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Selecting tools and computational strategies    In focusing on how to explicitly  incorporate the principle of selecting tools and computational strategies in my mathematics classroom, I would keep in mind that the primary role of learning tools is to help students develop a deeper understanding of mathematics.  In one of the first lessons in grade 11 Functions, students are asked to build on prior knowledge.  They explore translations, reflections and stretches.  For example, they are asked to graph y = x2, then translate it by graphing y = x2 - 2, then  y =- x2, followed by y = - 4 x2 and then y = -4x2 +3.  (All of the underlined 2's are supposed to be superscripts for x squared.)  I thought, wouldn’t it be great if there was an animation of the curve, illustrating the parabola translating down two units, then being reflected in the x-axis, stretched by a factor of 4 then translated up 3 units?  Seeing the curve change in these ways would deepen understanding.  There is a free online graphing calculator, DESMOS that does just that!  (There is even a button for squaring the variable!)  There is also a tool known as the Geometer's Sketchpad that animates the changing curve in a video format as the value of c changes! )I would incorporate the selection of these tools in my classroom and collaborate with colleagues to learn about other available tools as well.  Lesley Cameron

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Selecting tools and computational strategies    In focusing on how to explicitly  incorporate the principle of selecting tools and computational strategies in my mathematics classroom, I would keep in mind that the primary role of learning tools is to help students develop a deeper understanding of mathematics.  In one of the first lessons in grade 11 Functions, students are asked to build on prior knowledge.  They explore translations, reflections and stretches.  For example, they are asked to graph y = x2, then translate it by graphing y = x2 - 2, then  y =- x2, followed by y = - 4 x2 and then y = -4x2 +3.  (All of the underlined 2's are supposed to be superscripts for x squared.)  I thought, wouldn’t it be great if there was an animation of the curve, illustrating the parabola translating down two units, then being reflected in the x-axis, stretched by a factor of 4 then translated up 3 units?  Seeing the curve change in these ways would deepen understanding.  There is a free online graphing calculator, DESMOS that does just that!  (There is even a button for squaring the variable!)  There is also a tool known as the Geometer's Sketchpad that animates the changing curve in a video format as the value of c changes! )I would incorporate the selection of these tools in my classroom and collaborate with colleagues to learn about other available tools as well.  Lesley Cameron

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The solutions to complex problems can be arrived if they are analysed systematically by the learners themselves. I would make the students understand that merely imitating teacher wouldn't help that much the students to master the competency. Therefore, I would ensure that the students should ask the questions themselves and analyse the whole situations, to maintain the spirit of discovery.
In this way they will be made equipped to solve any kind of problems in their lives.
 
 add

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Connecting - In Gr. 11 Functions course, the concept of transformation is a recurring theme. Students already had some foundation from Gr. 10 course on the transformation of parabola. In Gr. 11, students further explore transformation in new functions in the course. With a good foundation, students can progress well and apply their skills to transform the new functions. However, if the students are weak in this area, then further review and scaffolding is required before they can proceed and have success.

Representing - In Gr. 9, 10 and 11 courses, finding a solution between linear systems or linear-quadratic systems can be represented in different ways. Students can solve by graphing and finding the intercepts, or solve algebraically for the solution. Often, graphing put concrete or real life context to the problem and helps students understand the use of math in a more authentic manner.

Communicating - In my classroom experience, I found that students who can verbally explain their method and their solution to perform better in the topic. I've used peer-to-peer help by allowing students the opportunity to explain homework with each other. I also encouraged students to explain their method or solution to me when they experience a difficult questions. Often time, they realize their mistakes when math is verbalized. This also provided a great opportunity for assessment through observations and through conversations.

Ken

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Hello All!

Hope everyone is enjoying our class so far!
I have a couple of ideas on how I would like to incorporate the mathematical processes into my classroom.  My first theory / action I would enforce in my classroom is having a google document for personal reflection, I would like for each of my student's to submit a weekly personal reflection on what they have learned and how they feel their growth of the subject has increased and moved forward. At the end of each week I would share my favourite moment and put it on the board for all the students to reflect on. This task touches on the connecting and reflecting aspect of the mathematical process.
My final process that I would implement into would be a combination of problem solving and tool selecting. Throughout every form of assessment and evaluation I will have a variety of questions for student's to select that best fits their specific needs as well as I will have real life application questions where students can utilize their skills for the future. With that being said I would also like to have numerous different manipulative and concrete tools surrounding my classroom so that student's can access them whenever they are necessary.

