“Math is too hard!” “When will I ever use this?” “I'm just not a math person.”
Comments like these are all too common in math classrooms. Students, parents, and sometimes even teachers often believe that complex mathematics is just beyond certain learners. But some educators are demonstrating that anyone can learn, understand, and use complex math. Their efforts are illustrated in the following three stories: A group of 6th graders realize that they can perform math that is more complex than the work usually expected of students their age; parents and middle school students in an Ohio community discover that math can be both fun and relevant; and a classroom of elementary school teachers in training come to understand the meaning behind the procedures they memorized when they were math students.
Building Confidence with Calculators
Lesa M. Covington Clarkson
Even though achievement in mathematics on the National Assessment of Educational Progress has steadily increased since 1973, the gap between black and white students has persisted. Reducing this achievement gap requires high-quality teaching and even higher expectations, culturally relevant mathematics environments, and 21st-century technology and curriculums. Experiencing such instruction in the elementary grades translates into more course options for high school, more choices for courses and majors during postsecondary study, and more career and economic opportunities in the future.
Marshall Elementary School (a pseudonym) is an urban charter school serving students from preschool through 6th grade. More than 99 percent of Marshall students are black, and approximately 70 percent qualify for free or reduced-price meals. I recently spent seven months with the school's 6th grade classes providing instruction in using graphing calculators. Students used the graphing calculators to discuss patterns and data before moving on to functions.
Many people assume that 6th grade students are not able to use graphing calculators, and most mathematics textbooks reserve the use of graphing calculators for high school mathematics. So why did I choose to use this sophisticated technology with these students? Because I felt it was imperative for students to become curious about more sophisticated mathematics and the questions that they could answer with technology. I wanted to engage, encourage, and motivate historically underrepresented 6th grade students in the study of mathematics. During the calculator project, these 6th graders participated in mathematics that went beyond the routine curriculum of their peers. My expectation was that they could do the math, so they assumed that they could do it, too.
During the first lesson, students got acquainted with the calculators. I guided students through entering numbers, performing operations, and accessing the secondary functions. Then students “wrote” their names on the calculators. I wanted them to know how to use the alpha key, but even more important, I wanted to get to know their names. After they had written their names on the calculators, they held up the calculator to show me. I thanked each student personally. The personal connections enabled me to build relationships that would allow me to challenge them in future meetings.
Before exploring actual functions, students used data to explore the various menus. The purpose of this initial exploration was to help them understand the relationship between the list (gathering data); the graph (displaying data); and the line of best fit (predicting data).
Students started with prepackaged data sets (bags of M&Ms) before moving on to real-world data. Students sorted their M&Ms by color and recorded the color distribution, then entered their information into the lists on the graphing calculators. Students used box-and-whisker plots on the calculators to compare the color distributions of their M&Ms with those of their classmates, moving beyond the descriptive statistics typically expected of 6th graders (such as finding the mean, median, mode, and range) to talk about outliers, inferences, and predictions regarding the “population” of M&Ms.
Students gathered real-world data by searching newspaper ads for the purchase prices of several used cars of the same make and model. They recorded and then graphed the data on their calculators. During the following meeting, we found the line of best fit so that we could predict the cost of a similar car. Students worked in pairs and then presented their findings to the rest of the class.
Once students saw how the calculators could help them explore data that were interesting to them, they were ready to move on to more abstract mathematics. Sixth graders are routinely asked to solve for x when given equations like 15 = 3x or 4 + 5x = 19. We discussed such equations in terms of their graphs and how the graphs were affected when the equation changed. For example, how is the graph of y = 3xrelated toy = 3x + 7? Students could then make predictions about other related graphs based on what they observed in families of equations.
Throughout this project, I attempted to incorporate the interests of the students while scaffolding them through sophisticated mathematics. Teachers have the responsibility of providing students with entry points into a sometimes unfamiliar academic world. Only then will students view rigorous courses like calculus as a possibility, and thus consider future career options that are rooted in mathematics and technology.
Community Connections
Gay Fawcett and Elaine Shannon-Smith
A recent report by the American Council on Education found that 70 percent of Americans acknowledge that math and science are “very important” for all graduates. Yet an alarming 44 percent of those surveyed believe students avoid math and science because the subjects are “too difficult.”
Unfortunately, negative math attitudes are fairly prevalent in U.S. society. The comment, “I'm not good at math,” is heard in parent-teacher conferences, in the guidance counselor's office, at the dinner table, at the income tax office, at the checkout—just about anywhere people need to consider math.
If math achievement is to improve, a change in attitudes will be necessary. Is there a way to convince students and parents that mathematics is not only necessary in their everyday lives, but also fun? Akron, Ohio, is trying to do just that by embedding fun, yet challenging, middle school mathematics into family outings.
The project, Learning 24/7: Community as Classroom, is a partnership among schools, parents, and community organizations coordinated by the Summit Education Initiative, a nonprofit organization formed in 1996 by the business community in greater Akron. Community organizations, such as the zoo, the park, a minor league baseball park, and a playhouse theater, agree to host Learning 24/7 activities. An educator works with someone from the organization to design authentic, engaging activities with educational value. All activities reinforce challenging mathematics concepts from the Ohio Mathematics Benchmarks for middle school.
