Robert Chaney, Mathematics
Overview and Context
About the Course
The course is called Technical Mathematics I & II, or Tech Math. I have taught the class for approximately ten years. What makes this course unique is the integrated lab component. The lab activities contextualize math by teaching students to apply algebra to solve specific, real-world problems. One tool I use to contextualize math is SAM, a calculator-controlled robot I co-created with a colleague in the Physics department.
Key QR Assignment Description (links to section in this page)
There is a departmental syllabus for the course. But the way I teach it is that instead of having it all written down into a rigid syllabus, I have a general syllabus of topics that the traditional course has and then I integrate activities. It is just-in-time teaching method. And I pick and choose as we go. It depends upon the class. It depends upon what I think they need. It depends upon time and other factors.
So there is no written-down syllabus, but there is a manual that has instructions for the integrated activities that I use in the course, and in some sense that is my syllabus.
How Quantitative Reasoning (QR) and Literacy are Approached
Quantitative literacy is having the skills to be able to look at a situation and calculate an answer; or all the things that you would learn in a traditional math class. Quantitative reasoning is taking those skills further so that you can make sense of the real world. You can use what you know in a real setting and look at real problems, quantify them and reason through them to draw a conclusion or solve the problem.
I think that being able to apply math to solve real-life problems is very important. I have taught math for many years, and the first part of my teaching was pretty traditional–standing up in front of the class, doing examples on the board, and teaching directly from the book. As I collaborated with others and had some other experiences, I became concerned about this way of teaching because students were not able to apply math outside the classroom.
Part of what changed my outlook was that early on I had opportunities to do contract work with companies as a mathematician. I started to get concerned that students weren't able to transfer the ideas that they were learning in math class to real settings. Math was an isolated subject with X's and Y's. It had no connection to anything else.
At first I wanted the students to have the appreciation that learning mathematics, in particular algebra, was valuable to them in the long run. I started to bring more contextual learning into the classroom along with the traditional presentation of math. When I was able to bring more contextual learning in, I could see not only that it started to motivate the students, but also that it was teaching them some things that weren't being taught in a traditional sense.
I respect traditional education because it teaches the language of math. I don't throw that out. Instead, I put it into a contextual, real-world setting while they're learning it. It builds a stronger foundation of understanding for the students.
I found that using contextual learning not only motivated students, but it also helped them learn more. They were better able to take what they were learning and apply it in many more areas of their lives. I could see that it would be useful to them in a practical sense. I do like to motivate students and get them excited, as was my original goal, but I find that it is less important than actually getting them to have a deeper understanding and skills to be able to apply math.
Students are used to seeing algebra as an isolated, disconnected subject and they don't know where to grab hold of it. They don't know how to make sense of it. However, because of their wealth of experiences, students already know more math than they think they do. Everyday life is filled with variables and constants and some form of logic as in algebra. What I do is to start with what they already know and bring in the algebra so that they can learn that language.
For example, in Tech Math, we use a calculator-controlled robot that runs off a graphing calculator (which students can buy at many stores). They can program it and it moves the robot around. I start them with a problem that I want the robot to do. They already have an understanding of what a robot is and what it should do, so this task makes sense.
They have a good grasp right away of the steps that the robot would have to take to accomplish that task. But the only way to make the robot run is to program it, and the only way to program it is to do algebra. Therefore, they must use algebra as a tool in the context of the problem.
They already know what has to be done, what the variables are and what the constants are. But they have to describe that in the language of algebra. It works very well, and I got very excited early on when I saw that, so I continued to move forward.
Motivation to integrate QR
Originally, I started out trying to answer the questions that you get from students in algebra such as, "What are we going to use this for? "Why am I spending so much time doing this?" and, "I don't see how this connects to anything." I was trying to motivate them to know that their efforts would be worthwhile to them in the long run. Later on I came to realize that using context-based learning not only motivated the students, but more importantly, it helped students learn algebra better.
Personal Statement on Teaching Philosophy and Methods (Acrobat (PDF) 73kB Aug12 13) by Robert Chaney.
Over time, I have changed my motivation and also how I motivate students. I used to make the argument to students that if you learn these things it will be like weightlifting to a football player. You don't see them lifting weights on the field when they're playing the game. But weightlifting is part of developing the strength and coordination so that they play the game better. But this argument was not strong enough to convince them.