Hopefully with implementing these different types of processes, my classroom can establish an inclusive learning environment.

Sabrina

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Communication
I like to give students the opportunity to work in pairs or groups if they like to work through math problems.  When students have the opportunity to have authentic conversations as they learn and develop skill I feel the gain knowledge from each other and gain proficiency by sharing with each other.

Communication
I also like the Frayer Model as Andrea described below.  I think it provides a "fresh look" for a mathematics graphic organizer. It is a great way to communicate understanding and for students to use when reviewing how and when to use a strategy.

The strategy I mentioned in the discussion fits into multiple areas of the processes - I love to present students with a challenging question but give them little to no instruction. Sometimes this question may not be something they can solve at this time. The purpose is to have them dissect the question - identify what they know, reason and prove what they can, select tools and strategies they might use to solve, communicate what they need to be taught or need to research to solve the problem. The process of this exercise in the point, getting to the answer if that happens is just a bonus. :) or a great extension for someone who wants to run with it.

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One idea that I think is a good idea for students to communicate and represent new ideas and concepts is through the Frayer Model. For those who haven't used the model it is a page cut into four boxes (much like a window) and then a circle in the middle with the name of the concept or idea inside it. The boxes ask for a definition of the term, the characteristics, examples, and non-examples. This shows understanding of the concept and can help students to get their ideas across in multiple ways in order to help support their thinking. If we can successfully fill all of the boxes we should be able to communicate the concept effectively and gain a strong understanding of the idea.
-Andrea Blewitt

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One of the ways I would incorporate these principles into my classroom, is through the design and implementation of my activities/assignments. First, I would want to create assignments that are connected to student’s lives. For example, in a grade 12 college prep courses students will have a unit on personal finance. Many students will be going off into the real world or to post secondary and the concept of budgeting will come into play. You could provide the students with a problem of trying to create a budget for their first year out of school, with the goal of going on a trip the next summer with friends. They would have to look at mortgages, or rent, tuition,car payments, food costs, entertainment costs, etc. You could also have issues pop up that happen in the real world, such as car problems, that the students have to think about and then show how they would adapt their budget. I would have students work in pairs, so they could not only discuss the personal finances they need to consider, but also to work through their thinking by talking it out with a pair. Half way through this type of task I would have students reflect not only what they found difficult or easy about the math, but the connections to the real world and the importance of budgeting. Students would need to problem solve and choose appropriate strategies for these types of assessments, they would have to reason as to why they took certain things out of their budget or why they need to leave something in. The students could represent their data in a couple different ways, but also decide which software or charts will best help them to complete the assignment. For this specific activity I would have students either right a report to represent and explain their findings, but if I were to do a case study or smaller scale, you could have students present their work to explain their procedure and thought process to their peers.

Another way to incorporate the strategy is by using the “think, pair, share” strategy. I believe this allows students to really focus on their own thinking first, allowing them to try to solve the problem, figure out what they need to solve the problem, decide where they are stuck and what questions they need to ask. Then working with a partner they may be able to find answers to these questions, enhance their understanding or find a new solution. Lastly, haring with the class allows students to hear other solutions, have their solutions validated, or see where they have gone wrong. This can easily be adapted too if it seems to juvenile.
-Jenny

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Problem Solving vs Mathematical Principles

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Problem Solving vs Mathematical Principles

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Problem solving is one of if not the main pillar of mathematics and it is the vessel through which most math teachers would like to teach their students. The problem arises when students need to learn the underlying mathematical principles of certain problems. A strategy I propose is a problem solving notebook separate from the mathematical principles notebook. As students encounter new problems and the corresponding new math principles they use to solve these problems they make a separate note of it. students have a separate notebook where they can record each math principle and give a brief explanation what the principle is used for. when students encounter questions they can use this notebook to evaluate what tool to use to solve the problem. This could act as a reflective journal, a method for selecting correct problem solving strategies and even help to make connections as students become aware of the utility of various math principles. I think this will help encourage the teacher and the students to allow for exploration and reasoning to occur on a new topic or problem, and then allow students to see exactly what tools and strategies they accumulated as they solve more and more problems

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Problem Solving, Reasoning and Proving, Communicating

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Problem Solving, Reasoning and Proving, Communicating

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I think teaching our students how to problem solve and communicate their thinking is one of the most important aspects of a mathematics class.

