Baker's Choice Portfolio

Cover Letter

The problem posed in Baker's Choice is about maximizing profit from selling two types of cookies, plain and iced.  There are several constraints in the problem including limited cookie dough (110 lbs), icing (32 lbs),  preparation time 15 hours, and oven space (holds 140 dan cookies.)  Iced cookies cost $5.00/dzn to make and sell for $7.00/dzn yielding a profit of $2.00/dzn.  Plain cookies cost $4.50/dzn to make and sell for $6.00/dzn yielding a profit of 1.50/ dzn.

Baker's Choice begins to scaffold student understanding by exploring specific combinations that work and ones that don't work with the constraints.  The unit then looks at a simpler problem where only the time is constrained.   As inequalities are introduced, manipulation is taught where students create equivalent inequalities using mathematical reasoning.  

In Picturing Cookies Part I, Students become familiar with what graphing inequalities actually means by plotting points that satisfy a given constraint in one color and don't satisfy the constraint in another.  This clearly shows the border of one inequality dividing the plane into two distinct regions (one that satisfies the constraint and one that doesn't.) Then students graph the constraints of the problem for Picturing Cookies Part II and construct a feasible region which meets all the constraints.  It is evident that this region will be some sort of a polygon due to being composed of straight lines with half a plane meeting the constraints.

Finally, the profit function is examined in Profitable Pictures.  Students are asked to plot points of a specific profit (first $1,000, then $500, and finally $600) in different colors for different profits.  By connecting these points, students are led to discover that each profit produces points that are on the same line.  Also, students conjecture that each profit "line" has the same slope (coffee stirrers can be used to move the profit line up.)  As the profit increases, the profit function moves up and to the right and the slope remains the same.  Consequently students discover that the last point on the feasible region is the one that must have the maximum profit.

After connecting that the maximum profit is at a vertex (or along the constraint line), students revisit the profit maximizing of making dozens of plain and iced cookies.  They look at the feasible region and point at which the profit line crosses the last vertex.  Here they find that the last place the profit line touches is at (75,50) when the profit=$212.50.  Thus 75 dozen plain cookies and 50 dozen iced cookies gives a maximum profit of $212.50.
Selected Papers (Projects)
(Including Homework 15 and Baker's Choice Revisited)

Steps to Solving a Profit Linear Programing Problem 

Homework 15

First identify the variables and write the constraint inequalities.  Graph the constraint inequalities shading the area that makes the inequality true.  Identify the feasible region that meets all the constraints (the part that is shaded by every constraint.)  Write the profit equation using the defined variables.  Find the vertices of the feasible region and test them in the profit equation to find the maximum profit.

Lesson Summary for Baker's Choice Unit

Picturing Cookies Part I and II
This introduces the discovery of the relationship between an inequality an the the equation of that line.  The idea is to plot many points to get the idea of why we shade a region when working with an inequality.  This was especially powerful to me because many students don't understand what they are actually doing when they are shading an inequality.

As more and more points are plotted, students begin to see the ones that satisfy the inequality and the ones that don't form a border (the equation of the line.)  

Using this information the Cookie constraints are revisited.

Profitable Pictures

Profitable Pictures was the key to understanding why the maximum must be at a vertex of the constrained region.  The profit equation (in this case P(x,y) = 40x + 100y, where x = # of pastels and y=# of watercolors sold) is a line.  As the desired profit changes the profit line also changes.  However, the ratio of pictures sold does not.  This rate is the slope of the profit line, no matter what the profit is.  As the line moves up the y axis it increases in profit.  Therefore, the highest point still in the constrained region is the maximum profit.  This example on from Desmos demonstrates profitable pictures.  Several coordinate points were graphed at a particular profit.  For a profit of $1000 the points  (0,10), (10,6) and (5,8) that met this criteria were graphed.  These all form a straight line with a slope of -2/5.  For a profit of $500 the points (0,5), (10,1), and (5, 3) met this criteria and were graphed (also having a slope of -2/5 but different y intercept).  Lastly, the graph shows what the profit line looks like at a $600 profit using the specific points (0,6), (5,4) and (10,2) which again are in a straight line with slope of -2/5 and a different y intercept.  The profit equation is the last equation rewritten to allow the graphing program to work as -40x + p= 100y.  p is put on a slider to illustrate the changing profit.  It stops at the maximum profit of $1240 occurring at the vertex (6,10).   So the maximum profit is $1240 with 6 pastels and 10 watercolors paintings.  
Click on the link then on the far right line. Push play where it has p=0 to watch the profit line change.





