Scaffolding Your Child’s
Math Education
Math is boring………………….zzzzzzzzzzzzzzzzzzzzzzzzzzz
Math is terrifying!
Math is a waste of time.
“We teach basic arithmetic because we want students to grasp the contours of numbers and look for patterns, to have a sense of what the right answer might be. But what happens next in most schools is the road-to-math-Hades: the single-file death march that leads towards calculus.
“We are pretty much the only country on the planet that teaches math this way, where students are forced to memorize formulas and procedures. And so kids miss the more organic experience of playing with mathematical puzzles, experimenting, searching for patterns, and finding delight in their own discoveries. Most students learn to detest — or at best, endure — math, and this is why our students are falling behind their international peers.”
Tara Holm, mathematician “The real reason why the US is falling behind in math” Boston Globe
School-ish math:
Heavy on Calculations (which can be easily tested)
Light on Concepts and connections
(“desituated” knowledge)
“Knowledge acquisition is ‘situated,’ i.e., it reflects how it was originally acquired and has been used -- it consists not only of abstract rules, laws, and formulas, but also of personal experiences. Becoming an expert, say in mathematics or physics, may be a process of ‘desituating’ one's knowledge to make it less context-bound, less tied to surface features.”
~ Giyoo Hatano, Japanese cognitive psychologist
12 ➗ 4 = 3
12 ➗ 4 = 3 12 ➗ 3 = 4
3 x 4 = 12 12 = 4 x 3
3+3+3+3 = 12 12
4
= 3
12 ➗ 4 = 3 12 ➗ 3 = 4
3 x 4 = 12 12 = 4 x 3
3+3+3+3 = 12 12
4
If a x b = c, then b x a = c, c ➗ b = a, etc.
= 3
“Most homeschool moms want a better mathematics experience than what they got, but when they try to make mathematics better they end up falling back into their own bad experiences, and have the vicious cycle passed on to their kids.”
~ Dr. Maria Droujkova of Natural Math
?
Scaffolding
To support learning as students gradually become more secure in their understanding
Real-life opportunities to use math principles
A vision of the big picture of math that keeps them moving forward with interest
Build Number Sense
Play! Many toys are really math manipulatives in disguise.
“Manipulatives provide a geometric context where students can broaden and deepen their understanding, which is often only mechanical mastery of algorithms.”
Henry Piccioto, math educator
We want kids to not just memorize facts and know procedures, but to become comfortable with numbers.
Yes, there are manipulatives for algebra! The blue rods are variables, the yellow rod is a number (5) and the gray thingy is either a multiplication visual or the x-y axis.
Three Stages of
Thought Development
Concrete mental image Abstract
Here’s a thought experiment for you...
Avoid symbols
(at first)
They just get in the way of the child’s dawning realization of what numbers are for.
Which Representation of six actually helps a child understand what it is?
x 3600 | x 60 | units | value |
| | | 1+1=2 |
| | | 10+1=11 |
| | | 60+60=120 |
| | | 10x60=600 |
| | | 3600+3600+600+1=7801 |
Babylonian Base 60 number system
“Four little cubes lived on Right Street, but their dinner was always sliding off their tables since the street was sloped downward.
So one of them decided to move to Left Street. How many houses away down the street does he need to go to balance with the other three, so they can all eat in peace?”
Ten Pairs
Estimate, then Measure
Discover the characteristics of numbers
8
Discover the characteristics of numbers
7
8
9
1 3 6 10 15 21 28...
1 4 9 16 25 36 49...
1 5 12 22 35 51 70...
What shapes do numbers make in 2 and 3 dimensions?
What angles do they have?
What Kinds of symmetry?
Build logic skills
“More than most other school subjects, mathematics offers special opportunities for children to learn the power of thought as distinct from the power of authority. This is a very important lesson to learn, an essential step in the emergence of independent thinking.” (from Everybody Counts, by the Mathematical Sciences Education Board)
sources of truth
Authority | 5 Senses |
intuition | logic |
Logic is sequential thinking
Generalize truth from specific facts (Inductive reasoning)
5 x 2 x 3 = 30 3 x 2 x 5 = 30
Therefore it doesn’t matter in which order you multiply numbers.
(There is a formal proof of this that goes beyond anecdotal evidence--but this is enough for kids.)
What do you think will happen next?
Why do you think you are correct?
The Child must come to an understanding on his own, through the logical use of his own brain.
Direct instruction doesn’t work!
