photo by ddluong via flickr
Feature photo (above) by Sphinx The Geek via flickr.
Most homeschoolers feel at least a small tinge of panic as their students approach high school. “What have we gotten ourselves into?” we wonder. “Can we really do this?” Here are a few tips to make the transition easier.
Before you move forward, it may help to take a look back. How has homeschooling worked for you and your children so far?
If your students hate math, they probably never got a good taste of the “Aha!” factor, that Eureka! thrill of solving a challenging puzzle. The early teen years may be your last chance to convince them that math can be fun, so consider putting aside your textbooks for a few months to:
On the other hand, if you have delayed formal arithmetic, using your children’s elementary years to explore a wide variety of mathematical adventures, now is a good time to take stock of what these experiences have taught your students.
- How much of what society considers “the basics” have your children picked up along the way?
- Are there any gaps in their understanding of arithmetic, any concepts you want to add to their mental tool box?
[Feature photo (above) by wonderferret, photo (right) by University of the Fraser Valley, both via flickr (CC BY 2.0). This post is the first of three in my Homeschooling with Math Anxiety Series.]
Our childhood struggles with schoolwork gave most of us a warped view of mathematics. We learned to manipulate numbers and symbols according to what seemed like arbitrary rules. We may have understood a bit here and a bit there, but we never saw how the framework fit together. We stumbled from one class to the next, packing more and more information into our strained memory, until the whole structure threatened to collapse. Finally we crashed in a blaze of confusion, some of us in high school algebra, others in college calculus.
photo by Scott Robinson via flickr
A comment on my post Fraction Division — A Poem deserves a longer answer than I was able to type in the comment reply box. Whitecorp wrote:
Incidentally, this reminds me of a scene from a Japanese anime, where a young girl gets her elder sister to explain why 1/2 divided by 1/4 equals 2. The elder girl replies without skipping a heartbeat: you simply invert the 1/4 to become 4/1 and hence 1/2 times 4 equals 2.
The young one isn’t convinced, and asks how on earth it is possible to divide something by a quarter — she reasons you can cut a pie into 4 pieces, but how do you cut a pie into one quarter pieces? The elder one was at a loss, and simply told her to “accept it” and move on.
How would you explain the above in a manner which makes sense?
Photo of Eeva times 6, by Eric Horst, via flickr.
The question is common on parenting forums:
My daughter is in 4th grade. She has been studying multiplication in school for nearly a year, but she still stumbles over the facts and counts on her fingers. How can I help her?
Many people resort to flashcards and worksheets in such situations, and computer games that flash the math facts are quite popular with parents. I recommend a different approach: Challenge your student to a joint experiment in mental math. Over the next two months, without flashcards or memory drill, how many math facts can the two of you learn together?
We will use the world’s oldest interactive game — conversation — to explore multiplication patterns while memorizing as little as possible.
Photo of Evil Erin times 5 (and dog times 2), via flickr.
Perhaps the biggest challenge for any middle-elementary math student is to master the multiplication facts. It can seem like an unending task to memorize so many facts and be able to pull them out of mental storage in any order on demand. Too often, the rote aspect of such memory work overwhelms students, eclipsing their view of the principles behind the math. Yet rote memory is not enough: A student may be able to recite the times tables perfectly and still be reduced to counting on fingers in the middle of a long division problem.
We will use the world’s oldest interactive game — conversation — to learn the multiplication facts one bite at a time. But first, let’s take some time to think about what multiplication really means.
[Photo by scubadive67.]
Help! My son was doing fine in math until he started long division, but now he’s completely lost! I always got confused with all those steps myself. How can I explain it to him?
Long division. It’s one of the scariest of the Math Monsters, those tough topics of upper-elementary and middle school mathematics. Of all the topics that come up on homeschool math forums, perhaps only one (“How can I get my child to learn the math facts?”) causes parents more anxiety.
