Should you assigned drill questions as homework? When is a good time to assign homework? Today, we discuss the contentious topic of what to focus on when introducing a new math topic to students.
Conceptual Understanding or Procedural Fluency?
There has long been two opposing philosophy of education when it comes to introducing new topics in math. The first emphasizes drilling practice as its core foundation and is built on the notion that students will eventually gain deeper conceptual understanding after seeing the pattern in their practice. This is still prevalent in many parts of the world and has been popular with parents due to the quick results they see in students. However, there is no telling when (or if) students will gain the conceptual understanding behind all the procedures, which are so critical when the technique is extended in higher grades or applied to solve practical problems.
In recent years, emphasis has been shifted to favor a conceptual understanding-first approach. Educators understand that math education is a long, integrated process incorporating multiple levels of overlaying concepts. Without firm conceptual understanding at each stage, students may not be able to ascend the learning ladder confidently. Hence, these teachers and parents spent a large portion of their time, showing various ways of looking at problems, introducing different medium, manipulatives, pictorial representations to let the students understand the concepts first. The key philosophy is “do not assign homework until the students are ready”. However, pressed for time and faced with a long list of teaching standards to meet each year, teachers may find that they have to skip certain topics or resort to the quick band-aid of drilling practice for the last few topics of the semester.
This dichotomy between conceptual understanding-first and procedural fluency-focused education often leads to confusion among new teachers and parents, who were brought up doing tons of drill practices and are now told to focus on conceptual understanding.
Closed Loop Instruction
We feel that both philosophy can be combined in a more efficient hybrid which we shall refer to as “Closed Loop Instruction”. Here is an example. Say we’re introducing the topic of fraction multiplication.
1. Introduce an Anchor Task
First we introduce a central word problem which we have to solve together as a class. This word problem is challenging to most of the students, except for the advanced learners who have already understood and internalized the concept at hand – fraction addition. An example of this problem:
Mimi’s market sold 24 heads of lettuce one morning. That afternoon 2/7 of the remaining heads of lettuce were sold. The number of heads left was now 1/2 of the number the market had at the beginning of the day. How many heads of lettuce were there at the beginning of the day?
It is important to make a “big deal” out of this problem. Introduce the problem using pictures, animation clips, stories, skits etc to get students excited about finding solution. Dr Yeap Ban Har calls this the anchor task of the module and is key to a successful introduction of new topics.
2. Lay Out the Plan
Next, after discussing and satisfied that the class do not yet have the “tools” or knowledge required to solve the problem, we announce that over the next few days, we’re going to learn a topic that will help us solve the problem – i.e. fraction multiplication. However, keep in mind we’re still trying to help Mimi solve her problem at the end of the week.
3. Introduce the Concepts
This is where teachers bring in their concrete manipulatives, pictorial representation and examples to introduce the topic conceptually, along with the procedures required to get the solution. Do not overwhelm the kids! They need time to absorb new concepts. It is important to note that kids may not have internalize the concepts after you’re done, but that is ok. Once most of the kids have a fair grasp of the concept, it is time to assign practice questions to drill in the procedures to solidify their understanding.
4. Practice the Procedures
Depending on the topic, there may be more than one concept and procedure to learn. If so, the practice should be introduced in stages to avoid confusion between the steps. The objective here is to firm up the understanding that is fresh in the minds of the students, through repeated exercise. At the end of the week, have student turn in their work and check that they have finished their tasks and have understood the required concepts and procedures.
5. Close the Loop
Finally, we enjoy the fruits of our week-long labor! Once students have understood the concepts, put in the time to practice the procedures and gained working understanding of the topic, it is time to come back to the anchor task and review the problem again. This time, students should be able to work through the challenge using the concepts learnt during the week, and come up with a satisfactory solution.
6. Conclusion and Further Exercise
Once the module is completed, sit down with the class and review all that we have learned over the week. We can also assign more questions that are similar to the anchor task for school or home practice to boost the students’ confidence in applying their newly acquired skill!
We found that while emphasizing conceptual understanding is important when introducing a new topic, sometimes kids need time to internalize new knowledge, and practicing procedures can give them the time and exercise required to achieve that. Moreover, with an interesting problem at the core, students have a sense of purpose when learning the new topic and feel motivated to learn.