A bi-annual magazine for the Hong Kong Academy community.
When students learn by doing in this way, they gain a deeper understanding of the concepts being explored and are then able to apply their understandings to a variety of situations. What comes after using manipulatives and models? After students have used manipulatives to solve enough problems, they are usually ready to start putting pencil to paper to model these problems and their solutions. As they model more and more of these problems, patterns begin to make themselves clear. Students often observe these patterns themselves, and conversations with a teacher add to that clarity. Once they are confident with their observations of the patterns, students use these patterns, usually with teacher guidance, to develop a more efficient, procedural method for solving the given type of problem. For example, one of the most significant concepts in Grade 6 maths is dividing with fractions. When doing a problem such as ¾ ÷ ½, the first thing we want students to understand is what the problem is asking. The students have prior knowledge with division that can help them here. For “It’s easier to find the value of the ratio when you can see what it looks like.” Jeremy, Grade 6 mathematician, on using models to solve ratio problems. instance, in earlier grades they may have learned that 9 ÷ 3 is a way of asking “How many 3s are in 9?” They should be able to transfer this knowledge to division with fractions. ¾ ÷ ½ by asking “How many ½ pieces are in ¾ of a piece?” Once they understand this, they are ready for using manipulatives and creating bar-models. After constructing a sufficient number of models, students will start to see a pattern for a more efficient procedure for division than what most of us learned when we were in school. In our collaborative learning environment, students often see the pattern because they are helping each other to make observations and identify patterns. Regardless of how students first see the pattern, the understanding gained enables them to adopt the more efficient procedure for solving the problem. By the end of the unit with Grade 6 students, they understand not only to flip over the second fraction and multiply (multiply by the reciprocal of the divisor), they also understand the concept well enough to explain why this will work. “People visualise differently and models or manipulatives can help people to understand a problem in a different way.” Jorja, Grade 8 mathematician, on using algebra tiles to solve algebraic equations. Whether in Primary School or in the early years of Secondary School, much of the process of learning maths is the same. When a new concept is being explored, students are encouraged to use manipulatives that they can see, touch and move around. After they become more familiar with this new concept, they can create models to show their understanding. Finally, students use what they learned through their use of 8
“Using manipulatives can help you because they’re more visual which is good because you can touch and move around the tiles.” Zahra pictured with Frankie, Grade 7 mathematicians, on using integer tiles to add and subtract positive and negative numbers. manipulatives and models to develop a more efficient procedural method for solving the given type of problem, usually with some teacher guidance. Over time, students draw on their growing tool box to tackle more challenging concepts. The exciting ‘aha’ moments happen all along the way! What can parents do at home to support conceptual understanding? For many parents, helping children with maths homework can be challenging. The following are some simple things parents can do at home to support their children’s conceptual understanding. Most of these involve asking your child some simple questions about what they are doing. When a learner has trouble getting started on a problem, trying asking: What do you know about this problem? Have you seen a problem like this before? What do you picture in your head for this problem? How could you organise what you know to help you figure out what this problem is asking? When we focus on conceptual understanding in the area of mathematics, we can see the importance of students building their own meanings when exploring new concepts. Procedural methods may be quicker and more efficient, but without a foundation of conceptual understanding, the students will not be able to transfer these skills to new and various types of problems. Procedural and factual knowledge are very important in the realm of mathematics, but just as important is a strong conceptual understanding. This conceptual foundation allows students to transfer their procedural knowledge to a variety of new and unfamiliar situations and to apply mathematics to real-world situations. When a learner has worked out a solution, consider following up with these questions: Can you show how your solution makes sense? Can you explain what you did to solve this problem? Why did you solve the problem this way? Can you do it in a different way to check that it still works? 9