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How Do We Build Metacognitive Thinking Through Self-Assessment

We can all agree that metacognitive thinking is obviously beneficial. The environment in which a student feels comfortable enough to be a self assessor can be tricky. What does that look like? What is the goal?

Getting away from the time-honored question, “Is this the right answer?!”

As former students, this is probably the question that motivated most adults as young learners. The response of, “Yes, you are correct!” or “No, you need to work harder.” always seemed to hold a finality of the lesson.

That is where we all decided that we were “great at math” or that we would be lifelong math strugglers. We all have those memories.

So how do we as teachers change that narrative? How do we facilitate students in becoming their own teachers? How do we give them a voice and the confidence to speak to their own capabilities and shortcomings and how is that beneficial?

Learning Goals and Scales

Reflecting back to John Hattie’s book, “Visual Learning”, there are questions that guide a student through the self-assessing process.

Relating to the content each student should ask themselves:

  • Where am I going?
  • How am I going to get there
  • Where to next?

Keeping those guiding questions in mind each student is given a Learning Goals and Scales chart for each chapter studied.

The goals and scales are based on a combination of Common Core Standards and State of Florida Item Descriptors.

Seen within the items are leveled abilities. This is important to guide the learners in their self-evaluation. It is also very helpful for the students to keep the chart throughout the chapter.

This is not a summative assessment or checklist to use at the end of the chapter. The students should have access to help guide their thinking and goals. The goals and scales are written in “kid-friendly” language as “I can” statements.

This is done to make ownership of the goals easily relatable to elementary-age students. In reading each statement the student is then able to designate the color that best fits their comfort level for that item.

As it relates to their feelings about a topic:

  • Red indicates that the student feels that they are not confident in this skill. They feel that they require more work in this area.
  • Yellow indicates that the student feels somewhat comfortable with the statement. They can be effective when leading in whole group scenarios, or with partner support, but may still have questions.
  • Green indicates that the student feels very comfortable with this skill. Not only could they complete the tasks at hand, they could also explain them to a classmate.

Support through Discussion

Learning Goals and Scales and Think!Mathematics US

The students’ responses to their Scales and Goals can be seen below. Those are then used to guide the students through self-assessment by applying the goals to different examples that have been either created in their Math Journal or through concrete examples that the students can demonstrate and explain.

Examples below show the students reflecting on their goals for their first chapter in their mathematics classroom this year.

Students Reflecting on Their Goals and Think! Mathematics US

The students were allowed to use their goals and scales to keep track of their thoughts and their own accountability and then relate those directly to examples.

These discussions are truly the cornerstone of metacognition for the students. Being told that their thinking is more valued than a simple score is the first step in supporting a student in seeing themselves as not just as a spectator in their education but as their own teacher.

Initial Teacher Takeaways

Having a specific routine for this method of accountability and reflection is crucial. There must be a procedure put in place that helps the student guide themselves independently. If these steps are not taken students become distracted in the implementation of the task and the quality of the actual reflection is lost. 

Providing the students with the Scales and Goals (I can statements) is the first step, but the teacher contribution in creating those goals must be instructionally sound and purposeful.

Something as simple as incorporating the math topic in the journal headings allows students to organize their thoughts for their accountability discussions with their teacher.

For example, if the first three “I Can” statements relate to place value and the student feels that they can prove that they are comfortable with that topic, looking back in their journal headings that address Place Value will give them a place to reflect and prove their thinking.

It should also be said that in some of the discussions and pictures seen below you can see that through the interactions the students actually change their ability levels from their initial thinking. That might be the most profound takeaway with these strategies early in the school year.

Accountability and Reflection and Think!Mathematics US

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Assessment: Shelf Work in Kindergarten

Melissa Williams


In these initial phases, we are teaching kids how to reflect. To look over their week and explain certain images where they learned, wondered, or were confused about. Many have never been asked to “reflect”, especially in math. As we begin this year, we are trying to introduce this and make it a habit. The hope is that the mathematical language becomes stronger and the thinking becomes deeper and meaningful to understanding concepts better. 

