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EDU 531 Mathematics Pedagogical Knowledge

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Math is more than just the facts! This blog will dig into teacher knowledge that will help improve student growth!

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Formative Feedback Cycle

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Research Paper written by Nicole Barton

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Contrary to what many believe, the Mathematical Content Knowledge (MCK) needed for effective teaching is not solely dependent on math knowledge. "Mathematical knowledge for teaching is a kind of complex mathematics understanding, skill, and fluency used in the work of helping other learn mathematics" (Thames and Ball, 2010,p228).

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Research by Katie Liscovitz

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Research by Denise Goulet

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Research by Angie Taylor

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By: Shulman

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By. S. Chapin

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S. Chapin Specialized Content knowledge EDU 530 Research Assign(1)

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Knowing and Teaching Elementary Mathematics

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Knowing and Teaching Elementary Mathematics

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Review of L. Ma - Knowing and Teaching Elem. Mathematics(1)

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Knowing and Teaching Elementary Mathematics 

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Knowing and Teaching Elementary Mathematics 

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By Richard Askey

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Askey - Knowing and Teaching Elem. Math(1)

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Helping every student build an understanding of mathematics is a challenging goal, but an important one. Early awareness of student difficulties and their sources, along with instruction designed to diminish them, is essential to achieving this goal. Formative assessment probes are designed to support teachers in identifying student misconceptions and making sound instructional choices. Mathematics education consultant Cheryl Tobey has put together a guide explaining probes and how to use them for furthering students' conceptual understanding. You can download the full guide at the bottom of this blog. Continue reading for an overview of probes, their purpose, and their benefits to students.

The Power of Misconceptions

Misconceptions are flaws in conceptual understanding of a mathematical concept that lead to an error in application. Misunderstandings are common, and it is impossible to teach in a way that avoids creating any misconceptions. But, a teacher can minimize the chances of students' harboring misconceptions by knowing the potential difficulties students are likely to encounter and implementing instruction to help students build conceptual understanding of the mathematics. The full guide provides several problems demonstrating common student misconceptions and responses.

What is a Probe?

A probe is a short, highly focused, quick-to-administer diagnostic assessment designed to pinpoint student misconceptions regarding a mathematics concept. Typically probes consist of three to six items that each require a two-part response. This helps to reveal underlying patterns in incorrect answers, and whether selected responses are supported by strong or faulty reasoning. Reference the full guide for examples of these probes.

When and How to Use Probes

Teachers need to be clear to students about the purpose of the probe and how they will use the information gathered. Although probes look similar to a traditional quiz, they are not meant to be graded. For students who are not accustomed to explaining their thinking, a probe can be challenging, so it's important to support students over time. The full guide provides a chart that lays out when to use probes, who to give them to, and what to do after collecting the probe.

For more information on probes, how to effectively use them, and their relationship with formative assessment, see the full guide.

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Math Probes: Identifying Misconceptions to Build Understanding

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Helping every student build an understanding of mathematics is a challenging goal, but an important one. Early awareness of student difficulties and their sources, along with instruction designed to diminish them, is essential to achieving this goal. Formative assessment probes are designed to support teachers in identifying student misconceptions and making sound instructional choices.

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Common Core State Standards

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Can Personalized Learning Actually Deliver?

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Can Personalized Learning Actually Deliver?

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A new report published by the National Education Policy Center looks at the current state of K-12 personalized learning and finds that there are many reasons for school districts to think twice about embracing this hot new trend. "Personalized Learning and the Digital Privatization of Curriculum and Teaching" was written by Faith Boninger , Alex Molnar and Christopher M. Saldana of the University of Colorado Boulder, and it lays out several areas of concern. As the model spreads- and concerns spread with it -this report provides a clear view of the objections to modern personalized learning.
A History Lesson
Imagine a technology "which gives tests and scores-and teaches." Or a call for a revolution in which science and technology would "combine to modernize the grossly inefficient and clumsy procedures of conventional education" as well as saving teachers time by freeing them from administering and scoring tests. All of that comes from Sidney L. Pressey , the inventor of the first real teaching machine, patented in 1928. Neither personalized learning nor the problems that come with it are new. In fact, the idea of personalizing learning through some sort of mass customization is almost 100 years old; ironically, one of the common pitches for current techno-privatized education is as an antidote to classrooms that supposedly have not changed in 100 years.

B. F. Skinner emerged in the 1950s with an alternative approach to Pressey's , but as the authors note, the two both claimed that their approach would provide students immediate feedback, allow them to work at their own pace, and provide them more personal attention from teachers . " Both Pressey and Skinner also assumed that a student's ability to provide the required response to a question demonstrated competency/mastery-and therefore ' learning. '"

Modern personalized learning has not left any of this behind.
The Modern Version
You would assume that personalized learning meant something like "a humane school and classroom environment and open, flexible teaching strategies," or "increasing students' agency over their own learning" or "addressing needs of the whole child," but a different sort of model has been spreading.

