example of synthesis paper about mathematics

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What is Synthesis?

Here are some ways to think about synthesis:

Synthesis blends claims, evidence, and your unique insights to create a strong, unified paragraph. Assertions act as the threads, evidence adds texture, and your commentary weaves them together, revealing the connections and why they matter.

Beyond the sum of its parts: Synthesis isn't just adding one and one. It's recognizing how multiple sources, through their connections and relationships, create a deeper understanding than any single one could achieve.

Synthesis isn't just about what sources say, it's about how they say it. By digging into assumptions, interpretations, and even speculations, you uncover hidden connections and build a more nuanced picture.

Whereas analyzing involves dismantling a whole to understand its parts and their relationships, synthesizing involves collecting diverse parts and weaving them together to form a novel whole. Reading is an automatic synthesis process, where we connect incoming information with our existing knowledge, constructing a new, expanded "whole" of our understanding in the subject area.

You've been doing synthesis for a long time, the key now is being aware and organized in the process.

• Sharpen your research direction: Be clear about your main objective. This guides your reading and analysis to make the most of your time.
• Build a strong foundation: Use trusted sources like peer-reviewed journals, academic books, and reputable websites. Diverse sources add strength and credibility to your research.

Then Organize your Research:

• Dig deep and connect the dots: While reading, highlight key ideas, arguments, and evidence. Mark potential links between sources, like overlaps or contrasting arguments.
• Organize ideas by neighborhood: Group sources with similar themes or angles on your topic. This will show you where sources agree or clash, helping you build a nuanced understanding.
• Build a mind map of your research: Create a table of key themes, listing key points from each source and how they connect. This visual map can reveal patterns and identify any missing pieces in your research.

Finally, Build your synthesis:

• Lay out the groundwork: Kick off each section with a clear claim or theme to guide your analysis.
• Weave sources together: Briefly explain what each source brings to the table, smoothly connecting their ideas with transitions and language.
• Embrace the debate: Don't tiptoe around differences. Point out where sources agree or clash, and explore possible reasons for these discrepancies.
• Dig deeper than surface facts: Don't just parrot findings. Explain what they mean and how they impact your topic.
• Add your voice to the mix: Go beyond reporting. Analyze, evaluate, and draw conclusions based on your synthesis. What does this research tell us?

Tips & Tricks:

• Let the evidence do the talking: Back up your claims with concrete details, quotes, and examples from your sources. No need for personal opinions, just let the facts speak for themselves.
• Play fair with opposing views: Be objective and present different perspectives without showing favoritism. Even if you disagree, let readers see the other side of the coin.
• Give credit where credit is due: Make sure your sources get the recognition they deserve with proper citations, following your chosen style guide consistently.
• Polish your masterpiece: Take some time to revise and proofread your work. Ensure your arguments are crystal clear, concise, and well-supported by the evidence.
• Embrace the growth mindset: Remember, research and synthesis are a journey, not a destination. Keep refining your analysis as you learn more and encounter new information. The more you explore, the deeper your understanding will become.

Demonstrates how two or more sources agree with one another.

The collaborative nature of writing tutorials has been discussed by scholars like Andrea Lunsford (1991) and Stephen North (1984). In these essays, they explore the usefulness and the complexities of collaboration between tutors and students in writing center contexts.

Demonstrates how two or more sources support a main point in different ways.

While some scholars like Berlin (1987) have primarily placed their focus on the histories of large, famous universities, other scholars like Yahner and Murdick (1991) have found value in connecting their local histories to contrast or highlight trends found in bigger-name universities.

Accumulation

Demonstrates how one source builds on the idea of another.

Although North’s (1984) essay is fundamental to many writing centers today, Lunsford (1991) takes his ideas a step further by identifying different writing center models and also expanding North’s ideas on how writing centers can help students become better writers.

Demonstrates how one source discusses the effects of another source’s ideas.

While Healy (2001) notes the concerns of having primarily email appointments in writing centers, he also notes that constraints like funding, resources, and time affect how online resources are formed. For writing centers, email is the most economical and practical option for those wanting to offer online services but cannot dedicate the time or money to other online tutoring methods. As a result, in Neaderheiser and Wolfe’s (2009) reveals that of all the online options available in higher education, over 91% of institutions utilize online tutoring through email, meaning these constraints significantly affect the types of services writing centers offer.

[Taken from University of Illinois, "Synthesizing Research "]

The Writing Center at University of Arizona showcases how to create and use a synthesis matrix when reading sources and taking notes. It is a great, organized way to synthesize your research.

You can find it here .

Creativity in researching begins with developing a thorough understanding of your research topic; this is fundamental to streamlining the process and enriching your findings. This entails delving into its intricacies—exploring both similarities and divergences with related subject areas. Consider the most appropriate sources (and types of sources) for your study, critically engaging with all perspectives, and acknowledging the complex interplay between its positives, negatives, and broader connections.

Embrace interdisciplinary exploration. Delve deeper through transdisciplinary analysis, venturing beyond the immediate field to parallel professions and diverse academic arenas. Consider comparative studies from other cultural contexts to add fresh perspectives.

For example, researching rule changes in the NFL demands a nuanced approach. One might investigate the link to Traumatic Brain Injury, analyze case studies of impacted players, and even examine rule adjustments in other sports, drawing insights from their rationale and outcomes.

Remember, librarians are invaluable partners in this process. Their expertise in creative thinking and resource navigation can unlock a wealth of information, guiding you towards fruitful discoveries.

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How to Write a Synthesis Paper: Key Strategies and Examples

• Dr. Huey Logan
• December 8, 2023
• Study Guides

Here’s how to write a synthesis paper . Whether you’re a student tackling an academic assignment or a professional seeking to improve your writing skills, this writing guide will provide you with key strategies and examples to help you succeed.

Here's What You'll Learn

A synthesis paper is a written discussion that incorporates support from multiple sources to examine a variety of viewpoints related to a thesis. It is commonly used in various types of assignments such as analysis papers, research papers, argument papers, and business reports.

To write an effective synthesis essay , it is crucial to establish a clear purpose, carefully select and evaluate sources, develop an organizational plan, and properly document sources to avoid plagiarism. Additionally, understanding different strategies for organizing a synthesis paper , such as climactic order, problem/solution, and comparison and contrast, can greatly enhance the structure and flow of your writing.

Remember that the tense to use in a synthesis paper depends on the citation style you are following, with MLA using present tense and APA using past tense.

Whether you’re new to synthesis writing or looking to refine your skills, this guide will take you step-by-step through the process and provide you with valuable insights and examples. Let’s dive in!

Key Takeaways:

• A synthesis paper incorporates support from multiple sources to examine different viewpoints related to a thesis.
• Important tips for writing an effective synthesis essay include establishing a clear purpose, selecting and evaluating sources, developing an organizational plan, and documenting sources to avoid plagiarism.

Strategies for organizing a synthesis paper include climactic order, problem/solution, and comparison and contrast.

• Choose the appropriate tense based on the citation style you are using, with MLA using present tense and APA using past tense.
• Mastering the art of writing a synthesis paper can help you convey your ideas effectively and engage with multiple perspectives.

Understanding the Process of Synthesizing Multiple Sources

A synthesis essay is an intricate task that requires the writer to bring multiple sources into conversation with each other. It involves identifying connections between ideas and points of view from different authors in order to develop a well-rounded analysis. To successfully navigate this process, several key steps should be followed.

Pre-writing Techniques for a Synthesis Essay

• Identify the debate or issue: Start by clearly understanding the topic and the various perspectives that exist.
• Determine points of agreement and disagreement: Analyze the sources to identify common ground and areas of contention between writers.
• Imagine a verbal discussion: Envision a conversation between the writers, imagining how they would respond to each other’s arguments.

These pre-writing techniques help provide a solid foundation for the synthesis essay, allowing the writer to have a clear understanding of the sources and their respective viewpoints.

Organizing Ideas in a Synthesis Essay

Organizing a synthesis essay is crucial for presenting a coherent and well-structured argument. Three main approaches can be used:

• Point by point organization: This approach focuses on discussing similar points among multiple sources. It allows for a comparison and analysis of ideas that are aligned.
• Source by source organization: Here, each source is summarized and interpreted individually. This approach is useful when the sources have distinct perspectives.
• Blended organization: A blended approach combines both point by point and source by source methods. It allows for a more sophisticated and comprehensive analysis.

By selecting the appropriate organization style, the writer can present the information in a structured and logical manner, facilitating understanding and engagement for the readers.