Problem Of the Week: Kick It
Problem Statement: A football game is scored in a different way.  Each field goal counts for 5 points and each touchdown counts 3 points.  What is the highest impossible score?

Click on link below to see my write up of the problem.
POW Write Up: Kick It

This problem was really interesting to see presented in class.  I appreciated the link to a past problem called "Lot's of Squares"  that several students made.  What especially fascinated me was the fact that both students who presented came up with basically the same way of thinking about the problem.


Baker's Choice Revisited

Baker's Choice allows for students to construct a method to solve linear programing problems rather than given them set ways to solve a problem.  Using all of the constructed information, a final examination of the problem can be demonstrated using Desmos below.  Click on the link to see the equations and watch the profit line animation by pushing play.




The first inequality on the left represents the condition on cookie dough.
The second inequality represents the condition on icing.
The third inequality represents the condition on oven space.
The last  inequality represents the condition on prep time.
This creates a feasible region with vertices at (30, 80) and (75,50), (0,80), and (110,0)
Using the profit equation p(x,y)= 1.5x +2y where x=#of dozens of plain cookies and y = #of dozens of iced cookies, a profit line is constructed with a changing profit (last equation.)  Click play to change the profit.  By doing this, you can see that the last place the profit line touches is at (75,50) when p=$212.50.  Thus 75 dzn plain cookies and 50 dzn iced cookies gives a maximum profit of $212.50.


In Class Assessment
#1

The first inequality represented the condition on cookie dough.  Therefore by removing that equation, cookie dough is no longer a constraint and is unlimited.
The second inequality still represents the condition on icing.
The third inequality represents the condition on oven space.
The last  inequality represents the condition on prep time.
Using the profit equation p(x,y)= 1.5x +2y where x=#of dozens of plain cookies and y = #of dozens of iced cookies, a profit line is constructed with a changing profit (last equation.)

In this case, a new feasible region is created and the vertex the profit line crosses last is at (120, 20).  Therefore when 120 dozen plain cookies and 20 dozen iced cookies are made, the profit is maximized.  By using the profit equation with those amounts, the profit is $220.  Below is a desmos graph.  Press play on the p variable to see the profit line move through the new constraints. 





#2  In this case the Woo's can't sell more than 60 dozen plain cookies.  This offers a new constraint, because the original highest profit occurred when 75 plain cookies were sold.  Re-graphing the constraints creates one additional inequality to create a new feasible region.

The first inequality on the left represents the condition on cookie dough.
The second inequality represents the condition on icing.
The third inequality represents the condition on oven space.
The last  inequality represents the condition on prep time.
The last inequality (in red represents the limitation on plain cookies).
Click to follow the new profit line to until it reaches the end of the feasible region (60,60).  So this means that to maximize profit the Woos should make 60 plain and 60 iced for a total profit of $210.  Below is a graph using desmos to illustrate this.  Click on the graph then on the play button next to the variable p.






#3  This time everything is the same in the original problem except the Woos make a different profit.  Thus the profit equation changes, but the constraints remain the same.

The first inequality on the left represents the condition on cookie dough.
The second inequality represents the condition on icing.
The third inequality represents the condition on oven space.
The last  inequality represents the condition on prep time.
This creates a feasible region with vertices at (30, 80) and (75,50), (0,80), and (110,0)
Changing the profit does not change the feasible region.

Using the new profit equation p(x,y)= 2x +4y where x=#of dozens of plain cookies and y = #of dozens of iced cookies, a profit line is constructed with a changing profit (last equation.)  Click play to change the profit.  By doing this, you can see that the last place the profit line touches has changed due to the change in slope of the profit line.  There for the last place the profit line touches a point in the feasible region is now (30,80) when p=$380.  Thus 30 dzn plain cookies and 80 dzn iced cookies gives a maximum profit of $380.  Below is a desmos representation of the new profit line.  Click on the graph then push play on the variable p for the new profit line.









