Saturday, August 15, 2020

# Can students make change without a cash register?

ALTON, Ill. (Oct. 21) — On my drive into Alton High School this morning, I stopped to get some eggs and an English muffin at a local McDonald’s. As I sat down to eat my breakfast, I overheard a conversation between two old men at an adjacent table.

One says to the other, “The problem with kids today is, they can’t make change.”

“I mean, if they don’t have one of those cash registers that figures out the change for them, they’ve got no idea what to do,” the one answers. “If I owe $2.52 and give him$3.02, he looks at me like I’ve got four heads. ‘You still owe me 50 cents,’ he says.”

Although I’m not sure how such a stare would look, I can certainly confirm that many kids behind point-of-sale counters can’t figure out the correct change. I once had a cashier pull out a calculator to compute the change from \$10 for my purchase of a 50-cent newspaper.

### The Common Core could improve this

Although standards in almost every state included the addition and subtraction of decimals, including money—usually somewhere between fourth and sixth grade—they were just raw standards of learning. For example, Illinois’s former standard for this topic said simply that students should be able to “solve addition, subtraction, multiplication and division problems using currency” (late elementary goal 7.A.2b).

Under the new Common Core, adopted in 2010 by the state, students are required to be able to solve similar problems (see fourth-grade math standard MD.A.2: “Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals …”).

The difference between Illinois’s old standards, which were adopted in 1997, and the new Common Core, is a set of overarching principles, called Standards for Mathematical Practice. Laura Lauschke, head of the Math Department and a classroom teacher at Alton High School, says the headline for each of the Common Core’s math practices hangs in every math classroom at the school.

Don’t get me wrong: Illinois had overarching practices in its set of learning standards before the Common Core. However, they included such ideas as problem solving, application of learning, and communicating, which were common to all subjects and not specifically applied to math.

One Common Core math practice Ms Lauschke wishes her students grasped more completely is, “Look for and make use of structure”: She writes on the board as she explains:

$2x + 4 = 2(x+4)$

“If students could look at (the expression on the left) and see that the structure is the same as (the one on the right), they would just know there’s no solution to this equation,” she said. “You can’t have the same variable and add two different numbers and get an expression that’s equal, no matter what a calculator or computer tells you the answer is.”

This kind of “algebra sense” is the same kind of awareness of mathematics that kids who can’t make simple change lack when it comes to numbers and money. However, recognition of the structure of these two binomial expressions is a skill most adults, even those who are good teachers, lack.

“I wouldn’t be able to teach something like that,” said Annice Brave, a fellow teacher at Alton, in the English Department, and Illinois’s 2011 Teacher of the Year. And if she wouldn’t be able to teach students this sort of math sense, it’s difficult to see how someone could be taken out of another profession, given a few weeks of training, and step into a classroom of sometimes ill-behaved teenagers to impart this ability.

Luckily, most teachers in US high schools are highly qualified to teach math. “I hope teachers at the elementary school level are teaching the Common Core as much as we are here at the high school,” Ms Lauschke said. This is important, since the standards build on each other. For example, in elementary school, kids learn to work with and solve simple problems dealing with money. By high school, they’re analyzing mathematical functions that describe how money is invested.

### Examples of equations Ms Lauschke’s students worked with

During today’s class, Ms Lauschke divided her six students into three teams of two. She had them pick little index cards with equations on them, fill in a worksheet with the solutions, and then fill in teachers’ names on the bottom of the worksheet to tell a little story.

Equations that have only one solution are those that can be reduced, through algebraic manipulation, to the form “x = {some number}.”

$\frac{x+5}{5} = frac{x+9}{7}$
$7(x+5) = 5(x+9)$
$7x+35 = 5x+45$
$2x = 10$
$x = 5$

Equations that have no solution are those that can be reduced to a false statement.

$6(x+4) = 2(3x+3) - 7$
$6x+24 = 6x+6-7$

At this point, we could continue to manipulate the equation, but we could also recognize that doing so is a dead end: 6x + 24 can never equal 6x – 1. Continuing, we find, after subtracting 6x from both sides,

$24 = -1$

Equations that have an infinite number of solutions are those that can be reduced to a true statement.

$-3(x-6)+5x = 2(x+9)$
$-3x+18+5x = 2x+18$
$2x+18 = 2x+18$

As students worked, Ms Lauschke darted from table to table, reminding them about such tricks as cross-multiplying fractions that are set equal to each other, remembering to distribute the negative (see the third example above), and so on. All of this is getting at a recognition of the underlying structure of these algebraic expressions, and having a sense of that is something kids will thank algebra teachers for, no matter what field they go into (see the investing example above).

The room was also filled with about a dozen students who had given up their lunch hours to get tutoring from one of the other math teachers at the school. “We like noisy rooms,” Ms Brave said.

Why is it better to invest in an account that pays compound interest, compared with simple interest? Use examples in your explanation. See Common Core math standard HSA.SSE.B.3.C for more information.

We extend our sincere thanks to the people at Alton High School for their kind hospitality in allowing us to develop this story.

Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.

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