Mathematics

Making Thinking Visible

For 2017/18, nine teachers from the math department will engage in a book study: Making Thinking Visible by Ron Ritchhart, Mark Church and Karin Morrison. The purpose is to help teachers promote engagement, understanding and independence for all learners.  Specific instructional practices that support learning will be determined as the book is read. Strategies will be learned, implemented, and then discussed and reflected upon. Student’s performance will be assessed to see if the strategies are improving performance.

Data will be gathered in Spring of 2018 to determine how teachers and students feel implementation of these strategies are impacting student learning.

MATH DEPARTMENT ACTION PLAN 2017 – 2018

Making Thinking Visible

This school year the math department engaged in a book study: Making Thinking Visible by Ron Ritchart, Mark Church and Karin Morrison led by Department Head and Instructional Services District Mathematics teacher Theresa Walker. Teachers explored and implemented four teaching strategies with the intent of increasing student engagement and understanding as they explore mathematics.  The four strategies were as follows:

  1. See, Think, Wonder
  2. Think, Puzzle, Explore
  3. Zoom In
  4. Chalk Talk

Applying the Strategies – Student Engagement

In each of these strategies students were encouraged to ask questions about and reflect on mathematics at various levels and applications. At first, students were hesitant to express their ideas. After some encouragement, teachers found students engaged in the process and were insightful in their thinking. Another benefit of these strategies is that they foster the Core Competencies of Communication, Critical & Creative Thinking, and Personal & Social Awareness.

Here is an example of questions asked in a Chalk Talk activity in a Pre-Calculus and Foundations of Math 10 – Graphing Unit.

Students Prompts:

  1. How can we represent data?  
  2. What information can graphs show?  
  3. Why do we use graphs?

Here are just a few of the insightful student responses:

  • What sort of information is not able to be displayed on a graph?
  • What graphs give clearer information than others?
  • We can use graphs for analyzing trends in athletic performance, scientific data, First Nations Populations, stock markets, motion data, statistics or advertising
  • Are there benefits in using graphs over representing data in other ways?
  • We can use graphs to compare/analyze/gather/visualize data/look for patterns/correlate data points in a simpler way
  • How common are graphs in the world?
  • Which information is better represented in graphs?
  • Why is this information often shown in graphs and not just with numbers?
  • Why do we need to visually represent data?

Here is one teacher’s reflection on a Zoom In activity applied to a Pre-Calculus Math 11 class.

To introduce a unit on Quadratic Functions, I took a picture of the McDonalds Arches and zoomed in to the middle of the arch.  I asked students what they think it was and what clues led them to that conclusion.  Many students thought it was a car at first and gave solid reasoning.  I polled the class and asked those who thought it wasn’t a car why they thought that.  The students gave reasonable justification for their thoughts.   I zoomed out till the point where you could see curves and multiple colours.  One student guessed the McDonalds arches.  I asked him to explain his reasoning and polled the class.  There were several students explained why this was incorrect. I expanded it one more time and the students all realized the student who suggested “McDonalds Arches” was correct. I then made the connection between the arches and Quadratics and Parabolas.

Applying the Strategies – Teacher Engagement

Throughout the year, teachers have had an opportunity to share the various activities they tried in their classrooms.  It has been a great source of professional development and collaboration as we consider what apect of the activities worked and what didn’t.  This additional focus on student engagement has been a powerful impetus for change as  teachers in the department have applied other strategies as well.

Here is one teacher’s response around Games.

Math 9 – Our focus was less “quizzing” and more engagement in math games that are relevant to the unit that we are working on.  For example we were working on slope-intercept.  We played “battleships” using slope. We also practiced on the I-Pads which enabled students to adjust their slope by recognizing what the mx or the y would cause that slope to do.  These are just two example of student engagement versus practice questioning.

As an introduction to the algebra unit, students were asked to come up with questions to they saw on the board. We started with a zoomed in screen of a number, was later revealed to be an exponent of a long algebraic equation. It was felt that this really sparked students’ curiosity, especially when they started to see different variables on the screen.

Students also received a picture of an exponential-looking figure, and asked them what they saw, thought, and wondered. Students used sticky notes to put on the different areas around the room (See, Think, Wonder). Their thoughts were, then discussed.

Engagement:  Students enjoyed taking the classroom outside and seemed more interested in solving questions that they had come up with.

