## Curriculum overview

**Documenting **

Mathematical Thinking

Mathematical Thinking

**Key Words:**cognitively guided instruction, math reasoning, visual representation, concept mapping, modeling, metacognition

**Curricular Context Summary:**

The curriculum aims to engage 3rd grade students in visually representing their thinking through problem-solving, concept mapping, reflection, metacognition, and discussion. They will explore a variety of documentation and visual tools to help show their thinking in the subject of math. This process aligns with the Common Core Standards (CCSS), which recognizes the need for children to make sense of problems and use appropriate tools to solve them. There is a greater respect for the inquiry process and letting students learn in a more heuristic, meaningful way. This curriculum will help students routinely practice showing their mathematical work in pictures and in words. The focus will be a cognitively guided instruction (CGI) approach in mathematics, applying their math skills to problems, and documenting their thinking as evidence that they are making sense of the problems.

**RESOURCES**

Number talks & Warm Ups

- http://wodb.ca/numbers.html
- http://www.fractiontalks.com/2016/11/mtbos-connected-fraction-talks-and-wodb.html
- http://www.setgame.com/set/puzzle
- Filing Cabinet of Warm Up Activities

Problems (CGI: cognitively guided instruction + standards-based tasks)

- South Dakota Counts Website
- http://gregtangmath.com/wordproblems
- https://www.engageny.org/resource/grade-3-mathematics-module-6
- http://www.dusd.net/cgi/
- https://docs.google.com/document/d/1WYsFuzRNPtmFZC6wyLhR5TX8m9opK4qWKdUb7NLRs6M/edit
- https://sites.google.com/a/orange.k12.nc.us/prek-5-mathematics/curriculum-mapping-meeting-resources
- http://robertkaplinky.com/lessons/
- https://bstockus.wordpress.com/numberless-word-problems/
- https://www.youcubed.org/tasks/
- https://hcpss.instructure.com/courses/97/pages/three-act-tasks
- https://www.georgiastandards.org/Georgia-Standards/Documents/GSE-Effective-Instructional-Practices-Guide.pdf

- Mindset & GREAT Resources if you scroll: http://www.nctm.org/Publications/Mathematics-Teaching-in-Middle-School/Blog/Promoting-Growth-Mindset-with-3-Act-Math/
- https://whenmathhappens.com/2014/03/04/cookies/
- Dan Myer (Desmos): https://docs.google.com/spreadsheets/d/1jXSt_CoDzyDFeJimZxnhgwOVsWkTQEsfqouLWNNC6Z4/pub?output=html
- https://gfletchy.files.wordpress.com/2013/09/3-act-recording-sheet1.pdf

Reflections & Mindsets

- http://cutaylor.weebly.com/uploads/3/7/5/8/37583977/math_homework_reflection_questions.pdf
- Boaler: https://tusd.haikulearning.com/kandre/buildmathematicalmindsets/cms_page/view/30057526

- Template for math reflection questions from Jo Boaler's Mathematical Mindset book
- https://www.youtube.com/watch?v=JUnhyyw8_kY Ewan MacIntosh - problem solvers
- youcubed: setting up positive norms & mindset (pdf) - Boaler

Common Core Standards

- http://achieve.lausd.net/cms/lib08/CA01000043/Centricity/domain/219/ccss%20docs/CCSS%20for%20ELA.pdf
- http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf
- http://www.corestandards.org/Math/Practice/
- http://www.insidemathematics.org/

- Screen Castify
- Seesaw App
- https://drive.google.com/file/d/0B72ZinW1ZU1deEdUTWhrYTZybUk/view
- https://goformative.com/
- http://digitalliteracy.us/math/
- https://awwapp.com/
- Padlet App
- http://braindoodles.net/files/resources/bd-lesson02-exercises.pdf

**CA Common Cores State Standards/State Standards Addressed:**

**CCSS Math Grade 3**

In grade 3, instructional time should focus on four critical areas:

- Developing understanding of multiplication and division and strategies for multiplication and division within 100
- Developing understanding of fractions, especially unit fractions (fractions with numerator 1)
- Developing understanding of the structure of rectangular arrays and of area
- Describing and analyzing two-dimensional shapes.

**Standards for Mathematical Practice 1 » Make sense of problems and persevere in solving them.**

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

**Standards for Mathematical Practice 3 » Construct viable arguments and critique the reasoning of others.**

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. Mathematically proficient students are able to analyze situations by breaking them into cases, and can recognize and use counter examples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that consider the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

**Standards for Mathematical Practice 5 » Use appropriate tools strategically.**

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

**Big Idea/Concept:**

*Documenting mathematical thinking visually builds students’ understanding and mastery of the content.*

**Enduring Understanding(s):**

- Everyone can learn math and show their thinking.
- Writing/documenting helps you understand mathematical concepts more deeply.
- One can make conjectures and find evidence and proof by referencing one’s work.
- There is more than one way to arrive at an answer.
- It is okay to make mistakes.

**What essential questions will be considered?**

- How can I create and model work that shows my understanding?
- Which strategies do I use to make sense of a problem and solve it effectively?
- Can I find patterns and relationships in math?
- How does reflecting and using metacognitive skills help me strengthen my understanding of the material?

**Appropriate Technologies and Tools:**

- Paper
- Math Journals
- A variety of pencils, pens, and markers
- Rulers
- Counters
- Blocks
- SMARTboard
- Chromebooks
- Apps: SeeSaw, GoogleClassroom, Goog

**What key knowledge, skills, and dispositions will students acquire as a result of this curriculum?**

Students will be given opportunities to creatively represent their work and will learn a variety of tools to help them document their thought process. Students will become more adept at representing their thinking, and refer to their written/drawn work to help them explain or defend their thinking.

KNOWLEDGE = Content

- Mathematical thinking and reasoning

SKILLS = Power Verbs

- Use what they know to make sense of math problems
- Generate solutions to a problem through multiple strategies.
- Create concept maps
- Organize mathematical thinking through sketching
- Document the steps of their problem-solving.

DISPOSITIONS = Attitude

- It is okay to make mistakes.
- I can attempt any problem using what I already know.

**Assessment**

Formative:

- Evaluating daily work and observing the strategies used and understanding of material
- Checking for accuracy and precision to detail in daily work/exit tickets
- Written math reflections showing evidence of metacognition
- Feedback & goal setting sessions with teacher and student

Summative:

- Interim assessments
- Post-tests

**Objectives from 6 Facets + Misconceptions:**

EXPLAIN:

- Students will be able to document and communicate their thinking in a clear, concise way- visually and verbally.
- Student work will be self-explanatory.

APPLY:

- They will be able to demonstrate their understanding by sketching, drawing, and visually represent their work.

INTERPRET:

- Students will be able to analyze and interpret math work and be able to compare their strategies to the work of others.

EMPATHIZE:

- Students will relate and empathize with their peers when discussing and comparing their approaches and solutions to math problems.
- They will appreciate the effort it takes to persevere, document, and revise one’s thinking in math.

GAIN PERSPECTIVE:

- Students will become more flexible in their thinking as they notice the variety of ways to figure out a problem.
- They will see that everyone makes errors and learning from mistakes helps build understandings.

GAIN SELF-KNOWLEDGE:

- Students will reflect, and learn to recognize their own mastery of the content.
- Students will see the value of effort in trying a variety of strategies and expressing their thinking visually.

OVERCOME THE NAÏVE VIEW OF: (Misconceptions)

- There is only one way to solve a problem.
- The algorithm is the goal (rather than relationships and flexibility in problem-solving).
- Not everyone can be good at math.