- Ask K…
A group including cognitive psychologist Elizabeth Spelke of Harvard University wondered if children could apply that ability, called nonsymbolic arithmetic, to Arabic numerals after learning to count but before they learned to add and subtract.
To find out, they gave several groups of children a laptop-based audiovisual test that asked whether one person had more or fewer candies or other objects than another person. The screen showed numbers to be added, such as 21 and 30, or subtracted, such as 64 and 13, followed by another number, such as 34, with which to compare the added or subtracted value.
The children answered correctly from 64 to 73 percent of the time, according to a report published online today by Nature. More affluent kids tested in the laboratory did better than their less well off peers tested in their classrooms, the group reports. The reason for the difference could be the testing environment, says Spelke, who adds that the important point is that kids from diverse backgrounds all showed the ability. “We never dreamed that you could simply give children the symbols and they will succeed,” she says.
Consistent with the use of nonsymbolic arithmetic, their answers were more accurate the greater the difference was between the sum (or difference) of the first two numbers and the comparison number. Kids also did better when adding than when subtracting. (Subtraction requires a larger number than addition to arrive at the same comparison, and larger numbers have fuzzier sizes in nonsymbolic arithmetic, Spelke says.)
Spelke says the link between number symbols and nonsymbolic arithmetic might help kids learn math more easily. Teachers were skeptical of the experiment because arithmetic lessons easily frustrate children, but “the kids really loved these problems,” she says. “It looks to us like a big part of the logic of addition and subtraction is already available to them.” – Scientific American
Teaching Math at Home
by Debbie Mason
As a homeschool mom with a math education background, I am seen as the answer person for homeschool math questions, such as:
a.. What is the best curriculum?
b.. When do I need to start teaching math?
c.. Do I need to drill the times tables?
d.. How much math do I need to teach?
Because I love math, it is my favorite subject to teach. My passion has caused me to develop some opinions about how to teach it. I do not claim to be a math education expert. My opinions come from twenty-one years of homeschooling four children, now ages fourteen to twenty-two, my interactions with other homeschooling families, and my personal study.
One of the reasons that I chose to homeschool was that it provided the freedom to tailor the education of my children around their unique interests, abilities and the family’s priorities. I didn’t want to do it the way the school did it; I wanted to do it differently and better. When homeschooling parents adopt the institutional approach to education, they miss out on so many of the beauties of homeschooling. If you are going to homeschool, take advantage of its advantages. Three of the goals that I had for my homeschool were that my children would love to learn, know how to learn and be allowed to learn at their own pace. I saw many problems that were caused by children being pushed to do something before they were developmentally ready. We often see children pushed in reading, but it also happens with math.
Math and the Young Child
In her book titled An Easy Start In Arithmetic, Ruth Beechick says there are three modes in which children think about math: manipulative, mental and abstract. These modes also correspond to the developmental stages of a child. First, young children learn through the manipulative stage. They need to touch, feel and move. When you, as an adult, see the problem 2+3=5, you think in the abstract mode. You understand the concept of two and three. You do not have to see and touch two blocks and three blocks. You don’t even have to picture two blocks and three blocks in your head. Preschoolers cannot do this; they are in the manipulative stage. Later, during elementary school, they develop the ability to do math in the mental mode. They can picture the number and the addition process, but they are still not able to understand the abstract concept of a number. This ability to understand the abstract concepts of math develops around age twelve.
It’s best for a homeschooling parent to keep these developmental stages in mind while teaching math. During the early years, math concepts need to be taught with things that the child can touch, feel and manipulate. This need usually corresponds nicely with the real life of the child. Children need a lot of real-world, concrete experiences before they can internalize the meaning of numbers, arithmetic operations, geometric shapes, proportion and all the other terms, ideas, processes and relationships that are a part of mathematics.
One of the best things a homeschool parent can do is to get a good elementary math book and read it themselves. Learn the terms and concepts, and then apply these terms and concepts to your child’s everyday life. Many homeschool parents hate math and do not feel very competent to teach it. If this is true of you, you need to do some homework. The more you understand the concepts yourself, the better off your children will be. Now I’m not talking about algebra; I’m talking about early elementary math.