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Assessments: Setting Routines for Self Reflection

Setting Routines for Self Reflection

MJ Kinard

Assessments and Mathodology

Precision teaching is the idea that relates to the book, Visible Learning by John Hattie. The premise behind Hattie’s research is the correlation between surface, deep, and transfer learning.

The findings go on to address the influence that different aspects of the education system have on our students and the math classroom. Basically, the idea of getting the “most bang for your buck”.

Every school district, school, and classroom teacher wants a magic formula for success. With limited time and funding, how do we ensure our academic goals are being met in the math classroom?

As math teachers, how do we know if our students are learning?

Setting Routines

Setting routines are one of the cornerstones of a successful classroom. It is important for teachers to understand the difference in a compliant classroom and an environment conducive to learning.

When beginning the school year it is crucial to ask ourselves a few questions when considering routines and procedures.

  • Do your students know what they are supposed to be learning? And more importantly why?
  • Are your students benefitting from your expertise?
  • Are your students capable of connecting their learning to what they’ve already learned and are they able to see where this learning goal will lead?
  • Are your students able to manage their own learning?
  • At the end of the lesson, are your students able to hold themselves accountable for what they have learned? (leading into self-assessment)
Pictures of students during our “admiration tour”. You can see them and the work they have produced.

Self-Assessment

Self-assessment has one of the largest effect sizes for student learning (Hattie, 2009). The process of self-assessment requires several different elements implemented in the classroom.

One example of self-assessments in the math classroom is the use of Math Journals. Beyond a journal prompt containing the “content goal” teachers have an amazing opportunity to go beyond a right/wrong answer by providing students with a voice for self-reflection.

Below is an example of a pre, mid, and post year self-assessment used in a fourth-grade classroom to gauge confidence and attitude toward mathematics.

This same questioning is also used in this classroom in the aforementioned Math Journal activity. The example seen below addressed the math topic of Place Value. The students were asked to describe the value of the 8 in the number 88,888.

They could use pictures and words. The prompt was actually shared at the beginning of class as an informal discussion. Students were asked to read the prompt and think about what they already knew about place value, what they remember from previous lessons that would help them with today’s goal, if they felt like the topic was going to be difficult and if so, why?

They were given time to think about their responses and write them in their journal, then after the lesson, they returned to their journal to work on the prompt and reflect on their thoughts from the beginning of class. Once journals were complete, they were allowed to share the journals with their group.

The outcome of this part of the activity is to foster a safe space for math discourse, more importantly, mathematical talk is proven to lead to the ultimate level of transfer of learning and metacognitive awareness.

Self Evaluation/Assessment Form

Metacognition

If self-assessment is the goal and math journaling is one tool, what is the payoff? How do these techniques help spark success in our math students? Metacognition is the ability to think about our thinking (almost sounds too simple).

While it may sound simple it is the cornerstone of all actual and true learning. Metacognition happens when students practice self-reflection on their level of thinking. In addition, when students can relate this to a target it becomes powerful enough to increase understanding and motivation.

This knowledge promoted in the classroom is invaluable. As teachers, we all dream of that day when our students are intrinsically motivated. A moment in the classroom when the value of the content rises above a state standard or an arbitrary notion but becomes a self-fulfilling desire to learn and grow. But students need guidance and tools to develop their metacognitive awareness and become confident in the ability to self-question.

Formative Evaluation

Formative evaluation is the process of gathering evidence to inform instruction. In simpler terms, it is the process for teachers and students to communicate about the learning that is taking place and the direction that instruction should go. Formative evaluation should drive instruction decisions, gathering real-time data is crucial in guiding how the teacher will proceed with the delivery of the lesson.

In the book, Visual Learning, Hattie speaks of several internal questions that drive learners:

  • Where am I going? What are my goals?
  • How am I going there? What progress is being made towards my goal?
  • Where to next? What activities need to be undertaken next to make better progress?

Consequently, these are the same three questions that teachers must ask about instruction as they make adjustments based on the data they gather from students. Some examples shown below, include response cards (whiteboards), and exit tickets, journal entries may also serve as a way to inform instruction.

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We are Community!

Why Should We Work So Hard to Build Community?