I wanted to motivate them in the way they wanted to be motivated, to answer, "Where am I ever going to use this? How do people use it in general? Where is it valuable?" I wanted to get it to dawn on them that math is a very powerful subject. I wanted them to see algebra in context–how it is used behind the scenes in daily life. I also wanted them to see how it will be useful in future courses. So I developed contextual learning as a way to answer those questions.
As an example, in Tech Math, I tell students that the robot's brain is a calculator and a computer. It collects data, analyzes the data, and makes adjustments on the motor to make the engine run better. It's like your computer at home, in your car, or in your cell phone. It is also like a video game, which has an input controller that puts values into variable locations that ultimately move objects on the screen. Much of it is algebraic function and similar logic.
When students have something that they already understand, and then you bring in algebra and lay it over top of that then they understand the algebra better. They say, "Now I can see it!"
For example, I wanted them to be able to take a situation where the path the robot takes to get to a cup to knock off a ball will change from time to time. I wanted them to be able to press a button on the robot and regardless of what the path is, the robot follows it and finds the ball. Well, if that path can change, that means there's a variable.
I want students to transfer what they're learning in the classroom into many different contexts to learn how to apply algebra and how to think of real things algebraically. I also want them to learn the general concepts of algebra and the language so they can still be successful in traditional mathematics and go on to higher-level math courses in engineering, etc.
In Tech Math they use algebra to program an Excel spreadsheet to do calculations that will make the robot do different things. This is how I am getting them connected so that they start to develop a conceptual understanding of variables and how to create functions and graphs and analyze those things to solve real problems.
I never want them to get stuck anywhere where they're just seeing math applied in one context. I want them to understand that the same reasoning–the same procedures and skills–can be applied in a whole lot of varied types of situations. I want them to develop a general understanding of problem solving. So they can look at real situations and start to think of them mathematically, and then be able to use language of mathematics to start to work towards a solution, and so they recognize math in real things. It's not just in the textbook, it's not just for a certain type of problem, it's everywhere. And it can be a valuable tool if they know how to use it.
Pedagogic approaches used
I think contextual learning is very powerful. But I don't see it as replacing traditional education. I developed the idea early on that you don't throw everything else out just because there's a new idea that seems so great. Sometimes I back up on the contextual learning a little and use other methods that are more traditional. But the important thing is to bring these other things in alongside and intermixed with it. It's not just one thing, it's lots of things in a very balanced approach. And that balance can be different from one class to the next.
It takes experience in being able to feel that out. It's hard to write a 'Here's how you do it' book. This is just where I'm at now. Who knows next week, I can change some of these ideas. You can at some level describe what you are doing, but there are so many other things that are important to make it successful that come into play. It's teacher training, it's people to believe in it and who are excited about it. That enthusiasm transfers to students.
Knowing the course is successful
Our efforts have been supported by the National Science Foundation, by Ohio State Grants, the college, and math department at Sinclair.
We have a lot of qualitative data that the students think it's good. The students who like mathematics think it's valuable and they learn more mathematics doing it this way.
In addition, we have quantitative data, including a study done by the math department chair and by the physics department.
Our evidence was compelling enough to convince the mathematics department to allow us to make every section we teach in statistics an activity-based section.
However, we have found that it really takes a full-time effort from somebody to measure improvements in a statistically valid way because it is so complex. Even though we have quantitative evidence, it's not enough to be scientific proof, although the things we are doing align well with the educational literature.
The evidence we have fits with my idea that contextual learning is something that should be done from the beginning, and these skills are developed over time. Over time they become measurable and significant. You can't just take one quarter and think that you've really made such a huge impact on students' reasoning abilities that it's going to be something you can measure.
The difficulty in measuring learning gains discourages me from time to time because I want to have proof. I perceive, as an instructor, that they are learning valuable things. I think that they learn to appreciate math more and to see it's worth.
The data from the physics department showed that the students who came out of the Tech Math sections with the lab (which had the activities) did better in physics than the students who were just taught traditionally. It was motivating, but it was a small study so there were limitations to the data.
The department chair also did a study and showed that the students who were getting less lecture time in Tech Math and more activity time, did just as well as students who only had lecture only. He couldn't show that they were significantly better in the next course or further on, but they did just as well. So I was bringing all these things in and talking a little bit less, and they were still learning what they needed to learn in a traditional sense. There was not a test on the additional things they were learning in my class such as the ability to solve problems, and that might have shown additional differences.
It did show that if the students gone through a whole year of Tech Math (2 courses), that they fared a little better in the courses beyond than the students who just had one section of it. This evidence reinforces my idea that it takes time for improvements to be measurable.