Most of the “helpful advice” I’ve seen focuses on mnemonics (“Dad/Mother/Sister/Brother” to remember the steps: Divide, Multiply, Subtract, Bring down) or drafting (turn your notebook paper sideways and use the lines to keep your columns straight). I worry that parents are too focused on their child mastering the algorithm, learning to follow the procedure, rather than on truly understanding what is happening in long division.
An algorithm is simply a step-by-step recipe for doing a mathematical calculation. But WHY does the algorithm work? If our students could understand the reason for the steps, they wouldn’t have to work so hard on memory tricks.
[Photo by *Irish.]
In my post Elementary Problem Solving: The Tools, I introduced word algebra as a way to help students think their way through a story problem. In the next two posts, I showed how the tool worked with simple word problems.
Now, before I move on to focus exclusively on bar diagrams, I would like to show how word algebra can help a student solve a typical first-year algebra puzzle.
A homeschooling friend who avoided algebra in high school, trying to help her son cope with a subject she never understood, posted: “Help! Our answer is different from the book’s.” Here is the homework problem:
Josh earned $72 less than his sister who earned $93 more than her mom. If they earned a total of $504, how much did Josh earn?
[Photo by Aaron Escobar. This post is a revision and update of How to Solve Math Problems from October, 2007.]
What can you do when you are stumped by a math problem? Not just any old homework exercise, but one of those tricky word problems that can so easily confuse anyone?
The difference between an “exercise” and a “problem” will vary from one person to another, even within a single class. Even so, this easy to remember, 4-step approach can help students at any grade level. In my math classes, I give each child a copy to keep handy:
[Note: Page 1 is the best for quick reference, especially with elementary to middle school children. Page 2 lists the steps in more detail, for the teacher or for older students.]
[Photo by woodleywonderworks.]
The question came from a homeschool forum, though I’ve reworded it to avoid plagiarism:
My student is just starting first grade, but I’ve been looking ahead and wondering: How will we do big addition problems without using pencil and paper? I think it must have something to do with number bonds. For instance, how would you solve a problem like 27 + 35 mentally?
The purpose of number bonds is that students will be comfortable taking numbers apart and putting them back together in their heads. As they learn to work with numbers this way, students grow in understanding — some call it “number sense” — and develop a confidence about math that I often find lacking in children who simply follow the steps of an algorithm.
["Algorithm" means a set of instructions for doing something, like a recipe. In this case, it means the standard, pencil and paper method for adding numbers: Write one number above the other, then start by adding the ones column and work towards the higher place values, carrying or "renaming" as needed.]
For the calculation you mention, I can think of three ways to take the numbers apart and put them back together. You can choose whichever method you like, or perhaps you might come up with another one yourself…
[Photo by Alejandra Mavroski.]
Myrtle called it The article that launched a thousand posts…, and counting comments on this and several other blogs, that may not be too much of an exaggeration. Yet the discussion feels incomplete — I have not been able to put into words all that I want to say. Thus, at the risk of once again revealing my mathematical ignorance, I am going to try another response to Keith Devlin’s multiplication articles.
Let me state up front that I speak as a teacher, not as a mathematician. I am not qualified, nor do I intend, to argue about the implications of Peano’s Axioms. My experience lies primarily in teaching K-10, from elementary arithmetic through basic algebra and geometry. I remember only snippets of my college math classes, back in the days when we worried more about nuclear winter than global warming.
I will start with a few things we can all agree on…
Photo by jetheriot.
One of the most common math questions on homeschooling discussion forums is, “How can I help my child master the math facts?” Unfortunately, when it comes to drilling facts, many children think math is spelled “B-O-R-I-N-G.” Worksheets are tedious, flash cards make them groan, and even the latest computer game is a yawner.
photo by MC Quinn via flickr (CC BY 2.0)
Paraphrased from a homeschool math discussion forum:
Help! My daughter struggles with arithmetic. I guess she is like me: just not a math person. She is an outstanding reader. When we do word problems, she usually has no trouble. She’s a whiz at strategy games and beats her dad at chess every time. But numbers — yikes! When we play Yahtzee, she gets lost trying to add up her score. The simple basics of adding and subtracting confuse her.