How to develop a Synthesis Essay Thesis Statement

A synthesis essay thesis statement acts as the foundation of the essay. It should be carefully crafted to acknowledge alternative points of view and demonstrate an understanding of complexity. A strong thesis statement will:

• State a strong position on the topic
• Briefly summarize the writer’s main points
• Introduce the sources that will be analyzed

By establishing a clear and compelling thesis statement, the writer sets the tone for the entire essay, guiding the analysis and providing a roadmap for the reader to follow.

How to Write a Synthesis Paper in 5 Proven Steps,

Do you find the process of writing a synthesis paper intimidating? Are you unsure of where to start or how to effectively organize your ideas? Don’t worry, you’re not alone. Writing a synthesis paper can be a challenging task, but with the right strategies and examples, it can become a much more manageable endeavor.

In this article, we will provide you with the key steps and techniques to write an engaging synthesis essay or synthesis paper. Whether you’re a student working on an academic assignment or a professional looking to develop your writing skills, this guide will equip you with the tools you need to succeed.

First, we will dive into understanding the synthesis paper format. By familiarizing yourself with the structure and requirements of a synthesis paper, you will be able to approach your writing with clarity and precision. Next, we will explore the importance of conducting in-depth research. You’ll learn how to gather and analyze relevant sources, ensuring that your synthesis paper is well-informed and well-supported.

Effective organization of ideas is crucial in any form of writing, and synthesis papers are no exception. We’ll provide you with strategies to arrange your thoughts and arguments in a logical and compelling manner. Additionally, crafting a strong thesis statement is essential for a successful synthesis paper. We’ll show you how to develop a concise and impactful thesis that serves as the backbone of your writing.

But that’s not all—we will guide you through the five proven steps of actually writing a synthesis paper. From outlining your paper to revising and polishing your final draft, we’ll cover each stage of the writing process. Furthermore, we will delve into writing clear and cohesive paragraphs, ensuring that your ideas flow seamlessly and your message resonates with your readers.

By following these five proven steps, you will be well-equipped to tackle any synthesis paper with confidence. Don’t let the writing process intimidate you anymore. Start unlocking your synthesis paper writing potential today and achieve the academic success you deserve! So, let’s dive in and discover how to write a synthesis paper that captivates and convinces.,

Understanding the Synthesis Paper Format

Understanding the Synthesis Paper Format is crucial when it comes to crafting a compelling and coherent piece of writing. By grasping the structure and components of this specific type of academic paper, you will be able to present your ideas in a clear and organized manner. In the following section, we will explore the key elements of the synthesis paper format and how to effectively utilize them to convey your arguments persuasively.

To begin with, the synthesis paper typically consists of an introduction, body paragraphs, and a conclusion. The introduction serves as a roadmap for your readers, providing them with a concise overview of the topic you will be discussing and the thesis statement that anchors your paper. This section should capture the attention of your audience and establish the relevance and significance of your chosen topic.

Moving on to the body paragraphs, these form the backbone of your synthesis paper. Each paragraph should focus on a specific aspect or subtopic that supports your thesis statement. It is essential to present information from multiple sources and critically analyze them, highlighting the connections, patterns, and conflicts among the sources. As you incorporate evidence and examples, ensure that you cite your sources accurately and consistently according to the required citation style.

Furthermore, maintaining a logical flow between paragraphs is crucial to ensure that your synthesis paper is cohesive and easy to follow. Use transitional words and phrases to smoothly connect ideas and demonstrate the relationships between different concepts. This will help your readers navigate through your paper effortlessly and grasp the central argument you are presenting.

Finally, the conclusion brings your synthesis paper to a close by summarizing your key points and restating your thesis statement in light of the evidence you have presented. It is also an opportunity to reflect on the significance of your findings and suggest potential avenues for future research or discussion. The conclusion should leave your readers with a sense of closure and reinforce the main ideas you have conveyed throughout your paper.

By understanding and implementing the synthesis paper format effectively, you will be able to create a strong and impactful piece of writing. In the next section, we will delve into the first step of the writing process, which is conducting in-depth research.,

Conducting In-Depth Research

Conducting in-depth research is a crucial step in writing a synthesis paper as it lays the foundation for the entire writing process. It involves gathering relevant and credible sources, thoroughly understanding the material, and identifying the key concepts and arguments presented. By immersing yourself in the research phase, you will be equipped with the necessary knowledge and understanding to effectively organize your ideas and create a cohesive synthesis paper that resonates with your readers.

To begin conducting in-depth research, it is important to first identify your topic and determine the specific areas or aspects you want to explore. This will help you narrow down your search and find the most relevant sources. Utilize a wide range of resources, including books, scholarly articles, reputable websites, and even interviews or surveys, to gather a comprehensive understanding of the topic.

As you navigate through the sea of information, critically evaluate the sources you come across. Consider the expertise and credibility of the authors, the publication date, and the reliability of the sources themselves. By doing so, you can ensure that the information you include in your synthesis paper is accurate and trustworthy.

While conducting research, take notes and highlight the key points, arguments, and evidence presented in each source. This will help you keep track of the information and easily refer back to it later when organizing your ideas. Additionally, make note of any opposing viewpoints or gaps in the literature, as addressing these in your synthesis paper can strengthen your argument and provide a more well-rounded perspective.

To enhance your understanding of the material, go beyond surface-level comprehension and strive to grasp the underlying themes, connections, and implications presented in the sources. Look for patterns, similarities, and differences among the various viewpoints and arguments, as these will be the building blocks of your synthesis paper.

By immersing yourself in thorough research, you will be able to gain a deep understanding of your topic and develop a strong foundation on which to build your synthesis paper. With this knowledge in hand, you can now move on to organizing your ideas effectively, ensuring that your paper flows smoothly and engages your readers from start to finish.,

Organizing Your Ideas Effectively

To truly capture the essence of your synthesis paper, it is essential to organize your ideas effectively. After immersing yourself in thorough research, gaining a deep understanding of your topic, and establishing a strong foundation, it’s time to structure your thoughts in a way that flows smoothly and engages your readers.

Start by identifying the key themes and arguments from your sources. Look for patterns, similarities, and differences among the various viewpoints. By organizing these elements, you can create a logical flow that guides your readers through your paper.

Consider creating an outline to map out the structure of your synthesis paper. Begin with a strong introduction that sets the stage for your topic and captures your readers’ attention. Then, arrange your main points in a coherent order that supports your thesis statement. Each paragraph should be dedicated to a specific idea or argument, and transitions between paragraphs should be seamless, guiding your readers smoothly from one point to the next.

Within each paragraph, use topic sentences to clearly introduce your main ideas. These topic sentences serve as signposts for your readers, highlighting the focus and purpose of each paragraph. Connect these ideas with supporting evidence from your sources, ensuring that your arguments are well-supported and persuasive.

Remember to consider the balance between your own voice and the voices of your sources. Aim for a synthesis that blends your own analysis and insights with relevant quotes and evidence from your research. This integration creates a cohesive narrative that showcases your understanding of the material while demonstrating your ability to critically evaluate multiple perspectives.

Ultimately, organizing your ideas effectively lays the groundwork for crafting a strong thesis statement. Through a thoughtful arrangement of your main points and arguments, you can guide your readers towards your central claim with finesse and conviction. So, let’s dive into the next section and explore how to craft a thesis statement that captures the essence of your synthesis.,

Crafting a Strong Thesis Statement

Organizing your ideas effectively lays the groundwork for crafting a strong thesis statement. Through a thoughtful arrangement of your main points and arguments, you can guide your readers towards your central claim with finesse and conviction. As we delve into the next section, we will explore how to craft a thesis statement that captures the essence of your synthesis, setting the stage for the five proven steps of writing a synthesis paper.,

Five Steps of Writing a synthesis Paper

Through a thoughtful arrangement of your main points and arguments, you can guide your readers towards your central claim with finesse and conviction. As we delve into the next section, we will explore how to craft a thesis statement that captures the essence of your synthesis, setting the stage for the five proven steps of writing a synthesis paper. These steps will provide you with a clear roadmap to effectively synthesize information from multiple sources and present it in a cohesive and compelling manner, ensuring that your readers can follow your thought process seamlessly. By following these steps, you will be able to not only convey your understanding of the topic but also showcase your ability to critically analyze various viewpoints and synthesize them into one coherent narrative. So, let’s dive into the five steps of writing a synthesis paper and unleash your expertise in crafting insightful and persuasive arguments.,

Writing Clear and Cohesive Paragraphs

Crafting clear and cohesive paragraphs is crucial when writing a synthesis paper. Your paragraphs serve as building blocks for your overall argument, allowing you to present your ideas in a logical and organized manner. By following a few key principles, you can ensure that your paragraphs flow smoothly and effectively communicate your message to your readers.

