From Arithmetic to Algebra: Variables, Word Problems, Fractions and the Rules

Model Method

I have had much success with teaching my students how to use the model method. The training I have received regarding the Singapore model method teaches children to pull out and label the facts or the quantities from the problem. Then students write out the question, draw a model, and the final step is to refer back to the question and write the answer as a sentence that answers the question. The last step is not only helpful to make sure the students answered the question asked, but also because it essentially forces them to label the units. To reiterate, the steps I have my students follow are:

1. Pull out and label the facts (quantities).
2. Write the question.
3. Draw a model to show the relationship between the quantities in the problem.
4. Reread the question and write the answer as a sentence.

I have found that following these steps helps my students succeed. In my experience, it allows the students to focus more on the numbers, thus reducing the confusion the students face when they are mixed in with a bunch of words.

Because of my interest in and success with this method, I conducted further research into the history of the model method as well as the pros and cons of the technique, and the importance of identifying the unit to which the numbers in the problem refer.¹⁰ It is informative and provides more insight into the technique.

The model method is a heuristic, or a practical method, used to solve whole number word problems representing part-whole and comparison problems. A before-after concept is used with both part-whole and comparison problems to solve more complex structures. The before-after concept involves drawing two bar model representations to represent different stages in the word problem. The model method allows students to solve higher level problems without the use of algebra. Additionally, the model method focuses on representation, which is the key to solving word problems.

According to Ng and Lee,¹¹ the model method can be broken into three phases. The first phase is the Text Phase¹², in which children read the problem that is presented in text form. The second phase is the Structural Phase,¹³ where the students transform the text into a model. The third phase is the Procedural-Symbolic Phase¹⁴.  In the final phase students have to use the model and develop the mathematical steps to solve it. Ng and Lee suggest that since primary students do not yet know how to work with equations, they must use the unitary method.¹⁵ The unitary method is a technique where students undo operations to find the value of one unit. Typically, that unit can be used to figure out other chunks or pieces of the model, which lead to a solution to the problem. This strategy requires the students to develop arithmetic equations to stand for the unknown units, which is essentially what students do when solving with bar models. Due to the success of the Singapore math techniques and the model method, the curriculum has become more widely used across the world, and is popular in the United States.

Pros of the Model Method

There are many benefits to teaching the Singapore model method. Typically, students tend to see the numbers in a word problem and do the first calculation that pops into their heads. The model method helps teach students to slow down, read the problem several times, and think about what is happening. Once a student has gone through these steps, a model can be drawn to represent the situation. One could say, that it is a device that enforces Polya’s four step method. I can attest to this from my own experience as I learned how to use the model method to solve word problems. I find myself reading the problem over slowly several times until I understand the problem, which is Polya’s first step. Then, I extrapolate the key information that is both known and unknown, and restate the question in order to devise a plan as stated in Polya’s second step. In this case, the plan is to draw a picture, in particular, a bar model. Then, I carry out the plan (step three) and figure out the answer. Following Polya’s fourth step, I check over my work to make sure it makes sense. According to Englard, the model method “Puts the focus back on the relationships and actions presented in the problem and helps students choose both the operations and the sequence of steps that are needed to solve the problem.”¹⁶

The steps required to understand and draw a bar model is the crux of the model method. Ng and Lee¹⁷ refer to a number of studies that found that visual and concrete representations improve performance in solving word problems. “The model-method affords higher ability children without access to letter symbolic algebra a means to represent and solve algebraic word problems.”¹⁸ In one of the studies conducted by Ng and Lee, the teachers reported that because students represent the problem visually, it affords teachers the opportunity to inquire about difficulties students have with the representation. Another advantage of the model method is that it is not an “all or nothing process.” The findings agree with other research that reports that children tend to make their errors in the representation of the model. The Singapore model method has become a popular math instruction method because it is proven to enhance problem solving skills.

Cons of the Model Method

Yan Kow Cheong with Math Plus Consultancy conducted a study of the model method. The difficulties noted pertain more to problems drawing the models. Cheong noted three main problems: 1) difficulty of an “accurate diagram,” 2) division in a block diagram, and 3) inappropriate use of the model method.¹⁹

It can be difficult for students to draw models accurately. While the models do not have to be drawn to scale, the model does need to accurately represent the problem. The division of the blocks is problematic particularly when a bar model requires additional partitioning. Another problem is that some curriculum books are overusing the model method when it is not appropriate. Cheong explains that some writers of curriculum books, while pushing the use of the model approach, are using the method when it is not appropriate.

Difficulties with algebra

Solving word problems with algebra is particularly problematic due to:

1) understanding the meaning of letters used in symbolic algebra,

2) translating natural language into equations,

3) understanding the semantic structure of word problems, in particular the nature of relationships between quantities and how they are linked, and

4) using text-based semantic cues in the construction of equations.”²⁰

Singapore students who have been taught the model method have exhibited challenges transitioning to algebra. “Studies (e.g. Ng et al.)²¹ have shown that poor bridging of students from the use of bar diagrams to the use of letter-symbolic algebraic methods can hinder their learning of algebra.”²² The bright students don’t seem to have problems transitioning to algebra, but others tend to hold onto the bar model heuristic. A software tool called ALGEBAR²³ was designed in an effort to provide better bridging between the two subjects. Researchers conducted several studies to develop not only software, but a larger instructional package to address the algebra difficulties. The research determined that secondary problem solving requires mixed schemas. Students who used the bar models relied on these primary school schemas and forward calculations. Basically, prior knowledge influences new learning, and in this case, hinders it. According to this study, “students need to be taught the explicit structural homomorphism between bar model representation and algebraic equations.”²⁴ Students tend to want to calculate instead of manipulating the letter-symbol algebraic equations. Teachers should provide scaffolding by using models followed by weaning once the students gain comfort with equations. Eventually, more difficult problems that can only be solved with algebra should be introduced, so the students will realize the importance of letter-symbol algebra and link it to their prior knowledge. While the ALGEBAR is not a part of my curriculum unit, this article provided insight into how the model method hinders students’ algebra acquisition.