Thursday, June 27, 2013

How to Choose a Curriculum Part 2

How to Analyze the Curricula

If you have not read my last two articles in this series, I encourage you to take the time to do so before reading this article.  This article only makes sense in light of those.

There are two main approaches to presenting mathematics concepts in textbooks.  One type is a spiral curriculum in which the assumption is that children understand mathematics through repetition.  Mastery of a concept is not expected until it has been reintroduced several times, sometimes over years.  New concepts continue to be presented and then the previous concepts will be reviewed with each progressive lesson.  A spiral curriculum touches on a wide number of concepts in a year in less depth with less time devoted to each new concept.  Saxon Math is an example of this type of curriculum.

The other type of curriculum is content mastery.  The assumption is that learning is sequential and concepts build upon each other.  Foundational to this approach is the idea that children learn mathematics through understanding and mastering concepts.  A concept or skill is presented in many different ways and is mastered before moving on to the next.  Fewer concepts are covered than in a spiral approach and more time is given for each concept because it is taught in more depth.   Mastery of a concept is required before moving on to a new one.  Making Math Meaning by Cornerstone Curriculum is an example of this type of curriculum.

In light of the way in which we know how children learn math naturally as explained in part one of this series, one of these two methods seem counter to that.  Can you identify which one?  If you said the spiral approach you are correct.  What this approach does is to take teaching the new mathematics concepts out of their proper real world context.  If you take a concept out of its context you are taking away the one thing that gives the idea real meaning.    Do you think your child would understand an abstract concept if you use another abstract concept to teach it?  No, the natural learning occurs when you use something your child already understands to explain something abstract.  Each lesson presents a new concept before your child has had the opportunity to grasp the one presented the day before.  When your child is struggling to understand a concept, she experiences the physical effects of anxiety.  If you ask any child how their tummy feels while they are in a state of confusion with a new concept or skill, they will tell you that they feel a knot or butterflies.  When there is mastery of that difficult concept or skill, the anxiety is gone and there is a rush of relief and joy.  Their perseverance is rewarded and their confidence is elevated.  I have observed this over and over again in teaching children mathematics.  With the spiral approach to mathematics, day after day the child is exposed to new concepts with no mastery of that concept.  The amount of anxiety and frustration they are feeling is rarely relieved.  After years of learning with this type of curriculum the student can lose heart and give up.  They believe that they can’t learn math.  It does not have to be this way.

Do you remember what Mason revealed as the practical value of a mathematics education?  Those were the training in reasoning and the habits of understanding, a willingness to work, accuracy, and being intellectually truthful.  Do you think this lofty goal can be achieved through the spiral approach?

So how do you know what a spiral curriculum looks like?  You have to look at how the curriculum is set up.  Look at the scope and sequence from kindergarten through 12th grade if it goes that high.  Look for concepts that are taught, year after year.  Look at a specific grade level text book.  You can look at any grade level textbook table of contents at the Saxon Math website to see an example of how a spiral approach textbook is set up.  Determine how often new concepts are presented and if mastery of that new concept is achieved after it is taught.  Look at the assignments to see if they are loaded with computation problems with little or no word problems? A quick search on the internet for spiral math programs will also be a big help to you in identification of this type of curriculum.

Not all mastery programs are created the same.  They all start with the correct foundation that children learn through mastering concepts sequentially taught, but the way the concepts are taught may be artificial and frustrate a student.  When looking at mastery curricula, keep in mind the foundation of teaching mathematics that it starts with a problem that requires understanding before moving to the abstract symbolic representation of that problem.  This is the way you help the child connect an abstract concept to something real that they already understand.  Be aware, that there are many mastery programs that teach in the traditional method of starting with the symbolic representation of a problem then using objects to make the symbolic real.  Remember that the objects themselves are a representation and can be abstract to the child.  The proper context for mathematics is the real world math problem that needs to be solved first, with objects if needed, and then using the symbolic representation to show what they already know from solving the problem.

