Flipping the Classroom

    At last Thursday's Board of Education meeting, we had a presentation on math education in our school district.  One of the presenters was a junior high math teacher who has been introducing classroom flipping in her advanced math class.   Flipping the classroom "is a form of blended learning which encompasses any use of technology to leverage the learning in a classroom, so a teacher can spend more time interacting with students instead of lecturing. This is most commonly being done using teacher-created videos that students view outside of class time. It is also known as backwards classroom, reverse instruction, flipping the classroom, and reverse teaching."

In the version our board of education listened to, the teacher records a video of her classroom presentation and posts that presentation on Edmodo, a free online teaching tool that allows teachers and students to communicate for teaching and learning.    The students' homework consists of logging in to Edmodo and viewing the video presentation.     Edmodo records which students have done their homework, so that the teacher knows who has come to class prepared.   The student can view the lesson as many times as she wishes, can stop the lesson and think about what she has learned, and then start it up again when she is ready.   The following day, the class does "homework," in the presence of the teacher, whose major responsibility is to answer questions, monitor students to make sure they are using their work time constructively, and develop individualized strategies for students who are struggling and students who are ahead of the class and need individualized challenges.  

   Classroom flipping has become a hot-topic in education circles, and we all know that education has a way of grabbing onto certain new ideas and turning them into fads, but this new approach seems to have tremendous potential.   Numerous online resources discuss how best to use the technique.  There are articles on why it works and others on potential difficulties.  One advocate of the Flipped Classroom explains that the technique is

• A means to INCREASE interaction and personalized contact time between students and teachers.
• An environment where students take responsibility for their own learning. 
• A classroom where the teacher is not the "sage on the stage", but the "guide on the side".
• A blending of direct instruction with constructivist learning.
• A classroom where students who are absent due to illness or extra-curricular activities such as athletics or field-trips, don't get left behind.
• A class where content is permanently archived  for review or remediation.
• A class where all students are engaged in their learning.
• A place where all students can get a personalized education.

The teacher who presented to our board of education convinced me that, at least in her hands, classroom flipping will make her a better teacher and will confer a significant benefit on her students.   I definitely would have liked to learn in a flipped classroom when I was a student?  

    For those of us who are policy makers, a number of questions are presented by the flipped classroom.   They include:

1. Is classroom flipping for gifted teachers (like the one who presented to us on Thursday) or for everyone?
2. If we believe in flipping, what are we going to do about access for students who don't have easy access to high speed internet?
3. Should a school district provide organized tech support to teachers so that they can produce quality online lessons?
4. Is the purpose of flipping to add instructivist or constructivist teaching, or both.  In other words, are we using flipping merely to improve the efficiency of what we are doing right now, or should flipping be a part of a transformation of our teaching philosophy?  
5. What if teachers start using pre-canned lessons from places like Khan Academy for their classes?  Do we expect teachers to develop lessons specific to their class, or is it ok to deliver a lesson created by someone else, that is not targeted to the specific class?  
6. Is teaching this way more work (I tend to think so), and if so, how do we accommodate that? 

I'll  write about these issues in an upcoming post.  Here are some resources on  Flipping. 

District 742 WebArticle on Flipping the Classroom
National Public Radio Discussion of Flipping the Classroom
Online Demonstration of Flipping
University of Northern Colorado Educational Vodocasting
Three Part Series on the Flipped Classroom
Before We Flip Classrooms, Let's Rethink What We're Flipping to.
Flipped Classroom, Pro's and Cons.  Edutopia
Edmodo
Flipping for Beginners  Inside the new classroom Craze (Harvard Education Letters)

Attacking the Global Mathematics Gap: "Knowledge for Teaching"

    In this post, I'm drawing from the materials of several outstanding thinkers in the field of elementary mathematics education who argue that elementary math should be taught by teachers with special skills and special knowledge, which might be called "mathematics for teaching."  This is the third in a series of posts in which I am arguing that we need to cure our global mathematics achievement gap -- the gap between American scores and scores in other industrialized countries -- by changing the way that we approach the teaching and learning of mathematics in the early grades.   I don't claim to have invented these arguments.   This fact:  that we must change our approach to the teaching of mathematics by improving the mathematics understanding of those who teach it, has been crying out for immediate attention for decades.  