Class Math Agreements and Community and Mathodology

Building community beginning on the first days helps to create new relationships and strong bonds that will last throughout the year. Creating a shared vision of the expectations, developing a common understanding of classroom limits, and fostering a love of learning are only a few of the characteristics you might have in mind as desired outcomes. Ultimately, achieving mutual respect and a spirit of collaboration creates an ideal working environment for the classroom.

When community exists, each child feels valued. A sense of shared purpose unites the group and working together to accomplish goals becomes a priority. Our goals are BIG and require the effort of all of our members. The uniqueness that each student provides as a member of the community must be valued and each individual strength will make the community stronger and better. As children develop a sense of duty to the community, self-discipline is likely to emerge more naturally and from the child’s (intrinsic) motivation rather than from external or reward-based methods (extrinsic).

Early in the year, creating purpose in the child’s movement and activity is desired and we balance the freedoms offered within the environment, the needs of the young child to move, and the constraints of the environment. Providing structures and routines will help to create order as well as ensure a safe environment for your children. A strong sense of community is one of the most effective ways to teach how to use individual freedoms.

How do we build community?

We play games and have fun together. We share lunch and work with each other, mixing-up our groups with an emphasis on getting to know new friends. We interview and find out more about each other by sharing experiences, stories, traditions, and the accomplishments we are proud to have achieved. We make time to appreciate each other and learn how to recognize others, as well as ourselves. 

In our community, we learn to problem solve, developing the skills necessary to take care of ourselves and others. When solutions are found and conflicts resolved with little or no direction or intervention by an adult, students feel great pride! Creating a class agreed-upon list of rights and responsibilities with the students allows them to partner in holding others accountable and enforcing your shared vision of community.

Grace and courtesy work also play a role in learning how to act in a community. A firm handshake and smile in the morning set a respectful tone for the day. Allowing students to have the role of a “class greeter” is a great way to have students serve in a leadership role as they create personal and inviting welcomes to the community. Practicing how to greet visitors with a cup of tea and a special chair or preparing a class snack are other ways students can assume responsibility. Modeling ways to ask for help, challenge other student’s ideas and even how to say “no thank you” respectfully are tools your students will need to have in order to work effectively in their community.

A natural extension of building community within our classrooms is to reach outward. The work that starts within our classroom might find opportunities in other areas within the school. Participating in the work of the larger community helps the students feel proud and invested.  Students experience, on a small scale but in a real way, that they can create change. We can act individually or as a group – and we DO make a difference! 

Specific Ideas to try at the beginning of the year might include:

  • Toss a ball in a group to help learn names
  • Learn a favorite food of a new friend
  • Create a scavenger hunt in the room to learn a new environment
  • Share with a friend something you like about yourself
  • Work together to line up without talking
  • Offer lessons on classroom jobs
  • Provide lessons and model grace and courtesy
  • Make a list of “Classroom Rights and Responsibilities” WITH your students and have them initial or sign
  • Have a procedure or place in the classroom for resolving conflicts –create a “Peace Table” or “Peace Corner”

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Getting Ready for Developing Roots

Developing Roots and Mathodology

We love seeing and hearing how you are getting ready for the start of the school year. What steps are you taking to be ready?

This was shared with us from a teacher in Georgia getting her kindergarten classroom ready for the start of Developing Roots!

“Organized the whole closet by chapters! So excited to get in front of it this year!!”- Melissa

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Lesson Study & Visualization

Lesson Study on Visualization with Mathodology

Follow our Journey. Lesson Study with a Focus on Visualization.

Jugyou kenkyuu, a Japanese phrase gives us the term “Lesson Study”. Introduced in the U. S. in the late 1990s, interest in Japanese lesson study remains strong in the education world throughout the United States. Our Lesson Study this year will focus on visualization and metacognition.

Lesson Study & Mathematics

Lesson study works well across education and in particular, in improving mathematics education. We will wrap up professional summer reading on visualization in September with a look into the routines we create in classrooms that promote visualization. During “Introduction to Lesson Study” in October, we will explore what lesson study is, how it works, how to use it, and best practices with a focus on creating metacognition in students.

Pre-Lesson Study Questions

We engaged our focus group from St. Edward School in Vero Beach by asking the following questions:

What attracted you to this Lesson Study?

