The story of each individual’s math education can be represented, in stages, by the sets of numbers which he or she has used as a universe of discourse. A universe of discourse is best thought of, in math terms, as the largest set of numbers being talked about — i.e., the set of numbers from which all solutions to problems and equations must be drawn.
In the beginning, we usually have only the counting numbers or the whole numbers, and if a number isn’t in one of those sets it doesn’t exist. That’s why, for example, a 1st-grader will solemnly assure us that, “5 doesn’t go into 19” and, “You can’t take 5 from 3!”
Once the rational numbers, including negative integers, have been revealed, many new problems become solvable. The process continues until the entire set of real numbers is on the table, including the somewhat mysterious irrationals. This is a “map” of the real numbers:
The next, and last, big step is the introduction of the imaginary numbers, when the square root of -1 is defined to be the imaginary unit, i. When the imaginaries are added to the reals, we have the set (or, more properly speaking, the field) of complex numbers. Any number of the form a + bi, where a and b are real numbers, is a complex number. If b happens to be 0, then a + bi is the real number, a. If b is not 0, then a + bi is an imaginary number.
The complete picture looks like this:
Sometimes there is confusion on this last point: Some contend that, for example, 2 + 3i is “complex,” while 3i is not. This is incorrect: 2 + 3i and 3i are imaginary numbers (try finding them on a number line!), but both are complex, because both can be written in the form a + bi.
Any of us around on that day in 2001 remember the emotion and uncertainty of those first few hours. Our students, though, have no such memories. All they know they will come from their parents, from us, from YouTube, or from elsewhere. So what to do? How to convey not just the heartbreak and bewilderment but – most importantly – the facts and the hindsight that time affords? Well, as professional teachers, you’ve found ways. You know what works, and what may need some tweaking. We’ve found a few good sites that may help infuse your plans with some new or different perspectives.
As stated on its website, this program is a collaboration between the 9/11 Memorial & Museum, the NYC Department of Education, and the NJ Commission on Holocaust Education. Lesson plans are divided by grade level, containing material appropriate to the age of the learners. Teaching guides are included, as well.
MCTSR also has many plans grouped by grade level. Of particular note is “The Second Day,” a video created by a 14 year-old student who lived blocks from Ground Zero.
This National Public Radio article offers less pedagogical insight, but does include perspective about the years after the attack. There are some useful links embedded in the report – including one to the aforementioned 9/11 Memorial & Museum.
Created on the 10th Anniversary of 9/11, these plans were created by a middle school language arts teacher in Ohio. She offers a number of strategies and mindsets to help with teaching the multiple perspectives of the terrorist attacks.
Offering the perspective of those indirectly and unjustly blamed for the attacks, this article provides information that could be used in the teaching of the subject. Though it offers no lesson plans, the first-hand accounts it contains could be fodder for deeper discussions about the reactions of people in the days- and years- after 9/11.
All of these can, in some way, provide teachers with new angles and additional information for teaching this very complex event. We hope this helps in even the smallest of ways.
Photo Credit to: https://www.flickr.com/people/themachinestops
About the Author
Kirby Spivey taught AP World History, US History, and numerous other Social Studies courses in Georgia. Mr. Spivey currently leads USATestprep’s Social Studies content team. He and his wife live in Atlanta. He was helping students with a project on Federalism in the school library when the first plane hit the North Tower.
With our national science scores remaining below those of many other countries, US states continue to look for ways to change the way we teach science. The newest set of science standards is the Next Generation Science Standards (NGSS), and the number of states adopting the NGSS standards or a version of them, is growing. Let’s look at the history and the future of the NGSS.
What are the NGSS?
The NGSS are a set of standards that cover every grade level and every scientific discipline. According to their developers, these are standards that go beyond a specific discipline and attempt to integrate all disciplines to the real-world. The focus is on a 3-Dimensional Model, which includes Disciplinary Core Ideas (DCIs), Scientific and Engineering Practices (SEPs), and Crosscutting Concepts (CCCs). The goal is for students to understand that science is more than just memorizing facts, and that science should be interwoven where it fits into the world.
How were the NGSS developed?
The idea of uniform science standards is not new. The National Research Council (NRC) was created over a century ago to focus on the use of scientific research in American industries.
Project 2061, created by Advancing Science, Serving Society (ASSS) in 1985, helped to define scientific literacy through its publication, Science for All Americans. In 1996, the NRC published National Science Education Standards, which were designed to enable the nation to achieve the goal of scientific literacy. In 2010, the NRC began the process of creating guidelines to change the way we teach science. A Framework for K-12 Science Education, released in 2011, provided the foundation to help develop standards that address what K-12 science students should know. This was the beginning of the NGSS. In the fall of 2011, 26 states with an 18-member panel of experts appointed by the NRCBlog Articles worked together to write the new standards. The final draft of the NGSS was released in April 2013, and Rhode Island was the first to adopt them in May 2013. This was separate from the development of the Common Core standards released in 2010, although the NGSS team worked with the Common Core writers to help with literacy connections.