Technology applications such as DESMOS were also popular as found in one grade 12 class:

Pre-Calculus 12 – TransformationsA strategy which was used repeatedly throughout the year was Desmos Classroom. The focus was on activities that introduced transformations of functions. It was felt that this was really helpful in solidifying their understanding of how to distinguish between different transformations. Students could pair up and work in groups with friends, it added a social element.  The teacher could see in real time how each student was doing and monitor their progress and could highlight the interesting solutions of one group of students to the whole class and they get to be more independent and not just listen to me lecturing. The marble slide activities also require very minimal background knowledge and can be accessed and completed by students at home.

 

 TEACHER RESPONSE

 See, Think, Wonder (Start with a picture)

This is a good teaching strategy to use an introduction to a unit or as a way to assess student understanding of concepts and ability to apply them.  Students are shown a picture or diagram and asked to write down everything they see – non-math information is included. Next, they are asked to think about their observations and come up with some conclusions. Finally students are encouraged to come up with questions that they wonder about.

Think, Puzzle, Explore (Start with a word)

This is a good method to start a unit.  In groups, students are given a topic or word and asked to write down everything they think they know about the topic. After sharing ideas between groups, each group comes up with any questions about the topic.  Students then come up with a question that they want to explore through the unit.

Pre-Calculus 12 – Think Puzzle Explore – Trigonometry

These are the explore questions that the groups of students came up with:

  • Why are triangles everywhere?
  • Do shapes really shape how everything is shaped?
  • What if a circle is just thousands of right angles?
  • What happens if we take the inverse of the side length of a triangle (does it still form a triangle)?
  • How many triangles can fit into another equilateral triangle?
  • When is standard position useful in any way shape or form?
  • What would life be like without trigonometry?
  • How different would trigonometry be if we used tau instead of π?
  • How do we apply trig in real life? When will we use it in real life? What fields of work use trig concepts?
  • Why are radians used instead of just degrees?
  • Where are trig identities derived from? How do we derive them?
  • How were the sin, cos, tan ratios invented?
  • What trig equations are most applicable to real life? Why? How?

Engagement:  This strategy showed where students were at at the beginning of the unit and increased student engagement throughout the unit.  Because students came up with their own questions, they were more interested in finding answers to those questions and finding out how specific concepts and skills related to their question.

A&W Math 10Think Puzzle Explore – Area. Students were asked to work in groups to Think, Puzzle and Explore concepts related to Area.  This task allowed the me to get an idea of what students remembered about the topic. They discussed what they knew, what they wondered about and then considered applications of the concept to real life.

Engagement: Student confidence improved as they generated ideas,  interacted in groups and came up with the following applications: Construction, Sports, Science, Design, Art.  They also came with some interesting questions to explore:  How will area help me in planning my living space?  How large does it need to be?  What will fit into the space?

Parking Lot Ratios. Students were shown a picture of a parking lot.  First they came up with observations around what they saw.  They then were asked to think about what math would apply to the photo.  Finally they generated math questions related to the situation.  After that students were sent out to the school parking lot to collect data and solve the questions we had wondered about.

Chalk Talk: In groups, students are given a chart paper with a starter question on it.  Students write their thoughts and/or questions without speaking.

Pre-Calculus and Foundations of Math 10 and 12 – Graphing Unit. This unit was used as a starter to access prior knowledge.  Students were given different questions about graphing on their chart paper and asked to write their ideas, opinions, and questions.  It worked very well to engage students in thinking more deeply and critically about the topic.  Some students who are normally very shy about sharing their ideas verbally were very involved in writing their ideas down.  Students walked around the room to see what other students’ had written.  Since doing this exercise, we have been able to refer back to some of the questions that the students came up with and I feel it has helped the students to connect better with the concepts.  They also seem to be more engaged in the lessons that have followed, where they are more apt to share their ideas about the problems we are solving.

The questions that were given to the students as prompts are:

  1. How can we represent data?
  2. What information can graphs show?
  3. Why do we use graphs?

Some thoughtful questions/ideas from the students that demonstrate their depth of thinking are:

  • What sort of information is not able to be displayed on a graph?
  • What graphs give clearer information than others?
  • What kind of graph would be best to show an increase or decrease in data?
  • Can a computer see the same data in the same way humans do?
  • We can use graphs for analyzing trends in athletic performance, scientific data, First Nations Populations, stock markets, motion data, statistics or advertising
  • Are there benefits in using graphs over representing data in other ways?
  • We can use graphs to compare/analyze/gather/visualize data/look for patterns/correlate data points in a simpler way
  • How common are graphs in the world?
  • Which information is better represented in graphs?
  • Why is this information often shown in graphs and not just with numbers?
  • Why do we need to visually represent data?
  • What is the most common way to represent data?
  • Is data just counting numbers?Is it “number” or “amount”?
  • Representing data is “a perspective view to create an illusion that gives an impression that one is larger than the rest”
  • How to know which graph to use?