One strategy I use is a "Think Aloud". When I model solving a problem I not only share the steps I take but my thinking that goes along with these steps. I strive to have the students hear my thought process. I also use this concept when a student asks for help with a problem they are stuck on, I have them go through a "Think aloud" with me and then I will have more information as to why they are stuggling with a certain problem. The student gets practice communicating their thinking and I know what they are having trouble with.

I often with get my students to work in pairs or small groups to work together to solve problems but something I started trying was after each group has solved their problem they switch answers with another group. They then have to look at the other groups answer and either agree or disagree with it and give support either way. This led the students into reflecting on their answer as well as possibly looking at different ways to solve a problem.

When we are discussing a problem as a class I find that if I give the students 2 minutes to first turn and talk with a partner about a problem I will get more class participation. I find it gives then a safe environment to put an idea out in.

Jennifer Fortnum

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Reflecting through reasoning and proving

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Reflecting through reasoning and proving

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the seven mathematical processes overlap and support each other far more often then they are found as distinct isolated skills. One skill I like to use and I would like my students to use is estimation. I think it is important that students be able to estimate what an appropriate answer might be based on a particular question. this estimation requires the student to be able use mental computational strategies to deduce whether answers are valid or whether they need to find another way to solve the question. teaching students how estimate and predict what solutions might be or what solutions should be is great way to encourage reflection. this can be accomplished by giving the students questions where you ask only for an estimation based on their mental computational strategies, or following the completion of problem ask how they know they have achieved a reasonable answer.

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Reasoning & Proving + Communicating •	Provide class a realistic word problem.•	Ask clarifying questions to students as they complete a problem. •	Refer to math word wall for appropriate symbols, terms, or conventions to be used. •	Ask the student to present their strategy to the class having them focus on the “How” (steps taken) and “Why”(why they chose the strategy used or why the method works). •	Model and teach students sentence starters and questions for them to begin a dialogue. 

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Reasoning & Proving + Communicating •	Provide class a realistic word problem.•	Ask clarifying questions to students as they complete a problem. •	Refer to math word wall for appropriate symbols, terms, or conventions to be used. •	Ask the student to present their strategy to the class having them focus on the “How” (steps taken) and “Why”(why they chose the strategy used or why the method works). •	Model and teach students sentence starters and questions for them to begin a dialogue. 

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See this poster for examples of language to begin math dialogue.

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Reflecting ...I like to check the students’ understanding by giving them one or two questions to work on at the end of the class. And give the students 10 minutes to solve them. Then, I ask them to answer the questions in the class as a pair or group depending on the type of the questions. I would also encourage the students to come up with their questions that they struggle with put them in the bulletin board and have them to analyze them and solve them as a group or in a pair first then solve it as a class. This strategy allows discussion and check for understanding.                                                                                                Connection.... I like to connect previous lessons and other courses materials such as physics, chemistry… to what they are studying. I also try to be realistic with my questions and connect them to the real world to deepen students’ understanding in math and show them the relevancy of it to the real world.                                                           Shatha

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Reflecting ...I like to check the students’ understanding by giving them one or two questions to work on at the end of the class. And give the students 10 minutes to solve them. Then, I ask them to answer the questions in the class as a pair or group depending on the type of the questions. I would also encourage the students to come up with their questions that they struggle with put them in the bulletin board and have them to analyze them and solve them as a group or in a pair first then solve it as a class. This strategy allows discussion and check for understanding.                                                                                                Connection.... I like to connect previous lessons and other courses materials such as physics, chemistry… to what they are studying. I also try to be realistic with my questions and connect them to the real world to deepen students’ understanding in math and show them the relevancy of it to the real world.                                                           Shatha

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Pair/ Group Work

Students have different learning styles and need to have lessons that help improve all styles of learning to get the best results. Group work is a simple strategy that allows students to work and problem solve with a buddy. When a teacher has provided the basic instruction, it’s helpful to split the class into pairs or groups to work on problems.

