Today I address the value of Khan Academy and where I see its use in the homeschool world. I like Khan Academy.  I think that it is a valuable and important resource for learning math on the internet, but the question remains, how do you use it in your life?

Math education is a multi-layered process that starts with an overall plan for teaching math.  For us the framework of that plan is now the Common Core Standards for Math which I am comfortable with. The Common Core Standards is the latest step in a 40 year evolving process by very sincere and educated people to improve math education. Next, teachers/ parents need to understand the broad concepts being taught in the standards and to have a mastery of those concepts.  The teacher needs to be able to keep an eye on the bigger picture while introducing the building blocks of math until the bell goes off in the child's head in terms of the bigger concept, how to perform procedures, and to use the concepts and procedures to solve complex, real life problems.

Furthermore, as teachers we need to be able to individualize to the needs of each child knowing their learning styles as the gateway to use their entire brain.  It is a big challenge.  The great thing about homeschooling is that we get to know our child's learning styles intimately and to teach or provide those resources that individualize to our child's needs. And, the big concepts of elementary math are very learnable even for the most math phobic parents.

That being said, I see Khan Academy as a great tutoring resource to provide children a visual and auditory way to learn how to perform math procedures. I like that they have virtual manipulatives to try out math knowledge and a video to learn how-tos. However, I don't always like each video. That is where I see the end of the value of Khan Academy. Khan Academy is not your teacher.  It does not customize to the needs of your child.  It does not provide a broad picture.  But, if you want your child to learn from a different resource or to have reinforcement on how to perform a procedure, Khan Academy is great. 

That is how I use Khan Academy.  So, use the Khan Academy, but don't use it as a substitute for a math teacher. Think of Khan Academy as a pretty good tutor. Preview each lesson before you give it to your child and make sure that it fits your program and your child's particular learning style.  
 
 
I enjoy being a member of various email groups and in particular I love the groups in the Homeschool community. People outside of this community have very little idea of the level of educational commitment that parents make across the country.  Starting out to be the primary teacher for your own child is a truly nerve wracking thing.  I see the regular, anxious, "I am new to homeschooling" introduction, can I really do it note!  It is great to know that all of the other parents who successfully homeschool starting with pretty similar feelings.

Over time these parents become educational experts in their own way and really drill down into becoming the experts of each of their own child's education.  It is a very different experience for professional educators.  The shear volume of children that float through a school is overwhelming on day 1 and it seems that by the end of the year, that just as teachers get to know their students, the students are on.

Right now I am enjoying starting my own mail group for parents to work with their children on learning math organically in the same way that parents share the world with their children in other topics.  http://groups.yahoo.com/group/organicmath/ In less than a week we are almost up to 60 people and the knowledge and willingness to share is astounding.  I am very excited about the future of this group
 
 
Here is the Punch Line:
  • Teaching math with concepts first is better than teaching procedures.  
  • Teaching concepts may eliminate the need to teach procedures.
  • Learning math from procedures can cause math anxiety.
  • Teaching and learning math conceptually is more efficient than learning procedurally which can lead to reduced math anxiety.
Does life get any better than that? Can it be much clearer? If you are teaching or learning math procedurally then you are wasting your time and making yourselves miserable!  Candidly, I have no idea how we got into this mess, but mathematics has a great tradition of building itself conceptually. 

Around 330 BC Euclid of Alexandria compiled and wrote Elements which is the ultimate work on elementary mathematics. Euclid did an incredible job of capturing logically the building blocks that make up math.  Euclid started from the definitions of points, lines, and line segments, eventually using the concept of a line segment to describe Arithmetic rules and Number Theory. His work inspires me to introduce the concept of proofs and developing theorems from day 1 in math education at any level.  The key is to find and explore truths that are self evident. 

When teaching Pre-Algebra (psst, its really teaching Arithmetic on a conceptual level), I like to start with the Base-10 Number System. There is really nothing more beautiful and elegant. When students learn the underlying concepts of our number system they get a new view on all of the procedures that flow accordingly.  Need for tricks and fairy tales of math are not needed.  Of course students learn and share short cuts later, but often these tricks and procedures become self evident. Along with the number system, introducing the basic Laws of Arithmetic such as Associative, Commutative, and Distributive properties can bring together a fast, bigger picture that is scalable throughout math education.

