# data[].attributes.body
1,4,5,8,9,10

# data[].attributes.body
2,3,6,7,11,12

# data[].attributes.body
1, 4, 5, 8, 9, 10

# data[].attributes.body
2, 3, 6, 7, 11, 12

# data[].attributes.body
1, 4, 5, 8, 9, 10

# data[].attributes.body
2, 3, 6, 7, 11, 12

# data[].attributes.body
1, 4, 5, 8, 9, 10

# data[].attributes.body
2, 3, 6, 7, 11, 12

# data[].attributes.body
1, 4, 5, 8, 9, 10

# data[].attributes.body
2, 3, 6, 7, 11, 12

# data[].attributes.body
1,4,5,8,9,10

# data[].attributes.body
2,3,6,7,11,12

# data[].attributes.body
1,4,5,8,9,10

# data[].attributes.body
2,3,6,7,11,12

# data[].attributes.body
2,3,6,7,11,12

# data[].attributes.body
1,4,5,8,9,10

# data[].attributes.body
1, 4, 5, 8. 9, 10

# data[].attributes.body
2, 3, 6, 7, 11, 12

# data[].attributes.body
1, 4, 5, 8, 9, 10

# data[].attributes.body
2, 3, 6, 7, 11, 12

# data[].attributes.body
1, 4, 5, 8, 9, 10

# data[].attributes.body
2, 3, 6, 7, 11, 12

# data[].attributes.body
1,4,5,8,9,10

# data[].attributes.body
2,3,6,7,11,12

# data[].attributes.body
1, 4, 5, 8, 9, 10

# data[].attributes.body
2, 3, 6, 7, 11, 12

# data[].attributes.body
10, 4, 5, 9, 8, 1

# data[].attributes.body
12, 11, 2, 6, 7, 3

# data[].attributes.body
1, 4, 5, 8, 9, 10

# data[].attributes.body
12, 11, 2, 3, 6, 7

# data[].attributes.body
1,4,5, 8, 9, 10

# data[].attributes.body
2, 3, 6, 7, ,11, 12

# data[].attributes.body
Unproductive beliefs

1,4,5, 8, 9, and 10

# data[].attributes.body
Productive beliefs

2, 3, 6, 7, 11, and 12

# data[].attributes.body
Unproductive Beliefs: 1, 4, 5, 8, 9, 10

# data[].attributes.body
Productive Beliefs: 2, 3, 6, 7, 11, 12

# data[].attributes.body
Unproductive Beliefs

1,4,5,8,9,10

# data[].attributes.body
Productive Beliefs

2,3,6,7,11,12

# data[].attributes.body
1, 4, 5, 8, 9, 10

Unproductive beliefs

# data[].attributes.body
2, 3, 6, 7, 11, 12

Productive beliefs

# data[].attributes.body
1. Unproductive: Walking students through every step keeps them from really thinking for themselves.

4. Unproductive: Students don’t need to completely “master the basics” before they can apply math in real situations.

5. Unproductive:  Math isn’t just about memorizing and using one method.

8. Unproductive: Teaching shouldn’t just be about telling students formulas and definitions to memorize.

9. Unproductive: Memorizing information and then repeating it on homework or tests doesn’t build deep understanding.

10. Unproductive: Focusing only on drills and memorizing facts limits students’ ability to problem solve when given equations outside of what they have memorized.

# data[].attributes.body
2. Productive: Students should be active learners who try different strategies and techniques like the ones listed.

3. Productive: Tasks that make students problem solve and reason are very important.

6. Productive: Students learn best when they explore and solve real world problems.

7. Productive: Making sense of concepts and procedures through problem solving and discussion allows students to have a deeper understanding of concepts.

11. Productive: Students need a variety of strategies to choose from when solving problems, not just one way.

12.Productive: Challenging students and encouraging them to keep trying is important. Support is important.

# data[].attributes.body
1. An effective teacher makes the mathematics easy for students by guiding them step-by-step through problem solving to ensure they are not frustrated or confused.

4. Students can learn to apply math only after they have mastered the basic skills.

5. Students need only to  learn and use the same standard computational algorithms and the same prescribed methods to solve algebraic problems.