Before deciding on a curriculum, read the scope and sequence.  Make sure that there is mastery of concepts before presenting a new concept.  Fewer concepts will be taught with more time given to teach each one.  Read sample lessons to see if you are comfortable with the teaching format.  Look for a focus on word problems first, rather than the symbolic representation of the math concept. There may be some lessons with mostly computation problems, but this should be the case only after the concept has been presented in the real world context with word problems.  At the website for Making Math Meaningful by Cornerstone Curriculum you can find a good example of what a mastery approach mathematics curriculum looks.  It also adheres to the teaching of mathematics that corresponds with how children learn naturally.    You can see the concepts and skills taught for each grade level as well as sample lessons to see how they are laid out.  There are more curricula available like this one, so you have options and can chose one that best suits you.

The curriculum should not require a lot of expensive manipulatives.  Dried beans or centimeter cubes work well as counters, Unifix Cubes work well for making groups and fractions, and some type of place value blocks are sufficient for teaching addition, subtraction, multiplication and division.  They can be found cheaply online, used, or in local stores so they don’t blow your budget.

With the growing popularity of homeschooling, there are new mathematics curricula entering the market every year.  Your choices are vast, but don’t be intimidated.  You do not have to be an expert mathematician in order to teach mathematics.  All you need is determination and the knowledge of how children naturally learn mathematics.  You have the knowledge you need to choose an effective math curriculum.  I wish you well on your search and wisdom for your decision making.  

Thursday, June 20, 2013

How to Choose a Mathematics Curriculum Part 1

Charlotte Mason’s Key to Teaching Mathematics

“Of all his early studies, perhaps none is more important to the child as a means of education than that of arithmetic.”  
Charlotte Mason vol. 1 pg 253-54

Mason’s statement is bold and elevates mathematics to the top of the priority list as a means of education.  This puts choosing the curriculum for mathematics something of the utmost importance and not to be done without considerable thought and research.  Can you make a mistake and choose an ineffective curriculum or one that is very difficult to work with?  Yes, I did in my sons first year of homeschool, kindergarten.  Like most American’s, I grew up and went to college in the public school system.  Although I originally went to college to be a chemist, I eventually switched majors to graduate with a teaching degree and a science major and mathematics minor.  I had been so trained in my thinking about mathematics by this history, that I had made my choice with that mindset. Fortunately, I realized early on, my grave error.  I hope to share with you, the insights I have gained by reading what Charlotte Mason had to say about teaching arithmetic, so that you are fully equipped to make a well informed choice.

“That he should do sums is of comparatively small importance; but the use of those functions which 'summing' calls into play is a great part of education so much so, that the advocates of mathematics and of language as instruments of education have, until recently, divided the field pretty equally between them.”  
Charlotte Mason vol. 1 pg 255

The ability to make a calculation has little importance according to Mason.  In today’s high tech, fast paced culture it is even truer than in Mason day.  Everyone has access to calculators on their phones, computers, tablets, or iPods.  If you can push buttons, you can make a computation.  It requires very little thought or ability.  She goes on to explain that it is the use of these calculations that hold the value to the education of the person.  While easily measured, computational skills are not the goal of mathematics education.

What I am about to share with you I learned, not from my teacher training in college, but instead from reading chapter XV- Arithmetic, in volume 1 of Charlotte Mason’s Homeschooling Series, which in turn caused a monumental paradigm shift in how I understood the teaching of mathematics.

Think about how you were taught to add and then how your child is taught the same concept in a traditional mathematics programs today.  Not much has changed.  If you open a math workbook today, you probably see problems like 2+5=___.  To a young child, addition is a difficult, abstract concept to understand.  He must first understand what the numbers, addition sign and equal sign mean.   In the workbook there may be two objects printed under the 2 and five objects printed under the 5 which the child then counts and finds the sum.  With the new hands on approach to learning, the student is given counters to represent each of the numbers and then he counts them all together.  Either way he is expected to find the sum of seven.  The pictures or counters are the means to make this abstract number sentence more concrete and to help the child understand what the number sentence means.  On an average math page you may see up to 20-30 computation problems.  The child labors and struggles to understand the idea of addition some with more difficulty than others.  This is the traditional approach to teaching abstract math concepts and how, you may agree, mathematics is taught.