    Before I get to the work of Dr. Loewenberg Ball and others, I want to begin with a parenthetical point.  Whenever one advances this idea--that we need to change our approach to elementary mathematics -- immediately someone steps forward to say that, yes, we need to use "manipulatives" and teach math to elementary students more concretely.  Using math manipulatives is often the centerpiece of reform, because installing manipulatives allows us to give the impression that we are doing something really different, and because children enjoy working with blocks, leggos and other concrete things.  But adding base-ten blocks, set cards, fish-counters, and cuisenaire rods, whatever their benefit in the classroom, completely misses the point.   When manipulatives are used by teachers who lack "mathematics for teaching", the manipulatives are not likely to solve the fundamental problem in American elementary mathematics....which is that we aren't teaching for understanding.

  The argument for fundamental reform begins with the this simple idea:  that in order to teach mathematics well, a teacher must know mathematics, but to know mathematics in a special way that makes teaching for understanding possible.     It is true that just about anyone can teach, and just about anyone can learn, tbasic math facts -- sums and times tables, how to subtract integers and do long division.   It is possible to learn these simple chores with a deck of flashcards and lots of repetition.  There was a time when a person who could do his sums, who could make change at the cash register, was considered to have mastered enough arithmetic to get by.    But these days, cash registers can do the arithmetic for you, and so being able to make change is barely enough to keep a minimum wage job.      More and more of us need to use mathematics for applications in the world of work that require understanding.   If we do not make the transition to concept based learning in mathematics, other countries are going to pass us by in engineering, computer sciences, telecommunications, architecture, banking, insurance, and a host of other areas that require application of mathematics to solve problems.  

    As I've been collecting information to write these posts, I ran across a great power point, online, that makes the points I've been advancing way better than I could ever do.   Its called "The Case for Elementary Mathematics Specialists", authored by Hung-Hsi Wu of the University of California at Berkeley (Cal) and Deborah Loewenberg Ball,  Dean of the University of Michigan school of education.  Dr. Hung-Hsi Wu, a research mathemetician and Ball, a teacher and now Dean, argue that its asking too  much of traditional elementary teachers to attempt to teach mathematics the way it needs to be taught.  The slideshow  summarizes ideas expressed in Ball's article in the American Educator, "Knowing Mathematics for Teaching."  (Fall 2005).  

Ball and Dr. Hung-Hsi Wu point out that a lot of people in the United States believe that it should be really easy to teach math --- that math is just a matter of learning to add, subtract, multiply and divide. If you're one of those people who believes that we just need to "get back to the basics", or even if you are not, I urge you to spend a few minutes reading through the power point, or the "Knowing Mathematics for Teaching."   Both are great supplements to the book by Liping Ma, that I discussed in my first post on elementary mathematics.   The point is, that"in order to really teach mathematics successfully, a teacher needs to explain the mathematical foundation of basic arithmetic, and "explaining mathematics...is not all that easy."    The point here is that you can do cash register math without understanding the mathematics behind elementary mathematics, but failure to understand elementary mathematics in a deeper way disables the teacher from teaching, and the a student from learning, enough math to serve as an appropriate foundation for high school and college coursework using math skills.  

Here is an example from the slideshow mentioned above.  

Many teachers can teach children to do multi-digit multiplication...that is the procedure commonly used to multiply with paper and pencil...but how many elementary teachers can explain why  his computation correct, and why it work?  

82 x 59 738 410   4838

In her book, Lipping Ma, discusses interviews of both American and Chinese teachers, in which she asks elementary teachers to explain multi-digit multiplication, and the answers are tremendously revealing.   When they were asked whether the answer would be different if a zero were added as a place holder after the 410 in this problem, many American teachers answered that adding a zero would be improper, because it would change the answer, or create mathematical confusion.  Some American teachers said that if a student wanted a place holder after 410, an X should be used, not a  zero, to make sure that students were not confused into thinking that 4100 was the same as 410.   In contrast, her interviews with Chinese teachers suggested that many had a much deeper understanding. They saw multi-digit multiplication as using the distributive law:   (82)*(59) =  (82) * (50+9)  = 9*82 + 50*82.  

