The following article is an example of using predominant learning styles and Bloom' Taxonomy to teach mathematical reasoning. It is the model all of our tutors are trained with. This has been very helpful to our instructors as they mentor adjuncts. The WHAT, HOW, WHY, and WHAT IF of Mathematics: Teaching undergraduates to think up Benjamin Blooms cognitive Levels By Carole Sullivan and Eldon McMurray of Utah Valley State College To fully understand mathematics, it is important to know three things: 1. WHAT precisely the problem is asking; 2. HOW to do the problem; and 3. WHY certain steps give you the correct answer. Then to consider this: WHAT IF the problem were a little different. Think about it. Can you really know how to work a problem if you don’t know what the problem is asking you to do? Can you really be sure that the how will produce the correct answer if you don’t understand why the steps work? It’s like trying to ride a bike for the first time without knowing what “ride” means. If you hadn’t seen someone ride a bike before, you likely would not understand this simple task: Ride the bike from point A to point B. You would first need to figure out what it means to “ride a bike.” So, let’s say this is what it means: To ride a bike is the act of making the bike move. That’s a good start, but how do you make the bike move? You might come up with the following steps: 1) Sit on the bike; 2) Put a foot on each pedal; 3) Push the pedals forward with your feet; 4) Grip the handle bars in your hands; 5) Keep the front tire straight until an obstruction compels you to turn, and so on. Now you know how to move the bike: Pedaling. Can you really be sure that pedaling will move the bike from point A to point B? You could jump on and give it a try. But could you be sure otherwise? No. You must know why the tires rotate when you push the pedals to understand how the bike moves (the pedals move the chain, the chain moves the tires, and the tires move the bike). To fully understand a problem, it is best to know the how, or the steps, and the why, the reason those steps work. “But I was so young when I learned to ride a bike,” you might say, “the hows or whys never even crossed my mind.” Fair enough. Like most of us, you just did what you saw everyone else doing and it worked. This is where the question what if comes in. At first, you probably didn’t wonder, “What if the bike won’t move when I push the pedals?” At some point, though, that very thing probably happened*the chain came off, and when it did you were obliged to think about how the bike moves and why so that you could fix it. Along the same lines, real success in math depends on more than just knowing how to get the right answer. Students arrive at correct answers all the time without really understanding mathematics. But if they get the right answer and don’t know what they were trying to accomplish or why their answer is right, have they learned the math? No, not really. “But all I’ve ever learned was how to do a math problem,” you might say, “and that’s gotten me through every math class just fine.” Point taken. Here’s the bad news, though. You can ignore the what, why, and what if for a while inside the classroom, but it’s probably going to catch up to you more quickly outside the classroom. Consider the following scenario outside of mathematics. Imagine that you have graduated from college and are working in your chosen field. Regardless of the occupation, your employer will want you to be able to solve problems. Maybe not algebra problems, but problems nevertheless! When you are given a problem, are you going to jump right into solving it? No, of course not. If you’re smart, you’ll analyze the problem first, make sure you understand what the problem is and what kind of answer is required. Once you fully understand what the problem is, then you are ready to tackle how to solve it. As you explore possibilities for achieving your goal, you will want to be aware of your resources. You may come up with a brilliant solution, only to find that the solution does not fit within the company’s budget. You need to know what you have to work with. Once you find a solution, your boss will ask you to explain why it will produce the desired outcome. No company wants to waste time and money on an iffy plan. You will need to be prepared to justify each step of your plan. And your boss won’t go for justification like, “Because I say it will work” or “Because my professor said it would work.” Your boss will likely ask many what ifs. What if questions help you consider and prepare for potential problems that may arise. Your boss will want to know that you have not only foreseen possible roadblocks but have devised a plan for how to deal with them. If you really want to impress your boss, you’ll be prepared with answers for every what if thrown your way. Do you see the importance of What, How, Why, and What If? Clearly, you can get in real trouble real fast in the real world when you ignore these vital questions. In the mathematics classroom, it may not seem to matter much until you get into a tough course. But it will catch up to you in here just like out there. Want to give the questions a try? The following is a basic example to get you started. EXAMPLE ONE Evaluate: What is the problem asking me to do? Evaluate What does that mean? Find the number that it is equal to. What rule will help me? Order of Operations What are operations? Addition, subtraction, multiplication, and division What is the Order of Operations? Work all problems in the following order: 1-Operations inside grouping symbols ( ) 2-Exponents*powers 3-Multiplication and division left to right 4-Addition and subtraction left to right How do I work the problem? Start inside the parentheses. Why? Order of Operations How do I do what is inside the parentheses? Do the division first then the subtraction. Why? Order of Operations Why is the 6 still in parentheses? Is this necessary? Yes. Parentheses can also be a symbol for multiplication. What does the problem look like now? What is the next step? Multiply and then add. How do I know that is correct? The Order of Operations was followed. What if the answer is a decimal or fraction? Decimals and fractions are possible answers, but if you haven’t worked with them previously in the class then the answer is probably wrong*go back and check your work. What if it doesn’t seem to follow Order of Operations? Remember that operations may be implied*meaning that you don’t see a specific symbol, but you’re supposed to know what to do. See the following