8. The role of the teacher is to tell students exactly what definitions, formulas, and rules they should know and demonstrate how to use this information to solve mathematics problems.

9. The role of the student is to memorize information that is presented and then use it to solve routine problems on homework, quizzes and tests.

10. Mathematics learning should focus on practicing procedures and memorizing basic number combinations.

# data[].attributes.body
2. The role of the student is to be actively involved in making sense of mathematics tasks by using varied strategies and representations, justifying solutions, making connections to prior knowledge are familiar context and experiences, and considering the reasoning of others.

3. The role of the teaching is to engage students in tasks that promote reasoning and problem solving and facilitate discourse the moves students towards shared understanding of mathematics.

6. Students can learn mathematics through exploring and solving contextual and mathematical problems.

7. Mathematics learning should focus on developing understanding of concepts and procedures through problem solving, reasoning and discourse.

11. All students need to have a range of strategies and approaches from which to choose in solving problems, including but no limited to general methods, standard algorithms and procedures.

12. An effective teacher provides students with appropriate challenge, encourages perseverance in solving problems, and supports productive struggle in learning math.

# data[].attributes.body
Unproductive: 1, 4, 5, 8 , 9 , 10

# data[].attributes.body
Productive: 2, 3, 6 ,7, 11, 12

# data[].attributes.body
1, 4, 5, 8, 9, 10

# data[].attributes.body
2, 3, 6, 7, 11, 12

# data[].attributes.body
1, 4, 5, 8, 9, 10

# data[].attributes.body
2, 3, 6, 7, 11, 12

# data[].attributes.body
1,4,5,8,9,10

# data[].attributes.body
2,3,6,7,11,12

# data[].attributes.body
1, 4, 5, 8, 9, 10

# data[].attributes.body
2, 3, 6,7 ,11 ,12

# data[].attributes.body
Productive Beliefs: 2, 3, 6, 7, 11, 12

# data[].attributes.body
Unproductive Beliefs: 1, 4, 5, 8, 9, 10

# data[].attributes.body
Unproductive:

1, 4, 5, 8, 9, 10

# data[].attributes.body
Productive:

2, 3, 6, 7, 11, 12

# data[].attributes.body
Unproductive Beliefs

4

5

8

9

10

11

# data[].attributes.body
Productive Beliefs

1

2

3

6

7

12

# data[].attributes.body
Productive Beliefs:

#2, #3, #6, #7, #11, & #12

# data[].attributes.body
Unproductive Beliefs:

#1, #4, #5, #8, #9, & #10

# data[].attributes.body
Unproductive Beliefs:

1

4

5

8

9

10

# data[].attributes.body
Productive Beliefs:

2

3

6

7

11

12

# data[].attributes.body
Unproductive Beliefs

1, 4, 5, 8, 9, 10

To me, focusing on memorization and step-by-step procedures without any more profound understanding is unproductive to the teaching and learning of math. When teaching is about giving students rules, definitions, and formulas to memorize, it feels more effortless in the short term but sets them up to struggle later. For example, if I show them how to solve an equation using one specific method and they memorize it, they might do fine on a quiz. But when the problem looks different, they freeze because they don’t know how to adapt.

I’ve seen this happen when students learn only the standard long-division algorithm and can’t explain why it works or when to use it. They’re just following steps without understanding what’s happening. It limits their independence and creativity because they rely on me or the textbook to show them what to do. They get stuck instead of thinking critically or trying out a different strategy. If they only focus on memorizing facts or practicing routine problems, they may forget what they’ve learned over time or fail to apply it when real-world math situations come up—like budgeting or comparing prices.

In summary, productive beliefs help students grow and think for themselves, while unproductive beliefs can hold them back and leave them unprepared for real challenges.

