The New Math Stumbling Blocks

The New Math

Stumbling Blocks

The change from solving equations to analyzing functions seems benign, but that has not stopped the Common Core from becoming a charged political issue. Currently 42 states plus the District of Columbia use the standards, with adoption motivated in part by financial incentives provided by the Obama administration’s Race to the Top initiative — a top-down tactic that has helped fuel blowback. There have been plenty of other complications, too, from parents complaining that they don’t know how to help their first-graders with their math homework, to concerns that the assessments that accompany the Common Core are too hard. As a result, even stalwart adopters are questioning whether the standards work. In December 2015, Governor Andrew Cuomo of New York announced that his state would undertake a “total reboot” of the Common Core math standards in the coming years.

 

 

 

 

 

 

 

 

The designers of NGSS, which came out three years after the Common Core without any kind of federal mandate, say they learned from the contentious rollout of the earlier standards. So far, 17 states plus the District of Columbia have adopted NGSS and 11 more states have implemented standards that are similar to varying degrees.

“The Common Core got people to sign on and implement standards before the standards were there, and I think that backfired,” Schweingruber said. “I feel like the intent of the standards is to improve what happens to kids in classrooms, and if that happens even before a state formally adopts, that’s fine with me.”

Still, NGSS has had its controversies. The document includes standards related to climate change and evolution, which has motivated opposition in conservative states. And, politics aside, the standards necessitate sweeping changes to the way science is taught.

Like Common Core math with its long-running development of core concepts, NGSS reframes science in terms of a small number of basic ideas that inform the scientific perspective. These include “structure and function,” “patterns,” “cause and effect,” “stability and change,” and “systems and systems models.”

“Even at a young age you’re going to have a workable knowledge of energy so you can apply it,” said Joseph Krajcik, a professor of science education at Michigan State and the lead author of the NGSS physical science standards. “At a third-grade level you might know that as something is moving, it has energy, and the faster it’s moving, the more it can do something. It’s a nascent idea of what energy is, and it builds across time.”

This slow-building approach is at odds with some aspects of public education. It’s not uncommon for districts to require that each class period address a discrete objective, and teachers are expected to measure whether students learned it at the end of the period. The authors of Common Core math and NGSS don’t see their disciplines fitting into that structure.

“One insight we got is that there’s almost no mathematics worth learning that breaks into lesson-size pieces,” Daro said. “You have a three- or four-week sequence and treat it with coherence. It’s about systems and structures, not small facts and small methods. It’s about how it all works together.”

Schweingruber agrees. “Some of these ideas in science are hard to get quickly,” she said. “It took humans hundreds of years, so why would kids figure them out quickly?” The same mismatch between the standards and the way public education is set up occurs in another major area: assessments. Because standardized tests often drive instruction, it’s hard to expect teachers to teach differently unless students are tested differently.

“Teachers are starting to make changes in their classrooms,” Schweingruber said, “but if they’re still looking out for a large-scale test their kids will have to take that is completely contrary to what they’re doing in the classroom, that can be problematic.”

There is progress in that direction. Two recent initiatives, the Partnership for Assessment of Readiness for College and Careers and the Smarter Balanced Assessment Consortium, are developing standardized tests that incorporate a greater variety of question types, like constructed response questions in which students are asked to explain their reasoning, and technology-enhanced questions in which, for example, students manipulate a line on a graph to make it match a given algebraic function. “You’re seeing a deeper push for conceptual understanding and the ability to apply mathematics, and assessments are on their way to becoming equipped to actually assess that,” said Robert Kaplinsky, a math teaching specialist and consultant in Southern California.

Chuck Reynolds
Contributor

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The New Math Less Is More

The New Math
Less Is More

How does one adjust the course of a curriculum that’s been gathering inertia for decades? The developers of NGSS and Common Core math started by reducing the mass of content that had accumulated over the years, often in haphazard fashion. “Mainly, the U.S. mathematics curriculum prior to the Common Core was a geological accretion of additions, mostly, and [some] compressions over 50 years,” Daro said. “There was a lot of mathematical junk food and traveling down rabbit holes and up cul-de-sacs.”

Schweingruber made a similar point. “The U.S. has a mile-wide, inch deep curriculum with tons and tons of things and ideas for kids to learn, but not an opportunity to go in depth,” she said. As the authors got down to work on Common Core in 2009 and on NGSS a year later, some of their first discussions were about what to leave in and what to take out. “It required some argument on the part of folks in the framework about what that baseline really would look like,” Schweingruber said.