Firstly, it’s important to start each paragraph with a clear topic sentence that succinctly introduces the main point you will discuss. This topic sentence acts as a guide for both you and your readers, setting the stage for the information that follows. It should be concise, yet informative, giving your readers a preview of what they can expect to learn in the upcoming paragraph.

Once you have established your topic sentence, it’s essential to provide evidence and examples that support your main point. This evidence can come from your synthesis of multiple sources, showcasing your ability to analyze various perspectives and integrate them seamlessly. Be sure to use credible sources and cite them appropriately to bolster the credibility of your argument.

Furthermore, as you develop your paragraphs, it’s crucial to maintain coherence and cohesion throughout. This can be achieved by using transitional words and phrases to smoothly connect your ideas. Words like “furthermore,” “in addition,” and “similarly” can help bridge different concepts, enabling a seamless flow of information. Additionally, using pronouns and referencing previously mentioned ideas can reinforce the connections between paragraphs, creating a sense of continuity.

In terms of paragraph length, aim for a balance between brevity and depth. While overly long paragraphs can become unwieldy and difficult to follow, excessively short paragraphs can lack the necessary substance to fully develop your ideas. As a general rule, try to limit paragraphs to around four to six sentences, ensuring that each paragraph focuses on a single main point.

Finally, don’t forget to wrap up your paragraphs effectively. Consider including a concluding sentence that summarizes the main idea and provides a smooth transition to the next paragraph. This helps create a cohesive narrative, allowing your readers to easily follow your thought process and understand the progression of your argument.

By writing clear and cohesive paragraphs, you can effectively convey your understanding of the topic and present your synthesis in a compelling manner. This attention to paragraph structure and organization not only enhances the readability of your paper but also showcases your ability to critically analyze information and construct persuasive arguments. So, with these principles in mind, let’s continue our exploration of the five steps of writing a synthesis paper, further honing your skills in crafting insightful and persuasive content.,

In conclusion, writing a synthesis paper may seem like a daunting task, but with these five proven steps, you can approach it with confidence and ease. By understanding the synthesis paper format, conducting thorough research, organizing your ideas effectively, crafting a strong thesis statement, and following the step-by-step process, you will be well-prepared to tackle any synthesis paper that comes your way.

Remember, the key to success lies in writing clear and cohesive paragraphs that seamlessly bring together different sources and ideas. This will not only impress your readers but also provide a comprehensive and insightful analysis.

So, don’t let the writing process intimidate you any longer. Start unlocking your synthesis paper writing potential today and watch as your academic success soars to new heights. As the saying goes, “The only limit to our realization of tomorrow will be our doubts of today.” Embrace the challenge, trust in your abilities, and let your newfound synthesis paper skills take center stage.

Now, armed with the knowledge and strategies outlined in this article, it’s time to put pen to paper (or fingers to keyboard) and let your ideas flow. Remember, your synthesis paper has the power to inform, persuade, and inspire change. So, use your voice, make your mark, and leave a lasting impression.

In the words of Albert Einstein, “The only source of knowledge is experience.” So go forth, gain experience, and let your synthesis paper be a testament to your growth, intelligence, and passion. Good luck on your writing journey, and may your synthesis papers shine brightly among the sea of academic excellence.

Concluding Thoughts on Writing a Synthesis Paper

Synthesis essays require a thoughtful analysis of multiple sources and the ability to synthesize their ideas into a cohesive argument or narrative. It is in the conclusion that we bring together all the threads of our analysis and present a final statement that encapsulates the main points discussed.

Effective writing

in a synthesis essay involves crafting a strong thesis statement that acknowledges alternative points of view while demonstrating an understanding of the complexity of the topic. This thesis statement sets the tone for the entire essay and guides the reader through the writer’s analysis.

Organizing ideas effectively is another vital aspect of synthesis writing. Whether using a point-by-point, source-by-source, or blended approach, the goal is to present a well-structured essay that showcases the writer’s ability to connect different sources and their perspectives.

In conclusion , writing a synthesis paper allows us to delve into multiple perspectives and gain a deeper understanding of complex topics. By employing the right strategies and examples, anyone can master the art of synthesis writing and effectively convey their ideas to readers. The journey of writing a synthesis paper may be challenging, but it is a rewarding process that enables us to engage with diverse viewpoints and contribute to the academic discourse.

What is a synthesis paper?

A synthesis paper is a written discussion that incorporates support from multiple sources to examine a variety of viewpoints related to a thesis.

What types of assignments require synthesis?

Synthesis is used in various types of assignments such as analysis papers, research papers, argument papers, and business reports.

What are the tips for writing an effective synthesis essay?

Tips for writing an effective synthesis essay include establishing a clear purpose, selecting and evaluating sources, developing an organizational plan, and documenting sources to avoid plagiarism.

What are the strategies for organizing a synthesis paper?

How should i analyze the position of each source.

Use different verbs to analyze the position of each source depending on its argumentative, research-focused, or emphasis-driven nature.

What tense should I use in a synthesis paper?

The tense to use in a synthesis paper depends on the citation style, with MLA using present tense and APA using past tense.

What is the purpose of a synthesis essay?

A synthesis essay brings multiple sources into conversation with each other, requiring the writer to identify the connections between ideas and points of view.

What are the pre-writing techniques for a synthesis essay?

Pre-writing techniques for a synthesis essay include identifying the debate or issue, determining points of agreement and disagreement among writers, and imagining a verbal discussion between writers.

How can I organize my thoughts for a synthesis essay?

Creating a grid of common points can help organize thoughts and identify categories for analysis.

What are the different ways to organize a synthesis essay?

Organizing a synthesis essay can be done in three ways: point by point, source by source, or blended.

How should I craft a thesis statement for a synthesis essay?

A synthesis essay thesis statement should be qualified to acknowledge alternative points of view and show an understanding of complexity.

What should be included in the introduction of a synthesis essay?

The introductory paragraph of a synthesis essay should state a strong position on the topic, briefly summarize the writer’s main points, and introduce the sources being analyzed.

What is the importance of analysis in a synthesis essay?

Synthesis essays require a careful analysis of multiple sources and the ability to synthesize their ideas into a coherent argument or narrative.

How should I conclude a synthesis essay?

The conclusion of a synthesis essay should summarize the main points discussed and emphasize the significance of the analysis.

What are the key elements of successful synthesis writing?

Crafting a strong thesis statement, organizing ideas effectively, and including sufficient evidence and analysis are key elements of successful synthesis writing.

How can writing a synthesis paper enhance understanding?

By understanding the process of synthesizing multiple sources, writers can create compelling and well-structured synthesis papers.

Why should I write a synthesis paper?

Writing a synthesis paper allows writers to engage with multiple perspectives and develop a deeper understanding of complex topics.

How can I master the art of writing a synthesis paper?

With the right strategies and examples, anyone can master the art of writing a synthesis paper and effectively convey their ideas to readers.

Source Links

• https://www.bgsu.edu/content/dam/BGSU/learning-commons/documents/writing/synthesis/planning-synthesis-essay.pdf
• https://writingcenterofprinceton.com/synthesis-essays-a-step-by-step-how-to-guide/
• https://www.lsu.edu/hss/english/files/university_writing_files/item35404.pdf

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• Synthesizing Sources | Examples & Synthesis Matrix

Synthesizing Sources | Examples & Synthesis Matrix

Published on July 4, 2022 by Eoghan Ryan . Revised on May 31, 2023.

Synthesizing sources involves combining the work of other scholars to provide new insights. It’s a way of integrating sources that helps situate your work in relation to existing research.

Synthesizing sources involves more than just summarizing . You must emphasize how each source contributes to current debates, highlighting points of (dis)agreement and putting the sources in conversation with each other.

You might synthesize sources in your literature review to give an overview of the field or throughout your research paper when you want to position your work in relation to existing research.

Table of contents

Example of synthesizing sources, how to synthesize sources, synthesis matrix, other interesting articles, frequently asked questions about synthesizing sources.

Let’s take a look at an example where sources are not properly synthesized, and then see what can be done to improve it.

This paragraph provides no context for the information and does not explain the relationships between the sources described. It also doesn’t analyze the sources or consider gaps in existing research.

Research on the barriers to second language acquisition has primarily focused on age-related difficulties. Building on Lenneberg’s (1967) theory of a critical period of language acquisition, Johnson and Newport (1988) tested Lenneberg’s idea in the context of second language acquisition. Their research seemed to confirm that young learners acquire a second language more easily than older learners. Recent research has considered other potential barriers to language acquisition. Schepens, van Hout, and van der Slik (2022) have revealed that the difficulties of learning a second language at an older age are compounded by dissimilarity between a learner’s first language and the language they aim to acquire. Further research needs to be carried out to determine whether the difficulty faced by adult monoglot speakers is also faced by adults who acquired a second language during the “critical period.”

