# 6: professor playground - Mathematics

• 6.1: 0.0 Special Symbols
Some symbols
• 6.2: The Derivative as a Function
• 6.3: Differentiation Rules
• 6.4: Product and Quotient Rules
• 6.5: Derivatives of Exponential and Logarithmic Functions
In this section, we explore derivatives of exponential and logarithmic functions. As we discussed in Introduction to Functions and Graphs, exponential functions play an important role in modeling population growth and the decay of radioactive materials. Logarithmic functions can help rescale large quantities and are particularly helpful for rewriting complicated expressions.
• 6.6: Limits at Infinity and Asymptotes
We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function f defined on an unbounded domain, we also need to know the behavior of f as x→±∞ . In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function ff.
• 6.7: originalThe Definite Integral
If f(x) is a function defined on an interval [a,b], the definite integral of f from a to b is given by [∫^b_af(x)dx=lim_{n→∞} sum_{i=1}^nf(x^∗_i)Δx,] provided the limit exists. If this limit exists, the function f(x) is said to be integrable on [a,b], or is an integrable function. The numbers a and b are called the limits of integration; specifically, a is the lower limit and b is the upper limit. The function f(x) is the integrand, and x is the variable of integration.
• 6.8: original The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus gave us a method to evaluate integrals without using Riemann sums. The drawback of this method, though, is that we must be able to find an antiderivative, and this is not always easy.
• 6.9: Original Integration Formulas and the Net Change Theorem
The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero. The area under an even function over a symmetric interval can be calculated by doubling the area over the positive x-axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative.
• 6.E: Applications of Derivatives (ALL Chap 4 Exercises)
These are homework exercises to accompany Chapter 4 of OpenStax's "Calculus" Textmap.
• 6.E: Derivatives (ALL Chapter 3 Exercises)
These are homework exercises to accompany Chapter 3 of OpenStax's "Calculus" Textmap.
• 6.E: Integration (Exercises)
These are homework exercises to accompany Chapter 5 of OpenStax's "Calculus" Textmap.
• 6.E: Limits (Exercises)
These are homework exercises to accompany Chapter 2 of OpenStax's "Calculus" Textmap.
• 6.E: Open Stax 4.1 - 4.5 Exercises
These are homework exercises to accompany Chapter 4 of OpenStax's "Calculus" Textmap.
• 6.10: Geogebra Calculus

## Number problem moves Morehouse professor: too few black math teachers

Nathan Alexander is on a mission to recruit and encourage more black men to teach mathematics, as studies suggest simply having black men as teachers has a significant impact on black students.

Alexander, a researcher and visiting professor in Morehouse College’s department of mathematics, founded the Black Male Mathematics Teacher Project, which connects black men who teach math across all grades and organizations, from K-12 classrooms to community-based programs and adult-education programs. U.S. Department of Education figures show just 2% of the country’s teachers are black men. Alexander said about .69% of those black men teach high school math.

“Seeing a black man who teaches math … is powerful,” Alexander said. He uses math as the project’s focus because of its many uses in everyday life, as well as it being a personal love of his.

In Georgia, the number of black teachers is slowly on the rise. During the 2017-2018 school year, there were 113,122 teachers in the state's workforce and approximately 21.4% of them were black, up from 20.2% during the 2015-2016 school year, according to Georgia K-12 Teacher and Leader Workforce Status Reports from the Governor's Office of Student Achievement. During the 2017-2018 school year, 24% of new teachers were black.

In metro Atlanta, 8% of teachers are black men. About 80 percent are women, and more than two-thirds of them white.

Research suggests the presence of black teachers in early grades can greatly influence a black student's future. A 2018 study by researchers from Johns Hopkins University and American University suggests black students who have one black teacher by third grade are 13 percent more likely to go to college. Two black teachers, and a black student is 32 percent more likely to go to college.

Alexander points to his own experience and the man who influenced him: a black high school algebra teacher named James Rivers who encouraged him to be more active in math events at Monroe High School in Monroe, N.C., just outside of Charlotte. Rivers, Alexander said, pushed him to enter area math competitions and to become president of the school’s math club.

“I believe that’s why I got into mathematics,” Alexander said.

The Black Male Mathematics Teacher Project, which began in 2016, hopes to identify, prepare and retain black men who teach math across different education settings. It studies professional practices, beliefs and attitudes of black men teaching math and wants to create groups to support and encourage black men to enter the profession.

