Math and Middle School Articles to Get You Thinking
Even with the advice I got as a 10-year old entering sixth grade, nothing prepared me for what I would encounter. I was a lion cub who had been thrown into the den. I would have to adapt and grow up fast if I wanted to survive the savannah that was middle school. As a cub trying to find my way around, to survive, and make it to class on time, I saw the cool kids pass by. I envied them, huddled in a pride with their fancy bags and clothing. I wanted to be cool.
Although it would be nice to say I didn’t care about being part of a group, that would be a lie. I don’t know any girl who doesn’t feel that way. Nobody, particularly in middle school, wants to be alone. We all want to have the security of being popular. But at what price?
Fitting in. Being popular. Being cool. What does that mean? The Oxford Dictionary defines cool as: fashionably attractive or impressive. It defines popular as: liked, admired or enjoyed by people or a group. Middle school gives cool and popular new meanings.
Two years later, I had grown from a sixth-grade cub to an eighth-grade lioness. I learned my way around the savannah. Although I didn’t rule, I knew my place. If I let the populars – the cool people – do their thing, I could survive in peace. Maybe I didn’t have to be cool.
I think the biggest lesson I took as I entered high school this fall was that it is much better when dealing with the cool kids to just stay out of the way and be cool with my own friends, who like me for who I am. I don’t need designer bags or “juicy” clothing.
I can thank the cool kids for one thing – showing me who I am. If I never went through sixth grade trying to be cool, chasing a mirage, then I would never know who I am or be as strong as I am today.
I had to learn as a cub how to hunt before I could go off and forget about fitting in with the popular pride. In order to find who you are, you have to find who you’re not. I know what I’m not. I’m not cool. I’m me. I’ll take my $10 bag from Target and some real friends, and I’ll show those cool kids how cool this lioness can be.
The short answer is “It can’t hurt.” The physical sciences, such as Physics, Astronomy, and Chemistry, all require a great deal of math to master. That is often why these disciplines are referred to as the “hard sciences.” When it comes to high school sciences, however, the level of mathematics knowledge required is relatively minimal. One could successfully complete AP Chemistry with only seventh grade algebra skills and an understanding of base ten logarithms.
High school and AP Physics: requires algebra and knowledge of trigonometric functions sine, cosine and tangent.
AP Physics C: requires Calculus
AP Physics 1 and AP Physics 2: requires Algebra and the trigonometric functions sine, cosine and tangent.
However, it has been my experience that a student who is not at, or above, grade level in mathematics, will struggle in these courses. This is not because she hasn’t been exposed to the prerequisite skills, but because there is either some aspect of number sense that has not yet been fully developed, or the perpetuated belief that they aren’t good at math. If at any point in her elementary years a child is falling behind in mathematics, get her the help she needs immediately. Chalking it up to “not being good at math” is the greatest disservice you can do to your child’s education, and will stunt the budding scientist within her.
There are some fields of science in which math is not paramount, such as many of the biological sciences. Whereas I firmly believe that mathematics facility can only serve a biological scientist well, high school Biology will place minimal demands on a student’s math skills. In college and beyond, where research is a necessary component to biology, mathematics competency will prove itself not only valuable, but necessary many times over.
“If your level of mathematical competence is low,” explains biologist E.O. Wilson, “plan to raise it, but know that you can do outstanding scientific work with what you have. Think twice about specializing in fields that require alternation of experiments and quantitative analysis. These include most of physics and chemistry. For every scientist, there exists a discipline for which his or her level of mathematical competence is enough to achieve excellence.”
Temple Grandin, the great animal biologist, barreled through her required finite math courses with the help of tutors, and devoted hard work in order to achieve her science goals.
In short, if math isn’t your thing, then make it your thing. After all, any skill can be mastered through diligence and hard work. Then, whether or not math is an essential component of your career, it will not be a stumbling block for you.
So much focus nowadays is on memorizing what’s necessary for standardized tests, that kids actually don’t learn any real skills, such as problem solving or critical thinking. High-stakes testing focus has put so much focus on what the right answer is that examining the question or appreciating the process by which you find the answer isn’t as important.
