LEARNING: Method Behind the Mathness


    If train A leaves the station going 60 miles per hour and train B leaves one hour later going 85 miles per hour, when will train B catch up with train A? More importantly, will the fate of either of these trains really prepare a high school student for the demands of college-level math classes?

    More and more, the answer is a resounding “no.” For K-12 math teachers across Aurora, word problems about trains are just one example of an increasingly outdated approach to instruction. It’s no longer a matter of multiplication tables, rote memorization and cut-and-dried algebra problems. From kindergarten to senior year, a modern math class is much more likely to ask students not only to find the right answer, but also to justify how they found their solution.

    Combined with a bigger stress on real-world applications, that “evidence and claim” approach to learning math is part of bigger shifts in statewide and national curriculum. Math standards under the Common Core system adopted in 2010 by the Colorado Department of Education place a higher emphasis on students explaining the logic behind complicated math problems. Updates to the SAT test take a cue from that approach; the title of one of the sections on the new test is “Problem Solving and Data Analysis.” The basic rules of math haven’t changed: 2 plus 2 doesn’t equal 5, no matter how impassioned the reasoning. Still, justifying the process behind arriving at the correct answer is a much bigger focus.

    That focus has had an impact in K-12 classrooms across the Aurora Public Schools and Cherry Creek School districts.

    “We took a big swing to constructivism — students taking the parameters of a problem and working at how they can best solve it,” says Amy Nichols, president of the Aurora Educational Association. In addition to her role as the head of the APS  teacher’s union, Nichols has been a math teacher in the district for more than 20 years. “We’ve also moved toward curriculum that is more project-based. You want to give a student an authentic situation.

    “It’s a mixture of skills and working to problem solve. That’s key,” she adds.

    In high school, the stress of problem solving combines the foundation of concepts like algebra, geometry and calculus with detective work.

    For example, students in a calculus class working to figure the volume of a soda can have to come up with more than a single figure. A typical project could see students figuring out the maximum volume per surface area for a cylinder. That would translate into a squat cylinder, a shape that looks nothing like your typical Pepsi can.

    That’s where students have to turn into detectives. They have to consider aesthetics. They have to consider how difficult it would be to hold such a can. Carbonation even figures into the problem — bubbles would escape a lot more quickly from such a can, and the soda would go flat a lot more quickly.

    “You have to be able to do the math to get the measurements,” Nichols points out. “Then they have to think about the practicality.

    “That’s really deep thinking,” she adds.

    High schoolers aren’t the only ones pondering math on a deeper level. Math instruction in kindergarten looks different than it did 10 or 15 years ago, according to Anné Cornell, a kindergarten teacher at Antelope Ridge Elementary School in Aurora.

    “Instead of having kids understand rote counting, it’s having a good sense of what the quantity means,” notes Cornell, who’s taught in the Cherry Creek district for more than 10 years.

    In a standard kindergarten class, that emphasis shows up in work with tools like the ten-frame, a graphic tool designed to help a student see numbers in a visual, tactile form.

    “That sense of number is really important so that as they begin to represent numbers and symbols in the later grades, they have a sense of what it really means,” Cornell explains. “It’s important to know that the expectations are just as big for these 5- and 6-year-olds as they are in the upper grades.

    “It’s a lot to ask them to explain their thinking.”

    The new approach under the Common Core standards has had its share of controversy. Critics have attacked the standards as a fad, and opponents like the conservative group Americans for Prosperity Colorado have a long list of objections. They’ve complained about the dangers of teaching to the test and the risk of watering down traditional curriculum.

    Even Nichols points out the AEA membership is split 50/50 regarding the Common Core.

    Even so, Nichols notes that the old problem about two trains leaving a station no longer meets the demands of college in the 21st century.

    “Those problems were always out there, but I think there’s a more strategic focus now,” Nichols notes. “The problems students are working with are more meaningful, that they relate to student experience. Every student can access something and create a visual in their heads.

    “That’s opposed to those two trains,” she adds before getting to one of the biggest shortfalls of that age-old math problem about trains. “Sometimes, who cares?” she admits.

    It boils down to getting students interested in math, and two hypothetical trains leaving a hypothetical station at different times don’t always do the trick.


     The New, Old Math

    A lot can change in 13 years. Comparing samples of math problems from the 2001 CSAP and the upcoming PARCC tests reveal two different teaching philosophies. These questions show just how much math instruction has changed in less than two decades.


    Sample 10th grade math problem from the 2001 Colorado State Assessment Program math exam: The problem is a traditional word math problem that sets out straightforward variables and constants. There’s no technology component here. The problem can be completed with a simple paper and pencil.


    Daniel owns a swimming pool cleaning service. He charges a flat fee of $75 per month which includes two cleanings per month. Additional cleanings are available for $25 each. Which of these equations represents the cost per month, C, to a customer whose pool is cleaned x times per month?


    Sample high school math problem from the Partnerships for Assessment of Readiness for College and Careers test, an exam that’s set to replace CSAP and its temporary successor, the Transitional Colorado Assessment Program test, next year. Technology and problem-solving are the emphasis here. This problem draws from the Common Core State Standards and features two parts, both of which involve technology. Students finalize the answers in Part A using a drop-down menu. In part B, students plot linear functions using graphing software.


    Myla’s swimming pool contains 16,000 gallons of water when it is full. On Thursday, her pool was only partially full. On Friday, Myla decided to fill her pool completely using a hose that flowed at a rate of 10 gallons per minute. It took her five hours to completely fill her pool.


    Type a number into each box to complete the sentence:

    • Before Myla started filling the pool, there were ____ gallons of water in the pool.

    • The rate at which the water is being added to the pool is ___ gallons per hour.

    On a coordinate plane provided, graph a linear function that represents the number of gallons of water in Myla’s pool given the amount of time, in minutes, she spent filling her pool on Friday.