A teacher's approach to lessons has great impact, Alexander has seen. While he was a University of San Francisco assistant professor, annual trips to Belize with students paired him and several students with a handful of black teachers keen on improving how they thought about teaching. The trips were part of Project Learn Belize — where students and faculty work to immerse themselves in the country's culture.

He noticed people were not traveling out of their communities to teach. "There was a certain tenderness and a certain mindset they possess," he said of the Belizean teachers. "When I looked at the teachers here, a lot of my work was convincing teachers to be of the community."

While working in San Quentin State Prison through the University of California-Berkeley, Alexander found himself co-teaching with Detroit, an inmate serving a life sentence. The man earned several associate degrees in prison and taught math for 15 of the 25 years he spent there. Alexander said he learned a lot during that experience, including not to underestimate the role — or awareness — of community-based teachers.

Detroit, who left San Quentin in 2016, said he enjoyed helping others understand the concepts around math and science, often using real world scenarios that fit their lifestyles.

“I was just a guy from the ‘hood doing a life sentence, and I tried to show them a different way of doing math … that they would be receptive to,” said Detroit, now a math tutor for a public charter school group in San Francisco. “In prison, I did it getting paid 15 cents an hour for 15 years.”

According to data from the project’s first year, black men are just as likely as other teachers to have math certifications, but are more likely to have only the minimum education needed when teaching high school math and more likely to have low to moderate success.

Recruiting men to the teaching ranks could be an uphill battle.

“Somehow, teaching has increasingly become an occupation of choice for women,” said Richard Ingersoll, an education and sociology professor at the University of Pennsylvania.

Ingersoll was speaking to reporters and educators in Atlanta Tuesday as part of the Critical Issues Forum organized by the Georgia Partnership for Excellence in Education, a series of lectures that discuss issues important to Georgia education. He said teaching jobs were initially created for women as a not-too-long-term role that women were expected to abandon as they got married and put more focus on raising families.

Ingersoll's research indicated that while the number of teachers of color was increasing, the number who quit their jobs was high as well. Teachers of color largely work in struggling schools, where turnover often is higher.

“There is big concern about the trend,” he said. “It’s not about the person. It has to do with the schools.”

Alexander said reasons so few black men teach math are rooted in discriminatory practices, such as the Negro Act of 1740, a South Carolina law that made it illegal for enslaved Africans, among other things, to earn money or learn to write. Though the law was voided in 1865, its impact remained for years as many black residents were denied fair education and access to higher education.

“Math was being used to keep us out of participating as a society through even voting,” he said.

He cited the need for Bob Moses' Algebra project — which began in the 1980s recognizing a lack of math literacy for black people — as evidence that repercussions from centuries-old laws persist.

Alexander said he is encouraged by his students at Morehouse, including several who say they plan to teach math after graduation.

## Scientists Say Child's Play Helps Build A Better Brain

Deion Jefferson, 10, and Samuel Jefferson, 7, take turns climbing and jumping off a stack of old tires at the Berkeley Adventure Playground in California. The playground is a half-acre park with a junkyard feel where kids are encouraged to "play wild." David Gilkey/NPR hide caption

Deion Jefferson, 10, and Samuel Jefferson, 7, take turns climbing and jumping off a stack of old tires at the Berkeley Adventure Playground in California. The playground is a half-acre park with a junkyard feel where kids are encouraged to "play wild."

This week, NPR Ed is focusing on questions about why people play and how play relates to learning.

When it comes to brain development, time in the classroom may be less important than time on the playground.

"The experience of play changes the connections of the neurons at the front end of your brain," says Sergio Pellis, a researcher at the University of Lethbridge in Alberta, Canada. "And without play experience, those neurons aren't changed," he says.

Our friends at MindShift have been looking at the role of play in learning. Play is as much a part of childhood as school and an organic way of learning. Check out these articles that dig into play:

Free, unstructured play is crucial for children to build the skills they'll need to be happy, productive adults.

At a school where free play and exploration are encouraged, children can educate themselves under the right conditions.

Many children in public school are getting less and less time outside, despite the documented benefits of free play.

It is those changes in the prefrontal cortex during childhood that help wire up the brain's executive control center, which has a critical role in regulating emotions, making plans and solving problems, Pellis says. So play, he adds, is what prepares a young brain for life, love and even schoolwork.