Overall the participants felt this lesson study would improve their ability to use visualization strategies in their own classrooms. They felt the experience would allow them to “dig deeper” into learning the best way to improve their teaching skills to build visualization.

What do you hope to learn from this Lesson Study

Participants generally responded similarly, wanting a deeper understanding of the science behind visualization, learning how to integrate visualization into their daily teaching, and using visualization to help students see concepts in a different way.

What is visualization to you?

It is creating a picture in your mind, being able to ‘see’ what you are hearing or reading to help you better understand the lesson, and it brings life to situations, assisting a student in understanding the concepts being taught.

What do you feel you already know about visualization? (before reading)

The response to this question was consistent with all participants. All felt that visualization was a way of seeing something in your mind to better or fully understand it and using it in math as well would bring life to situations and assist students in better understanding the concepts being taught.

Ideas on how to get kids to visualize math?

  • Using various concrete and pictorial models
  • Incorporating color in our board witing to connect ideas
  • Relating ideas especially in the operations
  • Have children create a short movie in their minds with each math concept so they can ‘see’ the process and verbalize it before computing

What questions do you have before we start the lesson study?

  • Can all students visualize?
  • How are other teachers using visualization?
  • Does the brain have any physical limitations with visualizing?
  • How do we teach visualization to students so they use it seamlessly when seeing a math problem?
  • What forces the brain to want/have to visualize?

We will be holding a private Lesson Study at St. Edward’s School, Vero Beach, FL in September.

Follow us through this Lesson Study.

We’ll be at Oak Hill High School in Nashville, TN, October 2, 2019 – October 4, 2019. Seats still open!

Click here to register for this event and for details on this Lesson Study.

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What Apps Does Sarah Use?

Apps Sarah uses at Mathodology

As schools begin to embrace technology in the classroom, one question that Sarah often receives is, “What apps do you use in your workshops and classrooms?”

With a multitude of apps available, it can be challenging to find the right app that can help teachers teach better and engage students more meaningfully.

Sarah shares her list of frequently-used apps below.

Drawing and Writing Tools

Algebra Balance (iOS $1.99)

Algebra Balance App and Apps Sarah uses at Mathodology

Bamboo Paper (free for iOS, Android and Windows)

Bamboo-Paper App and Apps Sarah uses at Mathodology

Turn your screen into a paper with Bamboo Paper! Draw and write effortlessly with the various pens and brushes available. The zoom function in this app allows you to write in small spaces and fit more notes onto a page. This app also allows you to add images to your notes and write on them. Notebooks created within the app can also be shared and exported to other mobile platforms and cloud services.

Noteshelf (iOS $9.99)

Noteshelf App and Apps Sarah uses at Mathodology

Personalize your notes with Noteshelf! Individual notebooks can be easily customized in Noteshelf using the extensive library of cover designs and page templates in the app. Writing and drawing tools are available to help you write notes and draw precise geometric shapes with ease. There are also several fonts to choose from within the app if you prefer to type.

In Noteshelf, notes can be written across pages, just like a physical notebook, making it easy to create sections. Noteshelf also allows you to insert photos and write on them. Another excellent feature is its audio recording function, which means you never have to worry about missing out on the nitty-gritty details. Making annotations and signing PDFs is also a useful feature in Noteshelf. Other features include scanning of documents, password protection of notes, and syncing with Evernote and cloud services.

Pencil Box (iOS $0.99)

Pencil-Box App and Apps Sarah uses at Mathodology

Pencil Box is a drawing tool that will help anyone who wants to draw with precision. To start, pick a drawing tool and begin drawing or import an image to draw on. Besides the many brushes and shapes available, Pencil Box also allows you to add and edit layers of your drawing, making it easy to add or delete parts of a drawing without having to start over.

Teaching Tools

Geometry Pad+ (iOS $6.99, Android $5.99)

Geometry-Pad and Apps Sarah uses at Mathodology

Geometry Pad+ is a dynamic app that allows you to construct geometric shapes, such as circles, squares, triangles, rhombuses, and parallelograms, with precision and ease. Beyond creating shapes, this app also allows you to mark out angles within shapes and draw arcs using the in-app tools. The app also comes with a measuring tool that can measure lines and other properties. Other shapes properties, such as perimeter and area, can be calculated using the in-app tools as well. All documents can be saved and exported, making it convenient for future use and reference.