However, to go completely in the other direction and say that memorizing anything isn’t true either. Because I don’t know about you, but I find that the memorization and retention of particular facts, processes, procedures, functions, etc. are vital in the real world and not just in trivia. When you learn anything, you very often commit the most basic parts of it to memory mainly because what comes next uses those basic parts or assumes you know them.
I think that schools have put way too much emphasis on memorizing certain facts and information and not enough about others. I feel that memorizing things are as important for exercising the brain as much as physical activity is important for exercising the body. Yet for both exercises, we want to do it in the most efficient and effective way so that we can reap benefits from this. Also, some exercises may be so boring and useless that I will give up before I ever see any benefits.
I think students do need to memorize certain things in order to function successfully on a daily basis. Personal information (name, address, date of birth, phone number) should be memorized at an early age. I think basic math facts of addition, subtraction, multiplication, and division should be memorized. Sure, you can use a calculator but memorizing these facts is more efficient than the time it takes to take out a calculator and input the numbers to find your answer. I know I memorized a lot of vocabulary words when I was learning a new language but unfortunately when I didn’t get to practice them or use them, I quickly forgot about them.
There are some facts that I found useless knowing and when I hadn’t needed this information, I have forgotten them. That makes me feel that it was a waste of time even learning the information. For instance, I had a teacher who made us memorize the Presidents in order. Now I can honestly tell you that I have never needed to come up with that information in order to function in daily life. It seems like we spent forever learning this information and it makes me wonder how much time was lost when I could have been learning something valuable.
I believe that when I have students memorize things, I need to think about the purpose for doing this. I need to be able to explain to them the rationale for memorization and have them understand that it will help them be more successful in life. If I can’t do this or even convince myself of this, I need to stop and rethink about having them memorize this information.
Many students cruise along just fine in math until fourth grade or so. Then, they hit a wall—fractions.
The wall is about to get taller. With mastery of the topic seen as a crucial stepping stone to progressing in math, federal standards are stepping up emphasis on fractions starting in third grade. National tests show nearly half of eighth-graders aren’t able to put three fractions in order by size.
The government is funding new research on more effective ways to teach the often dreaded subject. The new methods preface early rote learning of complicated fraction rules with more work on building a conceptual understanding of fractions. And instead of traditional pie charts, they rely more on tools like number lines, paper models and games putting fractions in context.
Teachers typically introduce fractions in third grade, explaining denominators—the bottom half of the fraction—as equal parts of a whole. Students study drawings of pizzas cut into wedges and label the fractional parts as fourths or sixths. Lessons then move into memorizing step-by-step rules for adding, subtracting, multiplying and dividing fractions. Some children have trouble grasping what fractions measure. When two pizzas sit side by-side, slices of one divided into sixths may not look that different from slices of another divided into fifths.
Fractions are especially confusing because they break rules third-graders have already learned. Whole numbers increase when multiplied, but fractions get smaller, for example.”Those are hard concepts” for children, says Lynn Fuchs, a professor of special education at Vanderbilt University. Teachers using the new method wait to introduce problem-solving until after students understand what denominators and numerators mean, and how fractions compare to each other. Fraction bars and number lines are considered easier than circles for children to draw and divide into parts. They also let students line up fractions in a row and see the difference in size, something they can’t do when dividing up a pie in the traditional approach.
Knowing how to place fractions on a number line in third grade is a better predictor of kids’ fourth-grade fraction skills than calculation ability, working memory or the ability to pay attention, according to a recent study of 357 children headed by Nancy Jordan. The effect continues at least through fifth grade, based on recent research, Dr. Jordan says.
Another recent study, led by Dr. Fuchs, shows working with number lines made a difference for struggling fourth-graders in 13 Nashville public schools. Using paper and kids’ own drawings is a way of making it easy for them to practice many approaches to problem-solving. A child’s knowledge of fractions in fifth grade predicts performance in high-school math classes according to a 2012 study led by Bob Siegler, a professor of cognitive psychology at Carnegie Mellon University.
