24 Jul 2011
Use algebra or arithmetic to solve
elementary math word problems?
Frank Ho
Founder and Teacher at Ho Math and Chess Learning Centre, Canada
July 24, 2011
www.mathandchess.com
Let me share some of my experiences and I think they are very insightful. I have been teaching a weekend math contestlike course at Vancouver Ho Math Chess Learning Centre in Canada for two years now and the reason I say it is math contestlike is because I do not just do math contests, often I give problems in some areas which I discovered in my regular teaching to student again by creating them myself in my presentation and I also give my created puzzles to students for them to work on, so the course is a combination of math enrichment, math contest and puzzles. Some of my students have followed from grade 7 to grade 8 and even learning grade 9 math so how can I teach all these students with different levels in the same room? So I know I must be creative. I started to do research by reading materials from Singapore, Taiwan, and China, and also North America.
While I was doing research, I noticed in China, some provinces start to teach algebra (simple model with only one variable on one side) at grade 4 and in British Columbia, Canada, and the children start to learn algebra at the end of grade 7. So here we can see when to teach algebra has lots of different opinions.
My research starts to get in to depth because I have found some model word problems in China keep coming up and these problems are in the category of Chickens and Rabbits word problems. I also noticed that in Singapore, there are some complicated word problems using bar charts to solve them but in some cases even drawing bar charts present challenges to students. So what shall I do to teach students in solving word problems? Use algebra, bar chart (line segment) as a universal way of solving word problems? I started to research this topic by reading all kinds of materials as long as I could get my hands on.
Because of a mixed background of students in my class so I have allowed my students to solve word problems to either choose algebra or arithmetic to solve the same problem, or they simply can use both methods. With this experiment it does shed some lights to my own research because it shows:
1. Most of the time, the word problem can be "easily" solved by using algebraic method, but not always especially the fraction word problems which require backward calculation to get answers.
2. In some cases, by doing word problems in arithmetic it actually require students to have deep understanding about the method instead of just coming up with an equation and the word problems then becomes a "procedure" type of problems by using equation.
So my feeling is students shall do lots of word problems using arithmetic method to train their thinking skills before using algebra. But by learning a bit algebra it also allows students to see why they are using the method or some steps in arithmetic. Often the reasons behind the arithmetic steps can be easily “seen” by using the algebraic equations. For example, Adam and Bob have 40 apples. Adam has 10 more apples than Bob. How many apples does each one of them have?
This problem can be solved in many ways, but the algebraic method way can be done as follows:
Algebraic method
A + B = 40
A  B = 10
So clearly 2A = 50
Arithmetic method
To split the total by averaging to get 20 for each and then to make the difference of 10 between them. Then 20 + 5 and 20  5 will make the difference to be 10 so Adam gets 25 and Bob gets 15. But we can see here that the arithmetic way actually requires more "thinking" skill.
If the above system of equations method is not used then the line segment method is taught to students but we can see the way of doing it is still the same as the method of systems of equations. This is the reason that I struggled myself, shall I teach systems of equation to students when there are so many contest problems actually use the idea of system of equations even in elementary math contests? (Especially in China). It is not just in model word problems students need to understand the concept of systems of equations, it also exists in the problems when there are more variables than equations. In this case, students also need to know how to eliminate variables by using the concept of system of equations. So this is one of the reasons that many North American math franchises cannot enter the market of China because all the Olympia math workbooks in China have much higher standard than the workbooks used in North America and this is also one reason that at Ho Math Chess, we are upgrading our math contest workbooks to the international level and standard. In Canada (British Columbia province) the students do not learn system of equations until grade 10 but I noticed that some math contest books in China are teaching students the systems of equations to grade 6, I was amazed.
My own thinking is that students shall still learn lots of arithmetic methods to solve word problems but at the same time we shall teach those (bright) students who are ready for algebra at very earlier grades so that they can understand how to use algebra to solve the same word problems in much quicker and "easier" way. In most cases, if the students are asked to do both arithmetic method and algebraic method then they benefit the most.
I am writing a math contest workbook. In that workbook, I present many examples to illustrate in some cases the arithmetic method is more powerful" and "quicker" than the algebraic method and vice versa and I also present the idea that by knowing the algebraic method often student understand better why they are doing a word problem in some steps by using arithmetic method, the algebraic method shed some lights on why the steps in arithmetic method are carried out.
The algebraic method is good because it is thought (and in fact it is true) as a universal way to solve all kinds of elementary math word problems, but is it always true? My research shows in most cases, it is; especially some word problems like Chicken and Rabbits, Age problems, Work problems, Travelling problems, % applications etc can be "easily" solved by using algebra but some word problems actually are awkward to use algebra, so the teaching of elementary math word problems can be raised to a bit higher level that is to show students in what situations, it is better off to use algebra and in what cases, the arithmetic method actually is superior than algebra.
Frank Ho