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To synthesize sources, group them around a specific theme or point of contention.

As you read sources, ask:

• What questions or ideas recur? Do the sources focus on the same points, or do they look at the issue from different angles?
• How does each source relate to others? Does it confirm or challenge the findings of past research?
• Where do the sources agree or disagree?

Once you have a clear idea of how each source positions itself, put them in conversation with each other. Analyze and interpret their points of agreement and disagreement. This displays the relationships among sources and creates a sense of coherence.

Consider both implicit and explicit (dis)agreements. Whether one source specifically refutes another or just happens to come to different conclusions without specifically engaging with it, you can mention it in your synthesis either way.

Synthesize your sources using:

• Topic sentences to introduce the relationship between the sources
• Signal phrases to attribute ideas to their authors
• Transition words and phrases to link together different ideas

To more easily determine the similarities and dissimilarities among your sources, you can create a visual representation of their main ideas with a synthesis matrix . This is a tool that you can use when researching and writing your paper, not a part of the final text.

In a synthesis matrix, each column represents one source, and each row represents a common theme or idea among the sources. In the relevant rows, fill in a short summary of how the source treats each theme or topic.

This helps you to clearly see the commonalities or points of divergence among your sources. You can then synthesize these sources in your work by explaining their relationship.

If you want to know more about ChatGPT, AI tools , citation , and plagiarism , make sure to check out some of our other articles with explanations and examples.

• ChatGPT vs human editor
• ChatGPT citations
• Is ChatGPT trustworthy?
• Using ChatGPT for your studies
• What is ChatGPT?
• Chicago style
• Paraphrasing

 Plagiarism

• Types of plagiarism
• Self-plagiarism
• Avoiding plagiarism
• Academic integrity
• Consequences of plagiarism
• Common knowledge

Synthesizing sources means comparing and contrasting the work of other scholars to provide new insights.

It involves analyzing and interpreting the points of agreement and disagreement among sources.

You might synthesize sources in your literature review to give an overview of the field of research or throughout your paper when you want to contribute something new to existing research.

A literature review is a survey of scholarly sources (such as books, journal articles, and theses) related to a specific topic or research question .

It is often written as part of a thesis, dissertation , or research paper , in order to situate your work in relation to existing knowledge.

Topic sentences help keep your writing focused and guide the reader through your argument.

In an essay or paper , each paragraph should focus on a single idea. By stating the main idea in the topic sentence, you clarify what the paragraph is about for both yourself and your reader.

At college level, you must properly cite your sources in all essays , research papers , and other academic texts (except exams and in-class exercises).

Add a citation whenever you quote , paraphrase , or summarize information or ideas from a source. You should also give full source details in a bibliography or reference list at the end of your text.

The exact format of your citations depends on which citation style you are instructed to use. The most common styles are APA , MLA , and Chicago .

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Ryan, E. (2023, May 31). Synthesizing Sources | Examples & Synthesis Matrix. Scribbr. Retrieved September 3, 2024, from https://www.scribbr.com/working-with-sources/synthesizing-sources/

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Title: a neural network solves, explains, and generates university math problems by program synthesis and few-shot learning at human level.

Abstract: We demonstrate that a neural network pre-trained on text and fine-tuned on code solves mathematics course problems, explains solutions, and generates new questions at a human level. We automatically synthesize programs using few-shot learning and OpenAI's Codex transformer and execute them to solve course problems at 81% automatic accuracy. We curate a new dataset of questions from MIT's largest mathematics courses (Single Variable and Multivariable Calculus, Differential Equations, Introduction to Probability and Statistics, Linear Algebra, and Mathematics for Computer Science) and Columbia University's Computational Linear Algebra. We solve questions from a MATH dataset (on Prealgebra, Algebra, Counting and Probability, Intermediate Algebra, Number Theory, and Precalculus), the latest benchmark of advanced mathematics problems designed to assess mathematical reasoning. We randomly sample questions and generate solutions with multiple modalities, including numbers, equations, and plots. The latest GPT-3 language model pre-trained on text automatically solves only 18.8% of these university questions using zero-shot learning and 30.8% using few-shot learning and the most recent chain of thought prompting. In contrast, program synthesis with few-shot learning using Codex fine-tuned on code generates programs that automatically solve 81% of these questions. Our approach improves the previous state-of-the-art automatic solution accuracy on the benchmark topics from 8.8% to 81.1%. We perform a survey to evaluate the quality and difficulty of generated questions. This work is the first to automatically solve university-level mathematics course questions at a human level and the first work to explain and generate university-level mathematics course questions at scale, a milestone for higher education.

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MATHEMATICS HELPS PREDICT THE BEHAVIOR OF NATURE AND PHENOMENA IN THE WORLD

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ابتسام الشهري

David L Abel

The F > P Principle states that “Formalism not only describes, but preceded, prescribed, organized, and continues to govern and predict Physicality.” The F > P Principle is an axiom that defines the ontological primacy of formalism in a presumed objective reality that transcends both human epistemology, our sensation of physicality, and physicality itself. The F > P Principle works hand in hand with the Law of Physicodynamic Incompleteness, which states that physicochemical interactions are inadequate to explain the mathematical and formal nature of physical law relationships. Physicodynamics cannot generate formal processes and procedures leading to nontrivial function. Chance, necessity and mere constraints cannot steer, program or optimize algorithmic/computational success to provide desired nontrivial utility. As a major corollary, physicodynamics cannot explain or generate life. Life is invariably cybernetic. The F > P Principle denies the notion of unity of Prescriptive Information (PI) with mass/energy. The F > P Principle distinguishes instantiation of formal choices into physicality from physicality itself. The arbitrary setting of configurable switches and the selection of symbols in any Material Symbol System (MSS) is physicodynamically indeterminate—decoupled from physicochemical determinism.

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The F > P Principle states that “Formalism not only describes, but preceded, prescribed, organized, and continues to govern and predict Physicality.” The F > P Principle is an axiom that defines the ontological primacy of formalism in a presumed objective reality that transcends both human epistemology, our sensation of physicality, and physicality itself. The F > P Principle works hand in hand with the Law of Physicodynamic Incompleteness, which states that physicochemical interactions are inadequate to explain the mathematical and formal nature of physical law relationships. Physicodynamics cannot generate formal processes and procedures leading to nontrivial function. Chance, necessity and mere constraints cannot steer, program or optimize algorithmic/computational success to provide desired nontrivial utility. As a major corollary, physicodynamics cannot explain or generate life. Life is invariably cybernetic. The F > P Principle denies the notion of unity of Prescriptive Information (PI) with mass/energy. The F > P Principle distin-guishes instantiation of formal choices into physicality from physicality itself. The arbitrary setting of configurable switches and the selection of symbols in any Material Symbol System (MSS) is physicodynamically indeterminate—decoupled from physicochemical determinism.

Research in Science Education

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Physics teachers’ approaches to teaching physics are generally considered to be linked to their views about physics. In this qualitative study, the views about physics held by a group of physics teachers whose teaching practice was traditional were explored and compared with the views held by physics teachers who used conceptual change approaches. A particular focus of the study was teachers’ views about the role of mathematics in physics. The findings suggest the traditional teachers saw physics as discovered, close approximations of reality while the conceptual change teachers’ views about physics ranged from a social constructivist perspective to more realist views. However, most teachers did not appear to have given much thought to the nature of physics or physics knowledge, nor to the role of mathematics in physics.

Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas " Einstein 2 The " Unreasonable Effectiveness of Mathematics " in Describing, Explaining and Predicting the Physical World 3 With a more or less implicit recall of Pythagorism, Leibniz once wrote that mathematics can be differentiated from music only because it a a form of conscious calculation, whereas music represents unconscious calculation. We can add that mathematics has a relationship to other sciences that is similar to music's relationship to other arts, as it is the most abstract but perhaps most effective instrument for understanding the world. From physics to biology, psychology to economics, there is no empirical science today which has not, in some way, been mathematized, and Immanuel Kant had already noted that Since in every theory of nature there can be only as much science, properly speaking, as there is a priori knowledge, it follows that the theory of nature can contain as much science, properly speaking, as mathematics that can be applied to it. 2 A few simple examples from the history of science will serve to illustrate the odd, and up until now, mysterious phenomenon on the basis of which entire parts of mathematics, which were initially invented and constructed without any applicative purpose, later proved to be highly useful in predicting, describing, and explaining new and unexpected natural phenomena, and therefore in bringing to light " areas of knowledge " which had been previously completely obscure. What renders the problem particularly difficult is that it does not seem easily resolvable by invoking one current philosophical position over another on the ontology of mathematics, given that prima facie, the applicability issue creates puzzles for all such positions. Within a constructivist philosophy of mathematics, for example, one must explain why mathematics – regarded as a creation of ours – 1 I would like to thank the audience at Copenhagen for the interesting questions raised during the discussion of a previous version of this chapter. 2 I. Kant, Metaphysische Anfangsgründe der Naturwissenschaft, Akademie Textausgabe, Bd. IV, Berlin 1968, p. 470.