Since I find math difficult myself, it’s hard for me to know what she needs. What’s missing to make it click for her? She used to think math was fun and tested well above grade level, but I listened to some well-meaning advice and totally changed the way we were schooling. I switched from using workbooks and games to using Saxon math, and she got extremely frustrated. Now she hates math.
Photo by powerbooktrance.
Paraphrased from a homeschool math discussion forum:
Help me teach fractions! My son can do long subtraction problems that involve borrowing, and he can handle basic fraction math, but problems like give him a brain freeze. To me, this is an easy problem, but he can’t grasp the concept of borrowing from the whole number. It is even worse when the math book moves on to .
Several homeschooling parents replied to this question, offering advice about various fraction manipulatives that might be used to demonstrate the concept. I am not sure that manipulatives are needed or helpful in this case. The boy seems to have the basic concept of subtraction down, but he gets flustered and is unsure of what to do in the more complicated mixed-number problems.
The mother says, “To me, this is an easy problem” — and that itself is one source of trouble. Too often, we adults (homeschoolers and classroom teachers alike) don’t appreciate how very complicated an operation we are asking our students to perform. A mixed-number calculation like this is an intricate dance that can seem overwhelming to a beginner.
I will go through the calculation one bite at a time, so you can see just how much a student must remember. As you read through the steps, pay attention to your own emotional reaction. Are you starting to feel a bit of brain freeze, too?
Afterward, we’ll discuss how to make the problem simpler…
From a recent e-mail:
Hello! I am on the board of a homeschool co-op. We have had requests for a math club and wondered if you have any tips for starting one. We service children from K-10th and would need to try to meet the needs of as many ages as possible.
There are several ways you might organize a homeschool math club, depending on the students you have and on your goals. I think you would have to split the students by age groups — it is very hard to keep that wide of a range of students interested. Then decide whether you want an activity-oriented club or a more academic focus.
When I started my first math club, I raided the math shelves in the children’s section at my library (510-519) for anything that interested me. I figured that if an activity didn’t interest me, I couldn’t make it fun for the kids. Over the years we have done a variety of games, puzzles, craft projects, and more — always looking for something that was NOT like whatever the kids would be doing in their textbooks at home.
[Rescued from my old blog.]
Spring cleaning has made my desk look worse than before. Nobody feels like studying. The kids would rather be outside, and their mom would rather take a nap. Sound familiar? It is our annual attack of homeschool burnout.
If you, too, are suffering from lethargy and can’t face another day of school work, here are some ideas that have helped me:
(1) Re-read the homeschooling books on your shelves, or get some new ones from the library. Try to read about one a month, if you can, to help get your enthusiasm back. And then read at least one new homeschooling book per year to help you stay inspired.
(2) Connect with other homeschoolers. Meet with friends for tea, or have a Mom’s Night Out while Dad babysits.
(3) Attend support group meetings. I find that after so many years, I let the meetings slide. I think, I already know everything they are going to say. But being with other homeschoolers is encouraging, and if you find out that you can help a new homeschooler with advice, that gives you a boost, too.
Image via Wikipedia
[Rescued from my old blog.]
I love Miquon math, but the program does feel odd to many homeschoolers, especially at first. It is so different from the math most of us grew up with that it takes time for the teacher to adjust. DJ asked for Miquon advice at a forum I used to frequent, but I thought enough people might find these tips useful to justify an expanded repost. If you have more advice on teaching Miquon, please chime in!
by d3 Dan via flickr
[Rescued from my old blog.]
What teacher hasn’t heard a student complain, “When am I ever going to have to use this?” Didn’t most of us ask it ourselves, once upon a time? And unless we choose a math-intensive career like engineering, the truth is that after we leave school, most of us will never again use most of the math we learned. But if math beyond arithmetic isn’t all that useful, then what’s the point?
If you or your student is singing the Higher Math Blues, here are some quotations that may cheer you up — or at least give you the strength of vision to keep on slogging.
We study mathematics…