Lesson Play: The Importance of Anticipating Student Response

Many teachers do not recognize the importance of anticipating students’ reactions to instruction and questioning.  However, this is a very important part of planning a lesson.  Teachers will often think about the questions they would like to ask, as well as the ideal answer they want, but will neglect to think about the responses that students may actually have.  This is vital for lesson planning because teachers should be aware of what students may struggle with and how to bring students to the desired answer without just telling the students.  This week in my course on Teaching Mathematics at the Intermediate/Senior level, we were required to make a Lesson Play.  What this entails is making a script that would align with your lesson plan and trying to anticipate the types of responses students would have.  This can assist us in determining how we would respond to certain responses or difficulties students may have.  The following is the script co-created with Laura Gravina that aligns with our lesson plan on linear relations and points of intersection.

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Teacher: 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: 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 one too but I decided to stick with it because I wouldn’t have to sell any hats in order to get paid. No matter what I’d get paid $16.

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: Uh I don’t know.

Teacher:  That’s okay. Can anyone help Jessica out? Bryan

Bryan: If the slope was greater, then 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 make.  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 all discussed in your groups what the meaning of the points of intersection are.  What do these mean for 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.

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As evident in this script, Laura and I tried to anticipate the types of responses students would have, the specific mistakes students might make, as well as how we would respond and get students to the correct answer.  Through creating this script, I am even more convinced of the importance of anticipating student responses.  Since we know the desired answers to the questions, it can be very difficult to guess what mistakes students might make.  However, this part of lesson planning is vital for being prepared and organized in your classroom.  It is important to keep in mind what you want the final takeaway to be, while still making sure you lead students to the correct answer rather than tell them the answer.  This will likely contribute to students better understanding the content.

Diagnostic Assessment Through Game Playing

In my last blog entry, I discussed diagnostic assessment using the online gap closing resources. One thing that I realized about these resources is that they still follow the general layout of a test.  Due to students’ negative views on tests, students may be immediately blinded by the test format of the gap closing resources, and thus not perform to their full potential.  Although I still believe in the effectiveness of the gap closing resources, I still want to discuss other possibilities for diagnostic assessment.  I believe that games can be very effective for diagnostic assessment.  Games have the possibility of letting a teacher know where his/her students are in their learning without the students truly being aware.

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            This past week in my Teaching Mathematics at the Intermediate/Senior Level course, we played two games that I believe would work very well for diagnostic assessment.  The first game was called Hedbanz.  For this game, students had a piece of paper on their forehead that had a quadratic equation on it.  Students were unaware of the equation on their own head and were required to ask yes or no questions to their classmates to try to determine what their equation was.  I believe that this game could be very useful as a diagnostic tool.  Teachers can listen to the type of questions students ask to determine their level of learning in regards to quadratic equations.  For example, if a student asks “Is my equation in vertex form?” the teacher can guess that the student has a general knowledge of what an equation looks like in vertex form.  Teachers can make informed judgements on students’ learning based on the language that a student uses and the questions that he/she asks. 

            A benefit of Hedbanz is that I believe that it can be altered in order to be beneficial in many different topics.  For example, instead of quadratic equations, there could be trigonometric identities on their foreheads.  The important part of this game is for teachers to pay attention to the type of questions the students are asking.

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            The other game we explored in class was a game called Polygraph.  This game was on desmos.com and also focused on quadratic equations.  However, this game focused on the graphs of parabolas instead of the actual equations.  For this game, students get teamed up with a partner and play a game very similar to Hedbanz.  One student picks a graph and the other student asks yes or no questions to determine which graph their partner picked.  Like Hedbanz, the teacher can assess students’ understanding of the different aspects of a graph based on the different questions they ask.  What is more beneficial about Polygraph versus Hedbanz is that for Polygraph the teacher can watch any of the students’ games from his/her own computer.  The teacher is also able to see a history of the questions that a student asked.  This can be very beneficial for diagnostic assessment.

            What I enjoy about these two games is that students are representing their learning in a fun way, without even necessarily realizing it.  I believe that this way of diagnostic assessment is more beneficial.  Diagnostic assessment is very important in a mathematics classroom.  I plan on incorporating different games within my classroom to make math more fun and engaging for my students.

Closing the Gap in a Mathematics Classroom

There is a clear gap in mathematics education.  Over time, students who do well in math continuously do better and better throughout their educational career; whereas, students who do poorly in math continuously decline.  The difference between the students in the former group and the students in the latter group creates a mathematical gap.  This gap is already very evident in younger grades and continuously increases as students move through the educational system and on to their secondary education.  Many people—sadly, some teachers—believe that this gap is inevitable.  However, I believe that as teachers, one of our jobs is to reduce this gap as much as we can.  As teachers, we want all of our students to be able to succeed.  Our classrooms should be beneficial and be engaging for all students, giving them the opportunity to succeed, no matter their current level of learning.

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One way to accomplish this is through proper use of diagnostic assessment.  Diagnostic assessment is part of Assessment for Learning.  It occurs before instruction, with a purpose of determining where students are in their level of learning.  Using diagnostic assessment in your classroom is very effective when wanting to differentiate students’ learning in order to reduce—or even close—the gap in your math classroom.