“Practical experience … shows that direct teaching of concepts is impossible and fruitless. A teacher who tries to do this usually accomplishes nothing but empty verbalism, a parrot-like repetition of words by the child, simulating a knowledge of the corresponding concepts but actually covering up a vacuum.” ~ Lev Vygotsky, Russian psychologist
Build operation sense
Operation sense is a feel for how numbers change when different things are done to them.
What things make numbers bigger? Smaller?
What happens if I write a zero after a number? What if I write a zero after a decimal number--does it also make the number ten times bigger?
What does addition do to negative numbers? What does it do to fractions?
Can you divide a fraction? How about squaring it?
Tell stories!
Draw pictures!
Get Creative!
Watch a video!
Play a game!
Make a hands-on manipulative! Or just get some apples out of your fridge!
Above all, have a CAN-DO attitude!
Use visuals when introducing negative numbers. This is 5 - 9 = -4
Tell a story about a beetle who can climb up trees and dig down into the dirt.
“This beetle went 2 feet up the tree, then went down 5 feet from there, so he dug 3 feet under ground.”
Look for patterns
(every kid should have a calculator and be encouraged to play with it)
Tricks with 9’s
Who knew?
Pascal’s Triangle with all the even numbers colored blue. Lots of cool patterns in here.
“Lost in Pascal’s Triangle”
--for the next time you visit Shanghai….
Number Squared Halved Difference
1 | 1 | 0.5 | |
2 | 4 | 2 | 1.5 |
3 | 9 | 4.5 | 2.5 |
4 | 16 | 8 | 3.5 |
5 | 25 | 12.5 | 4.5 |
6 | 36 | 18 | 5.5 |
7 | 49 | 24.5 | 6.5 |
8 | 64 | 32 | 7.5 |
Ask a question, Make a chart, Analyze, Find a pattern (Graphing it can help)
Name your pattern, it’s Yours!
“What happens if you square a number, then chop it in half?”
Build interest: PLay!
“All play is associated with intense thought activity and rapid intellectual growth. The highest form of research is essentially play. Einstein is quoted as saying, ‘The desire to arrive finally at logically connected concepts is the emotional basis of a vague play with basic ideas. This combinatory or associative play seems to be the essential feature in productive thought.’” Play is Education, by N.V. Scarfe
Real books! Humorous, Colorful, and clever is what we’re going for.
Build Interest
“Learning, creativity and problem solving are facilitated by anything that promotes a playful state of mind, and they are inhibited by evaluation, expectation of rewards, or anything else that destroys a playful state of mind….I suspect that many more of us would play in the realms of history, mathematics, science, and foreign languages were it not for our schools' attempts to encourage them through rewards and punishments, turning those potentially enjoyable activities into work.”
~ Dr. Peter Gray, Free to Learn
Build stamina
“In beginning work on a non-routine problem, the student must enter the space of the problem, which is a difficult thing to do at first. Imagine a time line with three intervals on it. The first interval is the time during which a problem is presented and the solver becomes clear about what is being asked for. Some people call this ‘booting up the problem in your head.’
The third interval is the time after the problem is solved, during which the student may enjoy a feeling of success, reflect on the solution and how it was obtained, and perhaps even discover a more elegant solution.
The second, middle interval, is the woodshed — the one in which the student actually grapples with the problem, seeking and usually finding a solution. It is typical that students want both the initial and middle intervals to be very, very short, or even of no length at all (just tell me the answer).”
~Rudd Crawford
In many games, a kid may do dozens of math problems...FOR FUN! Think about how much math is in Life, Monopoly, and similar games.
Factor Dominoes
Look at the diagonals in this visual number chart. What pops out to you?
Look at the diagonals in this visual number chart. What pops out to you?
Every Prime Number (excluding 2 aND 3)
is One less or One More than A multiple of 6
Visual-spatial intelligence
What if it’s not working?
Don’t move on, switch gears
System used in my daughter’s college physics class:
1-Reword the problem
2-Draw a picture of the problem
3-Solve with variables
4-Plug in the actual numbers and units and solve
5-Ask “IS THIS REASONABLE?”
“Can you write an easier version of the same problem?”
Use eyes and hands: Write it really big, illustrate it, or use manipulatives.
Use music
To Improve Working Memory and/or Decrease Math Anxiety
1 - Stress Reduction
2 - Healthy Diet
3 - Physical Fitness
4 - Memory Training
High Pressure
+ Low interest
= Superficial learning��Relaxed Environment
+ Meaningful/interesting
= Deep learning
School-ish math:
They Spend far too much time teaching far too little math
and far too little TIME
on all the other more awesome math