Here is my free advice: If you are just starting to homeschool your child in math, don’t get caught up with finding books that teach procedures and tricks. 
  • Count and add naturally, the skill seems to be instinctual. Keep the experience organic. 
  • Start with developing number sense through game play and working in the garden or building things. 
  • Build multiplication skills by working on area or number of squares that cover parts of floors. Make games by counting tiles.
  • Once you have basic number skills, start exploring together the Base-10 number system.  
  • Do a quick read on how the Chinese teach basic arithmetic in chapter 1 of Liping Ma’s landmark work, “Knowing and Teaching Elementary Mathematics." Examine how the Chinese use the concepts of composing and decomposing rather than borrowing and carrying.  
Let me know if you try this method, I would enjoy your feedback.

Here is a quick article from Science Daily to give you a further overview and connections to the researcher, Bethany Rittle-Johnson, Ph.D. on the efficacy of teaching with concepts. You Do The Math: Explaining Basic Concepts Behind Math Problems Improves Children's Learning
 
 
Math anxiety is a pain in the neck for the student and the teacher.  For anxious math students, I spend quite some time getting them to trust me and to feel safe, secure, and confident.  If a math teacher cannot deal with math anxious students, then they are likely creating the anxiety. Let me get to the heart of the matter and get it out of the way.  These are the facts as I understand them. 
  • Traditional methods of teaching math can cause stress for many students.
  • Math Anxiety is similar to other types of situational anxiety. 
  • Fear and anxiety flow down through the generations to compound the problem.
  • Elementary school teachers can and do pass it on to their students.
  • We have made it acceptable to be bad in math.
  • This type of anxiety is as treatable as other types of anxiety.
  • Math teachers often get anxious when the traditional methods don’t work and blame the students.
  • Math Anxiety is my biggest hurdle to teaching students math.
Full disclosure: I have never had math anxiety; I am not a psychologist; however, I can say that I have had plenty of anxiety.  I understand the feeling of avoidance and loss of concentration and memory.  I also understand the desire to flee an anxious situation.  Well, it turns out that math anxiety is pretty similar to other types of anxiety caused by stress. This article, “Researchers Say Math Anxiety Starts Young by Associated Press, May 17, 2011provides a very nice overview on the topic of math anxiety.

Learning math can be very hard in comparison to the humanities, the end result of math is objective.  And, with traditional teaching methods focusing primarily on the end result as black and white, there are many opportunities for a student to be wrong. And, being wrong enough times can be very painful and socially penalizing. Let's illustrate this point by following the attaboys.

The traditional method of teaching math is heavy in procedure and checking the final answer. Classrooms are hotbeds of right answers and a handful of kids getting attaboys for those right answers. For many, it feels like a game show with winners and losers where the kids getting the right answers are the attaboys for the teacher. As a result, teachers inadvertently gravitate toward kids who make the teacher feel good. But what happens to the rest of the kids?

For many children, the learning environment becomes stressful; they develop a negative self-image around math, and they become fearful and try to avoid math coming up with wonderful rationalizations about why they can’t do it. Now, imagine carrying that anxiety with you all of your life. How have you changed your life and what you do? If you are a parent, what messages are you sending to your child?  What if you happen to be an elementary school teacher who was never one of the math kids in the front of the room with the right answer; what message are you sending to your students? What if you were one of the “smart” kids; how does that affect your relationship to the rest of the learners.

The good news is that math education is getting better.  Newly trained math teachers are learning new techniques that incorporate the best parts of new math and old math.  The tide is turning.  I see it in my teaching candidate students who will be elementary school teachers.  Even though these candidates are capable of doing elementary math, they still need to learn for the first time, the actual concepts that connect Arithmetic to Algebra and Geometry across the early years of math education.  In the process these brave students finally defeat their childhood demons of math anxiety and get to see the chain of lifelong events that has effected their relationship to math.