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In my course on Teaching Mathematics at the Intermediate/Senior Level, we explored a resource website called gapclosing.ca.  This website includes many different resources that can assist teachers in closing the gap in their mathematics classroom.  There are nine different mathematic modules that focus on common areas of difficulty in a Grade 9 math class.  At the beginning of each module, students are required to complete a diagnostic assessment.  Questions on this diagnostic assessment are specifically made to identify key problem areas students are likely to have.  I believe that this is very efficient because not only does this diagnostic assessment help the teacher become aware of a student’s level of learning, it also provides an opportunity for differentiating instruction.  The teacher is given a chart that relates to the diagnostic assessment, telling the teacher which intervention materials a student should proceed with based on which questions in the diagnostic assessment the student had trouble with.  This is very beneficial because students are able to expand their knowledge on the specific topics that they are having difficulty in, rather than completing everything in the module.  This allows students who are falling behind to further their learning with some personalized interventions, whereas students who are doing very well in math do not get disengaged from their learning due to boredom; therefore, the gap in mathematics begins to close.

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One of my positive qualities of being a teacher is that I trust in any student’s ability to succeed in mathematics.  Everyone is capable of doing well in a mathematics classroom.  However, this success is largely dependent on the teacher’s instruction.  If a teacher does not believe in his/her students, or believes that the mathematics gap in education is inevitable, I believe students will be able to recognize this.  This situation would likely result in the teacher not only maintaining the large gap in mathematic learners, but also widening it.  Teachers’ belief in their students has a large impact on student success.  I plan on making students aware that I believe in them and that there is not a capped amount of success in my classroom.  I will definitely be using gapclosing.ca in my future classroom.  I hope to close the gap in mathematics as much as I can.

Differentiated Instruction: Open Questions and Parallel Tasks

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All students matter.  All students can succeed.  As teachers, it is important for us to facilitate our classrooms in a way that allow all students to succeed at their own level.  Differentiated instruction is vital for ensuring an inclusive classroom.  It also allows students to be engaged and successful in their learning by differentiating content, process, or product, according to the students’ readiness, interests, or learning profile (A. Lin, Personal Communication, Sept 28, 2016).  In my Teaching Mathematics I/S course, we learned three ways to differentiate the content in a mathematics classroom: open-routed questions, open-ended questions, and parallel tasks.  There are many other ways to incorporate differentiated learning in a mathematics classroom; these are simply some examples that I will use in my future classroom.

Open-Routed Questions

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            Open-routed questions have a single solution but multiple different ways to get to the final answer.  These questions are good for differentiating instruction because students are able to choose their own way of solving the problem.  Students can use the technique that resonates with them, as well as explore new and different ways of solving the problem.  This allows students to work at their own level and their own pace.  When using open-routed questions, the final result is not nearly as important as the process of getting to the answer.  I will be able to assess students’ understanding based on how they approached solving the problem.

Open-Ended Questions

            Open-ended questions are similar to open-routed questions; however, open-ended questions can have multiple final answers, as well as multiple ways of getting to the answer.  Similar to open-routed questions, open-ended questions allow students to succeed at their own level and be engaged by making choices based on their interests and skill level.  Open-ended questions recognize and support individuality by giving all students a chance to succeed.  This allows students to boost self-confidence in their own learning and also fosters independence.  This is true for all types of differentiated instruction.  Open-ended questions, in my opinion, are harder to mark than open-routed questions due to the infinite amount of possibilities; however, I believe that it is worth the extra effort to incorporate open-ended questions in a math classroom because it allows all students to succeed, as well as allows for better assessment of students’ abilities.

Parallel Tasks

            When using parallel tasks, teachers create multiple different tasks that focus on the same content but are at different levels of difficulty.  Usually two or three different tasks are produced, in which students can choose the task they wish to complete.  When thinking of this technique of incorporating differentiated instruction, the first thing I thought was that many students will always pick the easier task.  However, according to my facilitator Amy Lin, this does not actually happen as often as one would think.  More often than not, students choose the task that is more or less aligned with their level of knowledge or skill.  I am very interested in experimenting with this in my future classroom to see how students choose from parallel tasks. 

Question received from: Small & Lin, 2010, pg. 84,

Conclusion

            Differentiated instruction is very important to incorporate in a math classroom.  I believe that it is especially important in a mathematics classroom because many students have math anxiety or do not believe that they are capable of succeeding.  Some students believe that certain people have math brains and other people do not.  Differentiated instruction shifts their perspective into a growth mindset and lets all students feel capable of succeeding in a mathematics classroom.  I am looking forward to changing my future students’ perspectives of math, resulting in a more positive attitude in my classroom.

Reference 

Small, M., & Lin, A. (2010). More good questions: Great ways to differentiate secondary mathematics instruction. New York: Teachers College Press; Reston, VA: National Council of Teachers of Mathematics; [Scarborough, Ont.]: Nelson Education, c2010.