# data[].attributes.body
Productive Beliefs

2, 3, 6, 7, 11, 12

Productive beliefs, to me, are all about helping students build a deep understanding of math by letting them explore, reason, and solve problems in meaningful ways. It’s not just about getting the correct answer—it’s about getting students involved, thinking critically, and using different strategies to approach problems. Teachers taking on the role of guides, giving students the right amount of challenge, supporting them through struggles, and encouraging them to keep going when things get tough. For example, when I give students a problem and let them explore different methods to solve it, they gain confidence because they realize there’s more than one way to approach math. I don’t see my job as giving them all the answers—I’m there to guide, challenge, and support them when they struggle so they don’t give up.

I’ve seen how powerful it can be when students push through a complex problem and have that “aha!” moment. That struggle is part of the learning process. Instead of rushing to give them the solution, I ask guiding questions to help them think it through. This approach prepares them to face real-world situations because they know how to break down problems, reason through them, and apply their knowledge in new ways. For example, a student figuring out how to compare discounts using percentages and decimals won’t just remember the steps—they’ll understand the relationship between the numbers.

# data[].attributes.body
Unproductive Beliefs

1. An effective teacher makes the mathematics easy for students by guiding them step- by step through problem solving to ensure they are not frustrated or confused.

4. Students can learn to apply mathematics only after they have mastered the basic skills.

5. Students need only to learn and use the same standard computational algorithms and the same prescribed methods to solve algebraic problems.

8. The role of the teacher is to tell students exactly what definitions, formulas and rules they should know and demonstrate how to use this information to solve mathematics problems.

9. The role of the student is to memorize information that is presented and then use it to solve routine problems on homework, quizzes and tests.

10. Mathematics learning should focus on practicing procedures and memorizing basic number combinations.

# data[].attributes.body
Productive Beliefs

2. The role of the student is to be actively involved in making sense of mathematics tasks by using varied strategies and representations, justifying solutions, making connections to prior knowledge are familiar context and experiences, and considering the reasoning of others.

3. The role of the teacher is to engage students in tasks that promote reasoning and problem solving and facilitate discourse that moves students towards shared understanding of mathematics.

6. Students can learn mathematics through exploring and solving contextual and mathematical problems.

7. Mathematics learning should focus on developing understanding of concepts and procedures through problem solving, reasoning and discourse.

11. All students need to have a range of strategies and approaches from which to choose in solving problems, including but not limited to general methods, standard algorithms and procedures.

12. An effective teacher provides students with appropriate challenge, encourages perseverance in solving problems, and supports productive struggle in learning mathematics.

# data[].attributes.body
4-unproductive

5-unproductive

6-unproductive

9-unproductive

10-unproductive

# data[].attributes.body
1-Productive

2-Productive

3-Productive

7-Productive

8-productive

11-productive

12-productive

# data[].attributes.body
Productive beliefs

2

3

6

7

11

12

# data[].attributes.body
Unproductive beliefs:

1.

4.

5

8

9

10

These beliefs appear narrow minded and may not be reasonably expected of students.

# data[].attributes.body
Unproductive

1, 4, 5, 8, 9 and 10

# data[].attributes.body
Productive

2, 3, 6, 7, 11 and 12

# data[].attributes.body
Unproductive Beliefs:
Number 1: An effective teacher makes math easy for students by showing them each step so they don’t feel confused or frustrated.

Number 4: Students can only apply math after they’ve learned the basic skills.

Number 5: Students should only learn one method to solve math problems, like the standard way of adding or solving algebra.

Number 8: The teacher’s job is to tell students exactly what formulas, rules, and definitions they need to know, and show them how to use them.

Number 9: The student’s job is to memorize information from the teacher and use it to solve problems on homework and tests.

Number 10: Math learning should focus on practicing steps and memorizing basic facts.

# data[].attributes.body
Productive:
Number 2: The student’s job is to take an active part in figuring out math problems by using different strategies, explaining their answers, connecting to what they already know, and thinking about others’ ideas.

Number 3: The teacher’s role is to give students tasks that make them think and solve problems, and to guide discussions that help students understand math together.

Number 6: Students can learn math by exploring and solving real-world and math-based problems.

Number 7: Math learning should focus on understanding ideas through problem solving, thinking, and talking about the math.

Number 11: All students should have different ways to solve problems, including methods beyond just the standard steps.

