Last week CJ, who is thirteen, threw down his math book in frustration. He just wasn’t getting it. Even after I explained it. Repeatedly. In as many different ways as I could think of. Even after watching Khan Academy videos. Even after switching from Saxon to Life of Fred. Algebra was just beating him up.
Seeing the irreversible signs of frustration that would stop any learning for that day, I told him to put away the book and move to another subject. I pondered the problem. A few things occurred to me.
Math Understanding Does Not Come Automatically For Most People
Math is easy for me now, but it was by far my worst subject in school (except for that pottery class my sophomore year, but we don’t talk about that). I made Bs in all my math classes from Jr. High pre-algebra to college calculus by doing all the homework and getting Ds and Fs on all the tests. In other words I didn’t understand a single thing. From multiplication on I had learned math formulaically. I could plug numbers into a formula, but there was absolutely no understanding.
So why do I understand it now, but I couldn’t then? I reflected that I had learned math again as an adult from counting to complex algebra and precalculus with my two oldest children. When I learned it this time we were using Saxon Math, which I think does a stellar job of explaining the whys of math. Also, I was the teacher. I had to understand exactly what a square root was so I could teach it to my kids. Mostly I think I was motivated to understand the patterns and the reasoning. I was no longer plugging numbers into a formula.
Students Must Understand Why
Most kids, even at very young ages, can rattle off the times tables if they are drilled enough on them, if you set them to fun tunes, or create mental hooks to hang them on. But very few kids will ever understand the principle or concept of multiplication, the whys of multiplication, if they are not expected to learn it. Learning the times tables is important, but understanding them is more important by a power of ten. Some kids, math brain types, do intuitively pick up the whys behind math. But most kids don’t.
More and more math curriculums are moving toward manipulatives, not just for kindergarten and first grade, but for kids clear though high school. This is good. It hugely aids in understanding to be able to visualize the math. But at some point even this isn’t enough because math is not always concrete. Kids have to, eventually, be able to do math without a picture in their heads.
The second step up from using manipulatives is to get the kids to discover the whys of math on their own, or at least to understand it for themselves. They must have that ah-ha! moment. This involves not always giving them the answers.
To get kids to discover math principles for themselves, you can give them a situation, a story problem, and sit back and let them discover the math principle on their own. If you have more than one child, encourage a discussion. You can also prompt them by having them draw pictures, use manipulatives, or make guesses. Some answers may be fairly quick in coming and others may take several days, a week, or more to figure out. This is the way that I understood math the second time, when I was teaching my own kids. I understood the whys.
The ah ha! moment involves teaching in more than one way. Everybody’s brain is wired differently and the way one person understands may not be the way another does. Just because it makes perfect sense to the teacher does not mean it will make perfect sense to the student.
This is essentially why the Common Core method of math makes kids learn a half dozen ways of arriving at an answer. The problem with Common Core, in this instance, is that it makes all kids discover and master all ways, when kids ought to be able to chose the method that makes sense to them and creates understanding in them. Only the teacher needs to know all six methods, so she can guide her students. Forcing students to learn all methods creates more confusion than understanding, not to mention tedium.
I actually think most people hate math because they have never understood the whys, they never had those ah ha! moments and so they hate math. If you were still having to sound out words, how often would you read? How much fun would you think reading was? Schools, including homeschools, do a better job of teaching kids to be comfortable with words than they do of teaching kids to be comfortable with numbers.
Students Must Achieve Fluency of the Basics
You cannot do algebra if you do not understand multiplication, division, subtraction, addition, and fractions (which are really just division problems–one of my adult ah-ha moments was when I understood the nature of fractions). It is not good enough to plug them into a formula. You must understand the whys and you must have them down pat so they are automatic and take no thought. You must be completely fluent in basic math operations, including fractions.
My favorite novel is Pride and Prejudice. I love that book. But my fourth grader would think it was the seventh circle of Hades, because though he can read, he is not fluent. Reading still takes some degree of thought and effort for him. For me reading is automatic, easy, and effortless. This is how basic arithmetic must be for a student to do well at algebra. Math fluency is exactly like reading fluency. There is absolutely no point in struggling through Austen if you can’t understand a thing you’re reading. There is absolutely no point in struggling though algebra if none of the numbers make sense.
