By Stephanie Xiang
Factoring an expression, simplifying a radical, or graphing an equation can take students several steps by hand. With a NumWorks calculator, some of those same tasks can be completed in seconds.
Every Upper School student is required to have a NumWorks calculator, according to Upper School Mathematics Department Chair Ms. Vicki Greenberg. With graphing, equation solving, matrices, and other advanced features all built into a single device, NumWorks can be a powerful tool in math classes. But its capabilities also raise a question: When does a calculator stop assisting students and start doing too much of the work for them?
For Ms. Greenberg, the answer depends on what students are supposed to be learning.
“Calculators should do calculations so that students can focus on the problem-solving,” Ms. Greenberg said. However, she noted that calculators can also perform skills that teachers expect students to learn themselves, such as factoring. “The calculator can do almost every basic skill we teach in high school,” she said.
Upper School math teacher Ms. Becca Scott sees a similar tradeoff. She likes NumWorks because it is user-friendly and makes features such as matrices and radical functions easier to find. She also appreciates how the calculator displays answers in different forms. “When you type in an operation, it will, if possible, give you all of the forms that solution might be in,” Ms. Scott said, including decimals, fractions, and radical expressions.
Those capabilities can expose students to different ways of representing numbers, but Ms. Scott said some features can also take away important practice. NumWorks can simplify fractions and radicals and factor expressions, all of which are skills students learn in school.
“Factoring is something that is in every math course you take and is a really important skill to know,” Ms. Scott said. “Reliance on the calculator takes away that skill.”
Mrs. Greenberg pointed to a similar concern. A student who relies on a calculator for factoring may encounter problems later when more advanced math requires the same skill in a different form. “If students are reliant on a calculator to factor for them, they will struggle when they attempt to factor trigonometric expressions,” Mrs. Greenberg said.
Students recognize that tension, too.
Sophomore Sonul Mittal said NumWorks is “super advanced and has a lot of features,” and she especially likes its graphing feature because it makes checking her work easy. At the same time, she thinks the equation solver can sometimes go too far.
“I think the equation solver sometimes does too much work because it gives kids the actual answer without any work,” Sonul said.
Sophomore Zuri Underwood also identified graphing as a feature that can potentially replace students’ own work. “The graphing simulation makes it way too easy for students, and they could easily avoid actually graphing during tests,” Zuri said.
Still, both students said NumWorks can contribute to understanding rather than simply provide answers. Sonul said the graphing application helps her visualize problems, while Zuri said the calculator makes it easier to work backward when solving problems.
The calculator can also become something students grow accustomed to having nearby. Asked how she would feel taking a test without NumWorks, Sonul said, “I would be doubtful of my skills, but overall I wouldn’t feel too bad, especially if my unit didn’t require graphing.” Zuri said she would feel “uncomfortable,” although she added that she is not dependent on the calculator.
NumWorks is not the only option.
Both teachers pointed to the TI-84 as an alternative. Ms. Scott said the TI has “a history of being one of the best out there,” while Mrs. Greenberg said she would recommend the TI-84 because she is familiar with it, “not because it is any better.” Zuri, who used a TI calculator in ninth grade, said the two calculators are “about the same,” although she noted that the TI she used could not solve radicals.
Teachers are also considering how to balance access to NumWorks with assessments of students’ own mathematical skills. Ms. Scott said there is value in giving some assessments without calculators because teachers need to know whether students understand the mathematics behind the answers.
“We need to ensure that you understand the math behind what your calculator is doing, not whether you can type into your calculator correctly,” Ms. Scott said.
NumWorks includes an exam mode that can restrict some calculator functions during assessments. Mrs. Greenberg said the math department is exploring exam mode this year but does not yet have enough information to determine how it will affect cheating. She said teachers administer assessments with the expectation that students use calculators “to aid in their thinking but not to do the thinking for them.”
But a single testing rule may not fit every math class.
Despite concerns about individual features, neither teacher supported creating one testing restriction that applies identically to every math course. Mrs. Greenberg said each course “can and should decide what is best for the learning of that course’s content.” Ms. Scott similarly said different math classes have different standards and content.
Students also stopped short of calling for an overall restriction. Zuri said moving away from NumWorks could disrupt students who have become accustomed to using it. Sonul instead favored limiting particular capabilities when necessary. “I think they should restrict certain applications and features, but otherwise, they’re really helpful!” Sonul said.
NumWorks may be able to factor an expression, draw a graph, or simplify a radical in seconds, but the interviews suggest that the larger issue is not simply what the calculator can do. It is deciding when students should let the calculator do it.