At the Akron Zoo, families encounter TNT, the resident Komodo dragon. When they stop to have a bite in the zoo's café, families receive a colorfully illustrated tray liner with an activity that helps them determine the time it will take for TNT to capture a meal of his own. The liner lists the time in seconds and the distance in feet that TNT and his prey, Chicken Dumpling, are traveling. Over a sandwich, families can calculate how many feet TNT has to run before he captures Chicken Dumpling for dinner. When they get home, they can check their answers on the zoo's Web site.
While at the minor league baseball park, students receive an activity card on which they can chart the beginning and ending times for three innings. The activity card then suggests that students use the process at home to figure out how much time they spend actually watching a half-hour television show compared to the time they watch commercials during the show.
Other activities involve measuring walking strides to calculate distances students hike and climb, comparing and ordering decimals and percentages to complete a connect-the-dots activity, and determining how many bathtubs full of water it would take to fill a lock at the local canal. Sample activities are available at www.learning247.org.
- “The pictures were cute and engaging, making you want to take on the challenge.”
- “I liked it because of the weird but true facts about the animals.”
- “I liked it because I knew it wasn't for a grade.”
- “I liked this because it's something we could do together.”
- “We weren't actively looking for an educational experience like this, but it was nice to stumble on it!”
- “The individual hints made it easy.”
We realize the problem of poor math attitudes has a long history, is very complex, and will not be easily resolved. We do not claim that creating engaging mathematics activities for community sites will cause a huge turnaround. However, we do believe that every action taken to change math attitudes is worthwhile. Not only have we seen attitudes change in parents and students, but we are also sending a message to those who have bought into the belief that math is hard and not everyone can do it.
What can you do to involve your community and parents in changing attitudes toward mathematics? Could you design an activity for the local skating rink, bowling alley, or park? If you can work with a few others to jump-start such an initiative, attitudes can improve. Negative attitudes toward math are deeply entrenched and will take time and effort on our all parts to change. Let's get started!
Making Meaning for Math Teachers
Nancy T. Goldman
When I begin my graduate-level math pedagogy class each semester, I ask my students to raise their hands if math made sense to them when they were going through school. In seven classes during the past four years, only a few hands have gone up! Very few of the students who come into Long Island University's masters program in childhood education (grades 1–6) have strong backgrounds in mathematics. A typical student in the program has at least six credits in college-level mathematics and was exposed to algebra and geometry and some trigonometry and precalculus in high school. But exposure doesn't always equal understanding, and without understanding math, these students can't teach it successfully.
A recent study found that although 3rd grade math teachers in the United States are more knowledgeable about general educational theories and classroom skills than are their Chinese counterparts, Chinese teachers have a deeper understanding of math content. My students tell me that when they learned math their teachers told them what to do and then gave them practice problems that helped them memorize the procedures. They never felt they understood what they were doing. I have tried to build my students' understanding and confidence by teaching mathematics content along with pedagogy.
One activity that I've used involves the Pythagorean theorem. Generally, when they begin my class, every one of my students can recite the formula for it, but none can explain it. So I ask my students to illustrate the formula using tape on the floor, geoboards, tiles, or another material of their choice. One group of students in a recent class made a model on the floor with masking tape. First, they constructed the two adjacent sides (a = 3 feet, b = 4 feet) and then connected them to form the opposite side, or hypotenuse (c = 5 feet). Next, they turned all three sides into squares using tape. Finally, they computed the perimeters of each square. When they combined the measurements of the sidea square and side b square (a<SUPSCRPT>2</SUPSCRPT> + b<SUPSCRPT>2</SUPSCRPT> = 9 + 16), they realized that it equaled the squared measurement of side c (25). Their shouts signaled that they understood the Pythagorean theorem for the first time.
Another example involves the derivation of π. Students work in small groups to measure a variety of circular containers, recording both the circumference and the diameter of each. Then we make a large chart, listing each object, its circumference, and its diameter. Invariably, a student will notice that the circumference measurements are about three times the diameter measurements—that 3:1 relationship is π. Students now understand where π comes from and why C= πd, and they can approximate diameters in relationship to circumferences and vice versa.
Many students know the procedure for division of fractions without understanding what they're actually calculating. When I ask my students how to get the answer to 1 3/4 ÷ 1/2 they tell me to invert and multiply, but no one can explain why. So I show them two rectangles, the first one whole and the second one with 3/4 marked off.
Next, I ask students to divide the first rectangle into halves. They clearly see that there are two halves. Then I ask them how many halves there are in the 3/4 section of the second rectangle, and they say that there is one. So now we have three halves. I ask what portion of a half is left in the 3/4 section of the second rectangle, and students see there is only one-half of a half. The problem 1 3/4 ÷ 1/2 is actually asking how many halves there are in 1 3/4, and students can see that the answer is 3 1/2. Because students are halving each amount and counting the halves, the answer is 1 3/4 doubled!
How can teachers give students the mathematics experiences they need if they don't understand mathematics themselves? By engaging in fun, relevant math activities that build their confidence and understanding, my students are better prepared to guide their own students into deeper understanding of math. Mathematics is important. We can teach students well. We can't give up on this crucial endeavor.