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Synthesis problems

An aggregate of problems centered on the problem of constructing a control system with a prescribed way of functioning. A control system is usually made up of elements which are themselves simple control systems. Formerly synthesis meant that the structure of the elements, a rule for combining them and a means of using the structure of the control system to define the function which it realizes are given.

Every class of control systems gives rise in a natural way to a specific class of functions. A synthesis problem originally consisted of constructing a control system realizing a given function from this class.

Examples. 1) Given a Boolean function , construct a formula which realizes it. 2) Given a set of Boolean functions, construct a multipolar contact scheme which realizes this set. 3) Given an exact description of the behaviour of an automaton , construct the automaton itself. 4) Given a computable function , find an algorithm that computes it, or compose the corresponding program.

The above examples are concerned with discrete mathematics. However, the following are characteristic examples of a continuous nature. a) Given the logarithmic amplitude-frequency characteristic of an (automatic) control system (or, more generally, working conditions of it such as perturbation and control actions, disturbances, restrictions on working time, etc.), synthesize an (automatic) control system with these characteristics. b) Given a system of linear equations and inequalities, and also a linear function, construct a computing scheme which maximizes this function on the solution set of this system. c) Given a system of differential equations and a desired degree of accuracy for its solution, find an algorithm for solving this system numerically to within this degree of accuracy.

It is conventional to treat the last three examples as belonging to continuous mathematics, although the search for an approximate numerical solution already presupposes the approximation of continuous functions by discrete ones. These problems thus involve an interplay between continuous and discrete mathematics.

The mathematical formulation of synthesis problems presupposes an exact indication of the language in which a function is given. Which problems can arise in the solution of synthesis problems depend on this language. In the majority of cases there is a comparatively simple method for constructing a control system of a canonical form.

However, when the class of control systems is not too small, the problem can have many other solutions; the problem naturally arises of choosing one that is optimal in some sense. To this end one introduces a functional characterizing the quality of a control system, such as the number of elements in it, its cost, its volume, its capacity (that is, the greatest number of its elements in an active state), its working time, the probability of breakdown, etc. It is important that this functional, usually called the complexity, should accurately reflect the property required of the control system, and be readily computable from it. A refinement of synthesis problems consists of constructing, for a given function, a control system that realizes it with minimal complexity.

In fact, such a definite formulation of the problem is only possible for finite models of control systems. As a result, all synthesis problems are more clearly stated in this case, and so finite models are given most attention below.

For finite models of control systems a typical situation is that in which the problem has a trivial solution. For example, if one wishes to minimize the number of elements in a scheme, then the trivial method of solution consists in first listing all schemes containing one element and verifying whether or not one of them realizes the given function, then listing all schemes containing two elements and verifying again, etc. until one finds a scheme that realizes the given function. This scheme then has the minimum number of elements.

However, due to the large number of steps required, the trivial method is not very effective. Besides, if the complexity of the function to be realized exceeds a certain comparatively low threshold, then the trivial method becomes practically unworkable and the use of the fastest computers extends its scope only slightly. This means that a further refinement of the formulation of synthesis problems is required.

The tendencies listed here are illustrated below for the example of one class of control systems, diagrams of functional elements (cf. Diagram of functional elements ), for the special case where complexity is taken to be the number of elements in the scheme. The functions realizable by this type of control system are those of the algebra of logic (Boolean functions).

First of all, one must note that the non-applicability of solving by the trivial method is only one of the reasons leading to a revision of the formulation of synthesis problems. There is a conjecture that any algorithm which constructs for every Boolean function a scheme with a minimal number of elements inevitably contains in some sense a listing of all Boolean functions. As positive evidence for this there is a theorem which states that if one considers an infinite set of Boolean functions containing one most complex function for every number of variables, and if one takes the closure of this set under the operations of renaming variables (without identification) and substituting constants, then one obtains the set consisting of all general Boolean functions (up to inessential variable functions). This conjecture has much indirect support confirming the fact that even if one relaxes the requirement on the number of elements that are allowed for the construction of the schemes close to optimal, but retains the requirement that the algorithm be applicable to every Boolean function, then the listing cannot be reduced. The concept of the inevitability in principle of listing is another basic reason for modifying the formulation of synthesis problems. There are several possibilities here.

The Shannon approach (named after C.E. Shannon, who first suggested it for contact schemes) is connected with the function

$$ L ( n) = \max L ( f ), $$

where $ L ( f ) $ is the smallest possible number of elements in a scheme realizing a function $ f $, and the maximum is taken over all Boolean functions $ f $ in $ n $ variables. $ L ( f ) $ and $ L ( n) $ depend on the basis, that is, on the set of function elements that can be used in the construction of the schemes. Synthesis problems consist of finding an effective method of constructing, for any Boolean function in $ n $ variables, schemes with a number of elements essentially not greater than $ L ( n) $.

For a more general formulation of the problem, one assigns to every functional element $ E $ a positive number $ L ( E) $ as the complexity of this element, which depends only on the function realizable by the element $ E $, and ascribes to every scheme $ S $ the complexity

$$ L ( S) = \ \sum L ( E), $$

where the summation is over all the elements of the scheme $ S $. For every Boolean function $ f $ the complexity of its realization (in a given basis) is defined as follows:

$$ L ( f ) = \ \min L ( S) $$

where the minimum is taken over all schemes $ S $ realizing $ f $. Then, as before, one sets $ L ( n) = \max L ( f ) $, and for an arbitrary Boolean function in $ n $ variables it is required to construct a scheme of complexity not greater (or not much greater) than $ L ( n) $.

For this approach one needs to be able to compute $ L ( n) $ or bound it accurately, at least from below. It turns out that one can obtain a fairly good lower bound for $ L ( n) $ by the power method, which is based on the following consideration. The number of all possible schemes realizing Boolean functions in $ n $ variables and having complexity not greater than $ L ( n) $, cannot be smaller than the total number of Boolean functions in $ n $ variables. In the power method, one needs only obtain a satisfactory upper bound for the number of schemes under consideration.

One can then judge to a great extent the quality of an arbitrary universal synthesis method (that is, a method suitable for any Boolean function) by how many times the complexity of the schemes constructed by it for a given $ n $ differs from the lower bound for $ L ( n) $. At the same time, the complexity of these schemes serves as a constructive bound for $ L ( n) $ and is asymptotically equal to it as $ n \rightarrow \infty $. Thus, this is asymptotically the best synthesis method and enables one to describe the asymptotic behaviour of $ L ( n) $. Let

$$ \rho = \min \ \frac{L ( E) }{i - 1 } , $$

where $ i $ is the number of occurrences of an element $ E $ and the minimum is taken over all $ E $ for which $ i \geq 2 $. Then

$$ L ( n) \sim \rho { \frac{2 ^ {n} }{n} } . $$

For other classes of control systems, $ L ( S) $, and also the complexities $ L ( f ) $ and $ L ( n) $, are defined in a similar way. For a contact scheme $ S $, $ L ( S) $ is usually the number of contacts in it, and when $ S $ is a formula, it is the number of symbols denoting variables in it. For automata and Turing machines (cf. Turing machine ), $ L ( S) $ is sometimes taken to be the number of states. In these cases the corresponding asymptotic behaviour of $ L ( n) $

is shown in Table 1.

