Solving Mathematics Word Problems
In Singapore secondary schools, students are expected to use algebraic methods and equations to solve problems. It is widely acknowledged that students generally have great difficulty in formulating algebraic equations to represent the information given in word problems (Stacey, & MacGregor, 2000). Some of the reasons are the students’ weak understanding and poor algebraic manipulation skills.
Consider the following word problem:
Mrs Ooi is 36 years older than her daughter, Martha. In 10 years’ time, Martha’s age will be 1/3 her mother’s age. How old is Martha now?
Many students who are not confident and incompetent in using the Algebraic Method continue to use the Model Method to solve algebra word problems.
Based on the above step-by-step worked solution, we can see that both methods (model drawing and the algebraic equation) serve to represent information given in the question. The worked solution was made possible only after both representations have become explicit (Ng, 2009).
What then, makes it “easier” for secondary students to choose the model method over algebra to solve challenging word problems like this?
The Model Method explained
The Model Method involves drawing of diagrams in the form of rectangular bars to represent known and unknown numerical quantities, to show the relationships between various quantities and thus solving these problems. The configurations can be easily partitioned into smaller units whenever necessary. There are many ways of expressing different known and unknown quantities with these rectangular bars (Ferrucci, Kaur, Carter, & Yeap, 2008). Some of the more common models used are the Part-Whole Model (also known as Part-Part-Whole Models), the Comparison Models and the Before-After Models.
The introduction of the Part-Whole Model and the Comparison Models is an important element of the Concrete-Pictorial-Abstract (CPA) approach which is consistent with Bruner’s (1961) theory of intellectual development stages of enactive representations preceding that of iconic representations. Students make sense of the Part-Whole Model and the Comparison Models through the use of manipulative. Then they move on to draw rectangular bars as pictorial representations of the models, and thus using the drawn models to help them solve abstract mathematical word problems.
How does learning the Model Method help students understand Algebra?
Going back to the example above, the Model Method uses rectangular bars to represent the given information and show the relationship between the given quantities but the algebra method uses letters to represent the unknown quantity.
The Model Method focuses on the relationship between the ages of the mother and the daughter, which is the centre of the whole question while algebra focuses merely on the formation of the linear equations. In comparison, the models enhance the age difference between Mrs Ooi and Martha, thus making it easier for understanding.
Unlike the algebraic method which gives an overview of the entire situation of the question, the Model Method focuses on specific details.
The drawings of models allow students the opportunity to translate information from words into pictorial representations. Such representations help students to understand specific relationships. And in the process of understanding such relationships involved, students are ‘enabled’ to construct relevant algebraic expressions and equations.
Students’ experiences in using rectangular bars to represent quantities in the Model Method would enable them to appreciate better the use of letter symbols (variables) when they move on to learn algebraic method later (Kho, 1987). Learning to use the Model Method effectively to solve word problems is therefore important and essential for primary school students.
This article is contributed by Michelle Choo, co-author of My Pals are Here! Maths.
About the author
Michelle Choo has 25 years of teaching experience. She was a Level Co-ordinator at a local primary school offering the Gifted Education Programme and later took up the position of an Acting Head of Department at another local school. She shifted her focus to authorship and being a full-fledged education trainer. To date, her training experiences include conducting local and international workshops and seminars for educators as part of their professional development programme. Michelle is well-known for crafting creative, activity-based, exercises and assessments that are based on sound Mathematical Pedagogical Content Knowledge.
Bruner, J. S. (1961). The Act of Discovery. Harvard Educational Review, 31, 21-32.
Ferrucci, B. J., Kaur, B., Carter, J. A., & Yeap, B. (2008). Using a model approach to enhance algebraic thinking in
the elementary school mathematics classroom. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics, Seventh yearbook, (pp. 195-221). Reston, VA: National Council of Teachers of Mathematics.
Fong, H. K. (1994). Bridging the gap between secondary and primary mathematics.
Teaching and Learning, 14(2), 73-84.
Fong, H. K. (1999). Some generic principles for solving mathematical problems in the classroom.
Teaching and Learning, 19(2), 80-83.
Institute of Education Sciences. (2015). Retrieved from: https://nces.ed.gov/TIMSS/ on 20th July 2015.
Kho, T. H., Yeo, S. M. & Lim, J. (2009). The Singapore Model Method for Learning Mathemetics.
Singapore: EPB Pan Pacific.
Ng, C. H., & Lim, K. H. (2001). A handbook for mathematics teachers in primary schools.
Singapore: Federal Publications.
Ng, S. F. (2009). What is algebraic about the model method? 3rd Redesigning Pedagogy International Conference.
Singapore, 1-3 June 2009.
Ng, S. F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word
problems. Journal of Research in Mathematics Education, 40(3), 282-313. Reston, VA: National Council of
Teachers of Mathematics.