Example implementation reports

The GeoGebra file featured in the gif above was created by past participant Anne Wolkowitsch

Hi Richard, As you know I was fortunate enough to be teaching a unit on Coordinate Geometry while completing this course, so I was able to directly implement many of the strategies and principles I learned here. In fact, I gave my class your very same worksheet (Phase 3, Student Activity 2) in its entirety. My observations of the class while students were working on this sheet very much confirmed your predictions.

There was a mood of engagement and interest among the students that one learns to cherish as a teacher when it occurs. I heard many wonderful mathematical conversations between students as they discussed their ideas. This can only be because they were not routinely applying a demonstrated procedure, but because they were essentially using the basic geometric concept of gradient to creatively solve problems of a simple enough nature for them to experience success. I went on to design another worksheet of my own in which I tried to incorporate the principles of Conceptual learning. I have attached it to this comment. I also created three Geogebra files (the one you have seen relating to the Gradient-Intercept form of the equation of a linear graph) and two others which related to midpoint and the distance formula. I felt that the midpoint file worked really well. I did this lesson as an "assembly point" lesson where the entire class participated at once. In explaining the midpoint, I focussed on the concept of the midpoint as the "average" in both the x and y directions.

I hardly used the standard midpoint formula at all. Even when explaining how to calculate the endpoint, given one endpoint and the midpoint of a line segment, I did not use algebra at all, but showed the students how to solve such problems using strictly the understanding of an average, and co-ordinate geometry concepts. There is no doubt in my mind that students were able to understand the midpoint in this lesson far better than when a strictly procedural, and algebraic based routine is demonstrated.

Finally, I made extensive use of the "mini-lesson" pedagogy. where I retaught the same concepts quickly to small groups. This in itself is always a rewarding experience, because having done one 'mini-lesson" on a certain topic, the next one you give on the same topic to another group of students is always an improved version of the previous one. Because I taught this unit of work last year as a "flipped learning" unit of work, I was also able to integrate into the lesson videos for students who finished early to watch and go on independently. I was very fortunate for this convergence of timing and resources! What it meant was that the mini-lesson strategy was all the more powerful, because I was able to give mini-lessons to students who were choosing to work ahead of the class.

My summative assessment for this unit (otherwise known as a "Topic Test") is scheduled for this Monday, which is unfortunate since it is just outside the cut off date of the course, and it would have been interesting to report back on the results. Whatever the results are, my experience of teaching this unit using the principles I have learned here was a highly rewarding and enjoyable one, in which students showed a high level of engagement and motivation as well. I had two or three students say to me: "Hey miss, I understand this!", in the refreshing and excited manner of young people, that as teachers, we always are so gratified to hear.

I gave my class your very same worksheet (Phase 3, Student Activity 2) in its entirety. My observations of the class while students were working on this sheet very much confirmed your predictions - there was a mood of engagement and interest among the students that one learns to cherish as a teacher when it occurs. I heard many wonderful mathematical conversations between students as they discussed their ideas. This can only be because they were not routinely applying a demonstrated procedure, but because they were essentially using the basic geometric concept of gradient to creatively solve problems of a simple enough nature for them to experience success. I went on to design another worksheet of my own in which I tried to incorporate the principles of Conceptual learning.

I also created my own midpoint file. I felt that the midpoint file worked really well focusing on the concept of the midpoint as the "average" in both the x and y directions. I hardly used the standard midpoint formula at all. There is no doubt in my mind that students were able to understand the midpoint in this lesson far better than when a strictly procedural, and algebraic based routine is demonstrated.

My experience of teaching this unit using the principles I have learned here was a highly rewarding and enjoyable one, in which students showed a high level of engagement and motivation as well. I had two or three students say to me: "Hey miss, I understand this!", in the refreshing and excited manner of young people, that as teachers, we always are so gratified to hear.**Cathryn Marshall, Parade College, Bundoora, August 2019**

There was a mood of engagement and interest among the students that one learns to cherish as a teacher when it occurs. I heard many wonderful mathematical conversations between students as they discussed their ideas. This can only be because they were not routinely applying a demonstrated procedure, but because they were essentially using the basic geometric concept of gradient to creatively solve problems of a simple enough nature for them to experience success. I went on to design another worksheet of my own in which I tried to incorporate the principles of Conceptual learning. I have attached it to this comment. I also created three Geogebra files (the one you have seen relating to the Gradient-Intercept form of the equation of a linear graph) and two others which related to midpoint and the distance formula. I felt that the midpoint file worked really well. I did this lesson as an "assembly point" lesson where the entire class participated at once. In explaining the midpoint, I focussed on the concept of the midpoint as the "average" in both the x and y directions.

I hardly used the standard midpoint formula at all. Even when explaining how to calculate the endpoint, given one endpoint and the midpoint of a line segment, I did not use algebra at all, but showed the students how to solve such problems using strictly the understanding of an average, and co-ordinate geometry concepts. There is no doubt in my mind that students were able to understand the midpoint in this lesson far better than when a strictly procedural, and algebraic based routine is demonstrated.

