Will My Lesson Plan Be Successful?

We have been asked to create lesson plans in all of our teacher education courses this year. However, we do not have enough time for each and every one of us to facilitate our 75 minute lessons to our peers. Therefore, there is no way of knowing whether the lessons that we have created on paper will be successful in a classroom. Lesson plans are also a very brief overview of what will be completed in the class. There is usually no thought that the discussions that will take place after the learning has taken place. For example, writing that a reflection will take place in your lesson plan does not prepare a teacher for the essential questions that must be asked in order to properly reflect on an activity. Consequently, teachers will also not be prepared to explain the sections of the lesson that may be more difficult for some students.

The answer to this problem is to create “lesson plays”. Lesson plays are essentially a script that outlines potential discussions throughout a class in order to prepare a teacher for the parts of the lesson that students may struggle with. The following lesson play is a dialogue that was created by my partner and I. It is a discussion that would take place after a real world activity about points of intersection in a grade 10 applied class.

Teacher: Okay so now that everyone has presented their information to the class, we are going to discuss as a group. So would you still choose your payment method, now that you have done some mathematical research? And why? Billy.

Billy: Well I originally chose option 1 but now I think I would choose option 3.

Teacher: What made you change your mind?

Billy: Well I was thinking that the first one had the most money that I could get. But after making the graphs I realized that I could make more money using option 3.

Teacher: What aspect of the graph led you to the conclusion that option 3 was the best?

Billy: Well once we found the point of intersection I realized that as long as I sold 9 hats I would make more than $16. And I could do that no problem at a jays game!

Teacher: Okay great. Did someone have a different answer? Sarah.

Sarah: I chose option 1 too but I decided to stick with it because I wouldn’t have to sell any hats in order to get paid.

Teacher: That is true. Ok so what did you notice about the steepness of the lines? Jessica.

Jessica: They all had different slopes.

Teacher: How does this relate to how much money you would earn?

Jessica: Uhhhhhh I don’t know

Teacher: Can anyone help Jessica out? Bryan

Bryan: A greater slope meant that if I sold more hats I could make more money.

Teacher: Right! So because the slope is steeper for option 3, for example, if you sell tons of hats you can make more money. The amount of money you make increases more quickly with a steeper slope. Therefore, how hard you work will affect how much you earn. How is this different for option 1? Joseph.

Joseph: Option one doesn’t have a slope.

Teacher: So the slope is not increasing or decreasing but that doesn’t mean there is no slope. What would the slope be in this case? Rachel.

Rachel: Zero

Teacher: Great. So the slope of option 1 is zero. Therefore, in option one, the number of hats you sell doesn’t change the amount of money you make. You will always get $16. Okay so you you all discussed in your groups what the meaning of the points of intersection are. What do these mean in our real-world problem? Trish.

Trish: It’s where the two lines overlap.

Teacher: Okay good, so that is what it means mathematically, but we want to know the real world meaning. What does the lines overlapping mean in terms of money?

Trish: That when you sell that many hats you make the same amount of money no matter which payment method you choose.

Teacher: So I could choose any one of the three payment methods?

Trish: No, I think it is only the two lines that are intersecting.

Teacher: Awesome! So the intersection point of two lines means that if you sell x amount of hats, you will make the same amount of money for those two payment methods. Great job class! So now we are going to move on to our final activity.

I found this activity difficult because I did not necessarily know what the students responses would be. However, I now see the importance of lesson plays for beginning teachers. Since we do not have the opportunity to deliver our activities, creating lesson plays would prepare us for teaching this in front of a class. Using lesson plays also provides the teacher with an opportunity to figure out how to address the areas of a lesson that will most likely be the most difficult. It also allows the teacher to view the learning from the students perspective and to imagine seeing the new mathematical content for the first time. 

Mathematics and Technology

This past class, we were introduced to many mathematical programs that involve the use of technology. The program that I found most interesting was the game that we played on Polygraph. The game was about transformations of quadratic equations and the format reminded me of “Guess Who” which made it extremely fun! We were asked to join a class key in order to play against other members in our class. This added a friendly competitiveness to the game that would not have been achieved if we were simply playing against a computer. I am excited to use these types of games and activities in my classroom because they are truly engaging and the students are presented with a differentiated way to learn. Rather than drilling students with hundreds of questions about the transformations of graphs, why not have them understand this concept through a computer game. I believe the friendly competitiveness of this game is engaging for the students and allows them to uncover the main ideas of the topic much easier than through textbook questions. 

As a future educator, I plan on incorporating technology within my classroom. However, I want to ensure that I am using technology that will truly enhance the students’ learning. I want to try to avoid games that are still drilling the same type of questions that a textbook may be asking. Using technology in this way would simply act as a filler and would not help the students learn about the topic. The question now, is deciding which games are appropriate in a classroom and will ultimately enhance the mathematics that is being taught. 

SAMR Model 

We learned about the SAMR model which is a way to differentiate which games will be the most beneficial in a classroom. The four categories that a game can be placed into are Substitution, Augmentation, Modification and Redefinition. The two types of activities that you should strive to use in your classroom are the games under the modified category, or even better, games under the redefinedcategory. By using this model, educators are able to distinguish the games that could simply be done with paper and pencil between the games that the technology has redefined for the students.   

The conversation about mathematics and technology reminded me of an excellent Ted Talk that I had watched in an undergraduate course.