<tbody> </tbody> Class of control systems $ L( n) $ Remark Diagrams of functional elements $ \sim p \frac{2 ^ {n} }{n} $ $ p= \min L( \frac{E)}{i-} 1 $ Contact schemes $ \sim \frac{2 ^ {n} }{n} $ Formulas $ \sim \frac{2 ^ {n} }{ \mathop{\rm log} _ {2} n } $ Automata $ \sim \alpha ( n) \frac{2 ^ {n} }{n} $ $ 1 \leq \alpha ( n) \leq 2 $ Turing machines $ \sim \frac{2 ^ {n} }{n( k- 1) } $ $ k $ is the number of letters in the internal alphabet of the machine For other types of complexity, such as the probability of error, and for other classes of control systems, such as automata, the corresponding asymptotic relation may contain not one but two or more parameters, and, in general, the problem of finding asymptotics (say for automata) is algorithmically unsolvable. An analysis of the lower bound for $ L ( n) $ shows that, in fact, for almost-all Boolean functions in $ n $ variables, the complexity of realization is asymptotically not less than $ L ( n) $. This means that a given synthesis method for almost-all Boolean functions yields almost the simplest schemes. However, first, despite the fact that the class contains almost-all Boolean functions, for sufficiently large $ n $ none of the functions in the class can be clearly described, and secondly, it consists of the hardest functions to realize, and these are seldom encountered in problems of an applied nature. So, such functions are not of great interest, and it again becomes necessary to modify the formulation of the synthesis problem. It must be noted that functions for which the synthesis problem is actually important form a negligible part of all Boolean functions, and if the conjecture on the necessity of listing is true as stated, any effective universal synthesis method will automatically give in some cases schemes that are by no means the simplest. It is impossible to describe the set of these functions, and the problem arises of classifying Boolean functions for synthesis purposes. This approach to the formulation of the problem is concerned with the selection of important classes of functions, with the creation of special synthesis methods for them, each oriented towards a function of a specific class, and also with estimating the quality of the schemes thus obtained. An effective means for solving such problems is provided by the local coding principle. This principle presupposes a coding of the functions in the class under consideration in such a way that the value of a function can be computed using a comparatively small part of the code. If one succeeds in realizing certain auxiliary operators in a sufficiently simple way, then the local coding principle gives a synthesis method which is asymptotically optimal for the class considered. Moreover, the following relation holds with a fair degree of generality: $$ L ( N _ {n} ) \sim \rho \frac{ \mathop{\rm log} _ {2} | N _ {n} | }{ \mathop{\rm log} _ {2} \mathop{\rm log} _ {2} | N _ {n} | } , $$ where $ N _ {n} $ is the subset of those functions of the class that depend on a given set of $ n $ variables, and $$ L ( N _ {n} ) = \ \max L ( f ), $$ where the maximum is taken over all Boolean functions $ f $ in $ N _ {n} $( $ \rho = 1 $ here). The concrete form taken by this asymptotic relation is most conveniently illustrated on classes of Boolean functions which take the value 1 at a given number of points. The most typical examples are shown in Table 2. <tbody> </tbody> Number of points at which the function is $ 1 $ $ L( N _ {n} ) $ $ ( \mathop{\rm log} _ {2} n) ^ {c} $, $ c> 1 $ $ \sim n ( \mathop{\rm log} _ {2} n ) ^ {c-} 1 $ $ n ^ {c} $, $ c> 0 $ $ \sim \frac{n ^ {c+} 1 }{( c+ 1) ( \mathop{\rm log} _ {2} n ) } $ $ 2 ^ {n ^ {c} } $, $ 0< c< 1 $ $ \sim n ^ {1-} c 2 ^ {n ^ {c} } $ $ 2 ^ {cn } $, $ 0< c< 1 $ $ \sim 1- \frac{c}{c} 2 ^ {cn } $ $ \frac{2 ^ {n} }{n ^ {c} } $, $ c> 0 $ $ \sim c \frac{2 ^ {n} \mathop{\rm log} _ {2} n }{n ^ {c+} 1 } $ On the basis of the local coding principle good synthesis methods have also been obtained for very narrow classes of functions (such as symmetric functions), i.e. for those functions which are of most interest from a practical point of view. However, in the case under consideration, the lower bound obtained by the power method turns out to be too weak for judging the quality of schemes. As far as restricting the class of functions being considered at a given time is concerned, this is inevitable, since the limiting case of a narrow class is one containing not more than one function in each number of variables. This means in practice that one considers individual concrete functions. Two problems arise here. $ \alpha $) To construct, for a given concrete function $ f $, a scheme with the smallest possible number of elements, and thus to obtain an upper bound for $ L ( f ) $. $ \beta $) To obtain the best possible lower bound for $ L ( f ) $. Of these problems, the first is usually the simpler, since for its solution it is sufficient in the final analysis to construct just one scheme, whereas the second requires a survey in some sense of all the schemes realizing $ f $. This motivated a whole new trend in the search for methods of obtaining lower bounds for concrete functions. In finding lower bounds that are linear in $ n $, the number of arguments of the function (or, in general, the volume of input information) usually causes no great difficulties. Finding stricter lower bounds is a significantly more complex problem. The first examples of this type were a lower bound of order $ n ^ {1.5} $ for formulas in the basis $ \{ \&, \lor , \overline{ {}}\; \} $ realizing a linear function (with addition modulo 2) of $ n $ arguments, and a lower bound of order $ n ^ {2} / \mathop{\rm log} _ {2} ^ {2} n $ for Markov normal algorithms (cf. Normal algorithm ) applied to words of length $ n $. The best lower bounds so far (1984) obtained are of order $ n ^ {2} $, provided one ignores those obtained under very strong restrictions on the class of control systems (for example, when the system of elements is incomplete) or by the use of powerful means for representing a function that actually include a listing all functions. The non-triviality of the investigation into the complexity of realizing concrete functions is also indicated by the fact that many problems (such as the multiplication of integers, the multiplication of matrices, and the recognition of symmetry of a word on multi-tape Turing machines) for which comparatively high lower bounds were expected, turned out to be in actual fact simpler than at first supposed. In minimizing other parameters of schemes, such as delays, probability of a breakdown, etc. roughly the same problems arise. One can establish a definite connection between various parameters, and thanks to the simulation of certain objects by others, this can often be done for parameters of control systems in different classes. Thus, results obtained in synthesis problems are not isolated, but are closely interrelated, and can often be transferred from one domain into another. Much of the above carries over to infinite models of control systems. However, in the formulation of synthesis problems, and even more so in their solution, the situation is significantly more complex. [1] O.B. Lupanov, "On a certain approach to the synthesis of control systems - the local coding principle" , (1965) pp. 31–110 (In Russian) [2] S.V. Yablonskii, "On algorithmic difficulties in designing minimal control circuits" , (1959) pp. 75–121 (In Russian) Given a discrete- or continuous-time input-output dynamical system or control dynamical system, also simply called dynamical system, $$ x( t+ 1) = f( x( t), u( t), t) ,\ \ \dot{x} = f( x, u , t) , $$ $$ x \in \mathbf R ^ {n} ,\ u \in \mathbf R ^ {m} , $$ $$ y( t) = h( x( t)) ,\ y = h( x) , $$ $$ y \in \mathbf R ^ {p} , $$ and a starting point $ x( 0) \in \mathbf R ^ {n} $ at time zero, the system $ \Sigma $ defines a function $ b( \Sigma ) $ from a space of input functions $ {\mathcal U} $ to a space of output function $ {\mathcal J} $, sometimes called the behaviour of the initialized system $ ( \Sigma , x( 0)) $. Given a behaviour $ b $, one now has the synthesis problem of finding a dynamical system $ \Sigma $( of a suitable class) such that $ b( \Sigma ) = b $. One says then that $ \Sigma $ realizes $ b $, and in this setting the synthesis problem is usually called the realization problem and one speaks of realization theory. In the case of a time-invariant linear dynamical input-output system $$ \dot{x} = Ax+ Bu ,\ \ y = Cx, $$ $$ A \in \mathbf R ^ {n \times n } ,\ A \in \mathbf R ^ {m \times m } ,\ C \in \mathbf R ^ {p \times n } $$ (or its discrete counterpart), its behaviour is completely determined by its transfer function $$ T( s) = C( sI- A) ^ {-} 1 B = \ s ^ {-} 1 T _ {0} + s ^ {-} 2 T _ {1} + s ^ {-} 3 T _ {2} + \dots , $$ $$ T _ {i} = CA ^ {i} B, $$ which gives the Laplace transforms (or $ z $- transforms in the discrete-time case) of the input and output functions. In this case one speaks of realizing the strictly proper rational matrix function $ T( s) $ or of the series of Markov parameters $ T _ {0} , T _ {1} ,\dots $. Thus, a realization of $ T _ {0} , T _ {1} \dots $ is a triple of matrices $ A , B , C $ such that $ T _ {i} = CA ^ {i} B $, $ i = 0, 1,\dots $. The least $ n $ such that there exists a realization with $ A $ an $ ( n \times n) $- matrix is the McMillan degree of $ ( T _ {0} , T _ {1} , . . . ) $ or $ T( s) $. It is equal to the rank of the infinite block Hankel matrix $$ \left ( \begin{array}{cccc} T _ {0} &T _ {1} &T _ {2} &\dots \\ T _ {1} &T _ {2} &T _ {3} &\dots \\ T _ {2} &T _ {3} &T _ {4} &\dots \\ \cdot &\cdot &\cdot &\dots \\ \end{array} \right ) . $$ If one is looking for $ A , B , C $ such that $ T _ {i} = CA ^ {i} B $, $ i = 0 \dots k $, one speaks of a partial realization problem. There are realization algorithms that are locally continuous in the parameters $ T _ {0} , T _ {1} , . . . $. There exists a global continuous realization if and only if $ p = 1 $( one output) or $ m = 1 $( one input). (Non-existence of continuous canonical forms.) Realizations of matrix functions and, more generally, operator-valued functions have proved to be a valuable tool in, e.g., minimal factorization and Wiener–Hopf factorization of matrix and operator-valued functions, cf. e.g. [a11] – [a13] . Analogous realization questions arise for stochastic dynamical systems; cf. e.g. [a5] and [a6] for a discussion of two stochastic realization problems (state space models for Gaussian random processes, transmission line models realizing a given covariance function). For a survey of non-linear deterministic finite-dimensional realization theory, cf. e.g. [a10] . In control theory the phrase "synthesis problem in control theorysynthesis problem" mostly refers to the problem of finding for a given input-output dynamical system $ \Sigma $ a suitable feedback law $ v = k( z) $, where $ z $ is some (measured) output. (Or, more generally, a dynamical input-output system with input $ z $ and output $ v $ such that the resulting closed-loop system System Controller $ u $ $ z $ $ v $ $ y $ Figure: s091910a has some desired behaviour. For instance, that certain disturbances are suppressed or that $ u \mapsto y $ matches a pre-given behaviour.) For the specific question of the synthesis or realization of electrical networks, cf. (the editorial comments to) Network and the references given there. [a1] N. Pippenger, "The complexity of monotone Boolean functions" , (1978) pp. 289–316 [a2] A.A. Razborov, "Lower bounds for the monotone complexity of some Boolean functions" , (1985) pp. 354–357 ,  : 4 (1985) pp. 798–801 [a3] J.E. Savage, "The complexity of computing" , Wiley (1976) [a4] I. Wegener, "The complexity of Boolean functions" , Wiley & Teubner (1987) [a5] A. Lindquist, G. Picci, "State space models for Gaussian stochastic processes" M. Hazewinkel (ed.) J.C. Willems (ed.) , , Reidel (1981) pp. 169–204 [a6] P. DeWilde, J.T. Fokkema, I Widya, "Inverse scattering and linear prediction" M. Hazewinkel (ed.) J.C. Willems (ed.) , , Reidel (1981) pp. 307–350 [a7] M. Hazewinkel, "(Fine) moduli spaces for linear systems: what are they and what are they good for" C.I. Byrnes (ed.) C.F. Martin (ed.) , , Reidel (1980) pp. 125–193 [a8] A.H. Zemanian, "Realizability theory for continuous linear systems" , Acad. Press (1972) [a9] R.E. Kalman, P.L. Falb, M.A. Arbib, "Topics in mathematical systems theory" , McGraw-Hill (1969) [a10] B. Jakubczyk, "Realization theory for nonlinear systems: three approaches" M. Fliess (ed.) M. Hazewinkel (ed.) , , Reidel (1986) pp. 3–32 [a11] H. Bart, I. Gohberg, M.A. Kaashoek, "Minimal factorization of matrix and operation functions" , Birkhäuser (1979) [a12] K. Clancey, I. Gohberg, "Factorization of matrix functions and singular integral operators" , Birkhäuser (1981) [a13] H. Bart, I. Gohberg, M.A. Kaashoek, "Explicit Wiener–Hopf bifurcation and realization" I. Gohberg (ed.) M.A. Kaashoek (ed.) , , Birkhäuser (1986) pp. 235–316 [a14] W.M. Wonham, "Linear multivariable control" , Springer (1974) [a15] E.D. Sontag, "Mathematical control theory" , Springer (1990) This page was last edited on 7 June 2020, at 14:55. Privacy policy About Encyclopedia of Mathematics Disclaimers Impressum-Legal Exploring the Lesson Synthesis: When do I actually teach? By  Lizzy Skousen,  6–12 Curriculum Writer, IM Certified® Facilitator During a problem-based lesson, the teacher does a lot of listening while monitoring student learning. When teachers are introduced to a problem-based lesson structure, they may wonder: “Do we just hope that students will think about what we want?” “how will students learn the things they need to know”, “when do i actually teach”, “is this doable with the amount of instruction time i have”. One answer to all of these questions is a well-executed lesson synthesis. There is an overall structure to an IM lesson. We begin with inviting students to the mathematics of the day, then dive more deeply into concepts and procedures. After the lesson activities and before the cool-down, there is a lesson synthesis when we consolidate and apply the learning. What does it mean to consolidate the learning of the day? We need to bring together all of the learning that has happened with individuals and groups and make it explicit for the entire class. During each activity, students generate solution methods and then reflect and discuss how they solved the problems. There may be as many different ideas as there are students. Some students or groups will generate strategies or make connections that align with the learning goals of the lesson, but not every student will. Sometimes they may choose a more familiar but less efficient strategy or struggle to make a specific connection. I like to imagine that teaching is a bit like taking my students for a wilderness exploration hike. I want the hikers to explore, see new things, and get to play, but I also don’t want them to get lost or wander onto a trail going the wrong direction. As the leader of the hike, I need to start by knowing the area. What are the major landmarks? Where do we start, where do we end, and what are some major pitfalls we need to avoid? In a lesson, I need to know the same things. What is the purpose of the lesson? Where do I want students to end up? What do I want students to be able to accomplish? The learning goals and cool-downs are really clear signposts to help answer these questions. The learning goals help the teacher know what students should understand and be able to do, and the cool-down shows how students will demonstrate it. With this information, I can go into each activity with a clear idea of where the lesson is going. During the lesson synthesis, I check in to make sure all students have heard the important takeaways of the lesson so I know they are ready to demonstrate their learning on the cool-down. When students are exploring mathematical ideas, they are not all going to do it in the same way. On the wilderness hike, some explorers will carefully follow a pre-marked path, while others will try hopping on rocks or logs, and still others may follow an interesting bug. During the lesson, some students will want to use a familiar representation instead of trying a new one, others will skip using a diagram at all, and some may get focused on a different approach. One group of students may already be showing that they can meet the learning goal, while another group is still unsure. This is why it’s important to consolidate the learning of the day. The lesson synthesis gives teachers the opportunity to bring together students’ ideas and focus on the learning goals for the lesson. Targeted questions during the lesson synthesis highlight  the learning that has already happened during each activity and help make this learning explicit for all of the students in the class. When I have my wilderness explorers hiking with me, I think of myself as a scout leader. We may be exploring and having fun, but there is also an intended direction to travel. With my students, they explore and play with ideas, but there is a mathematical direction in a lesson as well. The lesson synthesis is a brief but vital component of the lesson where teachers can make sure that students have all heard and seen important vocabulary words, strategies, and connections. In this discussion with the students, the teacher asks targeted questions, and a few students will share their thinking. The teacher may revoice important ideas, highlight strategies that align with the learning goals, and explicitly state an important concept or vocabulary term. Most of the students will be listening during this part of the lesson. Even on the most relaxed hike, there are time limits. If I know that we have to meet back at camp before sundown, I have to plan ahead to make sure that my hikers all get there on time. Since I’ve already looked ahead at the map, I know about where we should be throughout the day and can set a pace for the hike. Pacing a lesson can be a challenge. It can seem like there is not enough time in a day to do the synthesis, but the lesson synthesis is critical to student learning. Each lesson synthesis should take about 5 minutes of class time before the cool-down. One way to pace the lesson effectively is to work backwards. Plan for the last 5 minutes of class to be the cool-down, and the 5 minutes before that to be the lesson synthesis. Then look at how much instructional time is left and adjust the pace for the other activities as needed. Since the lesson synthesis brings together all of the learning that happens in each activity and makes it explicit for the whole class, it’s not necessary for every student to have completed every activity perfectly before moving from one activity to the next.   