Calculus is often a major departure point in a student's math career. Applied problems from physical situations is now the norm instead of the exception. Furthermore, the strategies that a student would use, for instance, finding the inverse of a matrix are not always going to help in topics like integration and infinite series. Therefore it is essential that the instructor builds in opportunities for students to learn and practice problem solving strategies to ensure student success and confidence when learning the concepts of calculus.
There are 2 major problem solving paths in mathematics: procedural and, for lack of a better term, creative. Procedural or algorithmic problem solving is the more prevalent, and more familiar, form to teachers and students of math. Going back to the problem mentioned above, finding an inverse matrix, one can apply a procedure to achieve the result. The algorithm may look something like:
- Set up an augmented matrix with an identity matrix of the same size on the right.
- Multiply row 1 with a constant to produce a 1 at entry 1,1
- Add a factor of row 1 to row 2 to produce a 0 at entry 2,1
This would continue until the left side of the augmented matrix becomes an identity matrix, which produces the inverse matrix on the right side. If one can follow the steps individually, then one can solve this problem.
An added level of complexity occurs when a student has a set of algorithms to solve a problem, but must find clues, to choose the correct method. An example might be solving for the missing variable in a second degree polynomial. Direct computation, factoring, completing the square, solving by radicals and even guess and check may all be successful strategies, often with one choice being the most direct route. Being able to find clues is an experiential process, and therefore this added level is sometimes difficult for students to master as there is a high need for guided practice and personal success before students have command of these tools.
On the far end of the spectrum is what I call creative problem solving. These are challenging problems that may or may not have an algorithmic procedure available, and often give few clues for students to latch onto. A problem like ∫sin(101x)sin99(x)dx (a problem from a MIT integration bee) will require use of many tools and clues to find the solution method, or methods.
The problems in this guide are meant to provide enrichment for students to develop good problem solving skills not only for the problems in the section, but also to provide the framework for solving problems later in the text.
Writing in Mathematics
Writing in all subject areas is important, and while high school mathematics sometimes ignores the duty of writing, it is increasingly becoming an expectation of math students of all levels to express their thought process and reasoning in concise prose. Furthermore, all advanced mathematics requires solutions and conclusions to be presented in such a manner. It is for these reasons both the NCTM standards in the Problem Solving and Communication strands, and the California Math standards have writing required.
Teaching students how to write in math class can be a battle. It is sometimes unlikely students come into the class with experience with writing in previous math classes, therefore there will be the need to not only properly scaffold the necessary skills, but also to fight a little bit of the expectation “This is math, why are we writing?” It is useful to have clear expectations, regularly and consistently give the opportunity for writing, and give good critical feedback on student work.
Here are some general rules for mathematical writing. First, writing should be more than showing work. The temptation may be to simply narrate the steps needed to reach the same conclusion, and while some of this narrative may be part of a mathematical paper, it is insufficient on it's own. More critical than writing explanations of computations is guiding the reader through the writers though process and ideas. Therefore the reader can not only follow the work, but has an understanding of what is going on, but why those methods were chosen. Second, writing a technical paper with symbols and math expressions does not excuse the writer from the rules of grammar. Good writing has good grammar, and I recommend against the occasional habit of ignoring grammar and only grading technical content. Furthermore, there is also a grammar of mathematics that allows expressions to be implemented into text seamlessly. Complete math-sentences involve a comparative operator with two expressions, like clauses, on either side. The comparative operator can be an equals sign, greater than, less than, set element of, and so on. Expressions without a comparison to a concision should not be without text explaining what the expression is there for. It is bad form to start sentences with math expressions, but it acceptable to end a sentence with them, and should include a period. With all these rules, when it doubt: read the sentences out loud. More often than not, this will expose bad style immediately.
Setting up Computations
Often times papers will need to include a series of computations. There is a good way, but many less than perfectly clear methods. A couple of common errors: 3x+4=1−6x=3=−9x=−13=x. I have seen students mistake the symbol for “equality” with “therefore” or “giving”. Another less than clear method may be placing all work in line, such as: 3x+4=1−6x→3=−9x→−13=x. This is a better choice, but still not as clear as:
Notice how math computations are usually centered. If it is anything less than perfectly clear, (which depends on the audience which the paper is intended for) explanations of computations should follow each line in text.
Getting the variables both to the right by subtracting 3x, and the numbers to the left by subtracting 1 from both sides yields
Divide both sides by −9 to get an answer of
It is considered bad form to use the “two-column” method that is sometimes employed in teaching proof-writing in geometry classes.
Organization of Math Papers
Most math papers have a standard arrangement: Introduction, solution, interpretation/conclusion. The introduction should include a statement of the problem in the authors own words. It is useful at this time to interpret the significance or importance of the question if it applies. Also, it helps the paper to foreshadow the solution method used in the paper.
The solution can include the final “answer” either at the start of at the end. Sometimes it is clearer to present the answer and then present the method and reasoning after, sometimes it is clearer to follow the exact thought process, arriving at the answer at the end. The interpretation or conclusion will be included if there is some inference to be made about the question that required the answer. In social science and other applied questions the conclusion is often more important than the solution.
Typesetting mathematics can be challenging for students, but also provides great opportunities to teach some technological tools in the classroom. Like in other classes, the preference will always be to have the paper typed, and it should be depending on resources available to students. The challenge is how to put all of those math expressions in there. There are a few acceptable options. First, it is always acceptable to type a paper in a word processor, leaving space for math, pictures and graphs, and to draw them in neatly by hand after printing. Better is to use the built in equation editor in modern word processing applications. All the major programs have the option to insert mathematical expressions. The process, and the syntax required, can be accessed through the program's built in Help documentation. Some schools will own licenses to mathematics or scientific software like Scientific Notebook, Maple, MATLAB or Mathematica. While many of these are designed first for their ability to do computation and visualization, they have the capability to typeset some very nice looking math. The finest option, although the hardest to learn, is to use a TEX or LATEX typesetting front-end. A front-end will take your writing and commands (TEX, and to a lesser extent LATEX can look more like a programming language with it's commands) and set them in beautifully formatted documents. The learning curve is long, but this is what scientific papers are expected to be presented in at the university level. Also, it's free and can be used on 99% of all computer systems ever made. More information can be found from the TEX user group at http://tug.org.
How to Get the Students Started
A key to getting comfortable in writing math papers is getting used to metacognitively investigate solution methods and have other people reading about that process. Start with some problems that the students have some confidence with, but be careful not to make them too easy. Sometimes it is more challenging to write a good explanation of a problem where the solution appears to be obvious. Have students regularly take a problem, write an introduction and a clear solution. In small groups students should read their work to their peers. This acts as both a way to understand what is clearer as an audience, and gives the reader an enforced check on the quality of their work. Regular practice on “everyday” problems will equip students with the skills and confidence to tackle larger problems and papers later in the class.