Process Worksheets

My comfort zone in the classroom as a teacher is teaching the same way I was taught. I teach Mathematics, and the way I had it when I was a student was looking at worked examples on the board and then practicing (individually or in pairs). I would practice a learned skill or a method and finally I would practice solving problems involving these skills and methods. I liked learning Math this way and this is how I like to teach it. Some weeks before the end of the semester, Damilola Dauda sat in one of my S3 lessons and noticed this straight away. When we met to discuss her observation notes she suggested I could try to change the lesson structure that makes me comfortable, and see what happens. After having read Dan's blog post about the discovery method of instruction and the related article "Why minimal guidance during instruction does not work [...]", I decided to implement some of the highlights from the article in a lesson sequence for my S3 group in a different structure to the one I am used to and feel comfortable with. In the first week back form the holidays we had to look at Coordinate Geometry. Usually, for this unit I would give the relevant formulas at the start of the week and practice a wide array of problems throughout the week in the classroom, moving towards more complicated problems close to the end of the week. This approach supports students' acquisition of skill sets, and prepares them to access deeper problem structures. On the other hand, this approach strays away from giving students a solid understanding of the geometrical properties the formulas are meant to describe. In light of this conflict I chose a different approach: let students arrive at the formulas starting from basic geometrical constructs and previously acquired knowledge. To this end, I designed a series of process worksheets as described in the article in page 80. Below is a summary of the lesson sequence and my own observations of it. Overview of the lesson sequence The lesson sequence looks at basic fundamental concepts and formulas of coordinate geometry. Each concept listed below has a process worksheet associated to it (links to them can be found at the bottom of the page).

Distance between two points
Midpoint of a line segment
Relationship between gradients of parallel and perpendicular lines
The equation of a straight line

Objectives of the lesson sequence:

Understand the concepts described above
Be able to use the formulas correctly to solve problems of coordinate geometry
Be able to express ideas and communicate using appropriate language
Develop investigative work skills
Showing evidence to support claims
Testing their own generalizations with examples
Identifying patterns
Develop time management skills

Assessment and motivation At the end of the lesson sequence the students were given a test with three problems to solve they had never seen before. The problems addressed all the concepts; many of the concepts listed above were assessed in one single problem. The test did not have any kind of grade, only written and oral feedback. Also, students were told in advance that they would be able to use their process worksheets during the test to solve the problems. The aim of this measure was to ensure students would not be spending energy in memorizing formulas or forcing their brain to remember methods. In this way, they would be able to spend more energy in solving the problem instead. During the lesson sequence

Students responded positively to the process worksheets. Students were engaged most of the time completing the worksheets. Their questions were always relevant and contributed to constructive conversations with their peers. Some students worked faster than others, but they could move on to the next worksheet so they were never idle. Students who were working on the same worksheet allowed themselves to sit together to support each other. More able students provided help and support to those who were having difficulties moving forward through the worksheets. Students arrived at their own expressions for the formulas. They also discovered what their preferred approach to a concept was. Homework Since during the lesson students were busy working through the origins of common coordinate geometry formulas, homework assigned to them provided opportunities to practice skills based on the concepts seen in class. Guidance Students received plenty of guidance in different forms. Process worksheets are scaffolded so the breakdown of a process into many steps is a form of guidance; the more steps a process is broken down into, the more heavily guided it will be. Students also received feedback on each completed worksheet. The written commentaries serve as guidance towards improving aspects of their process work. I walked around the classroom to answer students' questions. Also, at the beginning of the lesson, students were given a short presentation (5 minutes) on the objectives, expectations and forms of assessment of the lesson sequence, so they would be informed as to what the plan for the week was. Self-Evaluation I made better use of homework by assigning a task that would relate to work being done in class and that worked towards a purpose. I spent most of my time working with students instead of lecturing them and trying to manage behavior. Process worksheets contained clear instructions so I did not have to repeat instructions very often for students who did not pay attention the first time.

In conclusion, the use of guidance in different forms helped me work with students when it mattered the most and I could spend more energy helping them than managing them. I saw evidence of constructivist learning when students arrived at the correct formulas but expressed in their own way using correct mathematical language. They were able to use previously acquired knowledge to create new knowledge and use it to solve problems they have never seen before. This suggests that constructivist teaching and learning does not need to fail, if appropriate guidance is given.