Using Bar Models for Problem Solving

During our Singapore Math Workshop at Columbia University Teachers College 2016 last Wednesday, on Mar 9th, 2016, we talked about the role of problem solving in our day to day classroom learning. In a recent survey by Program for International Student Assessment (PISA), “What 15-year olds know and what they can do with what they know”, the data tells us that:

“Students who are open to solving mathematics problems – who feel that they can handle a lot of information, are quick to understand things, seek explanations for things, can easily link facts together, and like to solve complex problems – score 31 points higher in mathematics, on average, than those who are less open to problem solving. Among high achievers, the difference between the two groups of students is even greater – an average of 39 score points.” – PISA 2012

While this finding might not be new to many teachers, our question is how we can encourage this “openness to problem solving” in students.

In one of my recent school visits, I was doing a lesson study on a Grade 3 class. The class is diverse, some are still struggling with multiplication and division while the rest have mastered the topic and are ready to move on. This is not uncommon in many classrooms, but how does the teacher introduce problem solving in such a diverse class? During the lesson study, we started by introducing questions that focuses on logic and reasoning, using only basic concepts on multiplication and division, keeping the numbers small and not overwhelming. This is one of the math question we used from Math in Focus, Workbook 3A Pg 175.

Lance and Alex have 70 pencils. Lance has 4 times as many pencils as Alex. Alex’s pencils are shared equally among 2 children. How many pencils does each child get?

Instinctively, students will think of division, using numbers 70 and 4. However, delving deeper, they understand that we are not dividing 70 by 4. Instead, we should be dividing 70 by 5!

Let’s explore this here using bar modeling.

Presenting the problem this way using bar model makes it easier to make sense of the seemingly complex set of clues, and many students were able to proceed to work out the actual solution.

Many students struggle with problem solving and bar modeling is a very useful visual tool for extracting and organizing information before moving to the next step. It gives students “something to hang on to” when they are stuck (and who doesn’t at some point or the other?). As one of our teacher reader from Cincinnati puts it,

The bar model is great for interpreting the information before you even look at the question. So many times kids toss the numbers around to come up with an answer before they even know what information they have. This year I started off by giving them the information without the question. That really makes them think!
-Penelope Oliver, Cincinnati, Ohio

It is important for teachers to understand the importance of problem solving and the tools to guide students effectively without introducing anxiety. This statement from the PISA survey sums it up nicely, “Teachers’ practices can promote students’ drive and willingness to engage with complex problems. Teachers’ use of cognitive-activation strategies, such as giving students problems that require them to think for an extended time, presenting problems for which there is no immediately obvious way of arriving at a solution, and helping students to learn from the mistakes they have made, is associated with students’ perseverance and openness to problem solving.”

What are your experiences in introducing problem solving? What are some of the challenges? Please leave your comments below, we love to hear from you!