Finally, I made extensive use of the "mini-lesson" pedagogy. where I retaught the same concepts quickly to small groups. This in itself is always a rewarding experience, because having done one 'mini-lesson" on a certain topic, the next one you give on the same topic to another group of students is always an improved version of the previous one. Because I taught this unit of work last year as a "flipped learning" unit of work, I was also able to integrate into the lesson videos for students who finished early to watch and go on independently. I was very fortunate for this convergence of timing and resources! What it meant was that the mini-lesson strategy was all the more powerful, because I was able to give mini-lessons to students who were choosing to work ahead of the class.

My summative assessment for this unit (otherwise known as a "Topic Test") is scheduled for this Monday, which is unfortunate since it is just outside the cut off date of the course, and it would have been interesting to report back on the results. Whatever the results are, my experience of teaching this unit using the principles I have learned here was a highly rewarding and enjoyable one, in which students showed a high level of engagement and motivation as well. I had two or three students say to me: "Hey miss, I understand this!", in the refreshing and excited manner of young people, that as teachers, we always are so gratified to hear.

I gave my class your very same worksheet (Phase 3, Student Activity 2) in its entirety. My observations of the class while students were working on this sheet very much confirmed your predictions - there was a mood of engagement and interest among the students that one learns to cherish as a teacher when it occurs. I heard many wonderful mathematical conversations between students as they discussed their ideas. This can only be because they were not routinely applying a demonstrated procedure, but because they were essentially using the basic geometric concept of gradient to creatively solve problems of a simple enough nature for them to experience success. I went on to design another worksheet of my own in which I tried to incorporate the principles of Conceptual learning.

I also created my own midpoint file. I felt that the midpoint file worked really well focusing on the concept of the midpoint as the "average" in both the x and y directions. I hardly used the standard midpoint formula at all. There is no doubt in my mind that students were able to understand the midpoint in this lesson far better than when a strictly procedural, and algebraic based routine is demonstrated.

My experience of teaching this unit using the principles I have learned here was a highly rewarding and enjoyable one, in which students showed a high level of engagement and motivation as well. I had two or three students say to me: "Hey miss, I understand this!", in the refreshing and excited manner of young people, that as teachers, we always are so gratified to hear.

During my Yr 10 Graphing Unit I used Geogebra to initially show the lines and equations of y=x, y=x+2, y=x-3. I then used the Socratic Method with students to have them consider the differences and similarities between equations.

I then removed those equations and put up the following lines and equations: y=2x, y=2x+2, y=2x-3. I again asked questions to check if the students’ responses were the same as for the original set of equations. With the 2nd group of equations I concentrated on getting answers from those who tended to ‘opt out’ with the initial questions. I then followed up with the question, ‘What would they expect to see from y=-x, y=-2x, y=-2x +2; and why did they expect to see that?’

I asked ‘what would they call the point/no. which sits on the y axis? Is there an advantage to having a label for this no./point? Also, what about the number in front of the x?’ It took some prompting but I was happy with their thought processes. Some recalled hearing the term y intercept and gradient and they all then gravitated to that.

I asked them to consider ‘why’ I was having them draw the line for the equation without creating a table. One of my favourite responses was “Because it takes less time and Mathematicians are always looking for shortcuts.”

We then moved into giving the y intercept and gradient and I asked them to draw the line, then construct an equation. Some flew through this, but of course some others needed more prompts. A no. of mini lessons occurred.

Students were also given the line on a Cartesian Plane and asked to find the y intercept and gradient and then to create the equation as a flip on the previous activity.

Most students were able to understand the concepts of linear equations, gradient and y intercept and quickly applied their knowledge to new situations. I also used this method when graphing parabolas; showing examples, comparing, discussing and then creating expressions. When it came to (x=3)2, I only showed them one example on Geogebra with the equation and asked them to consider where the parabola would be for (x-2)2 . The students had to prove to me that their answer was correct and then create their own example. I also had an extension activity of examples such as (x-2)2+3. These last 2 activities were intended as ‘brick wall’ strategies. I encouraged peer discussions at this time.

Student spread is considerable in my Year 10 classes, but I find that asking question of the students, giving them the opportunity to discover the answers really helps in their understanding and retaining of the concepts. As discussed earlier, I am happy to incorporate mini lessons as I feel this helps keep all students engaged and progressing rather than me stopping and starting the entire class. We do of course have ‘come together’ sections. I find I have to be very prepared with the challenging extension work for those students. Unfortunately I don’t always have enough of these tasks and it is an area I am working on.

**Tracy Pearson, St Mary MacKillop Catholic College Isabella Campus, May 2016**

I then removed those equations and put up the following lines and equations: y=2x, y=2x+2, y=2x-3. I again asked questions to check if the students’ responses were the same as for the original set of equations. With the 2nd group of equations I concentrated on getting answers from those who tended to ‘opt out’ with the initial questions. I then followed up with the question, ‘What would they expect to see from y=-x, y=-2x, y=-2x +2; and why did they expect to see that?’