The final documents omitted a number of familiar topics. The NGSS writers eliminated instruction in the rote formula for stoichiometry calculations (the process for quantifying elements at different stages of a chemical reaction) from the high school chemistry curriculum. Daro and his collaborators on Common Core math, William McCallum of the University of Arizona and Jason Zimba of Student Achievement Partners, decided the technique of “simplifying” answers didn’t add much to mathematical understanding, so they took it out.

By removing content, the creators of Common Core math and NGSS hoped to expose core disciplinary ideas. A good example of this is how the Common Core teaches proportionality. Before, proportionality occupied about 10 percent of math instruction in grades six and seven. The main outcome of all that instructional time was that given two equivalent fractions, students could cross-multiply in order to find a missing term.

“What they’re learning is: The way you find the fourth number is by setting up this gadget called a proportion,” Daro said. “That’s not really learning anything about proportionality, that’s learning how to get answers to problems in this chapter.”

Common Core math doesn’t mention cross-multiplying, and it cuts out the special case of finding a missing fourth term. Instead, it focuses on the idea of a ratio, which begins modestly in sixth grade and develops all the way through calculus. Students begin by looking at a table of equivalent ratios — also presented as a double number line — and progress to the understanding that the slope of a line is a ratio.

“[The Common Core writers] said, look, let’s figure out what’s important about fractions and choose a path through them, which leads to ratio and proportion, which leads to linear functions, which leads to aspects of algebra,” said Alan Schoenfeld, a professor of education and mathematics at the University of California, Berkeley.

The understanding of slope as a ratio feeds into an even more fundamental emphasis in Common Core math: the analysis of functions. By thinking about the slope of a line as a ratio, students get in the habit of analyzing the parts of a linear function so they can see how changes in elements of the function affect the relationship between inputs and outputs.

Daro sees this shift from solving equations to analyzing functions as one of the biggest conceptual changes in the Common Core.

“The important line of progress is the line that begins with the theory of equations, a 19th-century central focus, to calculus and analysis, which is 20th-century [mathematics],” he said. “It’s a move from spending almost all your time solving equations towards analyzing functions.”

Chuck Reynolds
Contrbutor

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God teaches His people to avoid sin and to live in His ways.

God’s Power in Your Life: Living God's Will
God teaches His people to avoid sin and to live in His ways.

Written by Hope on 18/10/2016
Series: Weekly Devotional
Tags: Life, Power
Everything in the Scriptures is God’s Word. All of it is useful for teaching and helping people and for correcting them and showing them how to live.

II Timothy 3:16
Do you know that God is calling YOU to live out an extraordinary life by following His commandments and seeking His will for your life? After you have experienced the power of God’s forgiveness, you can allow His power to remake your life in His ways.

Here are ways you can learn to listen for His teachings:

1.Obey God’s Word
“What God has said isn’t only alive and active! It is sharper than any double-edged sword.” (Hebrews 4:12)! The Bible is full of stories that tell of God’s love and hopes for His people. Reading and studying it regularly will help you come to know Him better. When you have questions about what you should do, use God’s Word as “a lamp that gives light wherever you walk” (Psalm 119:105). The stories of the people featured in God’s Word often present great examples of how we should behave (or how not to behave!). No example is better than that of God the Son, Jesus Christ.

2. Jesus Was Called “Rabbi,” Which Means “Teacher”
How can we know that God wants us to learn? Well, while He lived on earth, Jesus took the title of teacher — He was famous for it: even when potential enemies met with him, they said, “‘Teacher, we know that you are honest. You teach the truth about what God wants people to do. And you treat everyone with the same respect, no matter who they are.’” (Matthew 22:15-16). Jesus taught His followers, “If you love me, you will do what I have said, and my Father will love you. I will also love you and show you what I am like.” (John 14:21). Learn what Jesus has to say, and learn to obey those instructions, depending on His power to make it possible for you to do so.

3. Let God Teach You
“I will point out the road that you should follow. I will be your teacher and watch over you.” (Psalm 32:8). God’s power is so great that He knows you: He knows who you really are; He knows your situation; He knows what you want and what you need! Tell Him what you need, and listen for His guidance. The Apostle Paul advised, “Let the message about Christ completely fill your lives, while you use all your wisdom to teach and instruct each other” (Colossians 3:16). Ask trusted Christian friends, family, and leaders for godly support and prayer. Compare their advice with what you find in God’s Word (Acts 17:11) as you seek the power of God’s will in your life.

Pray this week:

O Lord, along with Psalm 25:4, I pray that You will show me the right path; and point out the road for me to follow. Amen.