"Clearly, being able to multiply correctly is essential knowledge for teaching multiplication to students.  But this is also insufficient for teaching.   Teachers do not merely do problems while students watch.  They must explain, listen, and examine students' work.  They must choose useful models or examples.  Doing these requires additional mathematical insight and understanding.

An elementary teacher should be able to:

• Analyze student work and understand what  misunderstanding has contributed to an error in the answer.
• Represent the meaning of mathematical algorithms or solutions that illustrate why they work.  
• Use the language of math in ways that are understandable to students, yet do not distort the fundamental mathematical principles. 
• Answer student questions like "why does this work," and "why can't we do it this way".  
• Find alternative ways to explain procedures, so that students have multiple windows on mathematical understanding. 

Ball and her co-authors in the 2005 article argue that there is a a body of mathematics knowledge that teachers need that they call "Mathematical Knowledge for Teaching."    Their research has demonstrated that students of teachers who possess this knowledge do better in elementary mathematics.    And in mathematics, students who do better this year, have a better chance to do better in subsequent years, because mathematics is a pyramid of skills and knowledge that must be built on a solid foundation.  

Those of us who want to "reform" education need to recognize that reforming education means reforming the way we deliver instruction.   Whether we organize the delivery education into charter schools, traditional public schools, or some non-unionized Walmart - style education factory, if math education is delivered by teachers who don't have "mathematical knowledge for teaching," the results are going to be the same. Too often, we believe that changing the organizational structure of education will magically transform what students learn.  But in mathematics, and in other fields, putting old wine in new bottles is not the solutoin.  When we discuss the reform of education, we need to focus, I believe, on the mechanism by which those reforms are going to actually translate into better math, better writing, stronger literacy skills, and better understanding of history, economics and civics.     

Blog Post Part II

Attacking the Global Mathematics Achievement Gap Part II

        In today's post, I want to talk about a really interesting article by James Stigler published in the journal of the American Mathematical  Association of Two Year Colleges, called What Community College Developmental Mathematics Students Understand about Mathematics."   This post is part of a series of posts I'm writing on reforming mathematics education.    In my last post, I cited a report by Harvard's Kennedy School  that warns that American mathematics performance is not up to par as compared to international norms.    You can find the report by clicking here:   Globally Challenged, Are US Students Ready to Compete.  Then, in that first post, I discussed the groundbreaking work by Liping Ma, Knowing and Teaching Elementary Mathematics:  Teachers' Understanding of Fundamental Mathematics in China and the United States.  Liping Ma provides strong evidence that elementary mathematics in America is being taught as a rote series of procedures.   She argues that in part this is because we don't require American elementary teachers to learn mathematics concepts, and that many lack the ability to attack math from a conceptual standpoint.  

    Stigler's article provides evidence that this problem still persists two decades after Liping Ma's book was first published.    Stigler points out that way too many American students arrive at Community Colleges unprepared to perform basic arithmetic.    As a result, they are placed in remedial math programs, and many must take two years of basic arithmetic and math before they reach the point that they can take an entry level math course in their Community College:

A student placed in basic arithmetic may face two full years of mathematics classes before he or she can take a college-level course. This might not be so bad if they succeeded in the two-year endeavor. But the data show that most do not: students either get discouraged and drop out all together, or they get weeded out at each articulation point, failing to pass from one course to the next (Bailey, 2009). In this way, developmental mathematics becomes a primary barrier for students ever being able to complete a post-secondary degree, which has significant consequences for their future employment.

As I have said, Stigler's article suggests that the problems identified in Liping Ma's 20 year old research continue to dog us today.    Stigler says:'

the fact that community college students, most of whom graduate from U.S. high schools, are not able to perform basic arithmetic, pre-algebra, and algebra, shows the real cost of our failure to teach mathematics in a deep and meaningful way in our elementary, middle, and high schools. Although our focus here is on the community college students, it is important to acknowledge that the methods used to teach mathematics in K-12 schools are not succeeding, and that the limitations of students’ mathematical proficiency are cumulative and increasingly obvious over time.

