Common Core Math (CCM) Practice Standard 8

Opinion – Learning math takes time

When I was Junior at UCLA, a classmate in my Dynamics (aka Mechanics) class asked me how I was able to complete the homework each week. My answer was “12 hours.” “That’s it?” he asked, surprised. “Yes, no secret,” I said, “I’m no smarter than you.” A week later, I talked to him again. “You were right,” he said, “that’s how long it took me to do last week’s assignment.”

When our children were learning to drive, I used our 30-year-old, 4-speed, manual-transmission Volvo to teach them.  You may ask why, when most cars, even in those days, had automatic transmissions, would I do that, and would they not be reduced to tears learning not to stall the engine on steep hills.  My reasoning was that a manual transmission gives drivers a better feel for engine speed, ground speed, and acceleration and that it requires two hands, two feet, and full attention to drive.  As it happened, our daughter was reduced to tears practicing starting from a dead stop on a steep hill.  But her tears were simply a sign of frustration that what we were doing was not working, not yet: the car stalled repeatedly because her coordination between clutch and accelerator was not yet strong.  After two days of practice and encouragement, the tears dried, replaced by capability and confidence.  It’s not easy to learn something really new, but it feels pretty good when you do, especially when your friends notice.

Learning anything new, I mean really new, is hard work. Mastering a new concept – one that you may never have thought about, let alone struggled with – takes time. Solving problems which apply that concept is challenging. Some teachers, parents, and students think that some students are just naturally “good at math” and that the rest will always be limited. It may be true that some students are gifted, but even those students can only acquire advanced math (and other academic) skills by putting in the time. Or, as Professor Dweck might say, “insert growth mindset here.”

I remember reading in a sports magazine about an amateur tennis player, in his 30s, who was complimented on his fine, “natural” backhand.  He thanked the observer, but then mentioned that his “natural” backhand was the result of about 2,000 of hours of lessons, practice, and competition.

A couple of other observations are in order. Although girls mature earlier and are often better students than their slower-developing male counterparts, some become discouraged by math. I believe that, in part, this is because, in the classroom, boys tend to speak up when they think they have a problem solved, whether they do or not. Without positive reinforcement and under the misconception that they are just “not good at math,” some girls may feel intimidated. As a coach, you can help once you understand the subject to be learned and the time required to learn it. To repeat, “apply growth mindset here.”

CCM Lesson of the day: CCM Practice Standard 8

Look for and express regularity in repeated reasoning

“Mathematically proficient students notice if calculations are repeated and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3.  Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x² + x + 1), and (x – 1)(x³ + x² + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.”

My Comment

This step is closely related to Step 7 and I have the same comment.

Definitions

Slope: the ratio of the "vertical change" to "horizontal change" of a line or a feature such as a road.

Applications and Examples

The slope of our street is positive to the north and (therefore) negative to the south. The slope is pretty steep right in front of our house, about 10% going north. For every 10 feet along the street, the altitude gain is about 1 foot. The concept of slope, one example of the “rate of change,” is fundamental to calculus, but it is useful in many applications. The roof of a house needs to have some slope to allow rain to drain off; the slope may need to be steep in areas where snowfalls are heavy.