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“When will I ever use this?” If you’re a math teacher, this is a question you’re very familiar with. As the math curriculum becomes ever more abstract in middle and high school, it’s difficult for students to see when and where the material they’re learning will be applicable.
Just telling them that the material will be useful for their following math courses or future academic career doesn’t cut it—and it shouldn’t. This is when the “unreasonable effectiveness of mathematics,” as explored through reading and writing, is the ace up the sleeves of teachers.
Reading and writing are complex, fundamental, integrative learning skills that should be used to their potential in math class. Reading is a two-part process that is similar to mathematics in that it requires first the transfer of encoded information to the reader and then the comprehension of that information. And writing is one of the most powerful modes of learning because it engages both hemispheres of the brain—one side generates ideas, the other structures them.
Effective writing also clarifies and organizes a student’s thoughts, and the slow pace of writing is conducive to student learning because it allows them to reason carefully to make sure they’re correct before they state their thoughts. Studies have shown that writing is valuable specifically for the math classroom—for example, it seems that a student’s ability to explain concepts in writing is related to the ability to comprehend and apply them.
And of course written explanations also allow the teacher to understand and assess the student’s thinking in a way that computational steps alone may not provide.
Writing in Math Class
Although writing prompts for the elementary—and less frequently for the middle school—math classroom have been available for some time, they generally focus on attitudes and dispositions, likes and dislikes, and thoughts, concerns, and feelings about math or math class. They include journals, math autobiographies, letters to the teacher, and freewriting activities. They involve writing about a concept or a process the students are studying, or center on metacognitive skills such as effort, goals, expectations, study habits, and so on.
These prompts can be adapted for high school students, but I’ve found it more useful to use reading and writing to illustrate how mathematics is embedded in every aspect of life. Reading and writing are particularly effective in developing a quantitative understanding of the world around us because they can be used to lead students to reflect on everyday experiences. One assignment that I’ve found effective is having students create a magazine of excerpts of articles about various applications of mathematics in real life.
A magazine about math in the real world, produced by high school students.
The magazine—I’ve included an issue here—is composed of summaries of published articles, so to get students started, I keep an updated list of articles that highlight the ways math is present in different facets of the real world—from the probability that a soccer team ends up in a particular World Cup group to gerrymandering, fractals in Jackson Pollock’s paintings, black holes, the fairness of social media, invisibility cloaks, and more.
I select articles from a variety of newspapers, journals, and magazines, such as The Washington Post, The New York Times (especially the Upshot section), The Economist, New Scientist, Scientific American, and The Atlantic, as well as from blogs, websites, podcasts, videos, and other online resources. The app MathFeed (downloadable for free in the Apple App Store), curated by mathematician Francis Su, is my most valued and trusted source—it allows me to find the most important news and views about math in the media in a single place.
After picking an article that has piqued their interest—from the list I provide or other sources of their choice—the students read it carefully, summarize it, and post it on the online magazine. (If they want to work with an article they have found, they have to ask me first so we don’t end up with repetitions in the magazine.) They know they’re required to understand the article content properly, edit their summary thoroughly, provide a visual to aid their readers’ comprehension, and cite their source.
I give them access to the draft of the magazine, and they also have to upload their article and take care of any technical issues that may arise.
For assessment, I’ve created a simple rubric that looks at content understanding, clarity of communication, editing, critical thinking, initiative, and creativity.
This has been one of my most popular projects—students are amazed to discover some of the myriad applications of math. They also enjoy being able to choose articles according to their interests, and appreciate that the end product, the magazine, can be easily shared with family and friends and stored in their own e-portfolios.
This is a valuable activity that encourages students who enjoy reading and writing more than the computational side of math and gives all a deeper understanding and stronger appreciation of the usefulness and effectiveness of mathematics.
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Class AB and Class C Power AmplifiersCross-over Distortion
In the push-pull configuration, the two identical transistors get into conduction, one after the other and the output produced will be the combination of both.