The finding is based on long-term studies of a total of 4,276 students in the U.S. and Britain comparing their scores on math tests at ages 10 to 12, and again at ages 16 to 17. Researchers believe the reason may be that to master advanced math, students must broaden their understanding of how different kinds of numbers relate to each other, and how they must apply different rules as needed when working with different kinds of numbers.
“If you don’t understand fractions, it’s literally impossible for you to understand algebra, geometry, physics, statistics, chemistry,” Dr. Siegler says. “It closes a lot of doors for children.” Common Core Standards, which are being implemented in most states, require students to be multiplying and dividing fractions by fifth grade.
Trouble with fractions is the most common reason parents seek math help for their fourth and fifth-graders, says Larry Martinek, chief instructional officer of Mathnasium Learning Centers. Many students are confused by the terms often used to describe fractions, such as “common denominator.” Denominators, for example, are “the name of the fraction,” rather than simply “the bottom number,” Mr. Martinek says. This helps kids understand why they can’t add 1/2 and 1/3 and get 2/5, he says. “One apple plus one apple is two apples. One banana plus one banana is two bananas. But one apple plus one banana isn’t two banapples.”
Mathematical Twitter is normally a quiet, well-ordered place, a refuge from the aggravations of the internet. But on July 28, someone who must have been a troll off-duty decided to upset the stillness, and did so with a surefire provocation.
It has to do with something that teachers call “the order of operations.” The latest blowup concerned this seemingly simple question:
Many respondents were certain the answer was 16. Others heard Yanny, not Laurel, and insisted the right answer was 1. That’s when the trash talking began. “Some of y’all failed math and it shows,” said one. Another posted a photo showing that even two different electronic calculators disagreed. The normally reassuring world of math, where right and wrong exist, and logic must prevail, started to seem troublingly, perhaps tantalizingly, fluid.
The question above has a clear and definite answer, provided we all agree to play by the same rules governing “the order of operations.” When, as in this case, we are faced with several mathematical operations to perform — to evaluate expressions in parentheses, carry out multiplications or divisions, or do additions or subtractions — the order in which we do them can make a huge difference.
When confronted with 8 ÷ 2(2+2), everyone on Twitter agreed that the 2+2 in parentheses should be evaluated first. That’s what our teachers told us: Deal with whatever is in parentheses first. Of course, 2+2 = 4. So the question boils down to 8÷2×4.
And there’s the rub. Now that we’re faced with a division and a multiplication, which one takes priority? If we carry out the division first, we get 4×4 = 16; if we carry out the multiplication first, we get 8÷8 = 1.
Which way is correct? The standard convention holds that multiplication and division have equal priority. To break the tie, we work from left to right. So the division goes first, followed by the multiplication. Thus, the right answer is 16.
More generally, the conventional order of operations is to evaluate expressions in parentheses first. Then you deal with any exponents. Next come multiplication and division, which, as I said, are considered to have equal priority, with ambiguities dispelled by working from left to right. Finally come addition and subtraction, which are also of equal priority, with ambiguities broken again by working from left to right.
To help students in the United States remember this order of operations, teachers drill the acronym PEMDAS into them: parentheses, exponents, multiplication, division, addition, subtraction. Still others tell their pupils to remember the little ditty, “Please excuse my dear Aunt Sally.”
Now realize, following Aunt Sally is purely a matter of convention. In that sense, PEMDAS is arbitrary. Furthermore, in my experience as a mathematician, expressions like 8÷2×4 look absurdly contrived. No professional mathematician would ever write something so obviously ambiguous. We would insert parentheses to indicate our meaning and to signal whether the division should be carried out first, or the multiplication.
I’ve come to appreciate that conventions are important, and lives can depend on them. We know this whenever we take to the highway. If everyone else is driving on the right side of the road, you would be wise to follow suit. It doesn’t matter which convention is adopted, as long as everyone follows it.
Likewise, it’s essential that everyone writing software for computers, spreadsheets and calculators knows the rules for the order of operations and follows them. For the rest of us, the intricacies of PEMDAS are less important than the larger lesson that conventions have their place. They are the double-yellow line down the center of the road — an unending equals sign — and a joint agreement to understand one another, work together, and avoid colliding head-on.