The future of the NGSS?
As of February 2016, 17 states and the District of Columbia have adopted the standards, while over 40 states have shown interest in them. With pressure to improve science scores and science education in the United States, many states see the NGSS as a way to bring about that change. At this point, the future of the NGSS remains to be seen. The NGSS are meant to serve as a guideline, and the decision to follow all or parts of that guideline is ultimately up to each individual state, but there must also be buy-in from the local and classroom levels.
Why should my state look at adopting the NGSS?
Some of the advantages of the NGSS standards are:
- Previous national standards are out-of-date
- Emphasis on how to use science in the real world
- Helps prepare students for STEM-related careers
- Helps students to solve problems as opposed to only learning facts
- States can save money by not having to develop their own standards
- Links the different science disciplines together
What are some of the cons of the NGSS?
There are also potential drawbacks to adopting the new standards. Some questions are:
- Will adequate teacher training be available?
- Are the standards too specific, and do they remove some of the creativity from teachers and students?
- What is the cost to implement the new standards?
- Will elected officials, students, teachers, and parents buy into the idea of uniform standards across state lines?
The NGSS are backed by research, and they were developed by both scientists and educators. As with any new development in education, many states are waiting to see how other states fare with the new standards. Only time will tell if NGSS are the answer to improving science education in the United States.
The College Board’s Advanced Placement programs have long been a staple of American high schools. For some students, the prestige of having an AP class on one’s resume draws them to the demanding courses, while parents may entrain visions of massive tuition savings once college hits. But dollars and status aside, the courses require students — and teachers — to strive to keep up with the rigors of the curriculum.
That curriculum, though, has garnered much attention in recent years. Updates take place regularly, of course, but rarely do they solicit any sort of notice outside of the teachers who have to revamp their plans every decade or so. No doubt you remember this was not the case in 2014 when the AP US History test experienced an overhaul. Critics, both in academia and in the general public, blasted the new standards for downplaying American exceptionalism and for emphasizing concepts over facts. Bowing to the pressure, the College Board quickly released a re-revamped curriculum in August of 2015. Since then, national media has resumed its “radio silence” of nearly all things AP.
Teachers, of course, understand that changes (likely less controversial) are always afoot. The 2016-17 school year saw major changes to Calculus AB, Calculus BC, and World History courses. The Calculus changes, according to some, were more tweaks than fundamental changes. World History, however, saw a massive reorganization of both the curriculum and the exam. Unlike APUSH’s first revision, the APWH curriculum changes were not only less (or “non-“) controversial but rich with details and specifics. Its testing format also emphasizes both facts and historical thinking skills. The efficacy of these course changes will only be known once the exams are graded and tabulated in July.
So what’s on tap for the 2017-18 school year? According to the College Board’s “Advances in AP” website, nothing new — something that, from personal experience, is welcomed news to AP teachers around the globe. The next major overhaul looks to be U.S. Government in Politics in 2018-19, the first in a decade. These changes promise a “deeper conceptual understanding of political processes” rather than a memorization of facts and specific Court cases. Students will be expected to interpret data and draw conclusions from those sources. In general, the expectations mirror many of those in the revised US History and World History courses. And as with those previous courses, teachers of this AP course will be expected to submit a revised course syllabus for an audit review. But again: this will not go into effect until the 2018-19 school year.
For AP U.S. Government teachers: you may want to get to work. For everyone else: we can hear your sigh of relief from here.
About the Author
Kirby Spivey taught AP World History, US History, and many other Social Studies courses in Georgia. He and his wife live in Atlanta
- Projector Resources
- Educational Games
- Practice Tests
- Vocabulary Flashcards
- Performance Tasks
To preview our NGSS-aligned content and test prep, request your 7-day trial. Our NGSS-aligned test reviews have been prepared by our subject area experts. In them, you will find questions that follow the course requirements, as well as items to reinforce the basic components of the course.
Dozens of authors, all certified teachers, across multiple states have been working all summer to:
- Update or create over 200 tests
- Add thousands of new items to meet new standards
- Make updates at all grade levels across 28 different states
- Update national products including NAEP and new AP Calculus and AP World History
May means the beginning of the summer development cycle for USATestprep’s content team. We’ll be making new tests and updating existing ones for next school year.
The performance task, video, and free-response item fill projects are just about wrapped up, and tremendous progress has been made. Below are the number of standards we have filled since January. These are elements which now have at least one of each item type in them:
- Science: +339 performance tasks; +242 videos; +1,118 free response items
- Math: +230 videos; +2,140 free response items
- Social Studies: +425 performance tasks; +160 videos; +2,300 free response items
- ELA: +333 videos; +116 free response items
If your state DOE makes changes to standards or assessments, USATestprep is committed to match these changes and provide you with an up-to-date product at no additional cost.
We are happy to announce the completion of our review for Alabama 3rd Grade Social Studies! This is available for immediate purchase, but if you would like a FREE 7-day trial, please email or call us.