When the signal changes or crosses over from one transistor to the other at the zero voltage point, it produces an amount of distortion to the output wave shape. For a transistor in order to conduct, the base emitter junction should cross 0.7v, the cut off voltage. The time taken for a transistor to get ON from OFF or to get OFF from ON state is called the transition period.
It is understood that this cross-over distortion is less pronounced for large input signals, where as it causes severe disturbance for small input signals. This cross over distortion can be eliminated if the conduction of the amplifier is more than one half cycle, so that both the transistors won’t be OFF at the same time.Class AB Power Amplifier
The cross over distortion is the problem that occurs when both the transistors are OFF at the same instant, during the transition period. In order to eliminate this, the condition has to be chosen for more than one half cycle. Hence, the other transistor gets into conduction, before the operating transistor switches to cut off state. This is achieved only by using class AB configuration, as shown in the following circuit diagram.
Therefore, in class AB amplifier design, each of the push-pull transistors is conducting for slightly more than the half cycle of conduction in class B, but much less than the full cycle of conduction of class A.
The conduction angle of class AB amplifier is somewhere between 180o to 360o depending upon the operating point selected. This is understood with the help of below figure.
The small bias voltage given using diodes D1 and D2, as shown in the above figure, helps the operating point to be above the cutoff point. Hence the output waveform of class AB results as seen in the above figure. The crossover distortion created by class B is overcome by this class AB, as well the inefficiencies of class A and B don’t affect the circuit.
So, the class AB is a good compromise between class A and class B in terms of efficiency and linearity having the efficiency reaching about 50% to 60%. The class A, B and AB amplifiers are called as linear amplifiers because the output signal amplitude and phase are linearly related to the input signal amplitude and phase.Class C Power Amplifier
When the collector current flows for less than half cycle of the input signal, the power amplifier is known as class C power amplifier.
The efficiency of class C amplifier is high while linearity is poor. The conduction angle for class C is less than 180o. It is generally around 90o, which means the transistor remains idle for more than half of the input signal. So, the output current will be delivered for less time compared to the application of input signal.
The following figure shows the operating point and output of a class C amplifier.
This kind of biasing gives a much improved efficiency of around 80% to the amplifier, but introduces heavy distortion in the output signal. Using the class C amplifier, the pulses produced at its output can be converted to complete sine wave of a particular frequency by using LC circuits in its collector circuit.
Why is the sky blue? I remember in my physical science class, our teacher showed us a possible reason why the sky is blue. He took a canister of liquid oxygen and poured it out on the table. I saw the blueness of the liquid as it flowed out and then disappeared. Then we talked about color, frequencies, and absorption, reflected and radiated light. I wondered how scientist ever figured these things out? Duh — math! How can you really teach science without math? It is impossible. Science is the application of math.
In science, geometric principles such as symmetry, reflection, shape, and structure reach down to the atomic levels. In science, algebraic balance is required in chemical formulas, growth ratios, and genetic matrices. In science, math is used to analyze nature, discover its secrets and explain its existence and this is the big problem. Science is so complex and getting more so each day. In order to In math class one of the biggest needs is relevance. Students want to know how they are going to benefit from being able to do calculations. Why not use science to teach math? Since one of the biggest uses of mathematics in science is data gathering and analysis, that is the best place to start. When a teacher gives students a real science problem to solve — one that requires math tools — the teacher is giving the students a reason to use math. Math then becomes something useful, not something to be dreaded.
Being able to teach math better and being able to teach science better are powerful reasons for the math and science teacher collaborate with each other. According to a case study conducted by Jennifer Dennis and Mary John O’Hair, another reason that math and science teachers should collaborate is that science helps provide relevance to math that is all too often abstract and isolated calculation operations. Instead of using the word problems out of the book that many dread, why not walk down the hall and talk to the science teacher to find out what math they are using in science class? Ultimately, as another study reported, the students’ increased conceptual understanding of math and science is the greatest benefit of math and science teacher collaboration. Conceptual understanding means the students know the bigger picture of why things work in math and science, not just how to make them work.
