Making Sense of Math Ed

by Julie Brennan

As a parent, I feel as most do a heavy sense of responsibility regarding my children’s mathematics education. The importance of a good mathematics education could not be emphasized more in our highly technological society. Demonstrable mathematical ability opens doors to opportunities today more than any other time in history.  Yet it seems that this societal emphasis and responsibility has not done much to help parents attempting to provide their children with a deep, meaningful understanding of mathematics. Home educating parents have an unprecedented (dizzying, really) choice of curriculum and teaching aids. Still, many parents watch their young children struggling to understand mathematics, developing negative attitudes toward mathematics, and there is often a valid fear of parents that their own negative attitudes toward math will be passed on to their children.

What is it about mathematics that creates such anxiety in parents and children alike at a time when the perceived stakes in mathematics education are higher than ever?  Multitudes of books exist on alleviating a national condition of mathematics anxiety. Do a search on the web of “math phobia” and you’ll get pages and pages of sites specifically addressing this problem. But try “language phobia” or “English phobia” – they don’t exist. We just don’t harbor the same fear over language that we do over math.

Math IS a Language

And yet, mathematics IS a language. Alfred Adler wrote, “Mathematics is pure language - the language of science. It is unique among languages in its ability to provide precise expression for every thought or concept that can be formulated in its terms.”

A majority of home educators recognize that a key to learning language is immersion. Workbooks and curricula that drill “fast is to slow as (blank) is to high” do not actually teach the concepts of opposites (or reciprocals, to use a mathematical equivalent), although they may be fun for language oriented children and may provide ideas for parents attempting to help kids having trouble grasping these ideas. Children learn the meaning of “fast” and “slow” from hearing these words used repeatedly in context – the rabbit is a fast runner and the turtle is a slow crawler we learn from Aesop’s famous fable. We provide children memorable stories and/or personal experience to attach these word meanings to. As a more advanced skill, we later attempt to define a whole group of these relationships as having a similar characteristic – “opposite-ness” - by these types of exercises. Children have fun when this is done in a light-hearted, natural or game-like fashion and many play verbal games like this with their parents long before they run into these formalized representations in workbooks or curricula.

Elementary math is really not much different than this in terms of how it can be learned. There is the basic alphabet – the natural, or counting numbers – the names and order of which are learned by most children as they are learning to speak. Like children who learn to chant the alphabet and are later able to attribute sounds and phonics to those letters, numbers are first learned as a chant, and later obtain meaning in terms of relative amounts as a child learns the concept of quantity attached to each number. 

Spelling and reading are the coding and decoding processes performed with the alphabet. Many parents see children learn this naturally without the need of formal curricula to put it all together for them. Further, parents recognize that writing is a separate, more advanced skill than reading or spelling. Even the most successful early literacy curricula rely on placing words in context as early as possible, rather than drilling. Most of us intuitively know that having a child see the words “the,” “and,” “is,” etc. in the context of an early reader or storybook is far superior to drilling flashcards to learn sight words. Thus many parents are not so fearful of veering off the path of an accredited reading program by exposing our children to vocabulary or books not listed in the curriculum.

However, when it comes to learning the language of math, traditional programs and teaching oppose these successful language learning lessons. There is a traditional tendency at an early age to remove most, if not all, of the context of math, and place numbers onto a workbook page for a child to begin coding and decoding. Most math programs suggest adding context to a child’s early arithmetic education in the way of supplementary optional activities, rather than treating context as a foundational part of mathematical learning. In fact, it is suggested or implied that a need for these activities to attain understanding may indicate a child is slow or remedial - they are only necessary if a child isn’t “getting it.”  Like a scientifically balanced vitamin pill, the curriculum can become distilled nutrients, no messy, bulky food package required. 

Some curriculums imply or state that presentation of math concepts or ideas out of the prescribed scope and sequence order can cause irreparable confusion, and therefore a child’s math experience should be limited to concepts encountered up to the current lessons. Drilling with flashcards, computer software and worksheets to obtain proficient and speedy recall of math facts completely devoid of any context whatsoever is the norm. Kindergarten curricula often require children to write number sequences repeatedly up to 100 or more when most children will not be able to comprehend quantities of this magnitude for years.

Many of the newer early math curricula rely on manipulatives to add concreteness to mathematical ideas, an improvement to older texts that consisted only of print exercises. But manipulatives are still artificial. Manipulatives can represent many things and have value as a stepping stone to abstract reasoning, but they still lack the real life context of our physical world that is the foundation of this language they are learning.

Language oriented children and math ability

Parental anxiety that a child’s math education can be forever doomed if they do not choose the right curriculum, follow the right sequence of introduction of concepts, do not use the right words and terms, etc. seems to drive even parents that allow their children to learn all other subjects naturally to cling to curricula for math. Methods such as Charlotte Mason that have found ways to breathe life into humanities and language arts subjects still seem to rely heavily on curricula for mathematics.