Through decades of observing children, Charlotte Mason discovered that this is not really how children learn math naturally. She discovered the missing key to teaching mathematics.

Imagine trying to teach a person to read music.  You show her the notation and tell her the names of each of the notes and symbols, but you never give her an instrument to actually play the notes.  You may be able to teach her to read the notes on the page with much struggle along the way, but does she really know music?  In the same way you may be able teach him to do math calculations with much struggle along the way, but does he really know math?  In the case of the music, the playing of the instrument is the proper context for the understanding of the abstract symbols that represent the music.  Presenting abstract ideas and concepts within their proper context is the key to understanding knowing the ideas or concepts.  Properly framing the abstract ideas or concepts in the real world setting is what makes learning something abstract so natural.  What then is the proper context for knowing mathematics?  Charlotte Mason went directly to real life application.

“Engage the child upon little problems within his comprehension from the first, rather than upon set sums.” 
Charlotte Mason vol. 1, p. 254

All mathematics starts with a real life problem that requires the use of numbers in order to solve it. You cannot go through a day without using math in one form or another.  We use it all of the time. This idea of starting with a real problem is a crucial foundational concept to the teaching of mathematics.  It is also the exact opposite of the traditional understanding of mathematics.    In traditional teaching of mathematics, the symbolic is the beginning and then manipulatives are used to make the symbols concrete.  Remember those long pages of ciphers with two word problems at the end?  This is not how we do math in the real world and this is not how children learn math concepts naturally.  First, start with the problem and offer the use of manipulatives to help solve the problem and make the real world math problem concrete.
For example:

You have two gerbils.  The mama gerbil gives birth to five baby gerbils.  Now how many gerbils do you have?

Next, give him counters of some sort in order to solve the problem.  It is after he has solved the problem and discovered there are seven gerbils that the symbolic representation of the problem he just solved is presented.


It is very easy at this point for the child to understand the meaning of the numbers and symbols.  Do you see the difference?  The symbolic representation now has meaning because it is representing something the child already knows.  The knowing (solving a real world math problem) must precede the representation (the corresponding number sentence).  Mason goes on to explain the need to demonstrate what needs to be demonstrated and here would require the skill of the math teacher to be able to present a concept in different ways if the child does not immediately grasp it in the way you initially presented it.

“The practical value of arithmetic to persons in every class of life goes without remark. But the use of the study in practical life is the least of its uses. The chief value of arithmetic, like that of the higher mathematics, lies in the training it affords the reasoning powers, and in the habits of insight, readiness, accuracy, intellectual truthfulness it engenders.” 
Charlotte Mason vol. 1 pg 255

Mason eloquently lays out the chief end of mathematics, the true value of it for the child.  The result of teaching children in the way they learn naturally as she has described is the training in reasoning.   The habits of understanding, willingness to work, accuracy, and being intellectually truthful are also developed.

When children are taught in a way that is compatible to the nature of learning, it is like paddling a canoe down a river along with the current.  The way is gentle and joyful.  There may be challenging problems to solve, but the struggle is with many rewards.  On the other hand, teaching in a way contrary to how learning is done naturally is like paddling that canoe against a turbulent current.  The journey is filled with frustration and confusion.  Some of you may have experienced this yourselves with the way you were taught mathematics.  The result of teaching contrary to nature is that you and many other people relate feeling that they are not good at math, but what is not good was the way in which you were taught.

Now that you understand the way in which mathematics should be taught and the goal of that education, when you analyze the curriculum choices you will be able to filter the choices by looking for a curriculum that matches this natural way of teaching children.