Homework 14 Reflections on Learning

My experience working on Baker's Choice has been very different than any way I have ever been taught math.  I often try to take information and connect it to what I already know.  Any algorithm I am given, I appreciate knowing why it works and will search that out.  I never thought about why teachers don't teach students to figure it out first.  I like the phrase "doing math" we heard throughout the process.  Attaching "doing math" to thinking through a problem instead of doing a bunch of calculations, is much more accurate description of what true mathematicians do.  Even though, I have learned and practiced linear programing, I found new ways of concretely connecting the material to what I intuitively knew.  I was especially interested in showing points that fit the constraints of the inequality and ones that didn't with two colors.  While I knew what an inequality represented, I appreciated the visual and the increased ability to use this when teaching.  I also thought the profit line exploration was valuable in visually explaining why the maximum profit had to be at a vertex.  The physical movement of the straw made this very clear to me.

I understood maximizing profit with constraints at a much deeper level during this unit.  By taking apart the steps needed to solve a problem with numerous steps, I was able to add greater depth to what I though was a pretty solid understanding.

I very much enjoyed the experience of working together with fellow math colleagues. I appreciated their contributions and especially their questions.  Being part of a group held me accountable for any statements I made as well as opened my mind up to new ideas.  The ability to make the connections was very satisfying and when I look back I can easily see the concept development of the entire process.  I feel more connected to this particular concept than before.

Statement of Personal Growth

I have never seen a high school level thematic math unit that was challenging enough in content.  The units I have seen often introduce a concept, but then fail to continually relate it using reasonable mathematics.  I must say, I was skeptical in seeing the Baker's Choice Book.  I thought it would include watered down mathematics.  Over this course, I have found this unit to require students to actually do more mathematics at a greater depth than any other math curriculum I have seen.  I can't wait to work on the next unit and have looked at IMP website for additional materials.

Working together in a group has been very rewarding.  When I was learning mathematics in my formal education, we never worked in groups.  Our desks were lined up in rows and we plowed through the problems.  I was very good at recognizing patterns and could calculate quickly but had no sort of passion for math.  It wasn't until college that I realized math was not just about solving random problems, it was actually about thinking reasonably and that is when I changed my major to math.  I am thankful that students get a chance to experience this "real" math before college.

This really allowed me to explore mathematics of a concept I thought I knew well.  For the next unit, I want to really put my mind in that of a student.  What kind of background would I have?  What would confuse me? What would be difficult to see?  What would be easy and difficult?  I would like to keep this at the forefront of my mind for the next unit.

Why I Flipped My Classroom

Flipped Teaching

I have decided to try the latest method of teaching for several different reasons.  I went through a summer training and created several videos with math content.  I found the video creation took much longer than I initially thought.  I also realized how many errors I inadvertently make when I am speaking.  Creating these videos required me to really think about a concept and how to break it down into important chunks.  I am interested to test this method out in the coming year.  

Below is my video on why I decided to flip my classroom.

Collaborative Technology

Using technology for collaborative group projects

Deal, Ashley (2009).  Collaboration Tools. Teaching With Technology, Carnegie Mellon, 1-8

http://www.cmu.edu/teaching/technology/whitepapers/CollaborationTools_Jan09.pdf

The days of students gathering on a Saturday afternoon to add the finishing touches on their “How Light Affects Bean Seed Growth” tri-fold display are gone.  Group projects are becoming integrated with technology, and student collaboration more often takes place in the virtual world.   Collaboration Tools, written by Ashley Deal in 2009, examines collaborative efforts in project-based learning and how technology plays an increasing role.

Deal categorizes the assessment collaborative project into three parts, the process, the product, and evaluation of learning.   The process includes how the group works together to create the project.  Assessment includes how and what ideas were shared as well as how students worked together to create the project.  The second part, the product, is the final result of the collaborative process.  Assessing the product includes grading how well the students accomplished the goal.  The final part, and the one most overlooked, is evaluation of learning.  Evaluation of learning includes looking at the overall knowledge the student was supposed to gain from the project.  Take, for example, a science project based on the amount of light affecting a bean seed.  This science project is designed to teach students the scientific method.   Just because the end result (product) is done well, doesn’t mean the group shared ideas and worked well together.  The finished product may look nice, but not represent that the students learned about the scientific process.  For this reason, Deal emphasizes the importance of looking at all three parts of the process for assessment. 

In the article, Collaboration Tools, Deal writes about how technology works with the collaborative process.  She explores the effect of using technology to accomplish this.  Interestingly, technology seemed to decrease “social loafing”, the tendency of one individual not putting in equal amount of effort.  Blocking, one student inhibiting contributions of the others in the group, was also reduced.  And while students communicating virtually didn’t share in depth information, the productivity of brainstorming increased in the virtual environment.