Much of the push for this wave, as with several previous education waves, comes from the Bill and Melinda Gates Foundation, which in 2014 funded a group of organizations to develop a "working definition" of personalized learning. The hard and fast definition still eludes experts and marketers, but the Gates definition (which arguably also informed the definition of Chan-Zuckerberg, now emerging as major players in the field) "offers a tech-friendly vision of an individualized, data-heavy, mastery-based education system." It is based not on research about teaching and learning, but presents itself as "common sense,' which, the authors points out, obscures several problematic assumptions.

It assumes that children are, and should be, focused primarily on their own goals, their own objectives-themselves--and not on community connections or goals. It's about a series of tasks, and not relationships. It assumes that learning involves moving along a linear track ( first learn A, then B, etc ) and that complex learning can be broken down and measured as small bits and pieces. First pass a punctuation unit, then a sentence completion unit, then a simple reading unit, and be declared able to write an explicative paper for a work of literature., because computer software can assess the first three, but not the last.

The modern Gatesian model implies constant assessment, feedback and record-keeping. But that means the "learning" must be doled out in small bites and bits that lend themselves to the kind of assessment and record-keeping that a software can handle
The Problems
The authors argue that personalized learning has been essentially taken over by a privatized corporate approach, because personalized learning smells like money. Lots of money. But because these programs value data most of all, "they reflect a restricted, hyper-rational approach to curriculum and pedagogy that limits students' agency, narrows what they can learn in school, and limits schools' ability to respond effectively to a diverse student body."

While personalized learning talks a big game about student agency, in fact most models are top-down instruction; the student may choose a speed or even a topic for an exercise, but it's the software writers who set the major goals, determine the sequence of units, and decides what will prove mastery. Needs and gaps are determined by someone other than the student, who becomes an object to be acted upon by software that is trying to elicit the desired response and behavior from her. Students may be able to move through their list of modules faster, but there's no support for the notion that learning has anything to do with learning better.

In fact, personalized learning has a limited idea of what an education actually is. Modern personalized learning envisions a series of discrete skills and scraps of knowledge, acquired in a particular sequence. This ignores everything we know about integrating learning into prior knowledge and the world at large. It stifles creativity and critical thinking; rather than forge paths and develop a personal relationship with a body of knowledge, personalized learning calls on students to just move down a path that has already been laid out with pavement, guardrails, and penalties for daring to wander.

Nor is that concrete path supported by research. The research base for modern personalized learning is weak, with little clear support for the idea that this approach can work any better than traditional methods. Too often the pitch is simply magical thinking tied to computers (It's on a computer, so it will be awesome).

Computer technology often comes with an presumption of unbiased objectivity. But software is written by humans, and it reflects their biases and assumptions, the culture that they breathe, and the culture of the tech world is overwhelmingly white and male. Nor do the venture capitalists who are doing much of the funding of these programs free from cultural biases of their own.
The Threats
The authors note one of the central ironies of modern personalized learning; though it claims to be about tailoring instruction for the individual student, it actually requires all students to get their education from a single one-size-fits-all delivery system. That system is not centered on a human teacher; they are reduced to the role of "coach" or "mentor," with little control over the education process. Where more control is given, the teacher spends more time on the computer, modifying, writing, adding, and otherwise maintaining the program.

The digitized approach to personalized learning involves collecting vast amounts of data. Even if the company honestly has no intention of ever putting that data to other uses, the fact that such a data bank exists means that it can be stolen. And since most of these programs come from businesses with investors and owners to keep happy, the pressure for monetizing must be huge. One need only look at Summit Learning, one of the most prominent personalized learning platforms : its software was developed with assistance from Facebook engineers, and the Summit Learning Program has been split off into Summit wants us to know that it has privacy policies in place, has signed the a separate company with a four-person board of directors that includes Priscilla Chan and the CFO of the Chan-Zuckerberg Initiative. Summit is free to schools that want to use it, but Mark Zuckerberg, the head of Facebook, knows a thing or two about how to get money out of a digital platform people use for free. He is not, however, known for his careful handling of user privacy. If data is the new oil, then digitized learning would bring in a data gusher of epic proportions. Student Privacy Pledge, and maintains a Privacy Center on their website that details all the ways they are protecting student privacy.

Personalized learning as currently pitched is not really new, and there are reasons to believe that it cannot deliver on most of its promises. This report lays out in considerable detail why school district leaders should think long and hard before making their students part of this new digitized version of an old revolution that has, for 100 years, failed to launch.

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Report: Can Personalized Learning Actually Deliver?

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A new report published by the National Education Policy Center looks at the current state of K-12 personalized learning and finds that there are many reasons for school districts to think twice about embracing this hot new trend. "Personalized Learning and the Digital Privatization of Curriculum and Teaching" was written by Faith Boninger , Alex Molnar and Christopher M.

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Maine Learning Results - Mathematics | Department of Education

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The Common Core State Standards, which are incorporated into the Maine Learning Results, is a single set of clear educational standards for mathematics that many states, including Maine, have voluntarily adopted and share.

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On April 26, 1983, President Ronald Reagan stood before the press and television cameras in the State Dining Room at the White House and held up a report titled A Nation at Risk. Eighteen months in the making and written by the blue-ribbon members of the National Commission on Excellence in Education at the behest of Secretary of Education Terrel Bell, the report examined the quality of education in the United States-and the findings were anything but stellar.

