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.

Common Core Math (CCM) Practice Standard 7

Opinion – CC Math Testing

Another argument against CCM standards is that they represent “teaching to the test.” Standards and tests are not the same thing. While standardized tests may be given at local, state, or national levels, most tests will be constructed by classroom teachers for the usual reasons, to measure and report student progress, to identify areas of difficulty and opportunities for improvement, and to assign student grades. Prior to testing, depending on grade level, teachers will ask students to solve problems in class and at home. Over time, the quality of CCM teaching methods and tools will improve, but as in any challenging endeavor, there will be misunderstandings. Rather than viewing poorly constructed or incorrect problem statements or solutions as proof that CCM is “worthless,” we parents would do well to offer constructive suggestions or meaningful questions designed to improve the process. This approach will help our children understand how issues are addressed and solved in the workplace of successful enterprises, where simply assigning blame seldom yields better products or services, improves customer satisfaction, or raises employee morale.

Some parents think that children spend too much time preparing for and taking tests. They may well be right, but how much is too much and what is the purpose of tests anyway?  If your son is taking weekly piano lessons, he will be “tested” every week. How he does on his piano test will be influenced by several factors including time spent practicing, whether he practiced the “right things,” and poise.

How much he should practice will in turn be influenced by his age, his interest, his aptitude, his attitude, your attitude, the teacher’s approach, the music, competing events – are all his friends out skateboarding? – and some level of agreement among teacher, student, and parents regarding how much practice time is “reasonable.”

What does it mean to practice the right things? In piano, fingering is critical – as in typing, there are known approaches to applying the correct fingers to keys efficiently – and that requires effective communication between teacher and student.  The teacher will need to choose pieces which are age-appropriate, or possibly re-written to accommodate small hands. And of course, timing and, for more advanced students, use of pedals, need to be practiced as directed.

Where does the poise come from?  While some confidence comes from external encouragement and positive reinforcement, most will come from preparation and experience. This is where the question, of how much testing is too much, gets a little murky. Preparation and experience are not the same thing: we can have a good command of subject matter and still be poor test takers. Your son can have practiced the piano to exhaustion and still be too nervous to perform well for his teacher.

Tests, whether they involve papers and pencils in class, piano recitals, or free throws in the big game, are a fact of life for most of us.  I believe that when students deal with their environment – unless that environment is openly hostile – they are preparing themselves for what lies ahead. That can include supporting the orchestra by playing their part as expected. It can also mean poise under the kind of pressure that fire fighters, police officers, paramedics, and trauma-center doctors and nurses face as part of their jobs.

Since testing is likely to be a part of their world, why not simply accept that more tests means more practice taking tests which in turn makes them better test takers, a useful skill. I strongly encourage emphasizing integrity rather than grades. Cheating is a terrible habit that can cause long-term damage. I strongly encouraged my children, if they were not well prepared, to do the best they could and accept the results.

Here are some themes to consider regarding academic test taking: 1) have a clear idea of what the test is about; 2) prepare by studying and practicing; 3) follow a process to cut through any nervousness; 4) be well rested and alert prior to the test; 5) read the test before starting to answer questions; 6) follow directions; 7) work the easy problems first; 8) do the hard ones in the remaining time; 9) check all answers; and 10) arrive on time and never leave the scene of the test early. Sound like common sense? Think back; did you follow all those suggestions when you were a student taking tests? If so, you are ahead of the game, coach. If not, but you have a better way, that’s fine too.

CCM Lesson of the day: CCM Practice Standard 7

Look for and make use of structure.

“Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x² + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers  x and y.”

My Comment

I believe that students may or may not go out of their way to “discern a pattern or structure,” depending on the questions being asked. The more problems they solve, the more they will recognize these properties.

Process implications

·         Practice using a process to solve problems to help uncover patterns and structure.

Definitions

 Square: the product of a number multiplied by itself.

Example

4² (4 squared) = 4 x 4 = 16