Fixing What Ails American Public Education: Plus and Minus
With math there are two kinds of people: those who think they can and those who think they can’t. The truth bares little resemblance to these extremes but if perception is reality then the notion of math competence is clearly delineated within the general population.
Which is probably why it exists in a similar state in our schools.
Nobody says this about reading, however. Some people read more rapidly than others. Some have larger vocabularies. Others read more and more challenging books. We recognize these differences, but we don’t say things like, “I just never understood reading.” Yet we say things like this all the time about math.
Reading, apparently, is an aptitude; math is an attitude. Our feelings about math may come from the fact that most of our work is either right or wrong; judgment, therefore, may play a role. By contrast, our work with reading is fuzzier.
Either way, our self-identification with academic skills probably begins as soon as we cam easily compare ourselves to others. Children’s attitudes about numbers probably form just as early on in school as their attitudes about text. The die may be cast in the early years for future math achievement just as it often is for literacy.
What’s odd here is that math is just as arbitrary, just as abstract, and just as confusing as phonics. Yet we seem to react differently to it. Reading is a universal; everyone must learn to read. Math is only for some kids, the ones who “get it”.
There’s a big difference in how we present reading and math to small children. In math, we give kids problems; in reading, we give kids answers. This may have something to do with how more kids self-identify as literate than as numerate.
Virtually all the words small children encounter in school are correctly spelled. Even the words they attempt to write on their own, if handled properly by teachers, will be corrected in short order. Much of the way kids learn to read and to spell comes from matching phonemes with correct letter patterns in correctly spelled words.
In math, however, this type of pattern matching seems to work out differently. Some kids appear unable to match meaningful patterns at all. They get off to a slow start in kindergarten and find it hard to keep up as time goes by.
As with reading, the end of third grade marks a crucial point in development. Kids who are doing well with math at this time will probably continue to do well throughout the rest of their school careers. Kids who aren’t doing will likely face years of struggle.
While math deficiency probably doesn’t lead to as many high school dropouts as reading deficiency, it makes school very hard and definitely robs kids of opportunities for post-secondary education because the standard “college track” in high school requires four years of challenging coursework typically beginning with algebra and ending with calculus.
Reading is Part of the World, Math is Part of School
As soon as kids begin to recognize words they start to see them everywhere. We live in a print rich environment; reading is part of the world and early readers make meaning from environmental print all the time.
Math, by contrast, seems hidden from view. A kid might see a sign for 49th Street but the number probably doesn’t register as a counter of some kind. Walking one block over to 48th Street and then up to 50th Street probably doesn’t help kids learn to count (though it might help them learn the word “street”).
Math is a foreign language spoken only at school. Just like the difficulties many of us have had studying foreign languages, many of us find that we either “get” them or that we don’t. Or we find, during an exchange year in France, that a few months of exposure to the real world is worth a few year’s exposure in the school world.
Another reason kids may struggle so much with math early on might be connected to how we introduce it—or how we don’t. A significant percentage of kids enter kindergarten with basic reading abilities. In math, however, many kids can count but they aren’t clear that numbers represent quantities, and they don’t know how to write them, much less add or subtract them reliably.
We enculturate kids to literacy far earlier and far more thoroughly than we enculturate them to numeracy. Reading is normal; math is not. Reading is for everyone: math is for a few, the few who are good at math. Reading is something everyone attempts and most come to understand; math is something some of us never quite feel comfortable with.
It may be that we unconsciously pass along to our children our own troubles with math, not overtly but perhaps in the lack of time we spend with our kids doing math activities relative to the amount of time we spend with them on language activities. This bias seems as prevalent in pre-schools as it is in homes.
In reading, we’re hyper-aware when kids start out behind; language deficits are often easy to spot. We don’t seem to feel the same urgency, however, about math. We should.
Of all areas in the curriculum, math is the most cumulative. Kids can make all kinds of skips and jumps through Language Arts or Social Studies, but what they need to learn in math one week depends a lot on whether they learned foundational concepts from the week before. Allowing children to fall behind and out of sequence presents not only an intellectual problem for students but also a structural problem for schools.
Bringing the Urgency of Early Reading to Math
Compared to mastering literacy, mastering numeracy by 3rd grade involves far less content. Aside from small amounts of trivial geometry, like learning the names of properties of shapes, much of what kids need to know comes down to basic whole number calculations and simple problem-solving.
Like successful decoding and fluent reading, these calculations must be completed rapidly, accurately, and all in one’s head. Here’s where we tend to let kids down. When it comes to basic whole number calculation, we do not strive for accuracy and fluency to the same degree that we strive for these things in reading.
This lack of accuracy and fluency in basic calculation is extremely costly as kids move up the grade levels. Where a single whole number calculation might comprise an entire problem in first grade, a student might need four or five such calculations to add two fractions in fourth grade and eight or ten such calculations to solve an algebraic equation.
If these calculations are slow, inaccurate, or otherwise difficult, students have limited mental bandwidth for understanding new math concepts. They may also move so slowly through problem sets, and commit so many errors, that they can’t develop accuracy and fluency at subsequent levels of development.
If we thought of whole numbers more like we think of letters of the alphabet. And if we thought of mastering basic math facts the way we think of mastering decoding, we might not strand so many kids in the upper elementary grades with trouble understanding fractions or solving simple story problems.
Fractions, percentages, decimals, probabilities, exponents, area, perimeter, volume, all of these more advanced concepts are worked through using small amounts of conceptual understanding and large amounts of rapid, accurate whole number calculation.