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The New Math The Old New Math

The New Math

The Old New Math

NGSS and the Common Core are a significant departure from the way science and math have been taught, but they did come out of nowhere. In fact, they’re consistent with a trend that’s been slow-boiling for a half-century.

In a 2010 paper, Baker and colleagues analyzed 141 elementary school math textbooks published between 1900 and 2000. They found that what kids were learning changed considerably during that period. Until the 1960s, basic arithmetic accounted for 85 percent of math instruction. By the end of the century, that proportion had dropped to 64 percent, with the balance of instruction devoted to more complex topics like advanced arithmetic and geometry.

“When you step back historically and sociologically, it’s clear education has really ratcheted up along these cognitive dimensions,” Baker said. “The idea that education is like men’s ties and just goes through this cycle of wide and thin is not true.”

Pedagogy has shifted as well. During the same period in which students began to learn more complex mathematics, leaders in science and math education launched complementary pushes to teach students to think more like real scientists and mathematicians. These efforts included the “New Math” of the 1960s and similar plans that decade to teach science as an “inquiry into inquiry,” as one leading expert of the time put it. Later manifestations of the impulse away from rote instruction include curricular standards created by the National Council of Teachers of Mathematics in the 1980s and the enthusiasm for “inquiry-based” science in the 1990s.

All of these initiatives had the right idea, but their implementation was off, say developers of NGSS and Common Core math. “Inquiry” is a habit of mind among scientists, but in the 1990s it was taught as its own curricular topic: Last week we learned about DNA, this week we’re going to learn about an inquiry.

“Inquiry became almost an empty word, where it didn’t really matter what the inquiry was about,” said Heidi Schweingruber, director of the Board of Science Education at the National Academies of Sciences, Engineering, and Medicine, which provided guidance for the development of NGSS.

The same problem happened in math. For the last 50 years, reformers have wanted to teach kids to reason mathematically, to think nimbly about topics like quadratic equations that otherwise come off flat. Instead, in programs that employed the New Math, students often ended up playing logic games.

“The push toward conceptual understanding and understanding rich mathematical ideas sometimes ended in practice with students just engaged in activities and messing around,” said Robert Floden, dean of the College of Education at Michigan State University.

It’s not surprising that ambitious changes like these would be hard to implement. After all, teaching kids to adopt a scientific mindset is a subtle and more complex task than having them memorize the parts of a cell. For one thing, it requires teachers who inhabit that mindset themselves, and they’re harder to find. For another, it takes a more patient perspective than the prevailing one in public education, which expects teachers to post a learning objective on the board before each class and end every unit with a multiple-choice test.

Chuck Reynolds
Contributor

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The New Math Unlike the Old Math

The New Math
Unlike the Old Math

The latest effort to overhaul math and science education offers a fundamental rethinking of the basic structure of knowledge. But will it be given time to work?

If we could snap our fingers and change the way math and science are taught in U.S. schools, most of us would. The shortcomings of the current approach are clear. Subjects that are vibrant in the minds of experts become lifeless by the time they’re handed down to students. It’s not uncommon to hear kids in Algebra 2 ask, “When are we ever going to use this?” and for the teacher to reply, “Math teaches you how to think,” which is true — if only it were taught that way.

To say that this is now changing is to invite an eye roll. For a number of entrenched reasons, from the way teachers are trained to the difficulty of agreeing on what counts in each discipline, instruction in science and math is remarkably resistant to change.

That said, we’re riding the next big wave in K-12 science and math education in the United States. The main events are a pair of highly visible but often misunderstood documents — the Common Core math standards and the Next Generation Science Standards (NGSS) — that, if implemented successfully, will boldly remake the way math and science are taught. Both efforts seek to recast instruction in the fundamental ideas and perspectives that animate the two fields.

“What we did in reorganizing the content of school mathematics was long overdue,” said Phil Daro, one of three lead authors of the Common Core math standards.

The changes go beyond the contentious new methods of teaching arithmetic that is grabbed headlines and threatened to blunt the momentum of Common Core math. Both documents developed out of decades of academic research on how children learn, and they reflect similar priorities. They exhibit an elegant rethinking of the basic structure of knowledge, along with new assertions of what’s important for students to be able to do by the time they finish high school.

“Overall, there’s a movement towards more complex cognitive mathematics, there’s a movement towards the student being invited to act like a mathematician instead of passively taking in math and science,” said David Baker, a professor of sociology and education at Pennsylvania State University. “These are big trends and they’re quite revolutionary.”

