Tuesday, February 2, 2016

Math as a language

If math is a language then we need to treat it as such and the best way to learn a second language is immersion.

Focusing on math in grades 1-6, children, when they come home, should be speaking this language.  Parents should ask their children what "words" they are learning.  These "words" could literally be English words such as "prime, composite, factor, multiplication" or it could be what Schwartz and Kenney, 1995, suggest are mathematical nouns such as numbers, measurements, shapes, spaces, functions, patterns, data, and arrangements.

Next, as a language it is important to learn it in a context.  Imagine if I asked you to conjugate verbs and nouns of a language through worksheets and repetition, how much of this practice would stick?  This is similar to children learning "naked numbers"; sitting and simply reciting their times tables.   Suppose there was one child learning math through flash cards and forced to recite their multiplication tables, while a second child plays board and dice games, counting activities while you shop, or even helps you count your change at the grocery store. Which child's experience would "stick" more?  Which child would grow to enjoy numbers and which would grow to hate it?

Keep in mind that the way a child learns a new concept is extremely important in how effective the learning truly is.

One common technique to learning math is through mnemonics or jingles.  However, I want to relate this back to learning a language.  I can sing the "Canadian National anthem" and "Happy Birthday" both in French due to the nature of how I learned it; through the song and rhyme.   Other than these two songs and some simple phrases, I remember nothing else in French.  Do I understand French?

Learning, or essentially memorizing, through jingles without connection to deeper meaning will not allow the child to retain the understanding needed to store this knowledge in their long term memory and ultimately not allow the child to extend this thinking for a new purpose later on.  (I can think of many multiplication songs, which don't really teach the idea of multiplication at all.)

Next, as we read math we must understand that students, not only have to attach meaning to previous knowledge, but also decode the language itself.  Barton and Heidema (2002) say:
In reading mathematics text one must decode and comprehend not only words, but also signs and symbols, which involve different skills.  
See the problem occurs that the symbols in math truly represents the unique alphabet of the language.  Not only does a child have to learn "plus, addition, more" and other words synonymous with addition but also the symbol "+".  Also, some mathematical concepts have multiple symbols associated with them.  For example multiplication could look like "x,X,*,  " or even simply a set of brackets.

For students to truly comprehend math, and find value in this language, we need to show how these symbols translate to English words.  This is where more confusion might occur as math word problems truly combine literacy with numeracy.  I am not suggesting this practice stops, but just pointing out the fact that students (especially ESL) could face another issue of dealing with a mathematical word problem.

The simple fact is how you read a math problem (or really any sort of reasoning, rational, logical, etc problem) is much different than how you would read a fiction text or novel.  The amount of information found in one sentence of a math problem is drastically higher than the amount of information found in a sentence in a novel.  Lastly, most math textbooks (and I would argue most school textbooks) are written above the grade level they are intended to be used in.

So a child/student is struggling with a word problem now what?

I remember back when I started teaching I would give hints such as "total means you should add" or "difference means you should subtract", however this is more procedural work for the child.  Memorize and output the math.  What is more important is for the child to actually hear the thought process out loud.  The first time a child encounters a math word problem, the adult (or teacher) could verbalize the actual thought process which is occurring as he/she reads the problem and truly illustrates how to translate the language of English to the language of Math.  In gradual release of responsibility model, this is called "I DO".

What we do not want from our students is simply knowing what "tricks" to employ when they see certain key words.   After hearing a child read a problem, a question which could be asked is "Are you unfamiliar with any of the words in the problem?"  Keeping in mind the meaning of a word, when it is used in math class, could imply something drastically different than when it is used in a different context.

When encountering an unknown word, simply giving the definition or asking the child to "look it up" usually is not sufficient in securing the understanding needed.

One great strategy is to use a Frayer Model, where the child would write a definition (in their own words), and provide examples and non examples of the words.

Next, ask the child if he/she is clear on what the problem is asking and to possibly read the problem out loud.  The idea of reading aloud, slows the child down and forces to not only see the words but also to hear them.  The worst thing you could do is simply tell your child what to do.  The best teachers have bite marks on their tongue to stop them from speaking.

Lastly, students need to know that certain ideas may have implied constraints.  For example a common question could be "How many ways can you arrange 3 books on a shelf?" with the common correct answer being 6.   If we named the books A,B and C then the arrangements would be


However, in reality these 3 books could be arranged in many more different ways; stacked horizontally, vertically, slanted to the left, to the right, forming shapes such as A, etc...

It may sound like difficult work, and it can be, but this work is valuable in advancing the knowledge of a child in mathematics.  The child must grow and learn to read math in a way they can internalize and ultimately distinguish information.  Simply telling a child the process is not only ineffective but can actually be detrimental to the development of his/her understanding of the concept being presented.

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