In his Ted Talk, Conrad Wolfram points out that mathematics education looks a lot like simply making calculations. Students often do not find any connection between what they are being asked to solve and how it can relate to themselves. Wolfram explains that we are spending too much time on calculations when we can have a computer do the calculations for us. Since technology is currently so accessible, it does not make sense to drill computations into students’ heads. Wolfram explains that through the use of technology, students are able to focus on more real-world examples since computers are able to do the calculations for the students. They will also find mathematics more meaningful when they can relate what they are learning to their own lives. Consequently, I look forward to using technology within my classroom and exploring online tools that will ultimately enhance the mathematical content.

Gap Closing in Mathematics

What are diagnostic assessments?

Diagnostic assessments are a way of assessing what students know in a particular topic. This is usually done before new material is presented in order to observe what the students know from previous learning. Diagnostic assessments also provide the teacher with a general idea of which students may need additional support throughout the mathematics instruction. Therefore, these assessments can assist the teacher when determining how to differentiate the learning within a classroom. 

Gap Closing

Gap closing is a type of diagnostic assessment that can be used in a mathematics classroom. It is usually given to students who may need additional support in a certain topic. However, a teacher may decide to give it to her entire class if he/she believes that many students are not where they should be in mathematics. On edugains.ca there are diagnostic resources and a set of intervention materials provided for each topic.

This past week, we were asked to complete one of the gap closing diagnostic assessments in partners. The assessment had questions that were very different from each other and this allows the teacher to observe which type of questions the student may be struggling with. For example, a student may understand all of the visual components of a topic but struggle with the mathematical procedures and computations. Once the diagnostic assessments have been assessed, it provides the teacher with useful information about which parts of the topic the student does not understand. For each topic, edugains.ca provides a chart in the Facilitator’s Guide that outlines the specific part of the topic that students do not understand. It also suggests which intervention materials to use with the student which personalizes the instruction based on the student’s diagnostic assessment. 

Small, Marian. (2010). Gap Closing: Number Sense (Junior/Intermediate Facilitator's Guide).

I especially like how this chart is provided for teachers because it pinpoints exactly where the student needs support, rather than simply skimming all concepts of a mathematical topic. Having all students that need additional support complete the same work is problematic because every student learns in a different way and has different strengths. For example, when using the gap closing resource, a student who only struggles with one aspect of a topic will not have to complete each and every ‘Think Sheet’ related to the topic. Think Sheets are provided on the edugains.ca website and are accompanied by solution sheets, questions to ask before the instruction, how to use the think sheet themselves, and consolidating and reflection questions. These documents demonstrate how the teacher plays a large role in the student’s success. The student must be guided by a facilitator to understand the concepts that they currently do not understand. Therefore, it is evident that this instruction must be differentiated for each and every students. 

As a future educator, I worry about how I will be able to reach the students in my class that may need additional support. However, diagnostic assessments, such as the gap closing assessment, are a great way to determine the specific area that those students may need support. I look forward to using this resource in my future classes because the diagnostic assessment results allow for personalized instruction.  

Resources

Small, Marian. (2010). Gap Closing: Number Sense (Junior/Intermediate Facilitator's Guide). Retrieved from: edugains.ca

Reaching Every Student

As a future educator, I had always worried about how I would be able to reach every student in my class. We have learned that every student learns in different ways and that it is crucial to differentiate learning in a classroom. It would be incorrect to assume that all students learn in the way that we, as teachers, have learned best in the past. Therefore, it is crucial to provide students with opportunities to demonstrate their understanding in different ways. Open-ended questions and parallel tasks are two ways in which students can demonstrate their knowledge of the mathematical content. 

Open-ended Questions

Mathematics is usually known as a subject that has questions with right or wrong answers. However, through the use of open-ended questions, we are able to eliminate the idea of right or wrong answers because the students have some freedom within the question. You may be asking yourself how a student could have any freedom in a subject that is usually very black and white. This question can be answered with open-ended questions because the students are able to create the question, as well as, the answer themselves. Since there is not one specific answer, these types of questions are excellent ways to guide conversations about the mathematical content. 

The following picture is an open-ended question that can be used for students in secondary school. 

More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction, p. 40.

This example shows how every student is able to find an answer to this question since there are many possible answers. For example, there are an infinite amount of lines with a slope of 2/3, therefore, students have the freedom to choose whichever one they would like to work with. Rather than solving for a specific point, this question allows students to become familiar with the different properties of a parallel line. This would also be an excellent opportunity to engage the class in a discussion about parallel lines.

Parallel Tasks 

Parallel Tasks usually involve two problems that are very similar in nature in order to address different developmental levels within a classroom. Since the questions are only slightly different, the same big idea can still be discussed as a class. Similarly to open-ended questions, these problems involve some freedom for the students because it allows them to choose which option they would like to complete. Using parallel tasks in a classroom creates an inclusive environment because students are able to engage in the discussions about the topic. An example of a parallel task is demonstrated below.  

More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction, p. 147.

In this example, the students are able to choose whether to calculate the area of a right-angled triangle or a triangle without a right angle. Since the two questions differ slightly in difficulty, this question demonstrates what content the student is comfortable to work with. However, both questions will be able to demonstrate the students ability to solve for the side lengths and area of a triangle. 

Using both open-ended questions and parallel tasks in a classroom allow students to have some choice in mathematics. It creates an inclusive environment that provides students with the opportunity to engage in discussions about the mathematical content. Both open-ended questions and parallel tasks are a great way to differentiate learning in a classroom. 

If you would like to see more examples of open-ended questions and parallel tasks, I would suggest that you read “More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction” by Amy Lin and Marian Small. The open-ended questions and parallel tasks are organized by grade level, as well as, mathematical topic. I hope that you find this resource as useful as I do!

Resources

Lin, A. & Small, M. (2010). More good questions: Great ways to differentiate secondary mathematics instruction. New York and London: Teachers College Press.