Children come into contact with math everyday. When children play with building blocks, puzzles, toy cars, when they have a need for counting, patterning, comparing, estimating, etc., they are building a repertoire of concrete experience. Helping mom in the kitchen or dad in the workshop offers many opportunities for real-life math. Gardening, playing a musical instrument, grocery shopping, setting the table, and playing board games are all examples of activities that provide children with context and a frame of reference for future math learning.
It is so important that your children have these experiences before they start a formal math program. I actually don’t start a formal program until third grade. Until that time, I do real- life math, read math books, do math activities and play math games while occasionally throwing in a few math lessons. A book that is fun to use during the elementary ages is Family Math. Also, the library is full of picture books with a mathematical theme. Once a child has the developmental maturity of a third grader he is able to cover K through second grade math rather quickly and can then move on to the third grade math. You could probably wait a little longer, if you’re patient and have the nerve to stand up against peer pressure. A student who has had a life rich in mathematical experiences will be better able to understand the math exercises that he is now asked to do. During this stage, he is able to do mental math. He doesn’t always need the manipulatives because he is able to pictu re them in his head. Manipulatives are sometimes helpful, especially when learning a new or difficult concept or process. However, be careful that you don’t expect him to be able to do the abstract thinking that is required for many math processes. He is not ready for that yet.
Flexibility is one of the greatest advantages of homeschooling. Your child can move as quickly or as slowly as necessary. If your child is getting the material quickly and seems bored or frustrated with doing all of the problems, then cut out some of the problems and move on. You have an advantage over the classroom teacher of knowing your student very well. You will be able to determine whether he is getting it or not. If he is having difficulty, then slow down, or take a break and come back a few weeks later. If you decide it is time to take a break from progressing in math, don’t just stop doing math altogether. I know you have heard the expression, “If you don’t use it, you lose it.” This is especially true of math. Keep doing review even during the breaks from introducing new material.
My oldest daughter, who is now a college graduate with degrees in math and music, was always good at math. I could tell from early on that she was good in this area. When she was going into the third grade, I started her in a sixth grade math book and spent two years in the book. Because this book started with so much review we were able to start in this book without having to do very much outside review. This worked, in a way. She did well academically, but she hated math. This bad attitude was a major red flag for me. One of my goals was for my children to have a good attitude about learning. I had to do something to fix this problem, especially since I knew she was gifted in this area. So, we took the next year off from progressing in math. I did consistent review for this year. However, the review did not take much time, so it really was a break. When she came back to learning math during the sixth grade, she had a much better attitude. (I need to add here that it was Mat hCounts, a math problem solving and competition club for junior high students that turned her into a math lover. To find out more about MathCounts go to http://mathcounts.org.
I tell you this story to show an example of the creativity you can have in the homeschool environment. You are not locked into the traditional program of conquering a math book a year and staying on grade level. Also, don’t feel that you have to use the same curriculum every year. You can alternate or switch books if you don’t like the one you are using. Different programs work better for different kids. I also recommend that you do math along with your student when at all possible, especially if this is a first child. It will help you review the math concepts, and is also good for your student to work closely with you. Remember that your attitude about a subject is more important than your knowledge about it. I remember pretending to be fascinated with the discovery of bugs and snakes when my children were little. If I can do this, you can pretend to like math. You don’t have to pretend to be good at it, though. It is good for your child to see you learning with him.
I am often asked about memorizing the math facts, such as the times table. I take a middle of the road approach on this topic. While I do think it is important that children know the math facts, I do not agree with stopping everything else until they are memorized. I usually took some time to work on memorizing the facts, and then I moved on before they were perfected. I found the more math the student did, the more these facts became a part of his knowledge. So I used a combination of some work on memorizing the facts and a lot of work on using them.
Problem Solving Skills
I usually teach problem solving skills while my children are doing their final years of elementary math and along with algebra. Many people tend to ignore these skills because they are not covered well in most textbooks, and covering them would take extra time and resources.