I was one of these parents anxious to do math right by my children. I was a successful professional working at a senior management level in an international CPA firm for many years, and I had no reason to suspect that what worked for me would not work for my child.  My oldest (read: guinea pig) child was highly language gifted and seemed to show early ability in everything he studied. I expected nothing less than a continuation of this prowess in all his subjects because I had been an academic star all my life. His troubles in math began near the end of second grade, although I now realize they were rooted all the way back to kindergarten with the emphasis on rote memorization and drill rather than contextual representations of mathematical ideas.

Hey, I survived this kind of math education, what was wrong with him? I cringe to admit it, but this was my feeling at the time and I was thoroughly frustrated with his inability to comprehend what the textbooks presented and what seemed easy to me. I had the good sense to take him out of the part time school he was in at the beginning of third grade and homeschool full time, but math continued to be incomprehensible to him, or at best, uninteresting, uninspiring, and dreaded in spite of multiple curriculum changes. A concrete, manipulative-based program like Math U See improved the situation, but clearly he was not learning to enjoy mathematics in a way that would motivate him without homework battles for years to come. And some ideas still didn’t stick. How can a child read a 250 page adult level novel in a day, quote lengthy passages from the book word for word, yet not be able to remember how to do long division?

Thus began my personal journey in researching learning styles, philosophies, methods, and motivation. Like many gifted kids, my son’s skill levels were bewilderingly asynchronous – off the charts reading, spelling and vocabulary skills, borderline dysgraphic and dyscalculia. He was fast developing a negative math self-image over the numerous walls he hit in computational activities. He demonstrated high problem solving ability in gaming and formal testing, yet his computational accuracy and ability was very low. My research began to show both my son and I that math ability comes in different packages, and an analytically gifted math mind does not tend to reveal itself in early elementary math exercises.

I realized from years of hanging out on homeschool elists and talking to other parents that this isn’t just a gifted kid phenomenon. Kids hitting walls at second, third and fourth grade level math is extremely common. Pick up a grade level workbook and it becomes apparent that the math moves from addition / subtraction to multiplication / division quite early. The little pictures of counting bears or other pictorial representations of the math disappear leaving solely numbers and symbols. Fractions are introduced and may or may not be linked to what the child already learned about multiplication and division. Long division shows up as early as second grade in some curricula, presumably to give kids more years to practice it!

The Jigsaw Puzzle of Arithmetic Concepts

Working arithmetic without context is the equivalent of working a jigsaw puzzle and skipping from one part of the puzzle to the other, without making connections to be able to see how the ideas are linked.  Unless a child is lucky enough to have a computational math gift that facilitates a leap into the abstract realm without bothering with this context, it may seem like each math idea is a separate hurdle to get over, a discouraging outlook for a child that does not obtain early mathematical reasoning ability. As mature adults who can now see the abstract connections between these ideas, it can be difficult to understand why children can’t see the same links we do. Two books that do a very good job explaining this to parents are Patricia Kenschaft’s Math Power (recently revised and reprinted) and Frank Smith’s, The Glass Wall, Why Mathematics Can Seem Difficult. 

Multiplication, division, fractions, decimals, percents, ratios – all of these require a type of reasoning to be developed in order for them to be truly understood. This can be referred to as proportional reasoning – the ability to understand relationships between quantities, how much of the whole is the part, and operations between various expressions of this. A math educator I conversed with on this topic quoted his father’s succinct summation: “When you grasp the concept of part vs. whole, you will be able to make sense of most elementary math and science.”  Most of the pre-algebra curriculum concepts eventually converge on this one very basic idea, proportional relationships. Deep understanding of this can be obtained rather quickly when the child is developmentally ready, and provides the child the ability to tie all these ideas together. 

Notice how the following list contains many ways a relationship of “three to four” can be represented, yet it can take years for a child to see the connection between all of them:

-75 cents out of a dollar (100 cents)

-Three quarters of a dollar

-$0.75,

-75%, 

-.75

-3/4ths

-The end of the third quarter

-45 minutes out of an hour

-9 months out of the year

-9 slices of a large 12 slice pizza

-3 out of 4 people have access to the internet 

-There are six milligrams to every 8 grams 

And on and on. Yet, each idea may be presented in elementary curricula as isolated and different concepts, proportional reasoning cut up into bite sized pieces since young kids can’t grasp the whole yet. Is the problem that there are so many varieties of ways to represent this proportional relationship that it is confusing to a child? I would suggest that, just as with spoken language structure, this is true only if the ideas are presented without their full context, and/or if a child is expected to master a concept entirely, as true and lasting mastery would entail understanding all these representations. 

Adherence to one curriculum publisher’s assertion of the best method and order in which to explain, represent and expect mastery of each concept generally limits a child’s experience in an artificial manner. We recognize that a child cannot be expected to master all the vocabulary and sentence structure he hears. Do we deprive that child from hearing conversations, or reading stories that include vocabulary and grammar we consider “over his head”? No, in fact, quite the opposite. We recognize that repeated exposure to ideas in a wide variety of contexts without requiring mastery is the feature of immersion that prepares children to understand something when it is either explained to them or learned deductively from the connections they make after multiple encounters with the same idea.