So on behalf of all math teachers, please excuse us for drilling your younger selves on this tedium. My daughters spent weeks on it each school year for several years of their education, as if training to become automatons. No wonder so many students come to see math as an inhuman, meaningless collection of arbitrary rules and procedures. Clearly, if this latest bout of confusion on the internet is any indication, many students are failing to absorb the deeper, essential lesson. Perhaps it’s time to stop excusing dear Aunt Sally and instead embrace her.
Better still would be to teach everyone how to write unambiguous math expressions, and then all of this would go away. For those students destined to become software designers, writing code that can handle ambiguous expressions reliably whenever they arise, by all means exhume Aunt Sally from her crypt. For everyone else, let’s spend more time teaching our students the more beautiful, interesting and uplifting parts of mathematics. Our marvelous subject deserves better.
If you’re playing basketball, forget the swish and go for the bank. Science has proven it’s a better shot, at least from close range. That’s because the backboard substantially increases your mathematical odds of hitting the target. “When the ball hits the backboard, it loses a lot of energy, which means you don’t have to be as accurate,” says Larry Silverberg, a mechacnical engineer.
Silverberg ran simulations of 1 milion shots and found that, when shooting from many angles within 12 feet of the basket, the bank can be 20% more effective then the swish.
Here’s how to beat your friend the next time at basketball:
1 READY From many angles-even straight on-the bank shot is the best approach. (Although baseline jumpers and straight-on shots from more than 12 feet fare worse with the banking method.) To help find the best spot to aim for, imagine a V with its point in the top-center area of the backboard square.
2 AIM Imagine a vertical line floating 3.3 inches behind the center of the backboard. Where it crosses the V is the optimal spot to bank. Near the baseline, the aim point is higher on the backboard and thus farther from the rim. Near the free-throw lane, the targets are lower and closer to the rim.
3 FIRE The optimal amount of bckspin is three revolutions per second- practice by putting a dot on the ball or watching the logo as you shoot the ball. Too much spin and the resulting increase in downward speed will keep the ball too close to the glass.
Who knew there was so much math and science in basketball?
Shoppers can skip the math. It’s about to get easier to compare prices on products from breakfast cereals to toothpaste at some of the nation’s major online retailers.
New York attorney general said yesterday that Costco, CVS, FreshDirect, Wal-Mart and Walgreens have agreed to list unit prices on their websites and mobile apps for shoppers nationwide in the coming months. Amazon.com, the world’s largest online retailer did not agree to participate, according to Attorney General Eric T. Schneiderman.
A unit price tells consumers how much a product costs for each 1 unit. It divides the full price of the item by the size of the packaging. For example, a 16-ounce bottle of shampoo costing $5.99 would have a unit price of 37 cents per ounce.
Because the same product can be sold in several different sizes, a unit price is a better way to compare whether a giant bottle of shampoo is really a good deal compared to a smaller one.
Retailers already list unit prices in stores, but they’re harder to find online.
The New York attorney general’s office said that 19 states and District of Columbia have some type of unit pricing requirement. New York requires that large retail stores clearly display the price per unit of measurement for most types of food, cleaning and paper products, pet food and over the counter medications. But before this initiative, unit pricing information online was rare. Among large retailers, full availability of unit pricing was limited to online grocer Peapod.
AMY SCHUMR GIVES 1,020% TIP TO PETER’S CLAM BAR WAITER
Rockville Centre-raised comedian Amy Schumer surprised a waiter at Peter’s Clam Bar in Island Park last week with a $500 tip.
Calling the act “low-key,” a spokesman for the 75-year-old restaurant said, “She just wanted to do a nice thing” and had empathized with her server because “she had been a waitress herself in the past.”
Amy had conversed with a waiter named Ryan, who did not want his last name used, the spokesman said. She learned he was a college student working two jobs and left a 1,020% tip of $500 on her $49 bill.
Schumer, who turned 34 on June 1, told E! News yesterday, “I waited tables for a long time, so when a waiter is sweet and does a good job I like to leave a really big tip.” She added, “I do it all the time. I can’t believe it’s a news story but het, everybody should tip a lot; that job is hard.”
