I am not, by any stretch of the imagination, a proficient mathematician; however, my self-taught mathematics come from the inside out, from a desire to understand the meaning of perceived patterns. In contrast, grade school kids are all-too-often force-fed the subject from outside in.
Depending on how children are taught, they will learn to love or loathe math.
When children are very young, they might play with Matryoshka dolls…
By second grade, a number line is introduced to teach addition and subtraction.
The equal spacing between the numbers on a number line tells that the difference between 1 and 2 is equal to the difference between 6 and 7. This is true arithmetically; however, when adults consider the difference between a Cat 4 and a Cat 5 hurricane, it behooves them to know that the latter will double the damage in dollars. At what point, are we expected to understand that the dangers of flooding cannot sufficiently be imagined when depth is measured in linear inches or even in feet. What is more relevant, and of critical importance, is a translation into cubic volume and the number of tons of water expected to force its way through avenues and lawns, inundating lobbies and living rooms.
Before memorizing addition and subtraction with flash cards, cubes can be used to compare and contrast linear distance and cubic volume. This concept is more basic, more natural, and more fundamental than the abstraction of adding and subtracting integers.
The flatness of the number line shown below gives the impression that all the evenly-spaced numbers are on the same level; it is not apparent that the larger numbers can enclose the smaller numbers. The numbers, as they are counted laterally, relate only linearly.
While teaching our children how things are enumerated/isolated; we can also show them how things in the world are related in hierarchical structures (atoms<molecules<cells, or neighborhoods<city<state<country).
The story of Fibonacci’s baby bunnies illustrates a summation series, wherein each number includes the numbers before it, somewhat like their first toys, the Russian nesting dolls and their stacking/hiding blocks did.
The two techniques of adding are complementary. The former produces linear thinking and the latter produces non-linear thinking. As early as possible, these two methods should be compared and contrasted so that whichever is appropriate can be used with equal ease.
Three dimensional interaction is essential for comprehension of patterns in the natural world, and of complex interconnected systems which are increasingly encountered in our modern age.
Compare these two learning tools:
The learning tool on the left focuses on naming parts in complete abstraction before the kids can relate to what the parts do. Instead, the model on the right puts the organs into perspective, where their names and functions can become meaningful.
Why do we teach young children to memorize the names of states and their capitals from an abstract puzzle?
What if, instead, we taught them first about regions of the country, discussing terrain, flora and fauna of that region in comparison to the one in which they live. Tell which foods they eat that are grown in that region. Tell them how long they would need to ride in a car, or fly in a plane, to get to the states in that region, and then show them pictures/video of some of the natural features of those states and some of their significant landmarks. Region by region, the children could learn to spell the names of those states and put them in alphabetical order, in order of size, in order of population, in order of which one they would most like to visit. Putting the puzzle pieces together should be in the context of seeing how the states share rivers, mountains, or plains. Until the kids are learning about local, state, and national government, the names of the capital cities is completely irrelevant.
Curiosity-led teaching addresses what is concretely relevant to the students, who will then more deeply appreciate and thoroughly assimilate what they are learning.
Mathematics is a science of patterns and relationships, but because it is so abstract we must get nature to nudge their curiosity.
In addition to cutting an apple into vertical segments to explain division, one could cut the apple horizontally and discuss the seed formation in the center.
Let them tile some irregular pentagons together in two dimensions and discover the six-sided secret of symmetry.
Allow them to handle a dodecahedron and notice how the pentagons fit together in three dimensions.
Once kids are ready to make the leap from nature-inspired, 3D constructs to more abstract maths, number lines, as visual tools to relate abstract numbers, become helpful. However, by no means should the old-fashioned, academic number line stand alone, without being counter-balanced by a nature-based number line, such as the one proffered by Quadernity in the parent chapter of this aside.
When learning arithmetic on the number line the student begins with a value; then an algorithm tells her what to do with it, whether to add by pushing further to the right, or to subtract by pulling away from the right and toward the left. The student is the agent who performs an algorithm; but is she apprised of her agency, or of the power that an algorithm gives her to discover an otherwise unknown outcome? It’s doubtful.
If math is seen by students as nothing more than pushing and pulling irrelevant numbers around on a line, or graphing them on Cartesian coordinates, the boring exercises are done only for the goals of pleasing the teacher and avoiding parental punishment.
If a teacher gives a bad grade to a student who has improperly manipulated the meaningless numbers, the humiliation and anticipation of adverse consequences at home may put this student into a negative state (qualitatively worse than neutral, but not quantitatively less than zero).
Knowing what makes us feel bad, or what gets us into a negative state, gives us something/someone to blame. Assigning blame generally makes us feel better because it isolates the bad feeling; however, blaming also builds a strong association. When a child, who blames math for feeling bad about himself, continues to be obligated, under duress, to pass math quizzes and tests, his experience becomes painful and traumatizing. A real-life problem has been created over an artificial problem.
Once all required math classes have been survived, a victim of bad math instruction will claim to have a persistent “math allergy”. Graduates who do not choose to become scientists, engineers, financiers or pure mathematicians often shudder to perform the simplest calculations required by their jobs or personal finance.
People with life-long disdain for math have no trouble finding others with whom to commiserate. Thus, it becomes rather fashionable to dish on the drudgery of doing math. When it comes time to assist their own young’uns with math homework, they are likely to express their hatred for the subject of mathematics, thereby promoting their disrespect downline.
The opposite of having someone to blame for what has gone wrong is having someone step forward who offers to move circumstances in a positive direction, toward gains and pleasure.
A positive outcome is the desired effect, and it is a big relief when a causal agent simply happens by and makes a helpful offering. It’s incredible, right? It should be. Advertisers and politicians count on us to adhere to hope, and to believe in quick fixes — at least long enough to buy what they are selling.
Those who are paid to say the sound-bites we want to hear, and paint the promises we wish will come true, assure us that help is on the way. In the meantime, they ask us, pretty-please, to:
- give them our votes,
- buy their products,
- perform their rituals, and
- fight their wars.
The very pretense of quick fixes belies the fact of inertia in complex organizations. With a better knowledge of hierarchies, we would have a logic that would intuitively warn us about fake ‘fixers’, who are embedded in an intricately-webbed system that is motivated more to protect and profit its overlord(s) than to assist those of us who are ‘lorded over’.
We certainly recognize the impact that advertising and political spin has on us grown adults, so we can only imagine how vulnerable our youth is to vulgarity, violence and the victim-victor dichotomy.
Our children are more likely to do-as-we-do than they are to do-as-we-say; therefore, we must intentionally, and with great determination, expose our kids at every opportunity to honest, wholistic, critical-thinking.
With every incident and reinforcement, their future grows more hopeful.
Included here is a brilliant article about Richard Schwartz, an exceptional mathematician and author of books for children. Please check it out; there are even sample pages from his book teaching kids about infinity!