Number 12: An effective teacher challenges students appropriately, encourages them to keep trying when things are tough, and supports them as they struggle to learn.

# data[].attributes.body
Unproductive Beliefs

Number 1

Number 4

Number 5

Number 8

Number 9

Number 10

Each idea within this list falls under the unproductive side of beliefs as these ideas target students' repetitive nature of learning and analytics, rather than their success and development beyond basic skills.

# data[].attributes.body
Hey Everyone!

Productive Beliefs

Number 2

Number 3

Number 6

Number 7

Number 11

Number 12

Each description breaks down the goals for the student's success and development within each specific topic.

# data[].attributes.body
1-frustration and confusion will happen regardless; productive struggle needs to happen in order to solve on their own

4-application can still occur even if mastery of basic skills hasn't occurred

5-students need more than just standard computational algorithms to be successful

8-telling vs. teaching; no learning occurring

9-memorization- no learning is actually taking place here

10-memorizing- again, no leaning taking place here; information is being stored for compliance and not to actually learn

# data[].attributes.body
Productive Beliefs:

2- making sense and making connections deepens their understanding of the material

3- discourse opportunities allow students to share and explain their ideas while also learning from another's perspective

6-learning through exploration and solving allows them to understand the reasoning behind how they are solving and why

7- learning through discourse, problem solving, and reasoning allows students to grow and engage in critical thinking and understanding

11- giving students various tools and strategies in order to learn; not all students learn the same and all students shouldn't be taught the same

12- a good balance with productive struggle; encourages students to learn and solve but still providing support and drive

# data[].attributes.body
1, 4, 5, 8, 9, 10

These beliefs are not achievable for all students and they make it seem like there is only one way to teach when that is not the case.

# data[].attributes.body
2, 3, 6, 7, 11, 12

These beliefs are very clear, achievable for majority of students and they allow for growth and success.

# data[].attributes.body
unproductive

4 5 8 10

# data[].attributes.body
productive

1 2 3 6 7 9 11 12

# data[].attributes.headline
The unproductive beliefs encourage students to solve mathematics problems by memorizing definitions, numbers, and formulas. This does not encourage students to build a long-term understanding of using previous knowledge of strategies and problem-solving skills to solve problems.  

# data[].attributes.subject
The unproductive beliefs encourage students to solve mathematics problems by memorizing definitions, numbers, and formulas. This does not encourage students to build a long-term understanding of using previous knowledge of strategies and problem-solving skills to solve problems.  

# data[].attributes.body
#1 An effective teacher makes mathematics easy for the students by guiding them step by step through problem-solving to ensure they are satisfied and understood.

#4 Students can learn to apply mathematics only after they have mastered the basic skills.

#5 Students must only learn and use the same standard computational algorithms and prescribed methods to solve algebraic problems.

#8, #9, and #10 These are beliefs that talk about the role of the teacher in telling students the exact definitions, formulas, and rules they can use and should know. The role of the students is to memorize all of the information presented and use it when needed to solve problems, homework, and quizzes. #10 Belief is focused on teaching students to focus on procedures and memorize basic number combinations.

# data[].attributes.headline
Productive beliefs can have a positive effect on the teaching and how students learn mathematics.   

# data[].attributes.subject
Productive beliefs can have a positive effect on the teaching and how students learn mathematics.   

# data[].attributes.body
All of these beliefs help students learn and be aware of multiple different ways to solve math problems by using previously learned strategies and recalling previous knowledge. These beliefs encourage teachers to stay actively engaged in sharing their experiences and knowledge to help students develop an understanding of concepts, procedures, and strategies and to learn how to use them when solving math problems.

#2 The role of the students is to be actively involved in making sense of mathematics tasks by connecting to previous knowledge and familiar contexts and experiences, using varied strategies and representations while justifying solutions.

#3 The role of the teacher is to engage students in tasks that promote reasoning and problem-solving and facilitate discourse that moves students toward a shared understanding of mathematics.

#6 Students can learn mathematics by exploring and solving contextual and mathematical problems.

#7 Mathematics learning should focus on developing an understanding of concepts and procedures through problem-solving, reasoning, and discourse.