Look for part 2, my next article on how to analyze the curricular.

Thursday, June 13, 2013

The Mathematics Curriculum: Should it Stay or Should it Go?

“I don’t like the math curriculum we are using.  Which curriculum would you recommend?”

Having taught mathematics in the public schools, I get this question every year.  I am sharing my thoughts on this topic to give you a perspective you may not have considered in the past.  When you hear a fellow homeschooling friend rave about the math program she is using in her homeschool, a lot of us can’t help having order envy.  Maybe we would be a lot happier doing math with that curriculum instead.  As a result many end up tossing over their current curriculum only to find that the grass was no green on this side of the fence.  This cycle may continue to perpetuate itself because you are looking for something that does not exist.

Every mathematics program has a scope and sequence or an outline of skills and information to be taught, organized by grade level.  If you switch the curriculum from year to year the unavoidable consequence is that your children will have gaps, concepts they did not master but should have, in their mathematics education.  The more often you switch the curriculum, the bigger and vaster the gaps.  As a result of these gaps, it is a lot harder for them to understand the more difficult concepts in the later years because they have not mastered the necessary foundational concepts needed.

My recommendation would be to stick with what you have.  I cannot stress enough persevering with a curriculum so that your children are taught all of the concepts they need, when they need them.  The bottom line is that there is no mathematics curriculum available that makes you want to cheer with excitement.  Yes, there are different types of mathematics programs, but ultimately math is math; the concepts to be taught are absolute.  2+2 will always equal 4 regardless of the curriculum.  Mathematics is not thrilling to most people no matter how the curriculum company presents it, but it can be enjoyable to master new concepts that were difficult.  There is satisfaction in that.

“My child is struggling with the concepts presented in the curriculum I am currently using.  Isn’t that a good reason to switch to a different curriculum?”  

First, the beauty of homeschooling is that you are not on anyone else’s timeline.  If you need to spend much longer on a difficult concept, then you have the freedom to do that.  If you want to skip pages in a section that has been quickly mastered, then you are free to do that.

Second, the struggle to grasp new concepts is not an indication that the curriculum is not working for you. It is also not a sign of academic weakness in your child or an indication that your child is not mathematically minded.  Struggle is the means in which your child learns hard things.  A weight lifter cannot start with the heaviest weight.  He must work his way up to it by regular practice with light and then gradually heavier weights in order to build and strengthen his muscles.  It is like this with mathematics as well.  It requires frequent practice of solving math problems with gradually increasing difficulty.  As a result, your child will develop the habits of accuracy, observation, and perseverance. Each struggle strengthens the child.  There is nothing more exciting for the child who has persevered with a difficult concept to finally understand it.  There is such an overwhelming sense of relief and joy in seeing the problem through to its successful end. In the midst of your child’s struggle, if you see him begin to lose heart, remind him of that last moment of success as a means of encouragement.

“Mathematics depends upon the teacher rather than upon the text-book and few subjects are worse taught; chiefly because teachers have seldom time to give the inspiring ideas, what Coleridge calls, the 'Captain' ideas, which should quicken imagination.” 
Charlotte Mason Vol. 6 p. 233

Finally, it falls upon the teacher to present the concept to the child in a way that he can understand.  The curriculum is only a guide.  If your child does not understand the concept after you have presented it as suggested by the curriculum, then you as the mathematics teacher must find a different way to present that concept.  Mathematics was one of the subjects that Mason agreed relied heavily on the teacher and she was correct.  This may seem daunting to some, but do not let this idea stop you from being a good math teacher.  There are so many resources online, that I have full confidence that you will be able to find the help you need.  If you are part of a homeschooling network, then there are many people within who could help you as well.

I would like to encourage you to persevere with your curriculum choice.  Make it work for you.  You have chosen the hardest thing already by choosing to homeschool, you can do this too.

Look for my next post on how to choose a mathematics curriculum.

Blogger Wordpress Gadgets