"Our nation is at risk," the report boldly declared in its first sentence.

Over its next 36 pages, A Nation at Risk lambasted the state of America's schools and called for a host of much-needed reforms to right the alarming direction that public education was seen to be headed.

"If an unfriendly foreign power had attempted to impose on America the mediocre educational performance that exists today, we might well have viewed it as an act of war," the report said. "As it stands, we have allowed this to happen to ourselves."

The commission found few signs of encouragement about the American education system. Test scores were rapidly declining, low teaching salaries and poor teacher training programs were leading to a high turnover rate among educators, and other industrialized countries were threatening to outpace America's technological superiority. The report provided mounds of statistical evidence -23 million American adults were functionally illiterate; the average achievement for high school students on standardized tests was lower than before the launch of Sputnik in 1957; and only one-fifth of 17-year old students had the ability to write a persuasive essay. Almost immediately, A Nation at Risk garnered massive media attention.

"When the report came out, it catapulted the issue of education onto the national agenda," says Mary Hatwood Futrell, professor and Dean of The George Washington University Graduate School of Education and Human Development in Washington, DC, and president of the National Education Association from 1983-1989. "As I was saying to my students not long ago, I can remember when it came out and it didn't matter if you were looking at the morning news, the afternoon news, magazines, newspapers, it was everywhere. And no one anticipated that it was going to have that kind of impact."

A Nation at Risk found that an "incoherent, outdated patchwork quilt" of classroom learning led to an increasing number of students who were subjected to a "cafeteria-style curriculum" that diluted the course material and allowed them to advance through their schooling with minimal effort.

In the 30 years since this scathing indictment, most schools have taken drastic steps to meet the report's challenge to adopt "more rigorous and measurable standards" for learning. All states have adopted academic standards, and forty-five states including the District of Columbia, four territories, and the Department of Defense Education Agency have all embraced the Common Core State Standards to ensure that students have the necessary knowledge and skills for success in college and careers.

Overall, however, despite the initial fervor around A Nation at Risk, the report didn't lead to many far-reaching changes. Many of the problems identified in 1983 remain unaddressed, and stagnant student achievement continues to challenge educators and administrators.

According to the US Department of Education (DOE), the average teacher, in today's dollars, earned $46,700 in 1983 and $54,900 in 2010. While A Nation at Risk recommended "professionally competitive" salaries for teachers, this has not become a finite reality-indeed, almost 20% of new public school teachers leave after the first year, and almost half leave after five years.

The DOE also found in a study that, of 20 children born in 1983, six did not graduate from high school on time in 2001. Of the remaining 14 who did, 10 started college that fall, but only five had earned a bachelor's degree after five years.

Even some problems that A Nation at Risk raised three decades ago have been made worse in the face of budget cuts and other reforms.

As the Cold War entered its waning years in the 1980s, education support programs that reached their zenith in response to the "Sputnik challenge" were being scaled back-much to the chagrin of the blue-ribbon panel's members, who saw this as a clear precursor to an ever more pronounced slide in academic achievement.

"Moreover, we have dismantled essential support systems which helped make those gains possible," the report read. "We have, in effect, been committing an act of unthinking, unilateral educational disarmament."

But if there was a growing concern in 1983 about the declining number of educational programs in schools, then education officials should be sounding the alarm bells with an even greater sense of urgency today. Budget cuts and the obsession with standardized testing have narrowed the curriculum, and handcuffed educators' ability to utilize creative supplemental programs to support and engage their students. After-school programs that once provided safe extended-learning environments for students and specialized programs to help struggling students are losing the funding and district support that they need to continue operating.

Educators like Ned Eisenhuth, a high school history teacher who retired in 2010 after a 35-year career at Minersville Area High School in Eastern Pennsylvania, saw firsthand the slow but steady disappearance of these types of programs over the past thirty years. Like many other teachers, Eisenhuth believes that over-testing severely hampers the ability of students to express their passions through alternative means.

"With the onset of testing to measure student achievement, many times at the expense of other disciplines, it is amazing what students can achieve when given half a chance," Eisenhuth says. "We feel that the students learned just as much, if not more, from the hands-on approach as compared to straight book learning and test taking."

NEA President Dennis Van Roekel credits A Nation At Risk with moving public education to the top of the national agenda, but adds that the report also became a catalyst for today's so-called "reform" movement.

"In some ways, A Nation at Risk was helpful, but what we've seen in the past 30 years are too many misguided efforts - most notably No Child Left Behind - that have only compounded the problems the report identified," Van Roekel said. "Educators across the country work hard to give their students the great education they deserve, but lawmakers cannot keep pulling the rug out from under them with bad ideas. We need to do what we know works. We need to fully fund our schools, invest in early childhood education, increase parental involvement and keep our class sizes small, especially in high-poverty schools at the lower grades. This is how we'll make our schools work for every student."

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'A Nation at Risk' Turns 30: Where Did It Take Us? - NEA Today

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On April 26, 1983, President Ronald Reagan stood before the press and television cameras in the State Dining Room at the White House and held up a report titled A Nation at Risk.

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