Reasoning & Proving Mathematics
The reasoning process supports a deeper understanding of mathematics by enabling students to make sense of the mathematics they are learning. The process involves exploring phenomena, developing ideas, making mathematical conjectures, and justifying results. Teachers draw on students’ natural ability to reason to help them learn to reason mathematically. Initially, students may rely on the viewpoints of others to justify a choice or an approach. Students should be encouraged to reason from the evidence they find in their explorations and investigations or from what they already know to be true, and to recognize the characteristics of an acceptable argument in the mathematics classroom. Teachers help students revisit conjectures that they have found to be true in one context to see if they are always true. For example, when teaching students in the junior grades about decimals, teachers may guide students to revisit the conjecture that multiplication always makes things bigger.
Ontario Curriculum Doc (Page 14)
The above paragraph is an excerpt from the Curriculum Document, a document most teacher, I included, read only to figure out the curriculum expectations for planning units or writing report cards. On pages 11-17 you will find the Mathematical Processes listed in detail. Have I read them in the past? Yes, I have skimmed through it. Do I integrate these in my planning for Math lessons or units? Yes, as much as I understand it through my experience and workshops. Have I spent time exploring the processes and thinking about them? No, this is the first time. At this point in time, I am going to focus on Reasoning and Proving.
Why is it important to develop reasoning and proving?
Learning Math can be very intimidating when students do not understand it. And when we do not understand it, how can anyone expect them to retain that learning, hence it is not permanent. In order to make learning permanent, learning has to be authentic, that is, related to real life. It needs to be logical so that the connections are made, conjecture is formed, reasoning is developed an argument is drawn. We, as teachers, need to plan for permanent learning that is transferable from one strand to another or one situation to another. It is integrated and is attained it by making connections, for which the students need to explore and investigate mathematical concepts and ideas.
Let us further expand our thinking and understanding by scaffolding the above excerpt of how it may look in a classroom.
Understanding any new concept requires collaborative exploration and investigation. For example, if you show students attribute blocks and ask them to sort them according to various characteristics. Different students will tell you various characteristics like colour, shape and/or size. Once they start exploring, they can sort them based on more than one characteristic, i.e, they will develop new ideas and make mathematical conjectures. In order achieve this, we, the teachers, should be flexible, ready for a constructive chaos in the classroom. We should be able to step back from direct instruction and facilitate the exploration and investigation by encouraging student dialogue and communication and asking direct, leading and open-ended questions, for example:
One idea leads to another. A developed idea encourages students to make connections. Making connections leads to challenging oneself to a more complex idea, which encourages students to make mathematical conjectures. For example, one attribute block can have more than one characteristic, hence it can belong to more than one group. Here is when students can use Venn Diagrams to explore and investigate. If they are successful, they develop an acceptable argument to support their conjecture, or continue to further explore their ideas and/or revisit conjectures.
How does it look like as children develop from K-6?
In my opinion, from K-2, the learners will work towards logical thinking in order to make sense of the mathematical idea. They explore mathematical situations and analyse them to draw a conclusion, further extend their thinking and communicate reasoning. Learners in grades 3-4 can go a step further and make and test conjectures and support their solutions by reasoning. In grade 5 and 6, the students can extend and connect their knowledge and experience of mathematical processes from various strands. They can refine their thinking and provide justifications and reach conclusions.
In order to conclude, I read what I wrote, and realized that it is impossible to isolate one mathematical process and talk about it. They are so intertwined. Let me think it out loud here. As a teacher, I feel that before I let my students plunge into exploration and investigation, I want them to have some knowledge and framework of Problem Solving Process. If I want them to be successful at the mathematical investigations, I need to prep them for it at the grade level they are. (Check out the Problem-Solving Model - Ontario Math Curriculum Document - page 15)
Reflecting is an integral part of successful investigation of a conjecture that one wants to argue about prove. The students need to look back, reflect and move forward with their exploration, have sufficient experience and knowledge about Selecting Tools and Computational Strategies that will move them towards being successful explorers. The students should independently be able to select and use the appropriately the calculators, computers, communication technology, manipulatives and computational strategies needed to solve the problems.
Continuing with the processes, I strongly feel that no learning is permanent without the authentic Connections that the students make during learning. Connections are authentic if they are drawn from one’s own experiences, and experiences they need the freedom to explore, investigate, make mistakes and learn from them. Their connections help them in Representing their ideas with concrete materials. This is when the ideas take a concrete shape which can be shared with other. For this, the students use pictures, numbers, words, manipulatives and technology. No learning is complete without Communicating. It is critical for collaborative exploration and investigation. The students need to communicate it by speaking, writing or drawing, which in other words is state in text as “using pictures, numbers and/or words”. For example, they can communicate their sorting by graphing it by simply colouring the bars. This helps the students to develop their ideas, reach a consensus and draw conclusions.
Resources I used:
Edugains - Mathematical Processes - explore it for many posters, bookmarks, tent cards that you can print or order.
Edugains DI Math Cards - This, I find is the most comprehensive resource for all Mathematical Processes.