But to produce this sort of brain development, children need to engage in plenty of so-called free play, Pellis says. No coaches, no umpires, no rule books.

"Whether it's rough-and-tumble play or two kids deciding to build a sand castle together, the kids themselves have to negotiate, well, what are we going to do in this game? What are the rules we are going to follow?" Pellis says. The brain builds new circuits in the prefrontal cortex to help it navigate these complex social interactions, he says.

Learning From Animals

Much of what scientists know about this process comes from research on animal species that engage in social play. This includes cats, dogs and most other mammals. But Pellis says he has also seen play in some birds, including young magpies that "grab one another and start wrestling on the ground like they were puppies or dogs."

For a long time, researchers thought this sort of rough-and-tumble play might be a way for young animals to develop skills like hunting or fighting. But studies in the past decade or so suggest that's not the case. Adult cats, for example, have no trouble killing a mouse even if they are deprived of play as kittens.

Where does play come from? Neuroscientist Jaak Panksepp gives a playful answer in this NPR animation.

So researchers like Jaak Panksepp at Washington State University have come to believe play has a very different purpose: "The function of play is to build pro-social brains, social brains that know how to interact with others in positive ways," Panksepp says.

Panksepp has studied this process in rats, which love to play and even produce a distinctive sound he has labeled "rat laughter." When the rats are young, play appears to initiate lasting changes in areas of the brain used for thinking and processing social interactions, Panskepp says.

The changes involve switching certain genes on and off. "We found that play activates the whole neocortex," he says. "And we found that of the 1,200 genes that we measured, about one-third of them were significantly changed simply by having a half-hour of play."

Of course, this doesn't prove that play affects human brains the same way. But there are good reasons to believe it does, Pellis says.

An overview of the Berkeley Adventure Playground, where children and their parents can paint, hammer, saw and run free. David Gilkey/NPR hide caption

An overview of the Berkeley Adventure Playground, where children and their parents can paint, hammer, saw and run free.

For one thing, he says, play behavior is remarkably similar across species. Rats, monkeys and children all abide by similar rules that require participants to take turns, play fair and not inflict pain. Play also helps both people and animals become more adept socially, Pellis says.

And in people, he says, an added bonus is that the skills associated with play ultimately lead to better grades. In one study, researchers found that the best predictor of academic performance in eighth grade was a child's social skills in third grade.

Another hint that play matters, Pellis says, is that "countries where they actually have more recess tend to have higher academic performance than countries where recess is less."

## Michigan Telecommunications & Technology Law Review

The entertainment industry has a history of framing new technology as piracy that threatens its very existence, regardless of the potential benefits of the technology or the legal limits of copyright rights. In the case of YouTube, copyright owners' attempts to retain content control negatively impact the public's ability to discuss culture in an online world. This implicates the basic policy behind fair use: to prevent copyright law from "stifl[ing] the very creativity which that law is designed to foster." The internet has become a powerful medium for expression. It is a vital tool in today's world for sharing original works, but is equally important as a forum for discussion of existing works. YouTube blurs the the line between publication and everyday conversation. It enables the sharing of culture, ideas, and debate in ways previously impossible, and therefore plays an important and progressive role in our society. In Part II, I will explain how YouTube works, how potential copyright infringement affects the website and its users, and describe the basics of the fair use defense. In Part III, I will argue that YouTube's open method of content distribution is important to our culture, and argue that fair use needs to be a flexible standard that protects the majority of YouTube content as non-infringing.

Get outside and explore geometry (and other math) all around you. A math trail is a walk with various stops where you look at math in the world around you, and ask questions about it.

KaBOOM!
Find a playground in your neighborhood, complete with reviews and photos. You can also upload photos of your own.

National Math Trails
Discover math in the world around you.

Voronoi Toy
This open-source program lets users play with adding points to a Voronoi diagram.

Geometry Games
A number of downloadable games that let you explore topology, polygons, tilings, and more.

Geometry Playground
This is a free ruler and compass application for multiple geometries. (Not related to the exhibition.)

subblue

Cinderella
Interactive geometry software. The current version is not free, but the older version is. Visit the &ldquoDownload&rdquo page to find it.