I asked ‘what would they call the point/no. which sits on the y axis? Is there an advantage to having a label for this no./point? Also, what about the number in front of the x?’ It took some prompting but I was happy with their thought processes. Some recalled hearing the term y intercept and gradient and they all then gravitated to that.

I asked them to consider ‘why’ I was having them draw the line for the equation without creating a table. One of my favourite responses was “Because it takes less time and Mathematicians are always looking for shortcuts.”

We then moved into giving the y intercept and gradient and I asked them to draw the line, then construct an equation. Some flew through this, but of course some others needed more prompts. A no. of mini lessons occurred.

Students were also given the line on a Cartesian Plane and asked to find the y intercept and gradient and then to create the equation as a flip on the previous activity.

Most students were able to understand the concepts of linear equations, gradient and y intercept and quickly applied their knowledge to new situations. I also used this method when graphing parabolas; showing examples, comparing, discussing and then creating expressions. When it came to (x=3)2, I only showed them one example on Geogebra with the equation and asked them to consider where the parabola would be for (x-2)2 . The students had to prove to me that their answer was correct and then create their own example. I also had an extension activity of examples such as (x-2)2+3. These last 2 activities were intended as ‘brick wall’ strategies. I encouraged peer discussions at this time.

Student spread is considerable in my Year 10 classes, but I find that asking question of the students, giving them the opportunity to discover the answers really helps in their understanding and retaining of the concepts. As discussed earlier, I am happy to incorporate mini lessons as I feel this helps keep all students engaged and progressing rather than me stopping and starting the entire class. We do of course have ‘come together’ sections. I find I have to be very prepared with the challenging extension work for those students. Unfortunately I don’t always have enough of these tasks and it is an area I am working on.

At present I am doing Quadratic equations with year 10 students. I used Geogebra to show them that when a quadratic equation is graphed, it presents a parabola. The students were visually able to see the solutions of the equations. They were able to understand better the concept of one solution, two rational/irrational solutions and no real solutions by actually looking at the position of the parabola.

They further factorised the equations to find out the solutions. Brick wall , mini lessons, socratic questioning are being used regularly in my lessons. I am doing representation of data with year 8 students. I got them to do a survey to find out their favourite animated movie.

I then got them to construct a divided bar graph on a 30 cm long strip of cardboard. The divided bar graph was then joined at the ends to make a circle. The students then placed it on their books and drew a circle and using the markings on the divided bargraph, made the sectors in the sector graph. The divided bar graph was then cut into sections and then pasted in their books as a column graph. The students enjoyed this activity and they were able to understand that the same information can be represented in many ways - divided bar graph, sector graph and column graph in this instance. I did this activity rather than use the textbook to do this topic in class. The textbook questions were given as homework for the weekend.

**Gousia Naeemullah, May 2016**

They further factorised the equations to find out the solutions. Brick wall , mini lessons, socratic questioning are being used regularly in my lessons. I am doing representation of data with year 8 students. I got them to do a survey to find out their favourite animated movie.

I then got them to construct a divided bar graph on a 30 cm long strip of cardboard. The divided bar graph was then joined at the ends to make a circle. The students then placed it on their books and drew a circle and using the markings on the divided bargraph, made the sectors in the sector graph. The divided bar graph was then cut into sections and then pasted in their books as a column graph. The students enjoyed this activity and they were able to understand that the same information can be represented in many ways - divided bar graph, sector graph and column graph in this instance. I did this activity rather than use the textbook to do this topic in class. The textbook questions were given as homework for the weekend.

During this unit I was able to implement some of the of the teaching strategies to my classes, however I am yet to implement it further due to lack of time. One class which I enjoyed working with has been my year 12 Methods, creating a lesson where they had to draw on their prior knowledge from year 11 and the some of this year to discover the derivative of tan. I only gave hints and aided ideas without giving too much away, which allowed most students to work through, in their own way, how you might work with rules from trigonometry and calculus to discover the derivative of tan.

From the coordinate geometry workbook I had the idea to try a similar approach to teaching parabolas, exponentials, hyperbola and circle graphs. I have created a, first attempt, workbook for students to learn about parabolas, though, will not be able to test it on my year 10 class until after this course has finished. I used my interpretation of the structure of the coordinate geometry booklet, being more fluid and student directed, to create my one, with some interactive GeoGebra files to aid in explaining different transformations.

I have attempted to take elements of the different strategies to help students learn about graphs in a more natural and explorative way.

Enjoy the attached workbook and files on parabolas. I know I have enjoyed making them.

**Naomi Aigner St Mary MacKillop Catholic College Isabella Campus, December 2015**

From the coordinate geometry workbook I had the idea to try a similar approach to teaching parabolas, exponentials, hyperbola and circle graphs. I have created a, first attempt, workbook for students to learn about parabolas, though, will not be able to test it on my year 10 class until after this course has finished. I used my interpretation of the structure of the coordinate geometry booklet, being more fluid and student directed, to create my one, with some interactive GeoGebra files to aid in explaining different transformations.

I have attempted to take elements of the different strategies to help students learn about graphs in a more natural and explorative way.

Enjoy the attached workbook and files on parabolas. I know I have enjoyed making them.

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