#11 All students need to have a range of strategies and approaches from which to choose in solving problems, including but not limited to general methods, standards, and procedures.

#12 An effective teacher provides students with appropriate challenges, encourages perseverance in solving problems, and supports productive struggle in learning mathematics.

# data[].attributes.body
Unproductive: 1, 4, 5, 8, 9, 10

These beliefs focused on teaching only one way of learning, memorization, and were not student-centered.

# data[].attributes.body
Productive: 2, 3, 6, 7, 11, 12

These beliefs all considered student-centered learning and encouraged teaching students the reasoning and logic behind mathematical skills.

# data[].attributes.body
Unproductive Beliefs:

#4: This belief implies a strict linear progression in learning, which may limit students' ability to apply mathematical concepts.

#5: This belief restricts students to a narrow set of approaches, hindering creativity and problem-solving skills.

#8: This belief promotes a more traditional, teacher-centered approach, which may limit students's autonomy and critical thinking skills in the learning process.

#9: Emphasizing memorization over understanding hinders the development of critical thinking and problem-solving skills, limiting students ability to apply mathematics in diverse situations.

#10: This belief neglects the importance of conceptual understanding and problem-solving skills, leading to superficial learning and a lack of deeper comprehension of mathematical principles.

#1: I think this one can fall between productive and unproductive: Productive because the teacher provides step-by-step guidance which can be beneficial for complex topics. It helps students build a foundation and reduces frustration. However, constantly guiding students step-by-step may hinder their independent problem solving skills. They may become overly dependent on the teacher's guidance causing them to struggle when faced with unfamiliar problems.

# data[].attributes.body
#2: This belief aligns with the principles of active learning and promotes deeper understanding by encouraging students to actively engage in the learning process.

#3: This belief supports a student-centered approach, emphasizing the importance of teachers facilitating discussions and promoting critical thinking to enhance students' understanding.

#6: Encouraging exploration and problem-solving fosters a deeper understanding of mathematical concepts and their real-world applications, making learning more meaningful for students.

#7: This belief emphasizes a holistic approach to learning, integrating both conceptual understanding and procedural knowledge.

#11: Recognizing the diversity of students' learning styles and preferences, this belief supports the idea that providing various problem-solving strategies allows for individualized and effective learning experiences.

#12:Acknowledging the importance of challenge and perseverance fosters a growth mindset, promoting resilience and a positive attitude toward overcoming mathematical difficulties.

# data[].attributes.body
I believe the following are unproductive beliefs because they seem as if the student's only goal is to complete the task at hand and not use their learning outside of the classroom. Their goal does not seem to apply toward overall growth on the subject.

#4 - Students can learn to apply mathematics only after they have mastered the basic skills.

#5 - Students need only to learn and use the same standard computational algorithms and the same prescribed methods to solve algebraic problems.

#9 - The role of the student is to memorize information that is presented and then use it to solve routine problems on homework, quizzes and tests.

# data[].attributes.body
I believe the following are productive beliefs because they are more concerned with a student's growth and development on the mathematical subject.

#1 - An effective teacher makes the mathematics easy for students by guiding them step-by-step through problem solving to ensure they are not frustrated or confused.

#2 - The role of the student is to be actively involved in making sense of mathematics tasks by using varied strategies and representations, justifying solutions, making connections to prior knowledge are familiar context and experiences, and considering the reasoning of others.

#3 - The role of the teacher is to engage students in tasks that promote reasoning and problem solving and facilitate discourse that moves students towards shared understanding of mathematics.

#6 - Students can learn mathematics through exploring and solving contextual and mathematical problems.

#7 - Mathematics learning should focus on developing understanding of concepts and procedures through problem solving, reasoning and discourse.

#8 - The role of the teacher is to tell students exactly what definitions, formulas and rules they should know and demonstrate how to use this information to solve mathematics problems.

#10 - Mathematics learning should focus on practicing procedures and memorizing basic number combinations.

#11 - All students need to have a range of strategies and approaches

from which to choose in solving problems, including but not limited

to general methods, standard algorithms and procedures.