How do I try to resolve issues with math anxiety? In Mr. Gelston’s One Room Schoolhouse, we start by creating a trusting, positive, supportive, and fun learning environment in small groups where children learn that it is safe to take chances.  The race is not to the right answer, but the focus is on communicating thoughts of how math works.  Students get to discuss and explore the self-evident laws of math that have been around before Euclid and to take this handful of rules and consistently reapply them in new creative ways.

One of the keys for me as a teacher is that I had to learn to see math ability in a different way.  Being competent at math is not an intrinsic skill that is limited to a few math people.  Being competent at math is a result of developing conceptual context, good skills, and perseverance in trying new solutions and taking chances.  In turn, it is important for me to believe that all students can learn math and that it is just a matter of both teacher and student putting in a sincere effort.  Both the student and I have to see our self worth in working towards the goal rather than feeling that our success is predetermined by genetics. 

Finally, it is important for a teacher to be aware of each student’s progress and thought processes in constructing math.  If the child is not building a sound foundation along the way, there will be a gap in the foundation that causes anxiety later on.  It is important to isolate the trouble moments and to arrange for private tutoring to focus solely on the student’s processes, taking chances, and finding success. 

At the end of the day, there is no greater cure than success.  If I can help a child to feel comfortable, take chances, and discover their own successes, I feel that they can start to learn math and define themselves as math people. 
 
 
Mathematics learning takes on new issues as children reach young adolescence.  Learning style differences and preferences begin to emerge in full force. Young children become increasingly independent.  Math gets harder and builds in complexity as gaps in prerequisite skills make for struggles. The red flags emerge regardless of whether the child is homeschooled or is in public school.  And, did I mention that our cute children start to become teenagers.

Homeschool education is a wonderful experience for many families where parents and children work together to explore the world anew.  Children learn to read and develop number sense.  Once children learn to read then the adage by Will Rogers can be quite true.  "A man only learns in two ways, one by reading, and the other by association with smarter people."   The world of the humanities and religion become open oysters for young readers to absorb and to begin to debate great thoughts.  Wise parents and their community become the "smarter people" to whom Will Rogers refers.

In comparison, math education gets started in a similar way, where parents guide their children in developing number sense. Children learn to add, subtract, multiply, and divide, work on their fractions; beginning to make sense of their world through games, the use of money, and working with everyday items in their life.  In contrast to humanities education, this is the fork where the roads diverge when moving to the next level of mathematics. This is when teaching our own child math becomes incredibly hard.

Candidly, American public schools have not done a very good job in teaching children math at the early levels.  Western teaching styles have been the cause of math difficulties which often lead to math anxiety for the group of students who do not jibe with traditional teaching styles. But, as homeschool families we have placed the load firmly on our own shoulders and we want it that way.  But, how can we do better? 

First, continue to examine how your child learns to identify the best learning experience.  My personal experience as a teacher and tutor is that many students who do not do well with math in public schools usually are from Missouri, in that they live perpetually in the “Show Me State.”  These kids usually have more questions than math teachers have time and answers with the traditional methods of teaching procedures rather than how things work.  Is your child a “Show Me Child?”

Second, don’t inadvertently repeat the mistakes of western teaching methods.  Do we really know the difference if that is how we were raised?  A great read is Liping Ma’s landmark work, “Knowing and Teaching Elementary Mathematics.”  This book alone can make a big difference for you as a homeschooling parent.  If you are not aware of this work, I promise that a light bulb will go on for you.  My college Elementary Teaching candidates are blown away by this book and how it highlights the difference in what is really happening in math versus what some of my students have called, “the fairy tales” of math education.

My biggest concern about many math education systems is that most programs are traditional systems that seem to go deeper into the fairy tales and tricks of math as a fun way to make it palatable for children rather than going to the simple and easy truths of math which children are ready to understand as young adolescents.

Third, your child’s math education needs to come from the base of constructing the basic rules of math that are as self-evident as the preamble of the Declaration of Independence. Classical Greek philosophies of logic and math are a tradition that is thousands of years old.  Euclid was the first to document the result of developing basic truths and building on those truths into a broader picture. Today we refer to these “truths” as properties and rules which children can discover in an exploratory environment, document, and then learn to apply creatively.