Since the pairs are working as a team, the students can discuss the problems and work together to solve the issues. The goal of pair work is to teach students critical thinking skills that are necessary for future math problems and real life.

Make Connections

To help students make sense of concepts, I provide them with connections to the real world or previously taught lessons. I always begin a new lesson with a reminder of the last. For example, I might say, “Yesterday, we learnt about how to factorize polynomials. Today, we will see how to use those factors to sketch a graph for the polynomial”. I will start with a fresh problem and to check if the students remember the previous lesson I will divide students into groups and ask them to find the factors of the polynomial on different whiteboards.

Focus on Strategies

Math is all about problem-solving using strategies. Sometimes, there's only one way to solve a problem, but many times there are multiple avenues to the answer. When teaching, model several strategies for understanding and exploring a concept. For example, to find the average rate of change or instantaneous rate of change I show students all different ways to calculate or estimate it, re, using tangent slope, secant slope, calculating derivative at the point, mid interval or left/right interval method. I leave it up to the student to decide which method they want to go for in a particular problem depending upon the data provided.

Communicate the Reasoning

Students need to explain their reasoning when solving problems. In order for a teacher to determine if every student truly understands the objective for the class period, it's necessary for each student to communicate both orally and in writing. By giving the class ten minutes to discuss their reasoning with each other while exploring multiple ways of solving the problems, you'll promote excellent engagement and learning. When I give students freedom to select a method of their choice to reach a solution I also ask them to communicate the thought process that went behind selecting that particular method.

Tools and Technology

One method that I use in my classroom is to ask students to sketch the graphs for the polynomial, logarithmic and/or trigonometric functions and then check it using a tool of their choice re, Desmos Graphing Calculator, Geogebra or Graphing calculator. Then I ask them to discuss the discrepancies between the two sketches if it exist so that they can figure out their mistakes on their own.

Preeti Matharu

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Problem Solving, Selecting Tools and Computational Strategies

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Problem Solving, Selecting Tools and Computational Strategies

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The use of graphic organizers is a great asset to assist students with problem solving. Students can organize the information they are learning to an appropriate reference sheet. For visual learners I find having different coloured graphic organizers helps to draw their focus to the material on the paper. There are some great ideas that can be taken from here. There are many ways to create graphic organizers and I find that when they have flaps with the information behind it becomes a very useful study aid for students.
I also find enjoy introducing the GRASS method to students as a strategy for problem solving. This method allows students to identify the appropriate tools or concepts to apply to each particular question depending on the information they are provided. Being able to organize the information that is being given to you may help students to recognize what is important to the end result and what information is unnecessary. Again for visual learners I often use different colours or shapes to highlight or reference the information being given in the problem that can later be pulled out and used to help solve the problem.
-Sara

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In my future mathematics classes I thought of two fun techniques which encourage students to use mathematical processes.

HISTORY AND CONNECT
I would like to use inquiries, into the history of mathematics, to help students discover why certain formulas had originally been created. Students can then use a modern formula and write about what they did compared to the original historic uses of that formula/ theorem. These studies could include formulas when learning vectors in Cartesian planes in grade 12(Calculus and Vectors) or even Pythagorean Theorem in grade 9 math. The students will achieve a greater understanding and improve their ability to reason through future problems.

Check this out:
For a complete view of the origins and divisions within the mathematics field see Dominic Walliman's fun cartoon MAP OF MATHEMATICS YouTube video below
https://m.youtube.com/watch?v=OmJ-4B-mS-Y
http://www.storyofmathematics.com/17th_descartes.html

REPRESENT IT CHALLENGE
When learning a new mathematical concept, that can be represented visually, I would like to challenge the students to depict it with materials or technological tools of their choosing. For example, students  can use chrome books to research various strategies. Some options include the use of physical representations, such as graphs to show their understanding, or manipulatives, like algebra tiles, to represent a given function. Below is a link to an activity that uses "spaghetti" to illustrate various angles,  create visual connections, and promote reflection and reasoning during the trigonometry unit.
https://pumphreysmath.wordpress.com/2013/05/08/spaghetti-trigonometry/

The real purpose of mathematical processes is to depart from using memorization and input strategies as a sole means to solve problems.  Instead of "a movement back to the basics, we are moving forward with the foundations" (Bruce Rodrigues, C.E.O. EQAO). We are challenging students to learn an array of strategies, and to use metacognition to decide which strategy is the best.  And furthermore, to learn  if that method can be supported, connected, modeled and communicated.