I have heard people (often parents and even teachers) express that higher math isn’t that important because they never use algebra, or calculus, or trigonometry in daily life. It is true that few people use much math beyond basic operations in daily life. But how much would you use reading in daily life if all you had ever mastered was the alphabet and sounding out three letter words. People who have mastered math use it often in their daily lives. It is impossible to use something that you do not know. If our children become fluent in math the way that our generation is fluent in language then the very world we all live in might be altered forever.
Mental Maturity Is a Real Thing
Finally, the second time I learned math I was an adult with a completely mature brain (in the biological sense, mind you). Sometimes I hear parents brag about how their little child is precocious because he can do the quadratic formula at age six or some other absurdity. I won’t discount that there may actually be some six year old out there who can actually understand the quadratic formula, but I will say that that person is a freak of nature (in a good way of course) and not the norm. As mentioned previously, anyone can be taught to plug numbers into a formula, just like anyone can be taught to sound out the words in Pride and Prejudice, but this does not mean that person has any understanding.
Every person matures at a different rate based on their biology. This is completely individual and it includes the maturation of the brain. That is why some kids can learn to read at age four and others don’t read until age ten. Some kids hit their growth spurts early and some hit them late. Some kids are ready to understand multiplication at six and others not until eight. One rate of growth is not better than another, they are just different. But it is important to wait until the mind is mature enough to grasp the concepts before expecting understanding.
For this reason, I am not a fan of the current trend for earlier and earlier education of children. It creates cycles of failure in more and more of our kids. We would never think less of a child because he was shorter than his peers and yet we do make kids feel like they are less because their minds have not matured as quickly as another child’s. Once a child feels stupid it is incredibly difficult to heal.
If we try to teach a math concept too early in a child’s development, she will not have understanding, and as we move on and require that early math skill in later tasks it will be missing, making further understanding impossible.
I’m not recommending delaying math, just doing it on an age appropriate schedule and if an individual child still struggles, take into account that his or her brain may not yet be mature enough to grasp the concept. You may need to slow things down a bit. In six months or a year it may click easily. Meanwhile you can move that child sideways in the math curriculum, for example from multiplication to telling time and counting money, or just have her get extra practice in addition and subtraction.
So what if you have a middle school or high school student who is struggling with algebra like CJ is?
After a lot of thought I decided to start over. At the beginning. Well . . . I didn’t think we needed to learn to count. He’s pretty good at that. But we are doing skip counting, spelling number words, relearning addition and subtraction properties, relearning multiplication and division properties, and relearning fractions. He, of course, has learned all of this before, but upon reflection I decided we had not achieved either understanding or mastery.
When he was in the early grades I was teaching four, then five, then six kids at all different levels all at the same time. Some people may be able to do that well, but I admit right here that some things were neglected during those years. Whereas I had sat there guiding my older boys personally though each math lesson at that age, CJ was often left to his own devices. This created some gaps that by the algebra years became chasms.
As a homeschooled child CJ has the freedom and opportunity to learn what he needs to learn when he needs to learn it and not what the “state standards” demand he must learn this year. At first I freaked out that starting over would put him so far behind. But then, quickly realized that there is nothing further behind than total ignorance. I would rather he thoroughly understands basic arithmetic than get clear through college, like I did, without even having the times tables memorized. Now that is sad. With his more mature mind I think the learning will go quickly and we will probably be able to start over with algebra in six months or a year. He will still get through algebra 2 by high school graduation if all goes well. But I’m totally fine with real understanding of basic algebra over a shaky-barely-there-getting-though-it of pre-calculus.
A Plan For Math Without A Curriculum
What I want for CJ over the next months is a quick and dirty, hands-on, mastery of basic arithmetic for older students. So I decided that instead of a curriculum I would create a math knowledge map. As we complete a skill, with total understanding and mastery, we will color in the circle for the skill on the map and move to the next skill. I plan to use lots of ideas for games, activities, and manipulatives from the internet as well as worksheets and lesson plans.
The method I plan to use is the single-topic mastery, rather than the spiral method (Saxon is spiral method, Math-U-See is topical). We will stick with one topic until he gets it down pat and then move on to the next topic. The knowledge map, above, will let us move from topic to topic in a non-boring, kid directed way, while still making sure we have prerequisites done in order.
I also decided that his younger brothers would join us. We are going to nail this arithmetic thing for good. (Listen to my optimism.)
More From Layers of Learning
Try an abacus for math . . . it might not be as outdated as you think.
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Try this to keep those columns of numbers in order when doing multi-digit multiplication.
Make math facts fun and memorable with this beach ball math facts game.