SketchUp
Free 3D modeling software from Google.
Then go here for polyhedron models to use in SketchUp.

## Welcome to the Department of Mathematics

Mathematics is the language of science and technology. It is the language used to translate real world problems into a form in which a solution can be found. It is the goal of the department to provide all students with the mathematical foundation they need for their careers and for lifelong learning.

The mathematics department offers a full range of courses, from non-credit classes for students whose mathematics preparation is insufficient for credit-level work to those designed for students in all degree programs at the college. The mathematics department also offers three degree programs, an associate degree program (AS) in Computer Science, a baccalaureate degree program (BS) in applied mathematics and a baccalaureate degree program (BS) in mathematics education.

The computer science associate degree program provides students with the frst two years of study leading to a baccalaureate degree in computer science, computer information science, computer systems technology, computer engineering technology or applied mathematics. Computer science is an excellent feld of study for those seeking career opportunities in the worlds of business, education, government and industry.

The applied mathematics baccalaureate degree program is a practical degree for students with an interest in mathematics. In the 21st century information is a resource. Our Applied Mathematics majors learn to process data and create valuable information. An applied mathematics degree is quite versatile. Our majors learn techniques for analyzing big data, such as from social media and finance, and small data, such as from a clinical trial for a new medication. Our program consists of a math core, free electives and two internships. These internships are crucial for employment after graduation. For more information on what our majors do for internships, please see Student Success. Please also see Applied Mathematics-Science - BS for degree requirements and samples of how to complete the degree in 8 semesters.

The mathematics education baccalaureate degree program is designed for students who wish to teach mathematics in middle school or high school. The program provides students with a strong mathematics background as well as the education courses that are required for teaching certifcation.

Employers value computer science and mathematics graduates, not just for their specifc technical skills, but for the broad analytic and problem-solving abilities that are developed in the study of these subjects. Both computer science and mathematics programs feature internship opportunities, where students can earn credits while working for an actual real world employer. There are also opportunities to work with faculty on research projects, attend seminars, present at conferences, and engage in social activities with other students with similar interests through the student run math club. Special scholarship programs are available for qualifed students, sponsored by the National Science Foundation and other organizations.

## Implementing the solution

The solution sounds so simple – shade. Whether it's from natural sources such as trees or construction shade sails over playground equipment, such as there are in Phoenix, doing so can dramatically improve the safety of playgrounds and reduce the burning potential for children.

Yet there are minimal guidelines for parks and recreation directors to follow on this issue. Vanos, became a Certified Playground Inspector with the NPPS over the summer to understand how inspectors were trained on safety issues and whether there was information in training literature about surface temperatures and materials when it comes to shade. There was not.

“There are dangers in a playground that are more important than just burning hands and feet and all that, but heat stress is a big issue, too,” Vanos said. “Kids are so much more vulnerable than adults for burning and overheating, and understanding heat stress is a big issue. Kids get overlooked a lot in that sense.”

This is where Vanos' work in climate change comes into effect. With the continued warming of the planet, temperatures are only going to rise, making playgrounds hotter and the need for shade greater.

“Urban climates also warm with growing urban areas due to the urban heat island effect. That's cumulative with the increasing temperature due to climate change,” Vanos said. “We need to be able to make sure kids can still play and not be stuck inside all the time, especially in warmer climates, because we know it happens. Providing shade is something Lubbock could easily do. We've started this pilot study in Phoenix because that's the hottest city in the U.S., but we want to expand it to other cities and climate zones.”

This study is one of the first done on the subject, so little information exists on playground surface temperatures. Vanos is hopeful this study begins the discussion to add heat stress and temperature-related guidelines to construction standards for current and future playgrounds.

Not all playgrounds will need year-round shade as the one in Gilbert, however. In Lubbock, Vanos said, shade would be necessary for many months, but the temperature variation of the South Plains would necessitate any artificial shade be removable to allow sunshine to blanket the playground during the cold or winter months.

Deciduous trees can provide shade in the summer and shed their leaves in the winter, providing the same effect. However, permanent artificial shade makes more sense in Phoenix, with deciduous trees for places that get cold in the winter, such as Chicago or New York.

Vanos and her fellow researchers also would like to expand the study in order to develop an algorithm using satellite imagery to predict what the temperature of playground surfaces or equipment will be at certain times of day in various climates in order to increase safety measures and awareness.

