Math connections in society

Summary: An integrated approach to mathematics stresses the importance of making connections among various perspectives and applications.

While mathematics in educational settings is often separated out into differing subjects, it is important to understand that mathematics is an interconnected field of study. While most individuals are aware that they must be familiar with basic addition and subtraction to ensure the proper handling of money, very few individuals give any thought to the multitude of deeper mathematical connections they experience daily. In fact, both the National Science Foundation and the National Council of Teachers of Mathematics have recently begun to strongly advocate for the use of an interconnected curriculum in K–12 mathematics education. An integrated approach to mathematics education stresses the importance of making connections among mathematical perspectives, as in algebra and geometry, making connections to other fields, as in physics or religion, and connecting mathematical concepts to society as a whole, as in applications and usefulness in daily living.

The purpose of an interconnected curriculum is to help students better understand how the various branches of mathematics are connected and how mathematics is connected to the real world. By teaching mathematics as a unified whole, rather than multiple discrete subjects, students may better understand that mathematics is not a set of indiscriminate rules and isolated skills; rather, it involves a rich interplay between mathematical concepts, as well as complex interactions with other academic subjects. It is this integrated approach to mathematics that seeks to answer that question, “When are we ever going to use this in real life?” When this objective is met, students often show an increased appreciation and enthusiasm for mathematical principles.

People use many different interrelated approaches to process ideas, analyze objects, make decisions, or solve problems. For example, one might calculate the optimal viewing distance of a painting in order to see the depth that the artist intended, examine the surface of the painting to appreciate the finer details and glazes, or stand back to appreciate the overall effect and balance of colors. Real-life situations are not divided the way they are in textbooks by their applicability to a certain topic or technique, like exponential models. In fact, throughout the twentieth century, employers, such as engineering firms, complained about the lack of connections made in school between different subjects. Mathematician Eliakim Moore discussed this problem a century ago in his 1902 address as the president of the American Mathematical Society. In 1989, the National Council of Teachers of Mathematics published a set of national standards for mathematics that included connections as a significant component.

Whereas traditional mathematics curricula in the twentieth century separated subject areas like algebra and geometry, an integrated approach involves presenting mathematical subjects as one interrelated whole that also connects to other subjects and real-world experiences. In antiquity, the square of a number was defined as the area of a square with the same side length. People with interdisciplinary interests were perhaps more common at that time—Greek mathematicians were also astronomers, inventors, engineers, and philosophers. Throughout history, mathematicians such as Carl Friedrich Gauss contributed to so many areas of mathematics and to other fields, like geodesy; but in the twenty-first century, researchers who specialize in a subdiscipline are more common. However, connections among multiple mathematical perspectives are still important in the development of mathematics. Algebra and geometry remain linked and the field of algebraic geometry is active today. Many researchers use techniques from a variety of different mathematical fields. Geometers may heavily rely on concepts from analysis, linear algebra, number theory, or statistics, for example. Other researchers work in the intersection of fields like statistical analysis.

Mathematics can easily be connected to other scientific disciplines, like physics or biology. Mathematics is sometimes referred to as the “foundation” or “language” of science. However, there are many other types of links between mathematics and the sciences. Some researchers work on problems at the interface of mathematics and a scientific field, while others translate ideas from the sciences to solve problems in mathematics and vice versa. Scientific disciplines like physics are often referred to as partner disciplines for mathematics. Researchers have met for a conference named “Connections in Geometry and Physics” that explores the interdisciplinary facets. In geometry and physics there is a concept called a “connection,” which is an operator that allows for comparison at different points in a space via parallel transport. Mathematics has been interwoven with physics since antiquity. There have also been historical linkages between mathematics and biology, but the interdisciplinary field of mathematical biology has grown rapidly in the early twenty-first century.

Students may have difficulty appreciating the importance of mathematics in nonscientific disciplines, but the connections between mathematics and subjects like business, art, music, or religion are multilayered and multifaceted. For example, mathematics has played a part in religious life since the earliest documented cultures. The ancient Mesopotamians, embracing a polytheistic faith, developed the time system we use today with bases of 60 (60 seconds make up a minute, and 60 minutes comprise an hour). Adherents of Christianity, Judaism, and Islam have all embraced elements of mathematics in the conceptualization of sacred time. Given the importance of religion today, this time is still of great value for humankind. Mathematics plays a key role in the calculation of religious celebrations around which many faiths flourish. The week and solar day provide a delineation of sacred days that are different from the others—Sunday for Christians, Saturday for Jews, and Friday for Muslims. In other ways, numeric or geometric symbolism plays a significant part of religious practice.

There are countless examples that highlight the importance of mathematics in daily tasks. In the twenty-first century, it is almost impossible to find a task that does not connect to mathematics, either directly or through the tools and technologies in which mathematics plays an important role. In turn, mathematicians formulate new theories and concepts in order to meet the needs of society.