I strongly recommend that you read Stigler's article.  That's why I've posted two links to the article, one at the beginning of this post and one at the end.  What Stigler has done  is to analyze more than just the rate of math proficiency, as most articles do.  He has drilled into the test results and used them to analyze why students seem to be making errors, and then discusses what those errors disclose about the way that we teach mathematics in the United States.  He is telling us, really, that although we have known for decades that we desperately need to change the way we teach mathematics in the United States, we still haven't made meaningful progress in implementing that task.

Our problem is that Americans learn arithmetic and problem solving by rote...by learning a series of procedures, as if we could learn math like learning to recite poetry.   We might trace this problem to the victory of the fundamentalists in the math wars  of the 70's and 80's in which we were bullied into believing that children who learned to do their sums and times tables and long division were somehow adequately prepared in mathematics.  That people who urged that students must learn to reason in mathematics were part of a socialist conspiracy of some kind.   Or, this problem might derive from our commitment in the education community to the concept that the same person can teach art, music, reading writing, social studies and science, without actually having demonstrated strength in all of those subjects.   Liping Ma's book drills into what American elementary teachers really know, or don't know, about basic arithmetic, and it is painful reading.  But Stigler displays the consequence of being taught under the American system of drill and kill.  

Perhaps some examples will whet your appetite to actually read Stigler's entire article.  When asked to place four numbers (two fractions and two decimals) in size order, only 22 percent of students entering community college got the right answer.    Their answers suggested that students believed that 0.53 is smaller than 0.345 because the number with more digits must be larger than the number with fewer.   How did they get through elementary school still believing that to be the case?   Only 19 percent of students could add a fraction to a decimal (as in 0.5 + 1/4 = 3/4).  Only 24 percent of students could add two improper fractions, and the most common error was to add the two numerators and the two denominators.   Many students don't understand that a fraction is a division because they were allowed to get by in fifth grade if they could memorize temporarily how to manipulate fractions, without understanding what they are.     They have been taught the procedures to use when confronted by pesky little problems like 1/4 plus 3/5, but because they don't have a clue why the procedures work, they soon forget how to apply them, and if they remember the rule incorrectly, they have no way of checking to see if they are correct.

Students had great difficulty solving any problem that needed two steps to arrive at the correct solution.   We know, for example, that in many classrooms in the United States, teachers despise having to teach solving what we used to call "word problems."    Solving problems with mathematics successfully requires building skills over time, learning the different approaches to problem solving (make a table, working backwards, drawing a diagram, estimating the answer, creating a number sentence) and so on.   When some teachers skip over the "word problem" curriculum, because they despise it, then later teachers don't have anything to build on.    "I hate word problems," the students say.   They learn distance problems (Dirt problems we used to call them) by trying to figure out from the word problem which number in the problem is D, what R probably is and what number in the problem is T, and then put them in the D=RT formula.  

We asked students, “Which is larger, 4/5 or 5/8? How do you know?” Seventy-one percent correctly selected 4/5 and 24 percent selected 5/8. (Four percent did not choose either answer.) Twenty-four percent of the students did not provide any answer to the question, “How do you know?” Those who did answer the question, for the most part, tried whatever procedure they could think of that could be done with two fractions. For example, students did everything from using division to convert the fraction to a decimal, to drawing a picture of the two fractions, to finding a common denominator. What was fascinating was that although any of these procedures could be used to help answer the question, students using the procedures were almost equally split between choosing 4/5 or choosing 5/8.

I want to conclude by arguing that this issue -- how to teach mathematics -- is as important to improving our American education system as any of the issues that so -called reformers want us to focus on.     We can fire teachers who have unacceptable results.  We can abolish the seniority system.   We can ramp up the voucher system, or expand charter schools without limit.  We can rate schools, rate teachers, or berate them.  But none of these solutions promise to change the current American approach to elementary mathematics, because frankly, the labor force available for hire have all been indoctrinated into the American way of learning mathematics.   If we want to reform the teaching of mathematics, we have some hard work to do and that means we need to start wading into mathematics.  The reform movement, or should I call it the reform fad, is trying to sell the idea that we can get something for nothing:  improve nationwide mathematics through strategies devised by people who don't understand mathematics or the teaching of it.

