

A Homeschooler's Guide: the Development of
Mathematical Thinking from Ages 48
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The more parents know about what to expect at each stage of
their child's growth, the better equipped they are with appropriate activities
that will further their child's learning and development. DreamBox Learning
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individualized math education.
If you find this Math Development Timeline useful,
click
here to learn more.


August 5, 2009   

 
The developing mind from ages 4  8
Homeschoolers know from experience that development is very
individual – patterns of growth can be unpredictable and don't always align with
a child's chronological age.
Young children possess a rich assortment of mathematical
cognitive abilities. Through play and everyday family activities, they have
spontaneously compared, sorted, arranged, and counted objects, explaining what
they did and challenging others' explanations.
Young children are intensely curious about their
environment and interact directly with it. What they know is filtered through
their perceptions, which are particular to them and can be very unreliable.
Children at this intuitive stage will believe that a quantity changes when the
arrangement is changed, even if they have counted several times. As children's
brains develop, they become less dependent on perception and the quality of
their thinking becomes more logical.
The Math Development Timeline
YOUR FOUR YEAR OLD
Math learning at age four
A four year old may easily compare sets of objects to know which is
more, but have difficulty figuring out how many he actually has. He may
recognize that how many? means to count, but may struggle to do
this. There are hurdles in successful counting: (1) each object is
counted only once; (2) the name of a number corresponds to each one
counted; and (3) there's a logic to the sequence: base ten has a
predictable pattern. Four year olds are also fascinated with collecting
and sorting objects. 
What you can do at home
Observe and listen while your child is playing to understand his
mathematical knowledge. How does she count? Is it a singsong or
meaningful? Does she touch each object once? Is her voice in sync with
her tag? Does she keep track of what's been counted? Ask questions to
help her develop counting strategies. Try to resist showing your child
what to do so you don't rob her of figuring it out on her own! When
walking, collect objects she likes. At home find different ways to sort
this collection. 
YOUR FIVE YEAR OLD
Math learning at
age five
A five year old is less dependent on matching
strategies to determine onetoone correspondence, and knows that for 5
kids she'll need 5 pencils. When she counts she knows how many?
but may not know that the last number counted means the total quantity.
Once they can count on, five year olds may know which set is more and
may sequence quantities from smallest to largest. But the question
how much more? can be difficult. They may struggle with how much
larger one quantity is than another. 
What you can do at
home
If your child counts accurately you can help him
think about the permanence of a set of objects. Put six pennies in a
row, then change the arrangement. Will he think the quantity changed?
Conservation of number is a big idea needed for addition and
subtraction. Five year olds love repetitive patterns, which help develop
mathematical thinking. Clapping patterns can help him discern sequences
and predict what comes next. Recognizing the unit in a pattern is an
important tool in his mathematical toolbox. 
YOUR SIX YEAR OLD
Math learning at
age six The six year old is developing a more complex
understanding of number. He knows that 6 can be 5 and 1, 3 and 3, etc.
He knows that all sets of 6, no matter what objects, are equivalent. And
the last number counted is the number of the set. These big ideas
underpin more efficient counting strategies such as counting on from
the larger number. He's also developing the idea that "nothing" is
represented by 0, and that any number in the system can be written with
the digits 09. 
What you can do at
home Dice, cards, and board games are
fun and can help a six year old gain fluency with addition combinations.
Engage her thinking by playing "Hidden Counters" in which part of a set
is hidden. Count out eight pennies (making sure she knows there are
eight). Hide four under a cup, leaving the rest visible. Ask, "How many
are hidden?" Notice her strategies for figuring this out. Does she know
automatically that four are under the cup because she knows 4 + 4 are 8?
Does she use her fingers to figure it out? 
YOUR SEVEN YEAR OLD
Math learning at
age seven Because seven year olds can
better understand space and quantity, a broader range of mathematical
ideas become more accessible. They now have a repertoire of basic
addition and subtraction combinations that they can use as tools in
computing. For example, to solve 19 + 21, a child might think of a
related combination, 20 + 20. Some may grasp reversibility, a big
developmental shift needed to understand how subtraction is the inverse
of addition (50 – 25 can be thought of as 25 + ? = 50). 
What you can do at
home Find ways to help your seven year
old build confidence in her mathematical reasoning ability. Asking,
how much money is six quarters? might help her realize that if she
knows that four quarters is one dollar, she can figure out what six
quarters is. To help prepare for multiplication, you can pose questions
that help her think in groups. A question like how many fingers do
five people have? may be hard or easy for her to solve. Your
challenge will be to find questions that support her reasoning. 
YOUR EIGHT YEAR OLD
Math thinking at
age eight At eight, children are using
numbers and quantitative methods in advanced ways, such as reversibility
— they understand subtraction as the inverse of addition. A major
developmental shift occurs when beginning multiplication. In addition,
130 is two or more addends that make a whole, but in multiplication 130
is related to a unit that can shift. If the unit is ten, 130 means 13
tens But if the unit is 100, 130 means 1.3 hundreds. This is unitizing,
the basis for future work with fractions, decimals, and percents. 
What you can do at
home Eight year olds may take on a lot,
then feel frustrated when accomplishments don't come easily. Help her
break a task down: (1) what do you know? (2) identify the problem to
solve; (3) identify one way to solve it. Help her develop confidence by
not correcting a wrong answer, but try to follow her reasoning. Focus on
the process, not the answer, to help her take risks. Eight year olds are
developing more complex ways of reasoning — they like strategic thinking
games like checkers, chess, Monopoly, and Clue. 
Parental participation in learning supports academic
success
By fostering a nurturing homeschool environment for math learning, and
through fun, everyday activities, parents can help their children become math
literate and prepared for success in a changing world.
According to research conducted by Greg Duncan of Northwestern University,
the single most important factor in predicting later academic success is that
children develop early mastery of math and literacy concepts. And math
proficiency is most important. Mastery of early math predicts not only future
math achievement, it also predicts future reading achievement! You can read the
full DreamBox Learning paper on our website: "A
Parent's Guide to the Development of Mathematical Thinking"
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