Mathematics as a Universal Language

Many people consider mathematics as the only truly universal language, regardless of gender, culture, or religion. For example, while the precise number of digits that are used in applications may differ, the ratio of the circumference to the diameter of a circle is still π, irrespective of the cultural context. Calculating the cost of groceries involves the same mathematical processes whether one is paying for those groceries in dollars, pesos, or pounds. With the universal language of mathematics, regardless of the unit of exchange, humans are likely to arrive at similar mathematical results. In fact, there are many examples of researchers in different areas of the world who independently arrived at the same theorems. Thus, mathematics as a universal language provides a common ground, creating the capacity for human beings to connect to one another across continents and across time.

Nutrition Labeling

An important way that mathematics can be found in our everyday life is on nutrition facts panels, which are mandated by the Nutrition Labeling and Education Act of 1990 to be placed on nearly all multiple-ingredient foods. The nutrition facts label on foods must list the fat, saturated fat, trans fat, cholesterol, sodium, total carbohydrate, fiber, sugar, protein, Vitamin A, Vitamin C, calcium, and iron content of the food. Other nutrients may be listed voluntarily. These labels also include a column that lists the percent Daily Value (% DV) to help consumers decide whether the nutrient content of a serving of the food product is a lot or a little. Mathematics is used to calculate the calories per serving and the % DV of a serving listed on the nutrition facts label.

As shown in Figure 1, at the top of the nutrition facts label, the serving size, as well as the number of servings per container, is listed directly underneath “Nutrition Facts.” In this case, a serving size is ¦ cup and there are eight servings per container. This means that there are four cups (¦ cup × eight servings = four cups) of food in this package. If a person consumed half the container, or two cups of food, he or she would have had four servings (the amount of food consumed divided by a serving size, or two cups divided by ¦ cup per serving = four servings).

Next, the calories per serving and the calories from fat per serving are listed. In this food, there are 200 calories per serving and 130 calories from fat in one serving. If the person consumed four servings and there are 200 calories per serving, then he or she consumed 800 calories (four servings × 200 calories/serving = 800 calories). Similarly, this person consumed 520 calories from fat (four servings × 130 calories from fat/serving = 520 calories from fat).

Following the calorie content, the nutrition facts label also lists the number of grams of total fat, total carbohydrate, and protein, which are calorie-yielding nutrients. A gram of fat contains nine calories, which is listed at the very bottom of the label. In this food, a single serving contains 14 grams of fat, which yields 126 calories (14 grams of fat × 9 calories/gram of fat = 126 calories from fat). This calculation was done to create the number of calories from fat listed on the panel (they rounded up to 130). As previously mentioned, if a person ate four servings, he or she consumed about 520 calories from fat.

The number of calories from carbohydrates and proteins can also be calculated. Both carbohydrates and protein yield four calories per gram, which is also listed at the very bottom of the nutrition label. In this food, there are 17 grams of carbohydrates, which provides 68 calories (17 grams × 4 calories/gram = 68 calories). In four servings, a person would ingest about 272 calories from carbohydrates (68 calories/serving × 4 servings = 272 calories from carbohydrates). There are three grams of protein in one serving, which means there are 12 calories from protein in one serving (3 grams × 4 calories/gram = 12 calories) and 48 calories from protein in four servings (12 calories/serving × four servings = 48 calories).

On the right side of the nutrition facts panel, the % DV is also listed. These daily values are based on a 2000-calorie diet, which is stated on the label next to the asterisk. Near the bottom of the label, it lists the maximum number of grams or milligrams of total fat, saturated fat, cholesterol, or sodium that a person should consume per day if on a 2000-calorie diet. It also lists the number of grams of total carbohydrate and fiber a person should eat if on a 2000-calorie diet.

If there are 14 grams of fat in one serving of this food and a person on 2000-calorie diet should consume no more than 65 grams of fat per day, then one serving of this food yields 22% of a person’s DV of fat (14 grams of fat/65 grams of fat = about 22%). If this person has consumed four servings, then he or she has eaten 88% of his or her DV of fat (22%/serving × four servings = 88%). The same calculations can be made for the saturated fat, cholesterol, sodium, total carbohydrate, and fiber. Similar calculations are also made for the vitamins listed on a nutrition facts panel.

As demonstrated, mathematics is used in the calculations surrounding calorie content and % DV on nutrition labels. The mathematics used can affect a person’s choice of foods and, in turn, a person’s health.

Sports

Mathematics is used in numerous other everyday activities, such as sports. It is common in popular sports to calculate statistics to measure performance. In baseball, a common statistic is a batting average. A batting average is a simple calculation: the number of “hits” divided by the number of “at bats.” This statistic is used to estimate an individual’s batting skills. In professional baseball, a batting average of .300 is considered an excellent batting average.