#12 An effective teacher provides students with appropriate challenge, encourages perseverance in solving problems, and supports productive struggle in learning mathematics.

# data[].attributes.body
Unproductive: 1,5,7,9

I belief that these are unproductive because they seem to be more of something that a teacher that doesn't care would do or a teacher that doesn't care if their students learn. I don't think that it's right to just expect a student to memorize equations or steps but not fully learn them because once they are done using what they memorized and they know they aren't going to be quizzed on it they will not remember it.

# data[].attributes.body
Productive: 2,3,4,6,8,10,11,12

I think that these are productive because they focus on the students actually learning and not just memorizing how to solve math problems but actually take time to practice them as well as the teacher teach the students different ways that they can solve a problem and not just let them to figure it out by themselves.

# data[].attributes.body
#1: An effective teacher makes the mathematics easy for students by guiding them step-

by-step through problem solving to ensure they are not frustrated or confused.

#4: Students can learn to apply mathematics only after they have mastered the basic

skills.

#5: Students need only to learn and use the same standard computational algorithms and

the same prescribed methods to solve algebraic problems.

#8: The role of the teacher is to tell students exactly what definitions, formulas and rules

they should know and demonstrate how to use this information to solve mathematics

problems.

#9: The role of the student is to memorize information that is presented and then use it to

solve routine problems on homework, quizzes and tests.

#10: Mathematics learning should focus on practicing procedures and memorizing basic

number combinations.

I was a bit unsure about some of these being unproductive, however based on their conservative approach I put them in this section. I believe that most of these teaching approaches are conservative and straight forward, and I believe that teaching has modernized and the students are learning much more quicker due to technology than before. So therefore, I believe teaching students to memorize formulas or steps wont be the best approach as it once use to before in the past.

# data[].attributes.body
#2: The role of the student is to be actively involved in making sense of mathematics tasks by using varied strategies and representations, justifying solutions, making connections to prior knowledge are familiar context and experiences, and considering the reasoning of others.

#3: The role of the teacher is to engage students in tasks that promote reasoning and

problem solving and facilitate discourse that moves students towards shared

understanding of mathematics.

#6: Students can learn mathematics through exploring and solving contextual and mathematical problems.

#7: Mathematics learning should focus on developing understanding of concepts and

procedures through problem solving, reasoning and discourse.

#11: All students need to have a range of strategies and approaches

from which to choose in solving problems, including but not limited

to general methods, standard algorithms and procedures.

#12:  An effective teacher provides students with appropriate

challenge, encourages perseverance in solving problems,

and supports productive struggle in learning mathematics.

These steps are all productive in my perspective because they encourage teachers to promote a deeper understanding of concepts and procedures of mathematics towards students. And I believe this approach will provide the strongest outcome of growth in their math learning skills and projecting that into real life scenarios.

# data[].attributes.body
Productive: 1,2,3,7,11,12

I found these to be productive, because they all help students build understanding of the concepts and practice strategies. This allows students to have a more complete understanding of mathematics. It also allows students to start to practice more real life situations with strategy they need to develop on the spot.

# data[].attributes.body
Unproductive: 4,5,6,8,9,10

I found these to be unproductive because they revolved around prior information or memorization. Instead of prior information and memorization, they should revolve around developing strategies and skills in the current class.

# data[].attributes.body
Unproductive: 4,5,8,9,10

I found these to be unproductive because they all seem to center around simply memorizing facts formulas and procedures. Just because you memorize facts and procedures does not mean you understand how you arrived at a solution which is why I consider these unproductive. A good example is number 9. Yes, students will have memorized how to solve a problem and may even get perfect scores on their tests but if the student is not able to explain why the answer is correct and explain their thought process then do they really understand mathematics?

# data[].attributes.body
Productive:1,2,3,6,7,11,12

I found these to be productive because they all center around helping students understand concepts by using critical thinking skills. number 11 for example states that students need a range of strategies and approaches to choose from when solving problems. I agree with this belief because mathematics is not just getting from point A to B its much more deeper than that and there are numerous strategies to solve a problem which means there are lots of opportunities for students to really think critically.