Kids who lack this basic ability often cannot survive the crucible of algebra. At this point, where the conceptual load increases dramatically, kids who are still wondering if 6 × 8 = 48 simply won’t have the cognitive surplus to understand variables, functions, or plotting equations on a Cartesian plane.
In the primary grades, we must bring the same urgency to math that we bring to reading. “Decoding” whole numbers and learning to calculate them “fluently” allows small children to process math in the world just like they process text in the world. Memorizing frequently used problems, just like kids memorize frequently used spelling patterns, ensures kids’ early success and gives them both the confidence and the know-how to tackle more complex concepts down the road.
To accomplish this, we need to focus on the following areas of the math curriculum during the primary years.
- Rapid, Accurate Mastery of Basic Facts. Unfortunately, many kids get off on the wrong foot here because of traditional tools like timed tests and flash cards. These approaches prejudice speed over accuracy during learning which encourages errors. Better approaches involve writing out fact families based on trios of numbers like 3-9-27. With that simple combination, a kid can quickly write out two multiplication and two division problems within a fact family without ever making a mistake. These kinds of error-free “recall” activities strengthen memory and improve rapid recovery of accurate information.
- Daily Mental Math. We all know that most of the math we do in our daily lives is done in our heads. Why not make mental math a staple of daily classroom practice? Mental math practice is valuable for many reasons. It helps kids hold more information in working memory for longer periods of time. It strengthens recall of frequently-used computations. It increases speed and fluency. It also builds confidence. Kids who can work problems quickly and accurately in their heads know they have a solid understanding of the math they’ve studied.
- Banish Calculator Use. In the early grades, there is simply nothing a child needs a calculator for. Even in middle school, calculators can probably be avoided. Until graphing calculators are needed for advanced math in high school, there’s simply no good reason to use them and many good reasons not to.
- Real-World Problem-Solving. Kids need to learn that math is all around them. They also need to learn that people solve math problems every day even when they’re not in school. So rather than focusing on contrived textbook problems, kids should experience solving math problems from their own lives. How many minutes before lunch? How many days left in school? How many chores do I need to do to buy a Nintendo? How many home runs does Alex Rodriguez need to beat Barry Bonds? If your dad’s car gets 30 miles per gallon and it has a 16-gallon tank, how far can it go? In general, these problems are far more challenging than the ones kids will find in their math books. But because kids develop them from their own life experience, they tend to be more tenacious about solving them and more willing to stretch their skills.
- Disciplined Work with Paper and Pencil. Manipulatives can be useful. Having kids draw pictures to explain their process can be helpful. Worksheets are a staple of the school experience. But none of these things is nearly as useful as basic paper and pencil work. Why give a worksheet when kids can write out all the problems themselves? Why solve a problem with manipulatives and neglect to write it down? We have many resources for helping kids express mathematical knowledge, but few are as good for learning math as a pencil, a piece of paper, and a brain.
- Early Introduction of Key Concepts From Algebra. It’s possible to introduce key concepts from algebra at a very early age, provided kids have a solid grasp of basic whole number calculation. For example, the idea of a letter representing a number can be introduced by doing problems with money in which letters represent the values of coins (3p + 2n = 13). These letters represent constants but this practice can easily be turned into working with variables (How much is x if 3 + 2x = 13?). Without needing to know how to balance both sides of an equation, kids can figure out these simple problems with basic whole number calculations. They can learn to understand the concept of a function in the same way.
If we know how important it is for every child to be a reader, why don’t we seem to know this for math? Math is an under-privileged subject. We look at our own lives and see that most of us have gotten where we’ve gotten without a lot of math. We have spreadsheets, calculators, bookkeepers, accountants, and banks to keep track of the important numbers in our lives. It’s natural to conclude that math just isn’t that important.
What we don’t consider is what it’s like to go through school with a poor understanding of math. Kids will take math almost every year they’re in school. Facing something one doesn’t understand year after year, day in and day out, diminishes the spirit. Frustrations and fatigue over math can easily bleed over into other subjects—especially science.
We also don’t see the value inherent in the process of learning math apart from the content. It’s reasonably accurate to speculate that after kids have had 10th grade geometry, most will never face another geometry proof.
Even if they never repeat the task again, however, practicing the task in school, and learning how to do it well, strengthens their ability to think logically, a skill they’ll use every day, and builds confidence for other more commonly encountered logic problems.
We also have to be honest with ourselves about the post-secondary prospects available to kids who don’t succeed in math. By the time third grade is over, it is possible to determine, in some cases, who is on track to college and who is not—a judgment that may affect dozens of opportunities for a child over the course of an academic career.
Finally, I think our entire society needs to reconsider the acceptability of “math phobia.” Far too many of our citizens dread doing math. It is tiring. It is mind-boggling. The results are often unsatisfactory and frustrating. This is understandable. But do we really need to pass this on to the next generation?
Would we ever let kids develop “reading phobia”? A few kids might, but we wouldn’t think it was OK. Telling ourselves that it’s OK if some kids don’t “get” math shouldn’t be OK either. Letting children’s math difficulties to go untended and unresolved is an easy way to shirk our responsibility and to short-change our kids.
The content of traditional math may not be important to every child. But the discipline and logical development is. So is the feeling of mastering a challenging subject, a feeling that may be just what some kids need to keep themselves moving through their education all the way to graduation and beyond.