Finally, at the end of the day, we have to remember that even though we need to put heavy emphasis on children constructing their own conceptualization of math, we do need to find our way back to practicing procedures.  The good news is that the procedures make much more sense once children understand the underlying rules.  And yes, practice still does make for perfect.

Hence, these are some of the ideas behind how I developed a math program this is built for small groups of young adolescents that is supportive, open to individual differences, and focuses on the conceptual basis of Mathematics.  In small groups, children develop trusting relationships and begin to learn Algebra by revisiting the fundamentals of mathematics and building the rules that we have discussed. By relearning the basics the right way, children develop new confidence and can start to view themselves as math people. For more on my programs please visit the Home Education page.

 
 
People often ask me why I only teach and tutor middle school math when I am comfortable with more complex math. Its funny, because I feel that I have a secret that I am now going to share with the blogosphere: it is the best age group and topics for math education!

There are many things that happen at early adolescents that make this age so exciting.  The first is that children go through an incredible transformation that makes them change dramatically over a three year period.  The physical changes are what the eyes can easily see, but the changes to the brain and style of thinking are dramatic as well. Without referencing neuroscience, we can look at the works of Bloom and Piaget to see that this is the age where children can build on concrete learning and can begin to work with symbols and abstract constructs more freely.  In elementary school level math, children begin to connect between the concrete world and the symbols to describe it. Just as children learn that words represent things, descriptions, and actions; children also learn that numbers and other symbols represent things and actions and eventually results.

The second big benefit is that is an age where children have a chance to catch up and/or fix misperceptions in math knowledge.  The curriculum for this age group is essentially to "finish up" Arithmetic and prepare for Algebra.  But I see it as more about getting a chance to relearn the fundamentals of Arithmetic and its connection to basic Geometry, grasping the broader concept of the base-ten number system, the various meanings of multiplication and division, the real meaning of fractions, and the rules that allow us to do what we do.  Then the students are ready for Algebra.

The third point is that a quality foundation in Arithmetic is directly transferable to Algebra.  Or better put, learning the rules of Arithmetic is learning Algebra before we introduce unknowns or variables.  My experience is that the introduction of letters/variables is anti-climactic after learning arithmetic.  Fractions translate to ratios and to proportions and to slope.  Solving for x is a continuation of solving a complex Arithmetic problem and using the same rules of Arithmetic.

Then, as students master Arithmetic and basic Algebra and Geometry, they can begin to solve problems by applying their knowledge to new areas of real life events in Geometry, Statistics, and Probability.  This third phase requires the highest level of conceptualization and is targeted for students close to 14 who are getting ready to ramp up for high school topics.  Adult problem solving begins to take place here and I get to build the launching point.  In my job with adults, it is amazing how many adults struggle with this level of math.  

So, I have the best job in the world teaching Elementary School Teaching candidates the beauty of elementary math and more importantly, I get to work with young adolescents with the knowledge that I am preparing them for a life of math competency and growth potential to higher math.
 
 
I had the pleasure of taking a course from Mahesh Sharma, PhD, at Cambridge College on the Psychology of Teaching Mathematics.  The presentation was groundbreaking for me.  Dr. Sharma presented a bridge between eastern and western math education within his own model that piggybacked on the work of Piaget.

Dr. Sharma described a method that he calls Vertical Acceleration where he introduces a simple math concept and then applies the same model for the forward levels within a particular strand.  By presenting a basic model of math, students are able to make simple connections from basic to more involved math.

My interpretation of Vertical Acceleration for Mr. Gelston’s One Room Schoolhouse is to create miniature courses that are accessible for tweens to adults who are looking to build a quality conceptual base of math.

The first mini-course to be launched is to build on the knowledge of addition and subtraction of whole numbers within the construct of Base 10, Measurement, Geometry, Patterns and Algebra, and Descriptive Statistics.  The value of such a program is to build knowledge of math along the Piaget hierarchy of understanding by taking concrete examples within the course, making connections to symbolic manipulations, and integrating these concepts.