Unfortunately, knowing that increased teacher collaboration in math and science will benefit students and teachers is not enough. Teachers are so busy that finding time to collaborate is difficult. Add to this, the structure of the school inhibits collaboration when math and science teachers are spread out in a large campus. How do you overcome this? Well, a simple request to the principal might do the trick. Let the principal know what you want to do and show him the evidence and if he can, he will help. Another solution is that even though geographically speaking the math and science teachers may be isolated, everyone has a cellphone. Just call them. Texting, Facebook, Google+, Skype, Google Hangouts, or even email can bring science and math teachers together. There are many options — just do it!
What are ways you work with your companion subject teacher (math or science) to help students understand math and science better?
If you’ve ever caught yourself spending hours sorting digital playing cards, trying to avoid mines, or exploding pieces of candy on a screen, you know just how engaging digital games can be. When used in educational settings, digital games have the power to immerse students in a rich learning environment.
Minecraft is one such game used by educators from kindergarten to higher education. It drops the player into an infinitely customizable world of blocks, animals, and monsters where players ( and teachers ) can set their own goals. And many students already play this game, which means teachers have the opportunity to transfer students’ skills into an academic context.
When I tell people I use Minecraft to teach high school English, I am most often met with incredulity: How is this possible? I explain that students can play through survival scenarios to create narratives, re-create settings and scenes from novels to demonstrate reading comprehension, and even analyze the digital game itself as a new media genre.5 Tips for Using Digital Games in the Classroom
1. Let students be the experts in the gameplay while you facilitate the learning experiences: Introducing video games like Minecraft disrupts the hierarchical model of teaching because students often know more about the medium than their teachers.
Instead of worrying about this, let the students collaborate by teaching each other and researching the game mechanics—the unique rules and features of a game. The classroom learning becomes reciprocal as the teacher designs learning experiences while the students teach the teacher more about the game.
2. Let it be a game, not a worksheet in disguise: Students are quick to pick up on activities that seem enjoyable on the surface but are no deeper than a worksheet. But teachers can use gameplay to achieve learning goals, by allowing students to play the game as the game is intended to be played while structuring the experience in a way that demonstrates learning goals.
For example, students who are charged with building a scene from a book in Minecraft might have to work together to collect the stone, wood, wool, and glass. Along the way, they’ll most likely take the initiative to go out into the online Minecraft community to research how to shear sheep or the fastest method of cutting down trees. While meeting the sole learning goal related to reading comprehension and imagery, the students have added to their research, critical thinking, and problem solving skills.
3. Embrace problems and challenge students to solve them: Digital games have the potential to introduce problems with technology and even behavior during the lesson. Fortunately, the standard premise of every game is that there is a problem that must be overcome, and it is up to the players to use their resources to overcome that obstacle.
Likewise, teachers should make students the central players in any problems that arise. If the internet goes out or a student’s laptop will not function, encourage the students to try to find solutions and work-arounds, like creating a mobile hotspot, moving to a different location, playing the game offline, or even playing the game in real life using manipulatives or by acting out parts. These complications have the potential to be valuable opportunities to create a classroom culture of active learning and problem solving.
4. Play the game first and then learn the teacher controls: Some games have an educational version that allows teachers to customize and control the learning environment. These versions may also include data tools, the ability to move players around in the game, and the option to modify the game’s difficulty.
However, I suggest that you start by just playing the game. The goal here is not to become a master of the game but to have fun while researching its potential for your classroom. As I play Minecraft and explore new games, I focus on matching the reasons I’m having fun with the learning objectives each challenge in the game can meet.