I tell students, "We don't want to tell the robot to go 500 clicks forward. What can we do to input something so that you can communicate to the robot in centimeters or inches or feet that you want it to move forward?" We have to find a relationship between the clicks–which is a variable. If you want the robot to go different distances, that's going to be a different number of clicks. If you want it to go 400 clicks, but you tell it to go 800 clicks, it would go twice as far.
But I don't want to talk in clicks, I want to talk to it in centimeters. So the variable of clicks, "X", which would be how many clicks it would take. It's a variable because we don't know how far we want it to go. Tomorrow we may want it to go another distance. So "X" will be a different number tomorrow. It's a variable, it changes from time to time. The path will change, the variable, the distances, the clicks will change.
We want to talk to the robot in centimeters, not clicks. How are centimeters related to the clicks? So they have to figure it out. If 400 clicks turns the wheel once, the wheel has a certain circumference. From geometry, circumference is related to the radius of the wheel. Circumference equals 2(pi)r, or (pi)d for the diameter. So 400 clicks would be related to so many centimeters of one turn of the wheel. Now if we can figure out how the centimeters would change based on how the steps would change, then we can develop a function, or an algebraic way of explaining that relationship.
Let's take some measurements to try to figure out how those two variables are related. Let's create a function, let's plot data, and look at a graph. Then let's put that formula into the calculator and let the calculator automatically figure out. Every time we want it to go a certain distance in centimeters, your formula will be in that calculator, and it will calculate the number of clicks automatically every time. So you just change the distance, the robot will figure out how many clicks it will take to go that distance, then it will do that distance–any distance you want.
Right away, they're realizing this language gets much more elaborate as we go along. The robot does all kinds of things. The robot draws with a marker. It has motors that it runs and it knocks on things, and it takes data from sensors in its environment. It has to analyze that data and figure out how to make a decision. How do you quantify that data? How do you write the program to quantify that data to help the robot make the right decision? By answering these contextual problems, students build their understanding of algebra.
- You don't always have success. I might do something in the classroom one day and it just doesn't turn out to be a success. The students weren't really excited, or what I was trying to do didn't work. And after all that work, was it really worth going on and doing more? If you really believe in what you're doing, you come back the next day and try it again. Try something different. Learn from failure. Keep going. The goal is good, the end is good, you just keep going.
- People can say things that make you discouraged. There are different philosophies about how people should be taught in different subjects. There are different teaching styles, and different learning styles. It's hard to prove who's right about lots of things. There were times when people with different teaching philosophies said things that were discouraging.
You have to learn from experience. The instructor is very important. Learning how to facilitate these activities is part of the learning process. You just have to believe in what you want to do and push through and give yourself some slack. Try something different the next day. Try to change it, see how it works. It takes time. You can also learn a lot from educational literature. Ultimately, I got to the point where I was helping a lot of instructors. The department wanted me to come in and work with instructors and help them to know how to do it. Maybe even have them sit in on one of my classes or a few of the classes just to see how the whole thing should go.
- Collaboration is important. I worked with Fred Thomas in the physics department right from the beginning who knew more about the electronics and we worked together to build the equipment. He has been a very, very important part. In addition to physics knowledge, he brought a lot of educational philosophy or ideas to the project. Working with people can bring a lot to any project. You can all build something bigger than what you can do by yourself.
- Learn what others are doing and decide what you want to do. What else is going on? What have other people done and been motivated by and found success in? Then figure out what do you want to do and what you believe in, realizing that there's a lot of room to be creative.
- Obtain support from the greater educational environment. People need support so that they can motivate themselves because at the beginning because there is a steep learning curve, especially for somebody who has taught traditionally. There is a learning curve on being able to work with students and facilitate their learning in contextual way. You need support in order to develop ideas on how you think about your teaching and how you want to work with those ideas and bring them into the classroom. It's going to be hard at first to figure out how to do that.
- Don't give up. There were many times early on when I was trying to do some of these things and feeling like, 'I don't think this is working.' There were frustrated days. It took extra work. What brought me back the next time was believing in what I wanted and having enough support to push through the difficult days. But I'd never go back. I can't teach a class without doing something like this. It has a number of rewards and enough good days early on that keep you motivated and keep you going.
Personal Statement on Teaching Philosophy and Methods (Acrobat (PDF) 73kB Aug12 13) by Robert Chaney.