 Abolishing something isn't good enough.   Turning something upside down won't make it happen.    We need a strategy to identify what needs to be done, figure out how we are going to deliver to American schools a program of teaching and a labor force capable of teaching a concept based curriculum, and then we need to deliver that program not just to a few lucky kids, but to all of them.

Blog Post Part I
Blog Post Part III  

Stigler, What Community College Developmental Mathematics Students Understand about Mathematics."     

Attacking the Global Elementary Mathematics Gap

        Lately, I've been reading quite a bit of material on elementary mathematics.   This post, and the following post, will weave together concepts from three works.  The first is "Globally Challenged, Are US Students Ready to Compete," an analysis of international comparison of math (and reading) proficiency results which argues that America's mathematics performance is well below international norms.      The second is a decade old classic work by Liping Ma, Knowing and Teaching Elementary Mathematics:  Teachers' Understanding of Fundamental Mathematics in China and the United States.      It suggests that one of the major reasons for our deficient performance is that American elementary school teachers don't teach mathematics the right way, and in part, that is because many American elementary school teachers themselves don't have adequate understanding of math.  

The third work, is a 2010 article "What Community College Developmental Mathematics Students Understand about Mathematics."  Its an excellent study of the mathematics proficiency (or lack thereof) of students entering Community Colleges.    MathAMATYC Educator ~ Vol. 1, No. 3 ~ May 2010.   Like Liping Ma's book, this article strongly suggests that the American approach to elementary mathematics is failing to provide a concept-based foundation in math, and that this approach is crippling students ability in later life to understand the math that they need to function in high school, post-secondary education, and in their careers. 

The "Globally Challenged" report by Harvard's Kennedy School is one of many that warns that American mathematics performance is not up to par:

U.S. students in the Class of 2011, with a 32 percent proficiency rate in mathematics, came in 32nd among the nations that participated in PISA (Program for International Student Assessment). Although performance levels among the countries ranked 23rd to 31st are not significantly different from that of the United States, 22 countries do significantly outperform the United States in the share of students reaching the proficient level in math. In six countries plus Shanghai and Hong Kong, a majority of students performed at the proficient level, while in the United States less than one-third did. For example, 58 percent of Korean students and 56 percent of Finnish students were proficient. Other countries in which a majority—or near majority—of students performed at or above the proficient level included Switzerland, Japan, Canada, and the Netherlands. Many other nations also had math proficiency rates well above that of the United States, including Germany (45 percent), Australia (44 percent), and France (39 percent).  Globally Challenged, Are US Students Ready to Compete....

Minnesota has ranked relatively high among the American states on measures of mathematics performance.  Of all the states, only Massachusetts had a majority of its students (51 percent) scoring at or above the proficiency mark on PISA.  Minnesota, the runner-up state, had a math proficiency rate on PISA of just 43 percent.

    What is the reason for our sub-par proficiency rates--what can we do to improve?   Some people want to argue that its poverty, or language barriers, or  other social and environmental problems.  Others want to argue that the problems are structural:   unions,  lack of competition, the seniority system, or our inability to fire bad teachers.   But what if, instead, the weight of the evidence compels the conclusion that we in the United States approach the teaching of elementary mathematics in the wrong way.  What if so-called "good" teachers and "bad teachers", private school teachers, charter school teachers, good schools and bad, are all approaching math education in the wrong way, because we Americans have just plain adopted the wrong approach to learning mathematics?   What if the focus on these non-academic solutions is a symptom of the underlying problem, that we Americans don't really understand the discipline mathematics, and so we try to find a solution that can accomplished without a deep understanding of mathematics?

   In short, what if we can't fix American shortfalls in mathematics unless we fundamentally restructure how we teach mathematics?

     In this post, and the post that follows, I'm going to summarize the arguments that if we want to close the global mathematics gap, we need to discuss mathematics itself, not the structure of education.