Deal gives a few examples of they type of computer mediation tools available for students, and cautions that the right tools must be tailored for each specific project.  Nowhere in the article, does it mention issues of privacy, which is certainly a factor when choosing tools for the classroom.  Some of the tools she suggests are collaborative suits, course management systems, wikis, and video and audio conferencing.  Since 2009, many other tools are available for such collaboration.  Google Plus allows students to collaboratively work and edit on a paper, spreadsheet, or slide presentation while video conferencing.   Prezi allows students to collaborate on creative presentations. Blogger is another tool students can use to communicate written, video, and visual information near instantaneously.  Had this article been written at a later date, these options would integrate with the rest of this article.

Technology based collaborative projects are significant to the field of education.  As teachers look at ways to engage students and expose them to the way the world communicates, learning to work in a group using technology is vital.  It is important for educators to understand technology and the ways it can be used to create collaborative projects.

Double Dose Algebra

Education Next WINTER 2013 / VOL. 13, NO. 1

http://educationnext.org/a-double-dose-of-algebra/

A Double Dose of Algebra has long-term benefits

“Algebra for all” is the common mantra for 9^th grade mathematics education.  Unfortunately, many students do not have the background to succeed in a rigorous Algebra course.  Simply making a student take Algebra doesn’t mean that student will be more likely to succeed, in fact the opposite is often true.  A promising approach for offering all ninth graders Algebra, is what is being termed, “Double Dose Algebra.”   This approach places struggling students in two periods of Algebra.  This allows them more time to learn and process the concepts.

In the article, “ A Double Dose of Algebra” by Kalena Cortes, Joshua Goodman and Takako Nomi explore the data from the Chicago Public Schools to look for any long-term benefits of increased algebra instruction time.  They find that while short-term benefits of double does Algebra are not apparent, students who take the double course receive some benefit in future academic achievement.

Chicago Public Schools implemented the double-dose algebra policy in 2003 for students scoring below the national median on the 8^th-grade math exam.  These students were required to register for two periods of math, Algebra and Algebra Support.  For the most part, the classes were taught consecutively, with the same instructor, and with the same set of students.

Interestingly, students who participated in double dose Algebra showed no improvement in 9^th grade algebra failure rates.  However, when long-term effects were examined, students who had the double dose of Algebra out performed those with similar initial 8^th grade test scores in standardized testing, trigonometry, graduation rates, and future college enrollment. ACT (a common college entrance exam) scores showed little improvement until 11^th grade when they increased by 1.5 standard deviations.  These students were much more likely to pass Trigonometry, and graduation rates for the double dose algebra students increased by %17.  All of the previous outcomes most likely led to the 30% increase in college enrollment for these students.

Giving students the time they need to digest higher-level math classes is an improved alternative to forcing student to rush through math concepts they may not understand without additional help.  With many districts requiring Algebra for all, educators need to make it reachable.  Giving students more time appears to benefit their long-term academic success.

Pythagorean Proof

Here is a simple proof of that right angle formula we all memorize in geometry and algebra!  Best of all, I was able to make this video using my new tablet toy and a free program called TuxPaint.

The Pythagorean Theorem

limit of (1-cosx)/x as x approaches 0 =0

A new video with another limit proof!

We can use the last proof of 

limit of sinx/x as x approaches 0 =1

 to help prove another "famous" proof

 limit of (1-cosx)/x as x approaches 0 = 0

I used my new tablet and an easy to use program called Skitch to create this.

Proof of limit of sinx/x as x approaches 0

Just bought a tablet to try making videos as part of my flipped classroom experiment.  Since I am working on studying for the math CSET III, I though I would try my hand at a well known calculus proof. 

Limit of sinx/x as x approaches zero

Learning about solids of revolution in Calculus can be confusing, especially if you have learned two different methods (often called the disk and shell methods).  Below is a concept map created in Bubbl.us to help students understand which would be easiest to use. You will have to zoom in to see all of the writing. You can use your mouse to move around the content.

Pythagorean Theorem

Go on a quest to explore the Pythagorean Theorem 
and learn that Pythagorus isn't the only one who discovered the beautiful connection between the sides of a right triangle.