GeoBoard (free for iOS, Windows, and web browsers)

Geoboard and and Apps Sarah uses at Mathodology

Learn about shapes in a fun and interactive manner! In Geoboard, bands can be stretched around pegs on a virtual board to create line segments and different types of polygons. Available on both mobile and web-based platforms, GeoBoard can also be used to illustrate area, perimeter, fractions, angles, and many other concepts. With three board types to choose from and multiple band colors to work with, this app is bound to bring a lot of fun to the classroom.

Number Lines (free for iOS and web browsers)

Number Lines App and Apps Sarah uses at Mathodology

Available as a mobile and web app, Number Lines helps you to customize number lines easily for your classroom. This app allows you to create number lines with whole numbers, fractions, decimals, and even negative numbers. Students can visualize number sequences and demonstrate strategies related to counting, comparing, adding, subtracting and much more. Elements of the number line can also be hidden to encourage creative thinking.

Pieces Basic (free for iOS and web browsers)

Number Pieces Basic App and Apps Sarah uses at Mathodology

This interactive app helps students learn about place value and develop their computation skills with multi-digit numbers. Number Pieces Basic is a great visual aid for teachers to use in classrooms to engage students as pieces can be dragged around to illustrate addition and subtraction problems. Equations and expressions can also be written using the text tool. The app can be used on-the-go on a mobile device or from a web browser on a computer.

is a great visual aid for teachers to use in classrooms to engage students as pieces can be dragged around to illustrate addition and subtraction problems. Equations and expressions can also be written using the text tool. The app can be used on-the-go on a mobile device or from a web browser on a computer.

Manipulative of the Week (free for iOS)

This app bundle provides a free manipulative every week and contains the 14 most popular manipulatives used by students and teachers. Some of the apps that Sarah uses frequently are listed below.

Algebra Tiles (iOS $1.99)

Algebra Tiles App and and Apps Sarah uses at Mathodology

Algebra Tiles provides an engaging experience for students to explore algebraic concepts. Using tiles to represent algebraic expressions, students can learn to add and subtract integers as well as solve algebraic equations.

Base Ten Blocks (iOS $1.99)

Base Ten Blocks App and and Apps Sarah uses at Mathodology

Base Ten Blocks are virtual blocks to help students learn about place value, addition, subtraction, regrouping and more. Students can work with blocks representing ones, tens, hundreds and thousands to explore addition, subtraction, multiplication and division strategies. The place value chart is especially useful when working with decimals as well.

Color Tiles (iOS $1.99)

Color Tiles App and and Apps Sarah uses at Mathodology

Use Color Tiles to build number frames, make shapes, find fractions and more. The user-friendly interface gives users the freedom to create and customize manipulatives to suit their learning needs. Color Tiles helps students to develop their computing skills while enabling them to learn how to sort and classify objects.

to build number frames, make shapes, find fractions and more. The user-friendly interface gives users the freedom to create and customize manipulatives to suit their learning needs. Color Tiles helps students to develop their computing skills while enabling them to learn how to sort and classify objects.

Cuisenaire® Rods (Number Rods) (iOS $1.99)

Cuisenaire Rods App and and Apps Sarah uses at Mathodology

Number rods can be placed along a number line to compare numbers and fractions. By adding and removing rods, students can visualize the addition and subtraction of integers and fractions. This app also enables students to view ratios and proportions at a glance.

Fraction Circles (iOS $1.99)

Fraction Circles App and and Apps Sarah uses at Mathodology

In Fraction Circles, fraction pieces can be moved, rotated, overlapped and put together. It is a versatile app as fractions, decimals and percentages can be used to label each fraction piece. Each fraction piece is color-coded, making it easy to identify and work with as well.

