I once had to accompany 160 students to a local pizza parlor, their reward for scoring advanced on their end of year testing. Although this restaurant was prepared for our arrival, the cashier was not. She took turns glancing at the register or glancing around her nervously as if looking for someone. I finally asked her if everything was all right. “The most I can ring up is 12 orders at once,” she said. “I don’t know how to ring up 160 orders and I don’ t want to press in 160 orders separately.” She seemed close to tears. “Yeah, that would be tedious. Just ring them up 10 at a time,” I suggested. “Oh, okay,” she exclaimed as if she thought this was a great idea, but she still continued to stare blankly at her register. Eventually she drew enough courage to ask me, “how many times do I need to put in orders of 10?”
Honestly, it didn’t surprise me, though it did make me sad. I have spent enough years in a math classroom to know that most students know very little “common sense” math. If you’re thinking this is an issue for only “lower level” kids, children from low income housing, or children whose parents work too hard to be involved in their child’s education, you’re wrong.
At the beginning of the school year, I typically asked my 7th grade Algebra students the following question to get a feel for their math ability: How many days does it take for a million seconds to go by. They are not allowed to use a calculator. Every year, at least half the class turns in a paper in which they are repeatedly adding 60 to their previous sum. They would have had to do this process over 16,666 times to get anywhere near a million, and even if they managed to get that far, this would only address minutes, not days. Remember, these are the advanced students.
By the end of the year, could my students identify a graph that went with a slope-intercept equation? Yes. Could they state the roots of a quadratic equation? Yes. Did they know when a system of equations had either no solutions or infinite solutions? Yes. But could they estimate 10% of $189.72. No, but given a pencil and paper, they could change 10% to a decimal and multiply it with 189.72 and get an actual answer. I showed them the concept that 189.72 rounds to about $190 and if you move the decimal one place over, you get $19. Most still had to write 189.72 on their piece of paper, underline the 7, draw an arrow that showed it rounds to $190, convert 10% to a decimal and actually do decimal multiplication to get $19. In other words, for most students, math has been reduced to following step by step procedures. The concept of doing math by thinking a problem through was foreign to them.
Could they figure out 1/2 times 50? Again, not without paper and pencil or a calculator. Most of my students were shocked when I told them the problem 1/2 times 50 is just asking “what is half of 50.” They all knew half of 50 is 25. They had never heard this concept related to multiplying fractions before. After showing them this, multiplying 1/2 times any (even) number was easy for them. Could they calculate 1/3 times 30 or 1/5 times 65? Not unless I specifically showed them those particular fractions.
I would like to say that by the end of the year, the students had learned enough in my classroom to use a different tactic for the million seconds problem. But, honestly, I doubt it. Am I a bad teacher? According to my Algebra test scores, no. 100% passed their end of course test, and over 70% scored advanced. But according to my gut, yes. The curriculum is so rigid and full. I tried to explain more basic math concepts as they came up, but in the end, I knew I had to move on to the next Algebra topic or else we would get too far behind.
The demand for high test scores is so great, the majority of teachers are teaching with one goal in mind. End of year testing scores. It makes for better scores, but not better learners.
There are always exceptions. I’ve come across a few students who knew to divide 1,000,000 by 60 and so on. And I’ve run across a few teachers who were able to spend the school year teaching students how to think, in spite of the curriculum and end of course testing. But those type of students and teachers are rare.
If these are the issues with advanced students, it only gets worse. In my experience in the high school Algebra classroom, students were comparatively advanced if they could state a multiplication fact within ten seconds. Most can’t seem to add without using their fingers.
Schools are under pressure to show progress with numbers. Pretty soon, there will be very few people in America who even understand what those numbers mean. But maybe then schools will be able to let teachers go back to what education really means: teaching students how to think for themselves to become life long learners.