by Laurie H. Rogers
Whenever I tutor students who were taught math via reform-math methods, one of the first things they have to do is learn a structured and consistent way to write down problems and calculations. Their experiences with reform math have left them with poor habits, leading to many errors and muddied understanding.
Repairing poor process isn't a small undertaking. By the time reform-math students get to middle school or high school, entire books of math content are missing and many poor habits are ingrained. Developing good habits, therefore, is Job One, and it takes months and months of reinforcement before an efficient process becomes habitual. (That's in addition to the actual math procedures, which also must be taught and learned.)
It's harder to "unteach" a poor process and replace it than it is to teach an efficient process from the beginning. The Law of Primacy says students tend to learn best what they learned first – even if what they first learned was wrong-headed. Once students learn something, they tend to go back to it, as a habit and an instinctive first reaction. This is one reason why proper process should be taught from the beginning. Unteaching requires extra dedication, patience, diligence and consistency. It's hard work to change bad habits, but it can be done. And with mathematics, it must be done. It's so important to instill good habits and efficient methods. Clarity is critical to accuracy; students who wish to be accurate in math must be focused on clarity as they write down their work.
How Things are Done in Traditional Programs
Traditional math methods tend to unfold vertically on the page because working vertically allows students to easily bring each aspect of an equation down to the next line. This is the clearest way to view work and to ensure that critical pieces are neither forgotten nor lost in a chunk of writing. The work is done incrementally to avoid confusion, just one or two steps per line. Mental math is done only for very basic calculations; other calculations are done on the paper so as to minimize error and allow for checking of work.
I teach students to solve math problems on the left side of the page and do calculations on the right. Space is left between problems so that we can clearly see the pairing of the problem and the work that went with it. We don't try to squeeze it all into some arbitrary snippet of space.
Pencils are used, not pens, so that mistakes can be erased and corrected. Students learn to check their work and to catch their own errors before I do. I don't allow a calculator until Algebra II because basic arithmetic skills should be practiced and reinforced. (If the textbook is good, with reasonable problems that focus on skills and not on excessively complicated problems, then calculators are largely unnecessary and can actually be counterproductive.)
The emphasis is on "showing work" in a tidy and clear manner so that students, parents and I can see how the answer was derived and where something might have gone awry. Like this:
As students progress, simple arithmetic and multiplying by -1 can be done in one's head. This approach is crystal clear and easy to check. Naturally, reform math programs tend not to do it this way.
How Things Have Been Done in Reform Math
Besides the nature of reform math programs – inherently confusing, word-heavy, picture-heavy, game-heavy, time-and-labor-intensive, and ultimately limiting – students also are encouraged to adopt poor habits and ambiguous notation. You wouldn't believe what I've seen.
Not only do students not know sufficient mathematics, but their work tends to be sloppy and riddled with errors. They aren't taught to write neatly, check their work or correct as they go. Their attitude toward accuracy is casual; toward math in general, it's negative and stressed. Motivating them to replace bad habits with good ones is a challenge that takes time, positivity, creativity and much intensive labor.
I don't blame the children. I've heard administrators, board directors and teachers do that by saying things like "They're just not motivated." Or "They don't care about math." Or even "This is a low group." I see a lack of motivation, yes, but I don't blame students for it. They learn what they're taught. If what they're taught is boring, incomprehensible, time-wasting, hard on their self-esteem, confusing, or stupid – they won't be motivated. Sadly, although the situation isn't their fault, it is their problem. For most of them, this early learning will haunt their lives forever. It's our problem, too. Graduates who have poor habits and insufficient academics are not capable of picking up the reins of the country.
I've been able to correct some or all of the bad process in a handful of students, but I am only one tutor and there are about 28,000 students in this district. Most will go back to their regular classes, where good process is not allowed and is even criticized, and where bad process is reinforced.
Here's what I've witnessed in students going through (or graduates of) reform-math programs.
- · They lack nearly all critical arithmetic skills. Some can't tell time or say how many days are in each month. They don't know their multiplication tables, long division, the number line, how to subtract negatives, how to work with fractions, how to convert between decimals, fractions and percentages, how to isolate a variable, how to solve an equation, how to read a problem, or how to show or check their work. Many count on their fingers.
- · They're all pretty much math-illiterate and math-phobic. They don't just lack skills; they also have zero confidence. One high school graduate panicked when I asked her to solve 6Ã8. Some cried over their math homework. They all tend to think the problem in math is them, and this embarrasses them. Some have actually apologized for taking up their teachers' time.