Conclusion  After students complete the cool-down, take a moment to reflect. If most students showed evidence of accomplishing the learning goals, what happened during the lesson synthesis that helped consolidate the learning for all students? If most students struggled with the cool-down, how could the lesson synthesis have gone differently? Connecting the day’s lesson synthesis with the cool-down can help you plan ahead for future lessons. A well-executed lesson synthesis is essential to accomplishing the learning goals of a problem-based lesson. This is the point where teachers make explicit what students should take away from the lesson and give students the necessary preparation for the cool-down. When preparing for a lesson, make sure to plan time for the lesson synthesis and work backwards from there to pace the other learning activities. to the IM Blog and Newsletter RECENT POSTS Fostering Teaching Practices to Nurture Opportunities for Students to Shine September 5, 2024 Reintroducing the IMplementation Reflection Tool August 29, 2024 Preparation Matters: Using Strategic Planning to Maintain the Magic in Math Communities All Year Long August 22, 2024 Annotate and Acknowledge August 9, 2024 Getting Ready for 2024–2025 Back to School with Illustrative Mathematics August 1, 2024 Back to School (5) Coaches Corner (1) Grades 3–5 (112) Grades 6–8 (122) Grades 9–12 (97) Grades K–2 (92) Grades K–5 (58) Partners (4) Professional Learning (80) Recent Blog Posts (10) Uncategorized (2) Popular Tags 211 IMAGES MA12 Synthesis Paper GED102-1 Synthesis Paper SOLUTION: Synthesis paper mathematics helps predict the behavior of How to Write an Effective Synthesis Essay? Tips MATH10 synthesis paper Synthesis Paper in mathematics VIDEO Synthesis Organic synthesis MSc objective question paper || MSc fourth semester objective based paper Organic Synthesis Example 4 HOW TO WRITE A SYNTHESIS ESSAY QUESTION PAPER OF ORGANIC SYNTHESIS (With Solution) M.SC IV SEMESTER (2023-24) Synthetic Division COMMENTS Mathematics in The Modern World: Synthesis Mathematics in the Modern World: Synthesis - Free download as PDF File (.pdf), Text File (.txt) or read online for free. Patterns are the core topic of mathematics and can be observed in both numeric and geometric forms in nature. Numbers patterns occur in things like calendar months and addresses, while geometric patterns appear in plant and animal features. Synthesis Paper Mathematics in the Modern World Mathematics: Organizing Patterns and Regularities in the World Mathematics plays a crucial role in our daily lives, encompassing numbers, shapes, equations, and measurements that contribute to making our lives easier. By organizing patterns and regularities, mathematics provides us with valuable insights and tools for understanding the world ... Synthesis- Paper SYNTHESIS PAPER: Section 1: Nature of Mathematics. In the partial fulfilment of the subject Mathematics in the Modern World. Submitted by: Angelica Eunice A. Lazaro Submitted to: Ms. Denise Briones. Introduction. The importance of mathematics in today's milieu is evident. PDF Bloom's Taxonomy Interpreted for Mathematics Bloom's Taxonomy Interpreted for Mathematics Lindsey Shorser This document contains a description of Bloom's Taxonomy, a educational tool developed by Benjamin S. Bloom (1913-1999) that ranks the relative cogni-tive complexity of various educational objectives. This taxonomy is often used as an aid when create test questions and assignments. PDF Synthesis of IES-Funded Research on Mathematics: 2002-2013 Synthesis of IES-Funded Research on Mathematics: 2002-2013 . Authors . Bethany Rittle-Johnson, Vanderbilt University . Nancy C. Jordan, ... This paper was prepared for the National Center for Education Research, Institute of Education Sciences under Contract ED-IES-12-D-0015. Meredith Larson was the project officer. MATH10 synthesis paper These two concepts that involves math are just a tiny dot of a whole lot of uses of math in our daily lives. Seeing things in the perspective of Mathematics, creates a new platform for new opportunities and strategies. A new way to do things with more confidence having math as a quantified proof. The uses of math will continue to grow its number. Synthesis in Research Designed for mathematics students, this guide takes you through the Library's resources to improve your research. ... Here are some ways to think about synthesis: Synthesis blends claims, evidence, and your unique insights to create a strong, unified paragraph. ... EXAMPLE: Although North's (1984) essay is fundamental to many writing ... A Neural Network Solves, Explains, and Generates University Math A Neural Network Solves, Explains, and Generates University Math Problems by Program Synthesis and Few-Shot Learning at Human Level Iddo Drori1,a,b,Sarah Zhanga,Reece Shuttlewortha,Leonard Tangc,Albert Lua,Elizabeth Kea,Kevin Liua,Linda Chena, Sunny Trana,Newman Chengb,Roman Wangb,Nikhil Singha,Taylor L. Pattic,Jayson Lynchd,Avi Shporera,Nakul Vermab, ... ORIGINS OF MATHEMATICAL THINKING: A SYNTHESIS U. Leron 1. ORIGINS OF MATHEMATICAL THINKING: A SYNTHESIS. Uri Leron. Technion - Israel Institute of Technology. <[email protected]>. Abstract: This paper synthesizes recent research from ... How to Write a Synthesis Paper: Key Strategies and Examples A synthesis paper is a written discussion that incorporates support from multiple sources to examine a variety of viewpoints related to a thesis. It is commonly used in various types of assignments such as analysis papers, research papers, argument papers, and business reports. To write an effective synthesis essay, it is crucial to establish a clear purpose, carefully select and evaluate ... Synthesis Paper: Mathematics Helps Predict The Behavior of ... Mathematics helps predict natural phenomena and the behavior of the world. Mathematical models correspond to real-world measurements and observations, allowing prediction of behaviors not yet observed. These models represent "laws of nature". Mathematics provides a framework to encode observations of the physical world. It can be used to explain recurring patterns in nature and predict events ... Synthesis Paper MTH Synthesis Paper MTH - Free download as Word Doc (.doc / .docx), PDF File (.pdf), Text File (.txt) or read online for free. This document discusses how mathematics can help control nature and occurrences in the world for human benefit. It provides several examples of how mathematics is used to explain natural phenomena, from snowflake formation to building construction. PDF Meaningful Learning in Mathematics: A Research Synthesis of Teaching For example, Hiebert and Wearne (1992) define learning with understanding as the construction of connections between mathematical ideas and their ... (qualitative meta-synthesis), using mathematics education as a framework of reference for methodological review and development. Nye et al. (2016), in turn, look at the corresponding methodology ... Synthesizing Sources Learn how to synthesize sources by combining the work of other scholars to provide new insights. See examples, tips, and a synthesis matrix tool to help you analyze and interpret sources. A Neural Network Solves, Explains, and Generates University Math We demonstrate that a neural network pre-trained on text and fine-tuned on code solves mathematics course problems, explains solutions, and generates new questions at a human level. We automatically synthesize programs using few-shot learning and OpenAI's Codex transformer and execute them to solve course problems at 81% automatic accuracy. We curate a new dataset of questions from MIT's ... (PDF) A Synthesis of Mathematics Writing: Assessments, Interventions To understand the mathematics writing of students, we conducted a synthesis to evaluate empirical research about mathematics writing. We identified 29 studies that included a mathematics-writing ... Math synthesis paper Mathematics synthesis paper mathematics helps predict the behavior of nature and phenomena in the world mathematics plays an important part in our daily life. ... Given all of the relevant examples, I conclude that mathematics can truly assist in the prediction of nature's behavior and phenomena in the world, as well as save many lives. ... Mathematics Helps Predict the Behavior of Nature and Phenomena in The World From physics to biology, psychology to economics, there is no empirical science today which has not, in some way, been mathematized, and Immanuel Kant had already noted that Since in every theory of nature there can be only as much science, properly speaking, as there is a priori knowledge, it follows that the theory of nature can contain as ... Synthesis problems Learn about the problems of constructing control systems with prescribed functions, and the methods and complexity of solving them. See examples of synthesis problems in discrete and continuous mathematics, and the Shannon approach to Boolean functions. Synthesis Paper in Mathematics SYNTHESIS PAPER IN MATHEMATICS - Free download as PDF File (.pdf), Text File (.txt) or read online for free. 1) Mathematics helps organize regularities and patterns in nature. Fibonacci sequences can be seen in flower petals and seed arrangements which ensures efficient use of sunlight. 2) Mathematics is also found in signals and sound. Radar uses the difference in time that a signal hits each ... Synthesis Paper in mathematics Synthesis Paper in mathematics. Synthesis Paper in mathematics in the modern world. Course. medicine (med101) 106 Documents. Students shared 106 documents in this course. University Ago Medical and Educational Center Bicol Christian College of Medicine. Academic year: 2021/2022. ... As an example. The number of tough words you input reflects ... Exploring the Lesson Synthesis: When do I actually teach? Lesson synthesis is a brief discussion where teachers consolidate and apply the learning of the day. It helps students hear and see important vocabulary, strategies, and connections, and prepares them for the cool-down. Learn how to plan and execute a lesson synthesis effectively. essay homework paper phd project resume review speech study thesis