5. Create a student contract for behavior in the game: Since Minecraft offers a creative world of possibilities, it’s important to set guidelines for students. For example, my first course of action when students try to blow up our shared Minecraft world with TNT is to ask the students to draft a constitution that governs game play. These collaboratively built contracts hold students accountable for their actions. Here are some ideas students might include in a constitution:
Whether you are a student, teacher, or working professional, solving mathematical problems is part of our daily routines. It could be a simple arithmetic addition and multiplication or complex calculus problem. Good part is that Windows 11 comes with multiple options to solve all your math problems. In this article, we will explain 6 different methods available in Windows 11 for solving your mathematical problems.Solving Math Problems in Windows 11
Some apps in Windows 11 come with in-built option to solve math problems while few other apps need integrating with external tools. The usage of these apps and tools completely depends on the type of problems you want to solve. For example, you can type a problem in apps like Calculator to get a quick answer. However, you need a full-fledged math tool for solving the values of variables in multilevel arithmetical equations. Other aspect is that whether you simply want to get the result as in calculator app or want to learn step-by-step instructions on how to solve a problem. At any case, try out all the below options and choose the best one that fits your need.1. Math Solver in Microsoft Edge
Microsoft offers Math Solver website which you can access from any device using a browser. However, Edge allows you to integrate this Math Solver feature within the browser window. You can either select problems on webpage or type custom problems with math calculator to get solutions. You can use this tool to resolve any type of mathematical problems like trigonometry, fractions, algebra, calculus, etc. The best part of using Math Solver is that you can also get the complete instructions on how the solution is derived so that you can use this for learning purpose. Learn more on how to use Math Solver in Edge.
Select Math Problem By Selecting in Edge
You should be connected to internet for using Math Solver in Edge. The tool will generate a unique link on Math Solver website for each solved problem. You can share this link with your friends and teachers for review.2. Math Assistant in OneNote for Windows 10
Add Steps to Problems from Math Assistant
The best part is that you can drag and drop the steps to your notes and share with others a guide. Though you do not need internet for using Math option in OneNote, you first need to connect the app to your Microsoft account and the connection is also needed for syncing and sharing.
3. Use Calculator for Math Problems
Calculator Functions in Windows 11
The scientific mode offers trigonometry, functions, log and many other options to get solution for your problems. Though this is an independent app which you can use without internet connection, functions like currency conversion still needs a connection. Also, you cannot solve custom equations as you can do with Math Solver and Math Assistant tools.4. Math Solutions Using Windows Search
You can use Windows Search or Cortana to find files, folders and apps in your computer. However, many do not know that it is possible to get answers to simple math problems using Windows Search when connected to internet. It will show the answers for simple multiplication of division while you type. For complex calculations, you will see a calculator in the result showing the answer. The best part is that you can edit the input in calculator and get the result within the search result.
Get Solution to Math Problems in Windows Search5. Get Solutions from Bing Search
In addition to searching keywords and getting results, you can use Bing search as a calculator. Simply, type the problem and Bing will suggest the solution in the prediction. If the prediction recognizes the problem wrongly, then press the enter key to view the problem as a result in a calculator. You can edit the input in the calculator if there are any mistakes in the inputs to get correct result.
Solving Math Problems with Bing Search6. Solving Math Problems in Microsoft Word
Insert Math Equations in Microsoft Word
This is useful when preparing instructional manual or guide for students in Word and not for learning or solving your own math problems. You can also easily convert the Word document to PDF and share the solution with anyone.Final Words
There are limitations while using Windows Search, Bing Search and Calculator app for solving your math problems. You can only use simple and basic calculations using these options. However, Edge Math Solver and OneNote Math Assistant are powerful tools to get solution to any type of math equation problems including integrals, trigonometry and algebra. Using Microsoft Word is a bonus option for those making documentation and wanted to show the problem with solution steps.
High school teachers can guide students to success in writing assignments by structuring tasks to account for working memory.
In high school, reflection essays, analysis papers, and literature reviews for English and other courses supplement more traditional summaries and narratives. Regardless of the focus, we’re familiar with the complicated writing process, which requires brainstorming, organizing, and translating ideas into words while using correct mechanics (punctuation, grammar, sentence structure, etc.). At the same time, writing a coherent and well-developed piece requires valuable working memory. Unfortunately, subtle working memory issues may increase these complex writing challenges.