A similar statistic to the batting average is in volleyball, which is called a hitting percentage. However, it is slightly different because it tries to measure an individual’s hitting or attacking skills and takes errors into accounts. It is calculated by taking the number of kills, subtracting the number of errors, and then dividing the difference by the number of attempts. A “kill” is when a hitter’s attack results directly in a point (the ball falling into the opponent’s area of the court, an opponent not being able to return the ball, or the opponent making a blocking error as a result of the attack). An “error” is when a player hits the ball and it goes into the net (does not cross to the opponent’s side) or out of bounds. An “attempt” is anytime the player tries to attack the ball. For example, if a player had 10 kills, 3 errors, and 17 attempts, the player’s hitting percentage would be about .412 ((10 - 3)/17 = 0.412), which would also be considered a good hitting percentage, similar to the guidelines to the batting average.

Mathematics is important in the calculation of college football Bowl Championship Series (BCS) rankings as well. A mathematical formula is used to calculate these rankings, which order the top 25 NCAA Division I-A football teams based on their performance during the prior week. At the end of the season, the top two teams play each other in the national championship bowl. Mathematical formulas are also used to calculate which teams will play in the other bowls, taking into consideration the conference the team comes from and how many fans and advertising dollars the team is likely to bring in as well.

More specifically, the main factors that go into these rankings are subjective polls, computer rankings, the difficulty of a team’s schedule, and the number of losses. The subjective poll numbers come from the average of two rankings from the Associated Press (AP) and the USA Today/ESPN Coaches Poll Ratings. Sports writers and broadcasters vote in the AP poll and a select group of football coaches vote in the USA Today/ESPN Coaches poll on which football teams they think are the best, and then these two rankings are averaged. The computer rankings are based on eight different computer rankings that are calculated based on a team’s statistics for that week (strength of the opponent, final score, win-loss record, and so forth). The strength of a team’s schedule is based on a cumulative win-loss record of its opponents, as well as their opponent’s opponents. The calculation of the number of losses is straightforward. Each loss that a team suffers corresponds to one point, which is added to its final score. Points from each category are assigned to the team, and then these values are added to create a team’s final score. The team with the lowest point total is ranked “number one” in the rankings.

Speedometers

Mathematics is also used in cars. All cars have a speedometer, which is a device used to calculate an instantaneous speed of a vehicle. It is important for a driver to know the speed of the vehicle at all times to ensure the safety of passengers and pedestrians and to abide by local traffic laws. In the United States, speedometers are read in terms of miles per hour. The calculation of the speed of the vehicle requires significant mathematics.

In many vehicles, an eddy current or mechanical speedometer is used, which is the speedometer with a needle that points to the speed that the vehicle is travelling. In these cars, there is a drive cable that runs from the speedometer to the transmission, which has a gear that tracks the rotational speed of the wheels. In other words, the gear tracks the number of revolutions the wheel makes within a certain time frame. Digital speedometers calculate miles per hour slightly differently, using a vehicle speed sensor. The vehicle speed sensor is in the transmission and also tracks the rotations of the wheels. From this information, the vehicle’s speed is calculated and displayed on either a digital screen or a traditional needle-and-dial display.

The calculation of a vehicle’s speed is dependent on the size of the tire as well. For example, if the tire rotates x times per minute, then the vehicle’s speed can be calculated in miles per hour. Knowing the diameter of the tire, the circumference of the tire can be calculated (diameter × π). Therefore, the vehicle travels the distance of the number of revolutions times the circumference of the tire, within a certain time frame. This ratio can then be converted to miles per hour by converting the units. Because all of these calculations are based on an assumed tire diameter and circumference, it is very important for drivers to ensure that the correct size tires are on their vehicle. If a car’s wheels are too large or too small, the speedometer will read slower or faster than the vehicle’s actual speed, which may lead to accidents, speeding tickets, or just slower driving.

Conclusion

Mathematics can be found in everyday situations that have a real and important effect on our lives. All areas of one’s life are in some way connected to mathematical principles. Only a small number of examples have been presented here—the list can be expanded infinitely. In fact, one would be hard pressed, in today’s technologically advanced world, to present even a handful of activities that do not involve some mathematical concepts, if even at the unconscious level. By bridging the disconnect between “school mathematics” and “real-life mathematics,” individuals gain a greater appreciation for—and curiosity of—mathematical applications.

By viewing mathematics as an integrated whole and understanding its connectedness to society, individuals become active participants, rather than passive recipients, of information. When one becomes aware of mathematical connectedness, rather than viewing math as a series of isolated and disconnected concepts to be learned though rote memorization, an individual develops the understanding of mathematics as a crucial and meaningful tool that can aid in the understanding, predicting, and quantifying of the world around us.

Bibliography

Brookhart, Clint. Go Figure: Using Math to Answer Everyday Imponderables. Chicago: Contemporary Books, 1998.

Cuoco, Al. Mathematical Connections. Washington, DC: The Mathematical Association of America, 2005.

Garland, Trudi H., and Charity V. Kahn. Math and Music: Harmonious Connections. Palo Alto, CA: Dale Seymour Publications, 1995.

House, Peggy, and Arthur Coxford. Connecting Mathematics Across the Curriculum. Restin, VA: National Council of Teachers of Mathematics, 1995.

Martin, Hope. Making Math Connections: Using Real World Applications With Middle School Students. Thousand Oaks, CA: Sage, 2007.