was one question in the chat room we can talk about a little later. It had to do with the fraction example. They were not able to hear so you might need to lean a little closer to the microphone when you’re talking about that one next time. We’ll get to it in the question and answer.
Our next speaker, folks, is Dr. Cary Supalo. He is going to cover the science area for you. For those of you who are wondering kind of why we are pausing here is that setting up slides sometimes takes a little while to get it going. And Cary I can see you’re up and rolling now so I’m going to turn the microphone back over to you.
Cary Supalo
Good afternoon, everyone. My name is Cary Supalo and I just finished my research work at Penn State University on the independent laboratory access for the blind project. This project was funded by the National Science Foundation Research and Disabilities Education Program for the last six years. We have been striving to develop a suite of access technologies that will help students who are blind or low vision has a more hands on science learning experience.