Fraction Manipulatives (iOS free)

Fraction Manipulative App and Apps Sarah uses at Mathodology

Hundred Board (iOS $1.99)

Hundred Board App and Apps Sarah uses at Mathodology

Linking Cubes (iOS $1.99)

Linking Cubes App and Apps Sarah uses at Mathodology

Linking Cubes are virtual multi-colored cubes that help students to visualize numbers in a pictorial manner. Students can learn how to add, subtract, multiply and divide using different colored cubes. Place value can also be represented using the place value background available in the app. The built-in graph background makes it easy to create graphs as well.

Little Bit Studio Bugs & Buttons (iOS $2.99)

Little Bit Studio Bugs & Buttons App and and Apps Sarah uses at Mathodology

Number Frames (iOS free)

Number Frames App and Apps Sarah uses at Mathodology

Osmo Tangram (iOS free)

Osmo Tangram App and Apps Sarah uses at Mathodology

Pattern Blocks (iOS $1.99)

Pattern Blocks App and Apps Sarah uses at Mathodology

Pattern Blocks is the closest thing you can get to physical pattern blocks. With a wide selection of two-dimensional shapes, including hexagons and trapezoids, students can learn about geometry, patterns, fractions, and decimals in a fuss-free manner.

Place Value Disks (iOS $1.99)

Place Value Disks App and Apps Sarah uses at Mathodology

Rekenrek (iOS $1.99)

Rekenrek app and Apps Sarah uses at  Mathodology

Software Smoothie Felt Board App (iOS $2.99)

Software Smoothie Felt Board App and Apps Sarah uses at  Mathodology

Two-Color Counters (iOS $1.99)

Two Color Counters App and Apps Sarah uses at  Mathodology

Two-Color Counters is a minimalistic app for those looking for a simple yet effective way to understand numbers, integers, and fractions. Only two colors are used in this app as its focus is on illustrating operations concepts. Teaching ideas aligned to Common Core are also featured in this app.

Are you inspired? Try out these apps in your own teaching or learning journey and let us know the results!

Note: All prices stated are in USD.



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Why do Students Struggle When Learning Math Facts?

Learning Math Facts

Math facts are simply the basics: addition, subtraction; multiplication; division. They are basic number combinations and calculations we do every day.  So why is it that learning math facts  creates such huge problems for teachers and students in our classrooms?

Subtraction mistake when learning math facts

Common subtraction mistake when children are learning math facts.

Is it that some of us are just naturally better at math than others?  Perhaps.  The good news is you as a teacher can help anyone improve their math skills.  You might be wondering how.  Well, rote memorization alone won’t get us there.

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Decompose Numbers Using Number Bonds

Number Bonds

Can you picture spending several classroom periods on the number 5?  Specifically, breaking it down into parts, and putting it back together again?  Sounds like a lot of time, right?  But, spending this amount of classroom time to decompose numbers using number bonds,  allows students to gain a deeper, more flexible understanding of numbers.  You might be wondering what those periods would actually look like, how you’d keep your students engaged and learning for that amount of time.  In this blog, we will explore exactly that.

What are Number Bonds  

Simply put, number bonds are the different ways we can break apart numbers. Number bonds are all about the relationship between numbers and quantities. The relationships of parts to a whole. Building the foundation for all mathematical operations. Building mental images of number relationships. While it sounds like a simple concept, it can be difficult for students to learn.  So, it is important to dedicate time for them to learn the concept.

How to decompose numbers using number bonds

Use of color and representation helps to connect ideas.

Why Number Bonds

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Journaling

How often are you journaling in your classroom? If you’re like most teachers, your answer might be, “Not often” or “Not enough”. Many teachers tell me, “I don’t know where to start”, or “I don’t have the time”. Sound familiar?

In this blog, we will focus on answering the following questions:

Why journal?
How do I start?
What will I gain from adding it in my classroom?

Why journal?

Did you know that Journaling is one of 5 non-negotiable learning experiences?
As John Dewey said, “We do not learn from experience… we learn from reflecting on
experience”.
As teachers, it is important to “see” inside the minds of our students in order to:
• Inform our delivery and instruction to prioritize what comes next
• Collect evidence of growth in their thought process and their ability to
communicate that on paper
Journaling enables us to “see” our students’ thoughts and strategies and it helps the students
build a habit of thinking. However, careful consideration needs to be given to the
implementation process because it is not a natural process.

How do I start?

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