Math doesn’t have to be hard. It is a tool: the tool of science and economics; the tool of problem solving and the algorithmic approach to issues. One of the many beauties of the tool is, it has been around for thousands of years, and although we find and develop new aspects, it is largely unchanged. I approach teaching math by trying to show the students that what they are learning is all within their current knowledge.

Number Sets
When I discuss number sets with the students I like to use a few analogies. I use a story about how numbers developed to discuss the natural progression of natural numbers to irrational. As I discuss each level, I like to the layer the discussion with geography: everyone in your neighborhood lives in your city, but not everyone in your city lives in your neighborhood. When I get to the final rational, irrational, real story, I link rational to the state we live in and irrational to the neighboring state. That leaves real to be the US.., no land of its own, but joining all of the states.

Adding and Subtracting Numbers or Like Terms
This seems like it should be easy; we all know our basic arithmetic, but this piece can leave more students dumbfounded and convinced that they cannot be successful in math. One big piece I highly emphasize is the idea of stacking like terms, or numbers that are being combined. When we are first introduced to math computation, we learn vertically, but once we get into algebra, not only do we add variables, we also include a horizontal approach. When the numbers are stacked, I put a rectangle around the sign. When I teach older students, especially when I taught in an urban setting, I would tell them that the rectangle was an ally. There are two rivals, the positives and the negatives. I ask them, “Are they going to party or are they going fight?” If they are the same sign, they party and we figure out how many are partying. If they end up in a battle, I ask, “Who is going to win and how many will survive?”

This topic is expanded with I discuss polynomials and multiplying them. Anyone who has survived an algebra class has heard FOIL – first, outer, inner, last. I have tried to avoid this acronym.  It is VERY limiting. It only discusses what to do if you multiply two terms to two terms, but what if there are more? It also emphasizes the horizontal approach to math versus a vertical approach, which is much more comfortable to students.

So what is the alternative? I like to reinforce the distributive property, which is what the acronym is reinforcing. I tell my students to use the “distributive property with stacking.” Distribute each term from the first parenthesis to every term in the second parenthesis and stack the like terms. It allows for easy completion of this type of problem. It also allows for flexibility of multiplying any number of terms to any number, making any problem approachable.

Math isn’t hard
Math is math. I remember my mother telling me that she could never understand Spanish; and consequently, I believed I could never learn the language. I made it through the high school requirements, and was relieved it was over with. In college I thought I would try it again, and give it a new chance. As it turned out, when I didn’t believe learning Spanish was impossible, the class became easy.

In other countries math isn’t hard because people don’t talk about it being hard. The brainwashing isn’t there. If we step back and give math a real chance, see it as a way to break problems down systematically to find a solution, it may not be the great stressor so many student see it to be.

Huckleberry Rahr taught high school mathematics for nine years throughout the Midwest and one year in Papua New Guinea. During that time, Rahr earned a Masters of Education from Cardinal Stritch University. She spent two years working in the private sector before returning to education as a full-time instructor for ITT Technical Institute in Madison, Wis. Rahr is a member of Inside the School’s editorial board.

Photo credits:
Math Class: attercop311 / Lauren on Flickr.com Creative Commons
Math Finals: ensign_beedrill / Rough Tough, Real Stuff on Flickr.com Creative Commons
for my math girl: crd! / Craig Dugas on Flickr.com Creative Commons

As a math teacher, I have struggled with what to do with homework. Homework is a conduit of communication between the students and the teacher. If a student knows the material and can show competency without homework, what is the use of the homework? Should every student have to complete the same amount of work for mastery? I guess we should break down homework types in mathematics. The most popular homework is rote practice. A student is given a list of problems broken up into multiple levels of difficulty, which allow for practice and mastery. A second type of homework is the story problem, which is a train wreck according to students everywhere. A third type of homework is experimentation and discovery.

Rote practice. Rote practice has become the four-letter-word of mathematics, yet remains one of the most common classroom assignments. It is frowned upon because rote practice doesn’t show how the mathematics is applied and used outside of the math class. Being able to add a list of fractions together is a great test of patience and determination, but will it translate to the kitchen when the student bakes a double batch of cookies, or worse, a half recipe?

Although many teachers frown on working on a list of problems, there is some validity to providing direct instruction practice so that when the student understands the application, the mathematics doesn’t stand in the way of finding an answer. With a list of pure mathematics to which the student has the answers, it is incredibly important to check for more than completion. If a student works all of the problems and gets full credit for completing the homework, but all of the work is wrong, what has she learned? When the student takes a quiz or test and does the same process as in the homework, and has the wrong answer, is it really the fault of the student? Was there a point in doing the homework? If the homework isn’t a vehicle of communication between the teacher and the student, then no, there wasn’t a point.