- · Asked to do some math problems, these students will often just plunk down an answer with no attendant calculations.
- · If they do calculations, the work tends to be indecipherable – scribbled along the side, in a corner, or wandering around the page, in tiny print, too small for anyone – including the student – to read.
- · Some students will automatically erase their work so no one can see it, or do it on a different sheet that is to be tossed out.
- · Equations, if there are any, are often written horizontally (not vertically), with many " = " signs, sometimes with arcs, lines or arrows drawn to connect math terms.
- · Many pictures and boxes are drawn because, in reform math, one correct and efficient method isn't enough and can actually lose the student points.
- · Because calculations often aren't shown or aren't done in a structured, vertical format, important pieces of an equation are neglected or forgotten, such as an all-important negative sign or a stray multiplier.
- · Incorrect answers tend to remain on the page alongside correct ones. Picking out the work and the answer is difficult.
- · Homework and worksheets often come from the teacher with no room for calculations because calculator use is expected. Extra worksheets often aren't even to be graded. They're handed out for students to do if they want, but no one plans to review them.
- · Students tend to reach for their calculator for the simplest of calculations, but the calculators don't consistently bring them correct answers. Fantastically wrong answers aren't questioned; students seem to have no idea of what a reasonable answer would be.
- · Students are expected to "discover" important concepts – such as the slope of a graph, the point-slope formula or the Pythagorean Theorem – at home with their homework. "I can't look for something if I don't know what it is," a student said to me, tearfully.
- · Students aren't taught to work vertically; show their work; check their work; or to value efficiency, logic, correctness, neatness or legibility. They're not taught to carefully assess the problem for what it's asking, or to see if their solution actually answers that question.
- · They're not taught to enjoy math, nor to enjoy the process of determining a correct answer. Instead, they learn to fret over math, to fudge answers, estimate, depend on the calculator, lean on "partners," give up, get it over with, and accept whatever the "group" says is right, before blessedly escaping out the door.
- · Students are taught that "close enough" is "good enough." One student said her teacher told the class that angles within five degrees of the correct angle were close enough. (But in the "real world," a mistake of five degrees can send you in a wrong direction or even kill you.)
- · Last, but certainly not least, students are taught that their parents cannot help them. "Don't teach your children traditional methods," parents are told in open houses or on the first day of class. "It will only confuse them." Imagine that – a failed education program actively interferes with parents helping their children. And then, that same program turns around and blames parents for not being involved enough.
Many people nowadays are dismissive of efficiency. I've heard that "Process doesn't matter; it's the results that count." But one doesn't consistently obtain good results without proper process. Those who prefer "deeper conceptual understanding" over correct answers have a flawed understanding of what math is and what it's used for. In the "real world," math is a tool used to get a job done. Correct answers are necessary. That means that proper process is necessary. In the real world, "deeper conceptual understanding" is reflected by being able to properly use a tool to get a job done correctly and efficiently. In math, that ability is gained through instruction, practice and mastery of sufficient skills.
Given proper instruction, a few people will come to love the field of mathematics and will want to delve more deeply into it, but for most of us, getting a deeper conceptual understanding of math is like getting a deeper conceptual understanding of a hammer. Math obviously is more complex than a hammer, but the principle is the same. For most of us, math will never be a philosophy; it's a tool, and we need to learn how to use the tool. Once we know how to use the tool, then we go about using it.
Understanding the basics of math cannot come without proper process and correct answers. Reformers don't appear to believe that statement, but their disbelief doesn't change its truth.
Laurie H. Rogers has a bachelor's degree in mass communication and a master's in interpersonal communication, emphasizing the evaluation of argumentation and logic. In 2001, she founded Safer Child, Inc., a nonprofit child advocacy information resource. In 2007, she narrowed her advocacy to public education, and in 2010, she founded Focus on the Squareâ¢, a nonprofit organization dedicated to improving American K-12 education.
Laurie is the author of the blog "Betrayed," located at http://betrayed-whyeducationisfailing.blogspot.com/. Her book Betrayed: How the Education Establishment Has Betrayed America and What You Can Do about It (Rowman & Littlefield Education, 2011) is now available from Amazon and Barnes & Noble.
Besides serving on the executive committee for Where's the Math?, Laurie has a background in finance, journalism and child advocacy. She has volunteered in schools – tutoring children in literacy and math, and teaching chess, argumentation and knitting. She lives in Spokane with her husband, daughter and two cats.
Contact Laurie Rogers at [email protected].