Watch on Khan Academy

Read on Wikipedia

Explore on pbs

Read about different cultures and the theorem

Chinese Proof

Hint

visualizing 1

visualizing 2

visualizing 3

pbs ladder and baseball problem

braining camp

History of Math
Timeline of a section of history using Time Toast http://www.timetoast.com/

Slope Intercept form of a Line

Web 2.0 Evaluation of GeoGebra

GeoGebra is a free program that integrates geometry, algebra, and calculus.  It runs on any platform that supports java.  GeoGebra can be used for teacher presentations, simulations, and both student and teacher authoring.  GeoGebra allows for students to explore a concept in depth and with a visual approach.  Time is reduced in modeling and exploring, and students are allowed to discover patterns and connections.  Students and teachers are able to use this a collaboration tool and can share what they have learned, although embedding a project onto websites or in blogs requires more technology knowledge.  GeoGebra is a powerful program and moderately easy to use.

For teachers, GeoGebra can be used to decrease modeling time, increase motivation, and engage them in critical thinking.  Teachers can show geometric constructions, calculating the area under a curve, and manipulation of parent graphs with ease rather than relying on tedious pencil and paper methods.  Students are able to receive immediate feedback as the equation changes (by changing the slope of a line or the vertex of a parabola.) Using visual representations gives teachers another method to reach students who are visual learners, and for others, it encourages full understanding of a concept.  Not only are students able to see the visual representation, but also they are able to discover patterns by manipulating variables and receiving immediate visual conformation.   

GeoGebra can also be used a student authoring tool.  GeoGebra allows students to explore real life applications and discover patterns in self-directed projects.  Since it easily models equations, calculations involving long decimals and complex calculations are not a deterrent.  The interactive and student led nature of the program increases the control the student has over their own learning as well as the motivation.

Since GeoGebra runs on a Java application it has a tendency to lag.  However, it can be downloaded to your computer and use it without connecting to the Internet.  GeoGebra is moderately easy to use and is fairly intuitive using buttons to represent actions (the text buttons have ABC written over them and the move button is represented as an arrow.)  Any questions on how the program works can be answered in the clear tutorials on the website as well as on YouTube.

Teachers and students can use created applications to share their knowledge.  GeoGebra has a site where projects can be downloaded and used by the online community.  The site is easy to use with a vast source of material available for a variety of math concepts.  If teachers or students want to embed a project into their websites or blogs, the process is more complicated, but achievable.

Overall, GeoGebra is a helpful tool when teaching mathematics.  It gives teachers a way to easily model concepts.  Students are able to be part of discovering patterns in mathematics, as well as receiving immediate visual feedback.  This tool will be especially useful with the problem solving emphasis of the Common Core Standards.  Both students and teachers can use the program to create their own projects with relative ease.  They are also able to access a variety of resources on the website using this program. 

I have created a new applet using Geogebra so that students can use sliders to change the slope and intercept of a line and receive immediate feedback. This applet requires that java is enabled. Use the sliders to change the slope and/or intercept of the line to see how it changes.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

The Sum of the Angles in a Triangle is 180 degrees

All of the triangles add up to 180 degrees.  We can do a rigorous proof of this (see below), but if you just want to illustrate this in a visual way, you can check out my newest You Tube video!

Before doing a rigorous proof of the triangle angle sum theorem (the sum of the angles in a triangle is 180 degrees) some background knowledge of geometry is needed.

1) A straight angle is defined as 180 degrees.

 

2) Congruent means the same size.  Also Congruent Angles have the same measure of degrees (or radians).  When we refer to the actual the degree measure, we use an m in front of the angle sign.

3) Parallel lines are lines are ALWAYS equidistant (equal distance) from each other.

The red segments show that the distance is equal.  

The red segments show the distance is not equal.

4) Understanding substitution:  If we say a variable x = z and we also know that y = z, then where ever we see an x we can substitute a y (since they both equal z.)  We can also substitute y in for x.

 5. When you have a line and a point not that line there is only one line you can draw through that point that is parallel to the original line.  This is called EUCLID'S PARALLEL POSTULATE. 

A line and a point not on the line.

Only one parallel line can be made through the point.
The pink lines are NOT parallel to the blue.

6. Vertical Angles are congruent.

7. When parallel lines are intersected by the same line or segment (called a TRANSVERSAL) the angles created are related to each other.

Adding another transversal to our parallel lines and using that vertical angles are congruent we can show numerous angle congruences.

For this particular proof, we only need to use the alternate interior angles (you can also construct a proof using corresponding angles.)