Pedagogical revolutions are chance endeavors, however. The Common Core math standards were released in 2010 and NGSS in 2013. Now, years on, even enthusiastic early adopters of the Common Core like the state of New York are retreating from the standards. While the ultimate impact of both the Common Core and NGSS is still uncertain, it’s clear these standards go beyond simply swapping one set of textbooks for another — to really take hold, they’ll require a fundamental rethinking of everything from assessments to classroom materials to the basic relationship between teachers and students.

Chuck Reynolds
Contributor

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Here’s Why You Should Never Throw Away Your Banana Peels… Who Knew They Were So Healthy?

Bananas are one of nature’s greatest treats. Not only do they have a distinct, delicious flavour, they’re also packed with vitamins and minerals, including potassium and magnesium. They’re also a great source of fiber and they’re the go-to food for athletes who need a fast-acting carbohydrates to get a quick energy boost. Awesome!

But what about the peel? Sure, you can use them to decimate your opponents in a round of Mario Kart, but outside of video games, you probably don’t really think twice about them and you just throw them away. But as it turns out, chucking your banana peels is a huge mistake because they may actually be the best and healthiest part of the fruit! How is that possible? Well, check out these surprising ways that you can be using banana peels.
Believe it or not, you can actually eat raw banana peels. But before doing that, make sure you clean them thoroughly. Some people eat the peels just like that, while others choose to blend them with other fruits. But if raw isn’t really your thing, you can try boiling them for about 10 minutes before eating them. If the idea of eating the peel is still too much for you, consider scraping out as much of the insides as possible with the spoon and consuming this portion. Benefits include…

reducing cholesterol and preventing cardiovascular disease and strokes; the peel contains more fiber than the banana itself!
boost your mood, as the peel contains amino acids to trigger your serotonin
reduce the risk of cataracts and macular degeneration thanks to the banana peel’s lutein, a powerful antioxidant that protects your eyes
You can also apply the banana peel to different parts of your body for some amazing benefits:

whiten your teeth by rubbing the inner part of the banana peel on your teeth on a daily basis
reduce wrinkles by rubbing the banana peel on your face, leaving the residue on for about half an hour before washing it off
eliminate warts by securing a piece of banana peel on the affected area with a bandage and leaving it overnight; repeat until the wart falls off
sooth the itch and pain of bug bites by massaging the peel on the affected area

Read more at http://www.metaspoon.com/peel-banana-benefits-health/?cat=exerheal#cEm0dolOvswHhksE.99

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Same-Sex Marriage | The Truth About…

What does the Bible teach about homosexuality? How does God feel about same-sex marriage? Have we been misusing the Bible when citing it to condemn gay and lesbian behavior? Did Jesus ever mention it? Is it time for Christians to finally accept that homosexuality is a civic right equivalent to racial equality? In this lesson, Don Blackwell shines the light of God’s truth on these and other questions that are trending cultural topics.

 

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The Amazing Power of Blogging

The Amazing Power of Blogging

Original article by Tony Cordingley

So what is a blog? After all, it just looks like a regular website to many people. In fact, it is one of the best ways of holding a conversation online. You can hold this conversation with as many people as you want and it is only limited by the amount of people who come to your blog.

What is a blog?

It is a great way to connect with people especially those with similar interests to yourself. Simply put up interesting or controversial thoughts and opinions and invite comment. If you have a particular passion you can easily attract interest from similarly passionate people and stimulate further interest in the subject.

Of course, it hasn't taken long for commercial interests to cotton on to this and see the money making opportunities inherent in this technology. The main object here is to establish yourself as an authority on a particular subject and to build a profile for yourself.

This can work for anybody from an internet marketer selling money making products to your local real estate agent or neighbourhood lawyer. However, building credibility has it's cost. You need to maintain contact with your readership and regularly write interesting and valuable content for your blog. This regular input is the most daunting aspect of blogging unless you are a skilled and competent writer. Fortunately, these written pieces do not have to be very long but they do need to be regular, especially if you are also hoping to get noticed and indexed by the search engines. Getting on the first page of Google is incredibly good for business.

Blog

Be careful to avoid blatant advertising in your blogging. Otherwise, it is all too easy to put people off. A casual recommendation is the best promotion. Building your profile and authority will have all the positive spin-off to bring people knocking on the office door. Encourage questions that can showcase your knowledge in your answers.

Google alerts can be emailed to your email address daily and can be a source of inspiration on what to write or discuss with your readership. Regular attendance at online forums on your subject will also give you ideas on what is of current interest on your subject.

How to start a blog

Blogging can be purely a platform to share your passion on a chosen subject or it can be a great way to build your business. If it is for building your business be careful not to be too obvious about it. Give good content, good advice and be open and conversational and you can't go wrong.