She told E! that in an earlier instance, “I left a chick $600 on a $00 tab. She was a great bartender in Boston at the Mandarin Oriental. I feel crazy lucky I’m in a position to do that.”
The comic, who was born in mManhattan, later moved with her family to Long Island, where she graduated in 1999 from South Side High School in Rockville Centre.
Minority groups will make up more than half the nation’s total population by 2044, and that change is expected to occur among children younger than 18 in just five years, according to new U.S. Census Bureau projections.
The non-Hispanic white population is forecast to remain the largest single group, but “no race or ethnic group is projected to have greater than a 50 percent share of the nation’s total,” the agency said in a report released Tuesday.
By 2060, non-Hispanic whites are projected to be 44 percent of the population, the report said. By comparison, non-Hispanic whites were 62.2 percent of the nation’s population in 2014, according to the bureau.
The segment of the population younger than 18 is “even more diverse” than the nation’s population as a whole and will “experience the majority-minority crossover in 2020,” it said.
The racial and ethnic shifts are occurring primarily within the native-born population, said the report, “Projections of the Size and Composition of the U.S. Population: 2014 to 2060.”
“We’re already a majority-minority for people under age 5. When we get to 2023, more than half the people under age 30 will be majority-minority,” said William Frey, a demographer with the University of Michigan’s Institute for Social Research and Population Studies Center and a senior fellow at the Brookings Institution in Washington, D.C. “It will quickly bubble up the age structure.”
What that means, he said, “is we really have to pay attention to this next generation.”
“They’re important because they are going to have to contribute to a labor force that is shrinking. The white working-age population is projected to decrease,” Frey said. He called the new generation of young minorities “every bit as important for us as the Baby Boom was for us back in the ’50s and ’60s.” Ensuring that these young people are prepared educationally should be a “big domestic policy for us,” Frey said, noting that many studies show the schools they go to are “under-resourced.”
Overall, the country’s population is projected to increase from 319 million to 417 million from 2014 to 2060, the bureau said. The report forecast that by 2030, 1 in 5 Americans will be age 65 and older, with that group growing from 15 percent of the nation’s population last year to 24 percent by 2060.
In addition, nearly 1 in 5 of the nation’s total population will be foreign-born by that year. “These demographic changes, in terms of aging, are happening on Long Island right now and over the next 15 years,” said John Rizzo, chief economist for the Long Island Association, the region’s largest business group.
Citing projections from Cornell University researchers for Long Island through 2030, Rizzo said declines are forecast in the 45- to 64-year-old age group, the segment expected to be employed, while the number of those ages 65 and older, with more retirees, is expected to rise.
“All of this underlines the importance of economic policies aimed at retaining working-age adults and planning for these demographic changes, some of which may be inevitable,” Rizzo said. The projections, he added, “show what might be, not what must be, if appropriate economic policies are taken.”
The bureau’s projections represent “one possible outcome for the future that would occur only if all the assumptions hold true” on births, deaths and international migration, the report said.
Los-Angeles – The case of actor Dennis Quaid’s 2-week-old twins, who reportedly were given 1,000 times the intended dosage of the blood thinner heparin at Cedars-Sinai Medical Center, underscores a major problem facing the health care industry: Medication errors.
At least 1.5 million Americans a year are injured – and 15,000 die – after receiving the wrong medication or an incorrect dose, says the federal Institute of Medicine. Such incidents have more than doubled in the past decade.
Causes include pharmacists stocking drugs improperly, nurses failing to double-check to make sure they are dispensing the proper medication and illegible handwriting by doctors that results in the wrong drug being dispensed.
In Quaid’s case, the babies were given concentration of heparin 1,000 times higher than intended – 10,000 units per milliliter instead of the correct dosage of 10 units per milliliter.
One way to avoid errors is to place bar codes on medications and swipe them into a computer programmed with information about correct dosages.
Ellen Venditti of Cape Cod Healthcare also praised systems in which the doctor enters the prescriptions directly into a computer, which raises a red flag if the dosage is not a normal one.