Factorisation

Teacher Despina uses a flipped classroom method to make formative conclusions on students’ mathematics learning, helping them become autonomous learners.

Summary

 Mathematics is a subject that students can feel like learning a new language. Each student has their own way in engaging with the abstract nature of math. Teachers can use a learning management system to collect data on students’ mathematics learning, and decide how to further help each student, even before coming to the classroom. Despina encourages teachers and students to use an open-source learning management system to learn about factorisation. Students can learn the content and assess themselves, work with their classmates and ultimately take control of their learning.

 

 

Download the scenario in PDF format

Keywords

Mathematics, factorisation, flipped classroom, self-assessment, inquiry-based learning

Quick reference
Objectives
Encourage students to autonomously learn factorisation in mathematics using a learning management systems
Country
Greece
Prerequisites
Home internet access or a computer lab, GeoGebra (optional)
Implementation level
Intermediate
Target group age
14-15
Digital tools
DFA tool
Dashboard/Monitoring tool
Duration
30-60 min for flipped classroom work, 1-2 sessions for classroom work

Context

In mathematics, factorisation is the method of breaking down a number into smaller numbers that should give the original number when multiplied together. Factorisation is an important method used to solve simple problems like splitting something to equal pieces or in complex applications like data encryption. However, as in all mathematical concepts, the abstract language of math can at first be intimidating for pupils.  Students can therefore learn mathematics better by learning in their own pace and in their own way.

Teacher Despina Plota is a lower secondary school mathematics teacher, who designed an algebra activity at the Experimental Gymnasium of Patras University. Her scenario introduces students to the concept and methods of factorisation (extracting a common factor in algebraic expressions), after they have revised the properties of the powers and the distributive property. Students must assess themselves on a digital learning environment. The teacher monitors the results, which she can use for formative assessment. 

Despina created a virtual class environment for teaching mathematics using LAMS, an open-source learning management system. Students need to sign up to use this platform (This tutorial in English provides an overview of this tool). LAMS allow for creating diverging learning paths and assign different learning paths to different students to adapt to their level. 

Teachers can search the Public Channel of LAMS Central Repository for free and open sequences of educational activities, download, adjust them to their own needs and use them in their classroom. They can also upload their own activities.

The Activity

Three days before the classroom activity, students visited the LAMS webpage and ran the activity sequence. They view the summary of the topic and the learning objectives (5 min.). The tasks were to: 

  • View 2 videos to revise prior knowledge on powers and distributive property (optional, 20-25 min.). The videos below are in Greek, but teachers can easily find an equivalent in English for the same 3 topics:
  1. The properties of the powers
  2. The distributive property (part 1)
  • View the Edpuzzle video to learn about factorisation and answer the questions (20 min.) (EdPuzzle is an interactive video tool that allows asking questions while watching the video).
  • Do the quiz (used for formative assessment), where they answer 3 multiple-choice questions related to the understanding of the basic principles of factorisation (5 min.).
  • Play a factorisation game, using GeoGebra (optional, 10 min.). GeoGebra is an open-source platform and software for teaching and learning mathematics.

Based on the answers from the quiz, the teacher does some preparation before the classroom session. She designs the differentiated lesson plan based on the students’ results accessible on LAMS. At this point students’ answers may reveal some common misconceptions regarding the principles of factorisation. Feedback regarding such misconceptions may be given to the students in 2 example ways. First, the teacher can add a feedback message explaining the correct answer to the quiz questions. Second, the teacher can do a recap in the classroom.

A typical error made by the students is that when they extract a common factor, they may confuse the involved mathematical operations, i.e., division vs subtraction. For example, when extracting a common factor in the algebraic expression 5x+5 they may write 5(x+0) instead of 5(x+1).

If one or more students does this, the teacher’s feedback can be to remind thinking about which operation they apply during factorisation, that is the division. Moreover, the teacher can ask them to validate the result by applying the distributive property. They can reach the correct conclusion if they compare:

5*(x+0) = 5*x + 5*0 = 5x + 0 (false)

5*(x+1) = 5*x + 5*1 = 5x + 5 (true)

The teacher began the class by a recap of basic points regarding factorisation techniques. She then assigned the students to 3 groups, according to their pre-class quiz results. In this case, the teacher can put students that have the same level of knowledge in the same group. Alternatively, she can ensure that each team has students with diverse levels of mastery of the topic. This way peers can learn from each other.

The 3 groups worked separately on the 3 sets of exercises and submitted their answers. Students here were free to choose subsets from among these sets. The three sets of activities (simple, difficult, and very difficult) pre-exist as part of the scenario. But the teacher, or even the students, may choose to add new resources for additional activities.

There are 3 variations on this scenario:

i)    Students work individually, and go sequentially through all 3 sets of activities, simple, difficult, and very difficult (Figure 1).

ii)    Students work in 3 groups (created by the teacher), working together in each group (Figure 2). Each group goes through only 1 of the 3 sets of activities. This means that there are 3 separate learning paths. 

iii)    Students work in 3 groups (randomly, or formed by the students’ own choice, or created by the teacher), working together in each group (Figure 3). Each group selects which activities they want to do from each of the 3 sets.

Figure 1 The sequence of factorisation activities designed by the teacher on LAMS.

Optionally, the teacher can moderate an online discussion and voting on the implementation and outcomes of the practice. Despina encourages students to discuss in the forum within the LAMS activity. Discussion topics can be:

  • The notion of factorisation.
  • What do you think about the presented simple form of factorisation?
  • Were you able to apply the distributive property and the properties of powers?

Figure 2 The concept of factorisation - Template 1 (with optional activities)

The discussion is also a good opportunity to collect feedback from students. The teacher can learn a lot to improve the activities even further. Students can be invited to vote on a poll: “How did you find the flipped classroom method and teaching factorisation with it?” with the following options:

  • I think it was easy
  • I think it was very easy
  • I think it was difficult
  • I think it was very difficult

The teacher can also collect more detailed thoughts from students about the learning activities in the forum provided by the LAMS.

Figure 3 The concept of factorisation - Template 2 (with branches based on previous formative assessment test results).

In a case that some students are joining the class remotely, students can work in teams in the classroom and in an online environment thanks to an online conferencing tool that enables breakout rooms.  This scenario is adaptable for the full online settings thanks to the flipped classroom model, LMS and online tools used. The class work can be done at home asynchronously if teachers provide students with clear guidance on how to get help when they need it. In a synchronous setting, the teacher can communicate with students thanks to an online conferencing tool that enables breakout rooms and students can cooperate on the tasks while the teacher monitors their work.

Outcome and lessons learned

A part of this scenario was implemented at the experimental High School of the University of Patras. Although this was a new way of learning for them, students reacted positively. According to Despina, the challenge is the fact that some students are accustomed to personal contact and they may have difficulty with this. 

Students might be used to having the teacher present the material and deliver immediate feedback to student questions. However, they will eventually discover that the teacher will have even more personal contact with their students. The flipped classroom design helps students prepare for the class and formulate their questions in a more specific way. In turn, the teacher acts as facilitator in the classroom and can invest more time in supporting the collaborative work of students. The groupwork allows more active involvement for each student. They can compare answers and discuss with peers.

This kind of activity requires some extra preparation from the teacher. The teacher needs to properly assess the students’ readiness for the various elements of the activity (digital and non-digital) and consider the characteristics of her class. But it is very rewarding because the teacher can easily adapt her upcoming lessons based on the results of students' self-assessment and peer collaboration.