Blogging is incredibly easy to set up. Google's Blogger.com or WordPress are very easy to set up even for those challenged by technology. And there is now also a new blogging platform for entrepreneurs called Marketive. There is no need to do anything fancy just blog regularly with good content and try to remain current with what is happening in your niche.

 

Start your free MARKETHIVE Blog today, join Markethive! >>> https://markethive.com/stephenhodgkiss

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 Don’t let anyone tell you what you can’t do.

Peter Kesaris
Director of Architecture & Design Customer Experience at Empire Office
6 years ago to date, I was hit by a car and nearly killed. After being stuck in a wheelchair for 6 months, doctors didn't know if I would be able to walk normally again. This week I ran my first half marathon.  Don't let anyone tell you what you can't do.  Follow my story on Instagram: @pakjan6 (edited) 

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How is life for self-employed workers?

Being self-employed is a way of life for almost five million people in the UK now, with some deliberately opting for a life of working independence and others being forced to adopt it.

Figures released by the Resolution Foundation this week said average earnings for self-employed workers are now lower than in 1994-95 but that the UK's self-employed workforce had grown by 45% in the past 15 years.

How has the life of a self-employed worker changed over the years – and what benefits or drawbacks does it bring? Some have shared their stories.

The session singer

Sam Blewitt

JANINE RASCH

"Payments to session singers have increased little in real terms in the past 20 years"

Payments to sessions singers have changed very little in real terms over the past 20 years, according to Sam Blewitt.

The Twickenham-based singer has been self-employed for 30 years and has sung with Madness, The Streets, Dizzee Rascal and Ultravox, as well as on soundtracks for films, TV shows and commercials.

"There have been good times and bad, but generally, I am working for the same as I was 10 or even 15 years ago," said Mr Blewitt. "I do believe I was earning more 20 years ago.

"Then the added impact of the change in people's music-buying habits have really changed the way self-employed musicians and singers earn money. My publishing income from music sales is virtually non-existent these days."

The cleaner

Darren Smith

DARREN SMITH

On his 40th birthday Darren Smith decided to make some major life changes including giving up his job in a large shop-fitting company to set up his own cleaning business.

"I was living my life out of a suitcase week in, week out, and though the money was good, when did I get the time to spend it?" said Mr Smith, from Eastwood, Nottinghamshire.

He got a part-time job packing computer components to ensure he had some money coming in while he worked on building up a client base, which involved designing leaflets and delivering them himself.

Nine years on Mr Smith – although earning less – says his decision to become a sole trader was the right one.

"I've taken a big drop in wages, but I have cut my cloth accordingly," he added. "My partner and I live a frugal, but in my eyes, a very rich life. My work-life balance is much better, and yes, there are times when I'd pack it all in tomorrow, but doesn't everyone have days like that?"

The homeopath

Suzanne Wright

SUZANNE WRIGHT

A patchwork career is a modern way of balancing motherhood with work

Suzanne Wright found that when she became a self-employed homeopath her hourly rate was actually higher than when she was employed as a part-time distribution manager. So while her income dropped, it was because she was working fewer hours.

The mum-of-two from Northampton decided to work for herself because she and her husband found it difficult to arrange flexible childcare for their primary school-aged children.

"A lot of people have started up as self-employed in recent years," said Mrs Wright. "Many, like myself, are mums seeking to work for themselves so that they can manage their working hours around school times, so that they can be with their children.

"A patchwork career is a modern way of balancing motherhood with work – it generally means less income, but also more family time."

Mrs Wright said she does have to spend additional time working on marketing and keeping her financial records up-to-date: "I get more career satisfaction now because I'm building something of my own. My husband is in full-time, permanent work, so I do know the mortgage will always be paid."

The shop owner

Chris Petterson

CHRIS PETTERSON

For the last decade Chris Petterson has owned and run three greeting card shops, but found he had to use his pension from a previous job to supplement his income.

Mr Petterson, from Wakefield, North Yorkshire, has had to work full-time in one of the shops himself but has only been able to pay himself less than the minimum wage.

He said: "I have eight part-time staff but the costs with pay changes and pensions has gradually risen as have rates, rent, utilities, whereas business has plateaued over the same period.

"One of my big frustrations is the unfairness of business rates. I have two shops of similar retail space, but one shop has a rateable value of £6,000 whilst the other is £13,500."

However, he said the positive side of being self-employed was being able to spend more time with his grandchildren.

If you believe that my message is worth spreading, please use the share buttons if they show on this page.

Stephen Hodgkiss
Chief Engineer at MarketHive

markethive.com


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