 
 
Today I would like to highlight the advantage of Standards Based Education: in particular Standards Based Assessment. The Department of Defense’s Domestic and Dependent Elementary and Secondary Schools did a very nice job of describing Standards Based Education.

Standards Based Education is not new and has been part of how the math education community has continuously reinvented itself the last 30 years.  The development of these standards has been at the heart of the discussion in popular circles around helping our children break the trap of learning only algorithms for problem solving rather than developing a deep conceptual understanding of mathematics and its application in our lives.

A bone of contention in education, not just math, is grading.  As simply as I can state it, grading offers a child a cycle of praise or punishment.  Grading can be great for kids who do well, get the “good grade” bug, and strive for more good grades in moderation.  These achievers identify themselves as good students and, for the most part, go on to achieve.  Outside of a central group who seem to do fine with grading, we have many kids who develop stress and mental health issues from a cycle of failure or hyper-perfectionism.

In general, I detest grades and grading.  We have failed the educational pursuits of too many students with traditional grading methods.

A trend that is beginning to take hold in education is a move to Standards Based Assessments.  This method of assessment is valuable in helping students to understand what they need to learn and at what level they need to learn. In this method teachers provide students with their learning objectives and options at a set of activities to provide evidence of learning the standard.  Teachers and students develop a partnership in finding methods for students to not just learn the material but to own and personalize their learning experience.

When teaching in an urban setting I applied standards based education to a geometry project and the end result was incredible.  Students who were previously considered to be weak students actually became the best students in the grade.  They demonstrated through projects of their own choosing that they could master the topic at hand.  These kids were creative and went places with their work that I could hot have predicted.  At the end of the project, when I was required to convert their work into conventional grades, the kids were astonished to find that they had earned A’s for the first time in their lives.  Learning was no longer a mystery for my under-achievers.  They identified what they had to learn, made a plan and worked together with their peers, they built on their knowledge and continuously moved up the completion level of meeting the standard with practical feedback.

 
 
Recently a respected leader in the MA educational reform movement gave me a copy of Liping Ma’s book, “Knowing and Teaching Elementary Mathematics” and it has been a very exciting read.  The impact of this book on elementary mathematics education is self evident.  Local math education in MA has incorporated cooperative learning strategies that emphasize a hands on approach that uses manipulatives and leads to structured discussions.  However, I have not had a chance to see the most important aspect which is the use of developing conceptualization of underlying mathematics.

In my tutoring and teaching of middle school students and working with preservice elementary school teachers, the same common theme has continued to arise throughout the populations that I have worked with: most students do not have the conceptual understanding of the underlying mathematics for their grade level.

In my graduate education at Cambridge College I was fortunate to learn middle school mathematics education from a team that emphasized the importance of students constructing their own mathematics, generating their own rules, and building their own procedures.

I had a reaffirming experience this week in my tutoring that supports the model that western teachers emphasize procedure over conceptualization. I am working with a very bright 8th grade student in an excellent school system whose teacher is a high achiever. This student is detail obsessed and is well trained to follow detailed procedures which are now starting to become too involved for this student’s memory. From my vantage point, the student is taught excellent habits that are necessary for a middle school student. The teacher prepares detailed documentation on problem solving and procedures and a majority of her students in this class succeed as they have been preselected as “advanced” students.  However, this bright young women is struggling. Why?  She did not understand the underlying concepts of her assignment.

My Approach:  With all of the good work that the teacher did in emphasizing procedure and good habit, I was allowed to take a perpendicular approach and spend a short amount of time introducing the underlying concepts from the bottom up.  We made pictures, used the USU NLVM virtual manipulatives library, and discovered the basics of the math moving from concrete to symbolic to abstract reasoning.  By the end, the student had created her own rules and then recreated the procedures on her own, taking ownership of the process, and finding multi-step solutions on her own. The result: Chapter Test 99%.

The bottom line is that the help that I gave this student went back to understanding base 10 and our decimal system.  This information needed to be developed in early elementary education, not in 8th grade.  Think of this: affluent neighborhood, top education system, healthy, well fed child with a supportive family.  What was missing?