example: Evaluate: How do I work the problem? Start with the operations in grouping symbols. Why? There don’t seem to be any grouping symbols. Remember that when the numerator or denominator of a fraction contains an operation, there are implied grouping symbols. So What next? Divide. Why? I don’t see a division symbol. Remember that a fraction is another way to represent division. This may seem like a lot of work for a fairly short problem, but understanding the problem thoroughly will help you with longer, more difficult problems that you will encounter later on. Don’t focus on finishing the problem in the shortest possible time. Focus instead on understanding all aspects of the problem. You’ll save time in the long run. Here are two more examples: EXAMPLE TWO Simplify: What is the problem asking me to do? Simplify What does that mean? Reduce the number of terms. What is a term? A number, variable, or a number and variable(s) being multiplied. How can I reduce the number of terms? Use the distributive property of multiplication over addition. What is that property? How does that apply to this problem? Using this property the problem can be written as The coefficients (numbers in front of the variables) can be grouped in parentheses followed by the x. Thus the rule is to add the coefficients and keep the same variable. How do I know that is correct? Check it by using a specific number for x. Say . (or any number you choose) Then Both expressions equal 60 so is correct. What if the variables are different? The distributive property cannot be applied and thus the expression cannot be simplified. Example: How can the terms be combined? They can’t because the distributive property cannot be used: Neither x nor y can replace the ? to make a true statement. So cannot be simplified. What if some terms have the same variable but others don’t? Example: How can I simplify this problem? Group the like terms. What are like terms? Terms with the same variable(s) raised to the same power(s). How can I group terms that aren’t already side by side? Use the commutative property of addition to order the terms in any way. What next? Apply the distributive property to the like terms. How do I know this is correct? Check it by using and (or any other numbers that you choose) EXAMPLE THREE Solve: What is the problem asking me to do? Solve What does that mean? Find the number that can replace x and make the equation true. How can I find x? First simplify. (Reduce the number of terms.) Then get x by itself. Why simplify? To make the problem easier to work with. How do I simplify? Apply the distributive property and combine like terms. How can I get x by itself? Add 2 to both sides and then divide both sides by -2. Why? The Properties of Equality say that adding the same number to both sides of an equation or dividing both sides of an equation by the same number will not change the solution. How do I know that is correct? Check it. When x is replaced with -4 the equation is true. What if the equation is not true? The answer could be extraneous, meaning you haven’t made any mistakes, the answer simply is not a solution. Or (more likely) you made a mistake. Either way, go back and double check your work. What if there is a variable on both sides of the equation? Example: How can I find x? First simplify. What next? Use the properties of equality to isolate the x. How? Add 3x to both sides and add 8 to both sides. What is x? because How do I know this is correct? Check it in the original equation. This is a true statement so is the correct solution. As you can see, after you determine what the problem is asking you to do, you can ask How, Why and What If in any order. And don’t get caught up in whether to ask what or why*just start asking questions! Have a reason for every step that you take in a problem, and imagine that you have to explain your reasons. (It will be good practice for explaining solutions to your future boss.) Try to anticipate every different kind of problem that your teacher could throw your way. And you won’t be stumped! By asking these key questions, mathematics can change from the daunting task of memorizing a jumbled mess of rules and formulas to a clearly marked path where every step you take has purpose and meaning. Eldon L. McMurray Director Faculty Center for Teaching Excellence Assistant Professor College Success & Academic Literacy Utah Valley State College 800 West University Parkway Orem, UT 84058 (801) 863-8550 --- You wrote: 5) Focus Also on Careers Two ideas we had here concern career-minded TAs. One idea we had was to contact former graduate students who have now gone onto academic jobs and ask them to share about the role of teaching in their current careers. Sharing these "testimonies" with the current TAs in some way might help them see the value of spending time developing their teaching skills. Also, we thought we might add some teaching-related career-oriented topics, such as writing teaching philosophy statements and building teaching portfolios. --- end of quote --- These are very good, because they hit the TA's where they are going to live. My memory is that you have a teaching evaluation system that applies to the graduate students. You can use it to motivate students to participate. We have one that we use for everyone who teaches, and though it is voluntary for senior faculty, most participate most of the time. Early on in our seminar, if I am involved in it, I mention to graduate students that the vast majority of our graduate students get overall teaching ratings higher than the department average. (This isn't a Lake Woebegone phenomenon; our visitors, postdocs, and most regular faculty are in those averages.) In the most understated way I can, I point out how easy it is for the writer to a teaching letter to say "So and So's average on the "overall how do you rate this teacher" question is 4.25 on a 1 to 5 scale with 5 being the best, while the department average is about 4.1," and mention the impact that has on someone's chances for a job interview at a liberal arts college or a university where teaching is the main faculty function. I don't want to push the idea too hard, because I have seen very good teachers with below average ratings. But everything you can do to positively link your seminar with students' job opportunities later in life is likely to have an impact on their commitment to the seminar. _______________________________________________ PSTUM-list mailing list PSTUM-list(a)lists.fas.harvard.edu http://lists.fas.harvard.edu/mailman/listinfo/pstum-list