Above all else, with the concern about childhood obesity in the country, the objective is to make playgrounds as safe as possible so children will utilize them more to exercise not only their bodies but their minds as well.

“Playgrounds are one of the only places kids get a chance to be creative and play and be kids,” Vanos said. “To be able to freelance and play and be creative, it's a really important aspect of kids' lives. We need to make sure we provide a good environment process for that.”

## Manga High Math

Manga High Math is a fantastic interactive math website consisting of 18 math games covering a variety of math topics across all grade levels. Users have limited access to all games, but teachers may register their school, allowing their students full access to all games. Each game is built around a particular skill or related skills. For example, the game "Ice Ice Maybe”, covers percentages, addition, subtraction, multiplication, and division. In this game, you help penguins migrate across an ocean full of killer whales by using your math skills to position floating icebergs that allow travel from glacier to glacier safely. Each game provides a different math challenge that will entertain and build math skills at the same time.

## 10 things teachers want to say to parents, but can't

Clockwise, from top left: let them get their own breakfast, John Terry's not such a good role model, be careful with video game age ratings and PE is compulsory.

Clockwise, from top left: let them get their own breakfast, John Terry's not such a good role model, be careful with video game age ratings and PE is compulsory.

Something I'm starting to hear with worrying frequency within the primary school setting is "my daughter's my best friend". Often, this rings alarm bells. Your kids aren't your mates. You're their parent, and your responsibility is to provide them with guidance and boundaries, not to drag them into your own disputes. Your nine-year-old doesn't need to know about your bitter feud with his friend's mother, or which dad you've got the hots for at the school gate. In the years to come he or she may realise that some of their own problems (social alienation, in its various forms, being a prime example) might have something to do with exposure to that sort of talk at an early age. Continue at your own risk.

2 Data levels aren't everything

3 Let them go a little bit

It's always tricky to bring up, as it's the child who dictates when this needs to happen. And that could be at any moment, regardless of year group or academic ability. And I empathise, as both a teacher and a parent. Our children are, of course, the most precious things in our lives and we will naturally fight to protect and provide for them. Independence, and the desire for it, however, comes to us all sooner or later and you would do well to recognise the signs. Is your child suddenly starting to produce independent pieces of writing or artwork, and then look to you for acknowledgement/praise? Or maybe following recipe or model-making instructions to a tee? Try setting a few tasks. Left to his own devices, you'd be surprised how well your 10-year-old can remember to pack his homework or get his own breakfast. Even seemingly basic routine chores will help foster his sense of worth and help him cope with life at senior school. In the years to come, he'll probably be more grateful than if you were still spoon-feeding everything to him at this age.

4 Video games carry certificates for a reason

I'm sure that XBox keeps your nine-year-old nice and quiet at home. But his last piece of writing featured SAS operations against Colombian drug cartels and was slightly disturbing. So too was the report from the four six-year-olds who were worried about being the bait in a make-believe drive-by shooting in the playground. I appreciate I can't control what you let your kid see at home, but until they can tell the difference between CGI and reality, would you mind if I just forwarded the complaints from the parents of those six-year-olds on to you?

5 John Terry is no role model

Ticking off a child for low-level disruption occurs at least daily for most teachers it's part of the job. Irritating as it is, it does actually help to establish or regularly reinforce boundaries and it rarely leads to escalation. That is, until your son goes into what I call "John Terry-mode" following said ticking-off: arguing back, gesticulating, rolling eyes, huffing and puffing, and so on. That's why he ended up getting the "hairdryer" treatment, and losing his lunchtime. The media might hold the likes of Terry up as heroes and let them get away with such histrionics every Saturday afternoon, but it's painful to watch eight-year-olds mimicking that sort of behaviour even in the playground. I'm not going to tolerate it in my classroom. Unfortunately, the odd lost playtime at school isn't going to go far in making this problem go away, so if there's any chance of you handing out a few red cards or match bans at home it'd probably enforce the point a lot more clearly.

6 Boyfriends can wait

"My daughter's really sad these days," isn't an uncommon thing to hear from a parent from time to time. I will then anticipate having to explain that, in my experience, girls' friendship issues do tend to drag on a bit whereas their male counterparts will just have a straightforward shouting match (or worse) and then get on with things. But when said mother then goes on to explain that her eight-year-old daughter's misery is due to the fact that she hasn't got a boyfriend, my klaxon goes off. Kiss-chase is all good fun, but it really is about as serious as playground romances tend to get at this age. Children are under enough pressure at primary school these days as it is, without having to worry about whether they're impressing Johnny SuperDry, or Billy Twelve-Mates. Let your child be a child.