Writing demands working memory capacity, retention time, and processing depth. For example, gaps in remembering and understanding information slow the process of manipulating and translating information. As a result, students may prematurely discard information they need. How can we engage students in maximizing their working memory functioning throughout the writing process?
Consider the following strategies: increasing capacity through note-taking, deepening processing with discussion and summarization, and extending retention time with review and revisions.
Setting Up a Writing Task to Account for Working Memory
Analyzing the writing task: Analyzing the assignment and identifying discrete steps creates a structure in working memory, easing the mental organization process. While doing this with your class, ask students for examples of relevant information. For example, if they are analyzing the Napoleonic Era, ask them to provide two decisions Napoleon made that led to his defeat. Examples provide students with brain priming and enable you to assess retention and comprehension. In this way, task analysis serves as a confirmation of students’ understanding of directions and their content knowledge.
Consider the following strategies: intermittent low-stakes testing to support remembering and understanding, student-generated teach-backs for knowledge review and rehearsal, student partnerships for reading directions, and use of step-by-step checklists.
Prewriting: Now that students have created a mental organization framework, they can begin writing. A structured approach is essential when considering the extensive working memory demands. For example, creating an organizer provides a review of information, thus increasing the depth of working memory processing. This way, information is more efficiently organized for easy long-term memory storage. Thus, rather than taxing working memory capacity, information can be accessed more easily from long-term memory as needed.
Start by activating prior knowledge with a 5- to 10-minute brainstorm. Then create an overall structure of subtopics, main ideas, and their logical connections, using outlines, mind maps, graphic organizers, or note cards.
Leave time between creating the organizer and revising it to allow for mental organization of the information and increased objectivity. During the revision, have students use notes to identify possible gaps. Be sure to recognize the need for processing time to facilitate decision-making. Avoid fatigue by establishing a work session of an hour at most, such as 45 minutes of focused work, a 5-minute break for processing, and a 10-minute review.
Planning: Executive functions such as attention, inhibition, and emotional regulation impact working memory functioning. Therefore, planning is a proactive step that can help students overcome future obstacles. Partner students to expand the writing process checklist they created during task analysis.
For example, have students enter work session appointments with alerts into a digital calendar. Have them enter interim due dates with a specific action step for receiving feedback. Finally, a growth step would be to include step-specific time estimates to encourage the development of accurate planning.
Translating ideas into words: Translating ideas into words requires self-regulation. Decisions regarding word choice, spelling, and grammar require persistence. Therefore, avoiding internal distractions impacts working memory’s ability to manipulate and organize information.
Have students consider the following strategies:
Cover everything in the organizer except the section guiding their current writing.
Lessen cognitive and physical demands with speech-to-text.
Write without editing by turning off spell or grammar check features.
Establish a cueing system to mark words or areas of uncertainty. Try highlighting or italicizing word choice to review, or adding a question mark to indicate uncertainty of ideas.
Editing: Allow at least an hour between writing and editing to let students focus on their actual wording versus what they think they wrote. Time also offsets the emotional attachment to their words. Finally, lessen the chances of students feeling overwhelmed by limiting editing to one or two specific areas. Their editing checklist might focus on writing mechanics, specialized vocabulary, or places they flagged as unclear during writing. Either partner students or consider using text-to-speech to ensure accurate reading of their draft.
Reflecting: Reflection provides a review of the student’s writing process. Emphasizing their goals and gains moves them from working memory to long-term memory.
To reinforce growth, ask students to identify a gain. Then establish a goal by focusing on a feedback suggestion. For example, perhaps they struggled to hold information in their working memory while writing an English essay. Ask them to identify a strategy, technology, or resource that would support their ability to decide what information to include in a future organizer.
When working memory is functioning effectively and efficiently, the complex demands of writing become steps in a workable process rather than obstacles of frustration.
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