Story problems. Story problems are a double-edged sword. Some problems that are written are pure gold. They show a valid application of the mathematics in a way a student may need to use in real life. For instance, given a recipe that serves eight with lots of different fractions, translate the amounts for 16 people or four people. A more challenging problem may be 12 people, but that makes the problem much less realistic. A train traveling at any speed is good for a quick warm up in class, but as homework the topic has too much of a stigma. I’ve had many students freeze up for no other reason but the word train appears in the problem. Once you get up to trigonometry, train problems are back on the table.

When creating story problems, less is more. Given a homework assignment with over five story problems, the students who need the practice will freeze up and be overwhelmed. The students who understand the math will either do the problems because they enjoy the math, or skip the homework because they understand the math, and don’t like “English.”

Creating a few story problems that get to the heart of the material is the best way to engage students in applying the math. At the beginning include several steps to help the students investigate the question. What information is pertinent? What is extraneous? What formula should we use? What concrete information are we given? What unknown are we looking to find? If this is started early on in the year and continued throughout the year, by the end the students should feel more comfortable with story problems and know how to solve them. Even if you teach the same students two year in a row, starting off simple and leading students through the problem solving will always benefit the learners.

Experimentation and discovery. New textbooks often start with experimentation and discovery, and, to me, that is a big problem. Students need to have basics to fall back to before they can discover a higher level concept. That said, if students can play with the math in with experimentation, they are much more likely to remember the topic later. They have invested time into the problem and are committed to the solution. A good extension to this is a project that brings life to the math and allows students to show their individuality and creativity.

Bringing it all together. Let’s see how all of this can be used together. I teach a class that has a unit on linear equations at the start of second semester. I want to make sure that once we get there, lines are the only challenge. (For more information about this activity, see next Friday’s post, which will provide teaching tips and a printable handout.) I try to use a form of sustained learning that focuses on the material throughout the year. Starting in the first quarter, I will hand out a packet with 10 to 15 problems that hyper-focus on a mathematical topic. I usually start with one-step algebraic manipulation. I demonstrate exactly how I expect the students to solve the problems from method of solution to checking the work. When the student hands in the packet, she either gets full credit or I hand the packet back to her for corrections. No score is recorded until the packet is perfect.

I move from one-step equations with addition and subtraction to one-step with multiplication and division. Before moving to two-step equations, I might add a packet combining all forms of one-step problems. After I moved through several layers of algebraic manipulation, I’ll move to formulas and their manipulation. Eventually I will have a packet on the formula for slope. I will have a packet on substituting into equations, and have a packet on point-slop form of a line. Then I’ll combine the two topics and have students take the point-slope form and solve for y. Throughout all of these packets I present the information and math basics, without saying anything about linear equations or algebraic manipulation.

When I get into the unit on linear equations, I can have application and story problems to figure out the equation, either directly or with scatter plots. I end the unit with a project. I ask the students to create a picture of a face using straight lines. If the class is upper level, I may include parabolas, hyperbolas and circles. Despite the number of lines they use for the creativity level of the project, I only grade 20 lines. Of that 20, only five can be vertical or horizontal lines. This allows for creativity, but doesn’t demand a huge amount of grading. I also give them a sheet that asks for endpoints, slope, point-slope and slop intercept form of each line. This really helps in grading and following their work.

Homework is for practice and communication. The point of homework is to give students a chance to practice on their own and communicate to the teacher what they do and do not know. If the theme of practice is rote, which has four letters is not a four-letter word; it is of immense importance that the teacher reviews the work and comments. Story problems are great for applying the math. It really highlights that math is just a tool, not an insurmountable obstacle. With story problems it is important to go over the work in some manner, be it by grading or just reviewing with the class. There are often several ways to approach a story problem so a class discussion becomes a learning moment where students can teach each other on how they approached the work. Experimentation and discovery is great, especially in a project. In this type of homework it is possible that a great amount of work has gone into it. Grading is important, but the real nit picky math check should be saved for the rote homework.

Huckleberry Rahr taught high school mathematics for nine years throughout the Midwest and one year in Papua New Guinea. During that time, Rahr earned a Masters of Education from Cardinal Stritch University. She spent two years working in the private sector before returning to education as a part-time instructor for ITT Technical Institute in Madison, Wis. Rahr is a member of Inside the School’s editorial board.

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