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.

https://teacher.desmos.com/polygraph

            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.

http://www.clipartkid.com/bridging-the-gap-cliparts/

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.

Relational Understanding vs. Instrumental Understanding

Saying that you understand something does not have just one meaning.  There are many different ways someone can understand.  In mathematics, there are two main types of understanding: relational and instrumental.  Relational understanding is “knowing both what to do and why”; whereas instrumental understanding is “described as ‘rules without reason’” (Skemp, 2006, p. 89).  In other words, instrumental understanding is simply memorizing the steps that get you to a final answer; whereas relational understanding includes knowing why you do each step.  

( http://www.clipartkid.com/
i-don-t-understand-cliparts/ )

Through reflecting on myself as a mathematics learner, I realize that in my education career I have been more focused on the instrumental understanding rather than the relational understanding.  I realize that I was not concerned about knowing why we do something or what it means; I was simply good at memorizing the steps to get to the correct answer.  Through learning this way I was able to more effectively and reliably reach the correct answer; however, this type of learning would not contribute to my overall understanding of the material.  This would often lead to me memorizing the steps, regurgitating the information onto a test or assignment, and then almost immediately emptying my brain of all the information.

For example, in second year university calculus we were taught how to find line integrals.  To this day, I could not tell you what a line integral is, and I would definitely not be able to solve one without looking over my notes from that calculus course.  I simply memorized the steps, regurgitated the information, and wiped it from my memory once it was no longer needed.  I strongly believe that if my course was tailored to a more relational understanding rather than instrumental, I would have a better understanding of what a line integral is.  In addition, I believe I would still have a general understanding of how to find a line integral, even without knowing the specific steps.  Skemp (2006) mentions that “instrumental mathematics is usually easier to understand”, as well as quicker to learn; whereas relational mathematics is more difficult to learn but “easier to remember” (p. 92).  A goal in mathematics is for students to truly understand and remember the material to use in their future, rather than forgetting the material the second they put their pencil down after writing their final test.

Through reflecting on myself as a mathematics learner and my past experiences in a mathematics classroom, I am able to better reflect on myself as a future mathematics educator.  Although I was able to achieve high marks through an instrumental understanding, I think it is important that I incorporate relational understanding in my future classroom.  I want students in my classroom to understand why we do certain things in mathematics rather than just following the rules.  Skemp (2006) mentions that there are arguments for and against both types of mathematical understanding (relational and instrumental).  I personally believe that it is important to incorporate relational understanding in a mathematics classroom in order to maximize learning and understanding.  Students should use steps to assist them in their mathematical process but make sure they are not simply memorizing the steps, again emphasizing the why. The following video further describes Skemp's article (2006) and describes the different combination of teachers and learners.

 

This idea of relational understanding can also support the concept of a spiral curriculum.  Contrary to a staircase model of education where students never return to a topic learned on a ‘lower step’, in education we want to emphasize a spiral curriculum in which you are constantly returning to the same concepts but you are building on your previous learning (Debra McLauchlan, personal communication, 23 September 2016). Relational understanding will allow students to more effectively build on their prior knowledge and participate in this spiral curriculum.

( https://www.quora.com/What-is-the-meaning-of-spiraling-curriculum )

Prior to reading Skemp’s article (2006), I would not have thought about these two different types of understanding, nor would I have recognized the importance of relational understanding.  I believe that this knowledge of the different types of mathematics will assist me in better teaching the students of my future classroom, and thus has helped me grow as a professional educator.

Reference

Skemp, R. R. (2006). Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 2, 88-95

Skyscraper Activity: Mathematical Processes and Constructing Your Own Learning

This week in class we participated in an activity called Skyscraper. For this activity students are given a grid with numbers along the outside (see example below) and a handful of unit snap cubes.  The purpose of the game is to fill the grid with stacks of cubes, each of which represents a building.  The buildings must be placed in a way that each number on the outside of the grid represents the number of buildings (or stacks of cubes) someone standing there would be able to see.  Similar to the game Sudoku, each column and row must have one each of each building height.
  

Example Grid

Solution to Example Grid

When we first were given the activity, the rules of the game were not explicitly given to us.  We almost immediately began the activity, with our facilitator Amy only quickly giving us a brief explanation.  These limited instructions and examples resulted in almost the entire class misinterpreting the rules of the game.  Instead of thinking that the outside numbers represented the number of buildings (or stacks) we can see, we were under the impression that the numbers represented the number of floors (or blocks) we could see.  This led to confusion and frustration when trying to solve the problem.  Amy walked around the room, giving hints to certain groups (for example, she told us that it is similar to Sudoku). However this initial miscommunication of the rules of the game caused a lot of problems when trying to solve the problem.

Misinterpretation (how many floors you see):
Front = 3; Back = 3

Correct (how many buildings you see):
Front = 1; Back = 2

Misinterpretation (how many floors you see):
Front = 3; Back = 3

Correct (how many buildings you see):
Front = 3; Back = 1

 

Misinterpretation (how many floors you see): 

Front = 3; Back = 3

Correct (how many buildings you see):
Front = 2; Back = 2

Initially for this activity, I was going to discuss the importance of being clear with the instructions you give to your students so they are fully aware of what is expected of them.  However, after reflecting on the activity in its entirety, I believe that the purpose of this activity was to experience the many different mathematical processes, as well as realizing the importance of letting students discover and construct their own learning.  If we were given the full and correct rules of the game, less mathematical processing would have occurred.

 

(image created by Lindsey Crawford)

Reasoning and Proving

Through misinterpreting the rules of the game, we were able to reason with each other and prove why either the rules given to us were incorrect, the grid numbers were impossible, or there was something else we were missing.  For example, I proved that with using the incorrect rules it would be impossible for numbers opposite to each other on the grid to be different.  This is because no matter which side of the buildings you are standing on (front or back) you will still see the same number of floors (or blocks).  This started to get us thinking that maybe we misunderstood the rules.

Communicating

Through discussing with my peers as well as the teacher, we were able to realize that the rules of the game were misinterpreted.  When Amy communicated to my group the hint about Sudoku, we were able to problem solve and realize the correct rules of the game.  Once we determined that the numbers represented the number of buildings we could see as opposed to the number of floors, we solved the problem together, communicating our ideas and reasoning.  When we determined where a stack of blocks would go, we would explain our reasoning to the other members in order to prove that it is correct, as well as contribute to our whole group's understanding.

Connecting

Connecting occurred in this activity through relating the rules of the game to a game that most of us would likely be very familiar with, i.e. Sudoku.  Making this connection assisted in our understanding that each row and column should have one of each sized building.  Connecting mathematics to previous experiences is very efficient for furthering one's understanding of a new problem and helping with one's approach to problem solving.

Reflecting

After completing a puzzle, we would look at the solution making sure it was correct.  We looked at all the numbers to ensure they accurately described how many buildings we could see.  We also made sure that each row and column had one of each building height.  This reflection of our solution would help us find any mistakes we may have made.  Reflecting is an important step in solving math problems.  When you reach an answer to a problem, it is important to reflect on it and make sure your answer makes sense before dubbing it your final answer.

 
 

Other mathematical processes occurred during this activity, but these four were the most prevalent in my opinion.  I realized many different things during the activity about being a math teacher.  One thing I learned is that as a math teacher, it is important to sometimes simply step back from the problem and let students problem solve on their own or with their peers.  Discovering mathematics is the best way to truly understand and appreciate it.  I also learned that the final result of a math problem is not nearly as important as how you got there.  The mathematical processes that occur during the problem solving and the discovering of mathematics are most important.