How to Choose and USE Math Resources

Okay, I like many parents recognized that a problem exists in traditional curricula’s approach. But what is the solution? Unschoolers or delayed formal education advocates would tell you, drop the curriculum and allow your child to learn naturally. From the standpoint of a traditionally-educated, professionally successful parent (dad and/or mom in this case) worrying their child will fall behind, this is not at all an attractive option. How can they possibly learn math if they aren’t taught? Wasn’t it hard enough with a teacher there every day?

There are oodles of books and articles that discuss the benefits and superiority of natural learning. My all time favorite is Frank Smith’s The Book of Learning and Forgetting, it was a life-changing read for me and affected how I view learning at every level. Still, for many families, reading material like this isn’t enough to calm a parent into trusting a child’s learning process in math. For starters, so many parents are, themselves, math phobic. Their homes are not obviously math-rich environments. It might be similar to expecting a child to learn to speak English in a non-English speaking home without instruction. That immersion feature of constant exposure to the language of math in context doesn’t exist. 

My solution over time was to create an immersion experience in our home so that I could relax and let go of the curriculum as the primary learning tool, knowing a multitude of tools for math learning were in place to facilitate natural learning.  I also opened my eyes and educated myself as to just how much math there was in our lives that I had never noticed before. Yes, of course everybody knows the classic examples of cooking and fractions, but I’m talking about relating time to fractions, seeing Fibonacci numbers in flower petals, seeing size and perspective comparisons in home projects, gardening, and children’s picture books, handling money, playing games, seeing the math in music, learning to speak mathematically in stating ideas like proportions, learning to restate common ideas mathematically, to emphasize synonymous relationships just as we do so naturally with language. 

Parents often say, “I don’t know how to do this.” But many parents naturally emphasize the sameness of word relationships in conversations without thinking about it and without any special training. Suzie says, “Mommy, look at this round rock!” and you say “What a nice, smooth stone!” Suzie replies, “I think it’s pretty.” And mom says, “Yes, it’s beautiful.”  Rock = stone, pretty = beautiful, each relationship is presented in a concrete context meaningful to the child, that provides an anchor for understanding beyond rote memorization. In this case, the parent is facilitating learning in an active way, without teaching or lecturing about synonyms, it is very natural to do. This mirroring of ideas with different words is a conversational technique even adults employ with other adults to communicate understanding and agreement without redundancy.

To do this with math only takes a degree of attentiveness, interest and possibly time investment on the part of the parent to be able to notice when this type of interaction is appropriate and how to express it. If the parent is rusty on mathematics themselves, brushing up through the use of curriculum or other resources for the parent’s benefit or other refresher activities are very effective. Many home educators find that curriculum was effective for teaching US, I have often said, “Wow, I never thought of it that way!” Using that newly-obtained understanding to present the mathematics in relevant and contextual ways to our children is the best use of early math curriculum I have found.  And as so many veteran unschoolers know, natural learning isn’t about finding a “teachable moment” and pouncing on it. It is natural to respond when a child asks for parental input, and to observe and file away what they are studying for future reference when opportunities to interact come up.

One of the most enjoyable and effective ways to both brush up on mathematics and explore many representations of mathematics with young children is reading aloud math picture books, and engaging in activities that are often suggested or naturally follow such a reading. It is enjoyable, low stress, and you can read the same book over and over again without it even once feeling like drill. The abundance of these books available today is unprecedented as more publishers seem to be willing to respond to interest in them. Check out the listings here for just a sample of what is available by mathematical conceptual areas: Reader Lists , Resources on how to bring out the mathematics in children’s picture books can be found in articles here and in a pair of outstanding resource books entitled Whitin & Wilde, Read Any Good Math Lately and It's The Story That Counts .

“Strewing” is another technique to create a mathematically friendly environment in your home. If you think about the early elementary schools of generations back that had lengthy periods of free time for children to explore books, build with blocks and other toys, and imaginatively free play, you realize that this is not a new idea. It entails giving a child many free hours to individually explore with objects and resources that can promote learning in very subtle as well as obvious ways. For mathematics, strewing is an extremely important component of my children’s learning experience. Ideas for what that might look like are included in this article on "spreading a learning buffet." 

When mathematics begin to make more sense to you, then you can make more sense of the bewildering choices in mathematics education. You can become your child’s mentor simply by staying one very short step ahead of them in mathematical understanding, and rejoicing the day they step beyond you in their own discovery of mathematics.

Julie Brennan 

March 2006

A. Adler quote taken from Mathematics and Creativity, in The World Treasury of Physics, Astronomy and Mathematics, (T. Ferris, ed), Little, Brown and Co, 1991, p. 435