One reason I take two years to cover algebra 1 is to have the student work on problem solving. In real life, math problems do not come with a label. You have to figure out what kind of problem it is and then how to solve that problem. Textbooks usually have exercises after a lesson is taught. The student exercises the skills that he has just learned by doing the exercises. Problem solving, on the other hand, teaches the student to look at a problem and determine what kind of problem it is and how it should be solved. This skill is much more helpful in real-life than the ability to work exercises at the end of a lesson. The textbook exercises provide skills and knowledge that are necessary for solving problems. The MathCounts competition is very effective in teaching problem solving skills. These skills will help your student in all areas of math, and it is great preparation for the SAT. This competition for seventh and eighth grade is especially good for children who are good at math. It is a challenging program, however, and can be discouraging for those who are weak in math.
Another math problem solving program is the Math Olympiad program. It is a problem solving competition for grades four through eight. The book, Math Olympiad Contest Problems for Elementary and Middle Schools, by George Lenchner, is a great supplement to your math curriculum. This book is also a good choice for junior high students who find MathCounts too difficult. I use the Math Olympiad book (it is an easy book to use) during the last couple of years of elementary math as a supplement, and the students love it. I remember one student, who had hated math before she started doing Math Olympiad problems, stating, “Math is so cool!” after just two days of doing these problems. I never participated in the Math Olympiad competitions, but they are available.
Beyond Elementary Math — Is Higher-Level Math Important?
Once elementary math is conquered, it is time to move on to algebra. I usually take two years to do algebra I. It is important that the student have a good understanding of algebra because it is the foundation of the rest of math. Don’t rush it, and don’t move on until each concept is grasped. I absolutely love Elementary Algebra by Harold Jacobs. All four of my children have used this book successfully. The more I use it, the more I like it. I also like Jacobs’ Geometry.
Many people have difficulty in seeing the practical application of teaching the higher levels of math. They don’t see that they use algebra in their daily lives, so they wonder why they need to learn it, or teach it. I am convinced that higher levels of math teach us to think more clearly and logically. How important is this to your daily life? The process of thinking that is taught in algebra and geometry teaches us to process information in a logical way. Other than my love for math and the connection with thinking skills, I do have other reasons for having my children learn higher math. I want to prepare my children to be able to do whatever they are called to do. Even if they are not called to do anything mathematical, they may be called to do something that requires college, and college requires three to four years of high school math. Until my children reach high school, I homeschool them the way I want. We study what we want, how we want, and when we want. When they ge t to high school, I make some compromises to prepare them for college. I make sure that they have two years of foreign language and three years of science, whether I feel that they need it for their life or not.
Now, I don’t completely cave in to the traditionalists. I do a lot of things in a homeschooly way. I integrate many of the subjects, emphasize math and science for those students with that bent, and emphasize history and government for the students with that bent. My children with a musical gift are active in music activities while the future politician will be in debate. So each student’s high school experience will be different.
Earlier I mentioned that many people don’t think that algebra and geometry are needed for every day life. I don’t believe that this is true. Because I am comfortable with this knowledge of math, I do occasionally find myself using this information to solve real-life problems. Also, if your children end up being homeschool parents, it will be very helpful for them to know these subjects. To sum up the reasons why I think higher level math is important:
a.. It teaches logical thinking.
b.. It prepares a student for his potential calling.
c.. It prepares a student for college.
d.. It teaches math skills that may be needed for real-life.
e.. It prepares future parents to pass along important math skills to the next generation.
Being a math major with a love of math, I place a high priority on learning math. However, I realize that all homeschoolers don’t have the same priority. Each homeschooling family brings its own set of priorities to the homeschooling situation. God knows this when he places children in families. He knew that I would place a priority on math and not on foreign language, for example. So, I will take all of my children at least through pre-calculus. However, it is hard to say what level of math is right for every student. Each student is different and has a different calling on his life. We as parents need to study each child, praying for wisdom in planning each one’s education. As we pray and plan, God will lead us to make the right decisions for each child. He knows the plans He has for each of them.