Generic Processes Rubric from EduGains
The links below are the most informative course resource for the process of Reasoning and Proving. It includes classroom observation videos, guiding questions, reflective question. Just pick the grade level that you need. Make sure to go through step by step as it scaffolds how this process looks like and feels like in a classroom.
Annenberg Learner K-2
Annenberg Learner 3 - 5
Annenberg Learner 6-8
Ontario Curriculum Doc (Page 14)
The above paragraph is an excerpt from the Curriculum Document, a document most teacher, I included, read only to figure out the curriculum expectations for planning units or writing report cards. On pages 11-17 you will find the Mathematical Processes listed in detail. Have I read them in the past? Yes, I have skimmed through it. Do I integrate these in my planning for Math lessons or units? Yes, as much as I understand it through my experience and workshops. Have I spent time exploring the processes and thinking about them? No, this is the first time. At this point in time, I am going to focus on Reasoning and Proving.
Why is it important to develop reasoning and proving?
Learning Math can be very intimidating when students do not understand it. And when we do not understand it, how can anyone expect them to retain that learning, hence it is not permanent. In order to make learning permanent, learning has to be authentic, that is, related to real life. It needs to be logical so that the connections are made, conjecture is formed, reasoning is developed an argument is drawn. We, as teachers, need to plan for permanent learning that is transferable from one strand to another or one situation to another. It is integrated and is attained it by making connections, for which the students need to explore and investigate mathematical concepts and ideas.
Let us further expand our thinking and understanding by scaffolding the above excerpt of how it may look in a classroom.
Understanding any new concept requires collaborative exploration and investigation. For example, if you show students attribute blocks and ask them to sort them according to various characteristics. Different students will tell you various characteristics like colour, shape and/or size. Once they start exploring, they can sort them based on more than one characteristic, i.e, they will develop new ideas and make mathematical conjectures. In order achieve this, we, the teachers, should be flexible, ready for a constructive chaos in the classroom. We should be able to step back from direct instruction and facilitate the exploration and investigation by encouraging student dialogue and communication and asking direct, leading and open-ended questions, for example:
One idea leads to another. A developed idea encourages students to make connections. Making connections leads to challenging oneself to a more complex idea, which encourages students to make mathematical conjectures. For example, one attribute block can have more than one characteristic, hence it can belong to more than one group. Here is when students can use Venn Diagrams to explore and investigate. If they are successful, they develop an acceptable argument to support their conjecture, or continue to further explore their ideas and/or revisit conjectures.
How does it look like as children develop from K-6?
In my opinion, from K-2, the learners will work towards logical thinking in order to make sense of the mathematical idea. They explore mathematical situations and analyse them to draw a conclusion, further extend their thinking and communicate reasoning. Learners in grades 3-4 can go a step further and make and test conjectures and support their solutions by reasoning. In grade 5 and 6, the students can extend and connect their knowledge and experience of mathematical processes from various strands. They can refine their thinking and provide justifications and reach conclusions.
In order to conclude, I read what I wrote, and realized that it is impossible to isolate one mathematical process and talk about it. They are so intertwined. Let me think it out loud here. As a teacher, I feel that before I let my students plunge into exploration and investigation, I want them to have some knowledge and framework of Problem Solving Process. If I want them to be successful at the mathematical investigations, I need to prep them for it at the grade level they are. (Check out the Problem-Solving Model - Ontario Math Curriculum Document - page 15)
Reflecting is an integral part of successful investigation of a conjecture that one wants to argue about prove. The students need to look back, reflect and move forward with their exploration, have sufficient experience and knowledge about Selecting Tools and Computational Strategies that will move them towards being successful explorers. The students should independently be able to select and use the appropriately the calculators, computers, communication technology, manipulatives and computational strategies needed to solve the problems.
Continuing with the processes, I strongly feel that no learning is permanent without the authentic Connections that the students make during learning. Connections are authentic if they are drawn from one’s own experiences, and experiences they need the freedom to explore, investigate, make mistakes and learn from them. Their connections help them in Representing their ideas with concrete materials. This is when the ideas take a concrete shape which can be shared with other. For this, the students use pictures, numbers, words, manipulatives and technology. No learning is complete without Communicating. It is critical for collaborative exploration and investigation. The students need to communicate it by speaking, writing or drawing, which in other words is state in text as “using pictures, numbers and/or words”. For example, they can communicate their sorting by graphing it by simply colouring the bars. This helps the students to develop their ideas, reach a consensus and draw conclusions.
Resources I used:
Edugains - Mathematical Processes - explore it for many posters, bookmarks, tent cards that you can print or order.
Edugains DI Math Cards - This, I find is the most comprehensive resource for all Mathematical Processes.
Generic Processes Rubric from EduGains
The links below are the most informative course resource for the process of Reasoning and Proving. It includes classroom observation videos, guiding questions, reflective question. Just pick the grade level that you need. Make sure to go through step by step as it scaffolds how this process looks like and feels like in a classroom.
Annenberg Learner K-2
Annenberg Learner 3 - 5
Annenberg Learner 6-8
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