 

Pre-Calculus Math 11 –  Zoom In. To introduce a unit on Quadratic Functions, a picture of the McDonalds Arches was shown and zoomed in to the middle of the arch.  Students were asked what they thought it was and what clues led them to that conclusion.  Many students thought it was a car at first and gave solid reasoning.  The class was polled and asked those who thought it wasn’t a car why they thought that.  The students gave reasonable justification for their thoughts. The photo was zoomed out until the point where you could see curves and multiple colours.  One student guessed the McDonalds arches.  The student was asked to explain his reasoning and the teacher polled the class.  There were several students who explained why this was incorrect. The photo was expanded it one more time and the students all realized the student who suggested “McDonalds Arches” was correct. The connection was then made between the arches and Quadratics and Parabolas.

 

Future Considerations

As the year comes to a close, the math department is currently collecting feedback from students  – How do they feel about their learning?

Possible goals for next year will be to explore alternative assessment strategies that enhance student learning and incorporate Core & Curricular Competencies and Content.

STUDENT FEEDBACK

Survey Questions (1 – Strongly Disagree to 5 – Strongly Agree)

  1. Discussions and questions around pictures with math increase my ability to inquire(e.g. picture of a building).
  2. Reflecting on my current understanding with others allows me to learn new math concepts more easily.
  3. Writing down my thoughts(diagrams, short notes, etc) as I’m solving math problems allows me to think more deeply on my current abilities.
  4. Discussing and practicing real-life applications for the concepts I’m learning increases my ability to problem solve.
  5. Making an action plan for how I can improve my math skills is useful.

 

 

 

 

 

 

 

 

 

Results 2017

Over the past year, the Math Department has been collaborating with the focus on Promoting Student Engagement in Mathematics. The following areas reflect where the majority of our work took place and give a sample of the activities that were explored.

Alternate forms of Assessment

Students demonstrated their understanding of mathematics by applying concepts to real life situations. Projects related to sports, environment and building were considered. These activities offer student choice of topic and flexibility in presentation. The depth of understanding and creativity demonstrated was rewarding. The criteria and marking rubrics were developed through collaborative efforts and used for personal and peer assessment.

Another evaluation of student understanding was through use of Student Interviews. Here is an example of how a teacher is changing how they look at their assessment of quizzes: “When I hand back the quizzes I give with them a little slip of paper that states the specific skills that were being tested, a mark and a check under one of three levels. The motivation behind this is two-fold for me and my students. For my students, the hope is that this makes it clearer as to what skills are required to solve certain types of questions as well as to give them an idea which specific skills they are already proficient in and which skills require more studying as they prepare for the test. For me, I have found that it helps inform me a bit better where each student’s strengths and weaknesses are and can see where they have made progress throughout each chapter/unit. By tracking progress in this way, I feel as though I can make a better judgment as to whether or not the earlier quiz marks should remain valid or not after doing the test/unit project. Furthermore, by me taking time to think about the skills required to solve specific questions I believe that I am able to build a better final assessment and am myself more aware of what we are working towards. When I have asked students about their thoughts on this they seemed genuinely positive and appreciative for this manner of reporting to them and when asked if they would like me to continue doing this at various points in the year the response has always been a resounding yes.”

Student Use of Technology 

Throughout the school year, staff have used a variety of graphing programs to enhance application of mathematics. Some of these programs include Desmos, Excel, and graphing calculators. This technology helps students apply mathematics to the world around them as well as developing transferable skills. Students have been encouraged to explore technology that can be accessed at home and not just in a school setting.

Physical Application & Use of Manipulatives

A variety of manipulatives have been used so students can explore new concepts and reinforce abstract concepts in a concrete way. Some of our lessons have been designed using stations where they are physically weighing, counting, flipping coins, etc. One class has introduced the ‘human number line’. Probability was explored in a number of classes through a carnival where students created their own games and shared their results with other classes. These physical activities are not only fun, but they help students experience mathematics in a tangible way.

 

R.E. Mountain Secondary

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