7 Yes, I would like help in the classroom – but not from you

To a primary school teacher, the offer of an extra pair of hands in the classroom is a truly wonderful thing, and 90% of the time any teacher would pull your arm off, so to speak. Helping with art and craft afternoons, listening to readers, making classroom decorations, putting up displays and being a friendly face on school trips are all an essential part of classroom karma, and the children love it.

However, teachers do talk to each other, and if you've got a track record of snooping through children's writing folders, checking maths corrections or questioning styles of delivery to senior management behind closed doors, I'll be keeping you very much at arm's length. Could your motive be to do some undercover snooping? You're not welcome.

8 Sorry – your kid's just lazy

When it comes to progress, every teacher wants the best for every child in their class – and not just for the sake of their own performance review meeting. It is actually why most of us do what we do. But there sometimes comes a point where we start to think we are pushing an immovable object.

If your 10-year-old isn't making the progress that he could be, and it's not because he's tired – it might be because he's, well, lazy. It's not just the flopped-across-the-table body language that tells me this. Compared to others of a similarly high ability, he's moving backwards – making frequent, basic errors.

It's difficult to teach someone who doesn't want to learn, but it's near impossible to teach someone who thinks they know it all already. Conversations about effort and attitude aside, it would be worth reminding them that they'll soon have senior school expectations to cope with. If any of this sounds familiar, could you maybe think about what you might do to help deal with it?

9 Fine. Don't do the homework

Homework is – and always will be – a tug-of-war between parents and teachers in primary school. A lot of parents complain when there's too much of it. Or when there's not enough of it. Or when it's too easy. Or hard. You complain when parents are expected to help with it and you complain when it's designed to be completed independently and your child struggles with this. You complain when your kid has "mislaid" it and it hasn't miraculously reappeared in her book-bag, the night before it is due. I will be sure to forward all these complaints to the school governing body, which wrote the homework policy in the first place. In the meantime, I'll just get on with all of the piles of marking.

10 PE is a compulsory subject

There has been a big rise in children saying they can't do PE, or bringing a note from home, and some excuses are dubious to say the least (for example, an ankle problem that seems to go on for months, or a cold that only afflicts the child on one particular day of the week). Just like maths and literacy, PE (swimming included) is part of the national curriculum and I'm afraid your child doesn't have a choice about whether he or she takes part in it or not. Regularly "forgotten" kits aren't a problem. Once we give up sending "forgotten kit" letters home each week we can always dip into the lost property bin, where there are countless substitute items ready and waiting for a good airing. Please don't forget: PE is good for them, after all, and doing it is in their best interests. As is homework, and most of the above. Thank you for reading and see you in school.

## Long Beach, Los Angeles win grants to develop ‘learning mindsets’

How does your work relate to Common Core math standards?

My book and our website provide many mathematics tasks and advice on teaching that help teachers with Common Core mathematics. The tasks and advice are all consistent with the Common Core and the changes teachers need to make.

Excerpt from “Mathematical Mindsets”: “One of the most important contributions of the Common Core State Standards (CCSS), in my view, is their inclusion of mathematical practices – the actions that are important to mathematics, in which students need to engage as they learn mathematics knowledge. ‘Modeling with mathematics’ is one of the eight Mathematics Practices Standards … (using such tools as diagrams, two­-way tables, graphs, flowcharts and formulas).”

“Modeling happens all through mathematics, but students have not typically been aware that they are modeling or asked to think about the process.” (Chapter 9: Teaching Mathematics for a Growth Mindset – pages 195­-196)

“I speak to many groups of parents about Common Core math, and I am often asked, especially by the parents of high­-achieving students, ‘Why should my child discuss his work in a group, when he can get the answers quickly on his own?’ I explain to parents that explaining one’s work is a mathematical practice, called reasoning, that is at the heart of the discipline. … Mathematicians propose theories and reason about their mathematical pathways, justifying the logical connections they have made between ideas.” (Chapter 9: Teaching Mathematics for a Growth Mindset – pages 205­-206)