Okay, now we are close to being ready to construct a formal proof.  It is best to go through the steps informally first, and then write them down.   Here is a review of the background information we will use.

Draw a triangle.

Add a line parallel to AB.

Extend line AC so that you can recognize the transversal.

Mark the alternate interior angles as congruent.

Extend CB and mark those alternate interior angles congruent.

Now all the angles line up so they must be 180 degrees.
We will have to use substitution when we get to this part in our formal proof.

 Now, on to the formal Proof!

Given: Any triangle ABC

Prove: The angle sum is 180 degrees

Auxiliary line just means one that was not given in the diagram.

Angles are defined a a point between two rays.

I added a pink color so that < 3 stood out.

Remember if we actually want it to equal 180 degrees, we need to use measure (m) 

Beautiful!

Watch the video by clicking the link below.

Proof Video

Mindset Influences Learning

Krakovsky, Marina (2007). The Effort Effect. Stanford Magazine, 25-32.

http://alumni.stanford.edu/get/page/magazine/article/?article_id=32124

The key to ensuring that a student is successful is to tell them how smart they are, right?  Wrong.  In fact, telling a girl she is smart can actually hurt her ability to reach her potential.   Marina Krakovsky explores how mindset affects success in “The Effort Effect” published in the March/April 2007 edition of Stanford Magazine.  Her article is based largely on research conducted by Carol Dweck, a renowned scientist and professor at Stanford University, with regards to how people view intelligence.  Dweck’s research strongly indicates that students who view intelligence as fixed or inborn are less likely to succeed than students who believe you can improve and grow your intelligence. 

In addition, Krakovsky emphasizes that teachers and other adults are pivotal in how students view intelligence.  Numerous studies were conducted to show the different responses and achievements of students who had or were encouraged to have a native-intelligence mindset (you were born smart) or a growth mindset (you can grow intelligence with practice).  Students who naturally had a growth mindset or who were encouraged to think with a growth mindset were more likely to view problems and failures as opportunities to learn rather than as attacks on their individual worth and self-esteem.   They were also more likely to take risks, which are essential to expanding knowledge.  Students with a fixed mindset were more likely to prefer easy tasks where they felt competent and able to show that they were “smart” and live up to the adult’s expectations.  Below is a video of one such experiment directed by Dweck.

Looking at the effect of mindset is significant to the field of education, and especially relevant to teachers.  Children’s views of learning are heavily influenced by the primary people who teach them.  Teachers with a fixed mindset, or ones that encourage a fixed mindset, set students up for discouragement and a lack of ambition to tackle challenges.  These students will even lie about their scores to keep appearing “smart.”   This type of student values appearance rather than the acquisition of knowledge.   Teachers who encourage a growth mind set are able to help students reach their full potential by allowing them the gift of working through a tough problem without linking the outcome to their intelligence.  Students are much more likely to rise to a challenge when they don’t interpret a task’s difficulty with their own low ability. 

One of the most encouraging assertions in the article is that students can be taught to change their mindset to a growth mindset if they are given the opportunity.  The article references a very powerful study by Dweck and Lisa Sorich Blackwell of low-achieving seventh graders.  These students participated in one of two study skills classes.  One class taught students important study skills needed to succeed.  The other class taught these same study skills, but in addition, taught that the brain and memory operated like a muscle that grew stronger with mental exercise.  The students in the second class not only improved their grades, but their motivation as well.  The first group (the control group) showed no improvement despite the valuable study skills taught.  Krakovsky encourages “learning goals” rather than “performance goals” to emphasize the growth mindset.  Before educators think about what to teach students, it is a good idea to look at the students' mindsets first.

Krakovsky provides abundant studies to back up her premise that mindset affects success.  Her main source of information is the extensive research by Dweck in her book Mindset: The New Psychology of Success, written after 3 decades of research examining why some people reach their potential while others don’t.  With this background, Dweck and Blackwell used their research conclusions in the study of low-achieving seventh grade students to launch a new software for educators and students called Mindset Works.  Based on their findings, they invented a series of lessons to help students change their mindset and achieve their potential. 

Below is a link to Mindset Works.

http://www.mindsetworks.com/default.aspx

Overall, the article has a powerful message to educators about the importance of mindset, and contains sufficient research to support the claims.  Students who are able or encouraged to see learning as a growth opportunity are more likely to succeed than students who believe intelligence is an innate ability.  Instead of praising children by telling them how smart they are, it is more beneficial to praise how much effort they put in.