One of the most persuasive arguments for fundamentally changing the teaching approach in the United States is an older book, Knowing and Teaching Elementary Mathematics:  Teachers' Understanding of Fundamental Mathematics in China and the United States.  The author, Liping Ma, studied the teaching approach and mathematical knowledge of Amercan and Chinese teachers.   She argues, anecdotally, that well trained Chinese elementary math teachers are significantly better prepared to teach mathematics than their American counterparts.    She argues that in the United States:

• Many elementary teachers are poorly prepared in the mathematics of arithmetic.  In other words, they have a poor, or even an incorrect,  conceptual understanding of the foundations of arithmetic. 
• Elementary arithmetic in the United States is taught with an emphasis on learning procedural rules, rather than an understanding of the mathematical meaning of those procedures.  The result is that when students progress to algebra or more complicated arithmetic (e.g. fractions and problem solving) they lack the building blocks that they need to succeed
• That the teaching of arithmetic rushes through basic concepts so rapidly that students don't have time to understand what they are doing.   That most elementary teachers don't have strategies to develop mathematical understanding or to make connections between the inter-related building blocks of mathematics.   
• That teachers in China spend more time preparing, more time collaborating with other teachers.   
• That many teaching elementary mathematics in the United States cannot themselves solve simple math problems, and cannot explain the reasoning behind the solution of those problems.  

The American approach to elementary mathematics is based on learning procedures that arrive at the right number answer, not learning concepts, that lead to a deep understanding of the core principles of mathematics.   We Americans want our kids to learn math facts and the procedures of arithmetic, but we don't demand that our children understand the meaning of what they are doing.    

If  Liping Ma's book is correct, our problem can't be solved with a quick-fix or with a structural reform.   We have a deep-seated structural problem in the way that students are prepared in the United States, and that problem is infecting our schools of education, our supply of elementary teachers both good and bad, and indeed our entire culture. The problem is our approach to mathematics, not the structure of our public education system iteself.  I'll give some specific examples of how this problem manifests itself in elementary school in a future post.   But I want to mention that this is not an issue that should be characterized as good teacher versus bad teacher:  arguably, the American approach to teaching mathematics in elementary school permeates public and private education alike, and the American approach is used by both good teachers and bad. 

     Most of the popularly discussed reforms involve changes in compensation rewards, implementation of accountability systems, or the introduction of choice and competition.   But what if Lipping Ma's insight is telling us that if we want to improve performance in mathematics, we need to focus on what we are teaching, how we are teaching it, and the preparation and qualifications of those who are teaching it?

A recent conference on the status of teacher development in mathematics is reported in the National Academies Press article that I've listed at the bottom of this post.  The section on Teacher training states:

....most elementary teachers in the United States are trained as generalists and do not have [appropriate] mathematics training. In most states they have taken 6 credit hours of mathematics, with that number rising to 12 credit hours in some states. “There’s not a lot of math there.....and many are very uncomfortable teaching mathematics.”  

" It's not just the number of mathematics courses that teachers have taken:  its also the content and the focus on concepts and understanding.  For those teachers who don't like mathematics and would really rather not teach it, taking mathematics in college preparatory to teaching may be more about overcoming a hurdle necessary for certification than preparing to teach." See  National Academies Press Teacher Development Forum   

Charter schools, breaking unions, changing seniority rules -- whatever the merits of these ideas -- don't speak to the great American failure in mathematics, because it is a systemic problem that is passed along in schools of education and in public and private schools alike.    We have decided in America, to try to learn mathematics without understanding, by rote, by learning procedures instead of by learning concepts. We've decided that anyone can teach the foundations of elementary mathematics whether they themselves understand it.  And as long as this belief permeates our culture, no matter what framework we choose for delivery of education, we are bound to fail. 

Blog Post Part II

Resource Links
 National Academies Press Teacher Development Forum  
Liping Ma,  Knowing and Teaching Elementary Mathematics:  Teachers' Understanding of Fundamental Mathematics in China and the United States.
 Globally Challenged, Are US Students Ready to Compete
What Community College Developmental Mathematics Students Understand about Mathematics."