Back to the Basics Discourse, Bruce Rodrigues
http://thelearningexchange.ca/projects/eqao-assessment-and-mathematics/?pcat=1080&sess=0

Eileen Kauk

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Problem Solving Strategies:- Flowcharts are very helpful to explain the steps for  problem solving. I like to use a flowchart to explain how to find the limits of a function. Graphs can also be used as a visual aid to understand the concept of limits of a function. 

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Problem Solving Strategies:- Flowcharts are very helpful to explain the steps for  problem solving. I like to use a flowchart to explain how to find the limits of a function. Graphs can also be used as a visual aid to understand the concept of limits of a function. 

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Reflecting:- Give cue cards to students and ask them to write down the points that they learnt from the lesson and post it on the bulletin board. Then teacher and students together discuss the points clarify anything that was unclear and also give them feedback. This also gives teacher an idea to change the strategy,if needed.

Shylaja

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Ideas on how to incorporate mathematical processes into a classroom: 

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Ideas on how to incorporate mathematical processes into a classroom: 

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Reflecting

Using a “bump-up wall” that the students work on throughout the unit; this allows students to:

Build exemplars

Add feedback

Bumping up the work on each level

Create success criteria

Selecting Tools & Computational Strategies

Within a finance unit and/or a data management unit

Teaching the use of google sheets to analyze, compute, represent data and communicate in various representations of the data

Easily created, shared and built upon from google sheets

You can use google forms as well to create surveys and collect the data to analyze

Communicating

Building a word wall throughout each unit and semester to refer to for proper use of numbers, symbols, pictures, graphs, diagrams and words

this can be present physically in the classroom and/or using technology such as Padlet to refer to over the semester

Kelly

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How to incorporate Problem Solving into Mathematics

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How to incorporate Problem Solving into Mathematics

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During university one of my favourite ways to study was to get into a group and discuss a problem on a board. I believe this is a great strategy to use as it is very helpful for problem solving. So what I would like to do in my classroom is create opportunities for students to get into "Math Squads" and have the opportunity to collaborate and discuss a problem. I believe this is beneficial as it allows students to share all their different methods on solving the problem and also teach each other how they view the problem too. It also teaches them to work with students they may not normally work with and this is a great skill to take into a workplace.

Daniel

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How to incorporate Reflections into Mathematics

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How to incorporate Reflections into Mathematics

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One idea that I recall doing when I was in high school but which can also be helpful in our classes for reflections is math journals. I believe that math journals are a great source to be used in our classrooms as they allow the students to reflect on the material we learn over the week and then write a journal about what they learned, what they struggled with and how they overcame those struggles.

Daniel

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Principles to incorporate in Math Classroom

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Principles to incorporate in Math Classroom

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Mathematics emerged as a science out of the society's needs that necessitated immediate solution. It always developed as a means of problem solving , leading to solutions. The problems widely varies in their nature, utility and extent of complexity. According to its convenience it can be divided into two types  (I) Practical Problems & (II) theorectical problems. The first type of problem is related to day to day life such as finding sums, differences, ratios areas etc... and the second type involves deducing new theorems and finding the proofs etc...

As I deal  with Problem Solving as the  central part of Mathematics Class room , I would always keep in mind the following.
1. Make the practical problems realistic
The learners would be able to appreciate the relevence of the problem if they are connected to the real life situations such as teaching the addition or subtraction when pupils go to a store and purchasing few items for their own family will make them find the total amount to be paid and subtracting in cases where they have to give more amount and have to collect the balance.
2. Adopts steps for increasing speed and accuracy
I would always keep in mind to accelerate the speed and accuracy as a competency through the class room instruction. The students should be able to master the tables, calculations, formulae and other principles throughly.
3. Mastering the technique of Analysis
The solutions to complex problems can be arrived if they are analysed systematically by the learners themselves. I would make the students understand that merely imitating teacher wouldn't help that much the students to master the competency. Therefore, I would ensure that the students should ask the questions themselves and analyse the whole situations, to maintain the spirit of discovery.
In this way they will be made equipped to solve any kind of problems in their lives.