• Old habit to eliminate: Defining equality as
“same as.”
The
problem: This is mathematically incorrect and leads to misconceptions.
New habit
to adopt: Defining equality as “same value as.”
For
example, 3 + 4 tells a different math story than 4 + 3, but they yield the same
value of 7.
• Old habit to eliminate: Calling digits
“numbers.”
The
problem: Failing to distinguish between digits, numbers, and numerals
New habit
to adopt: Clearly distinguishing between numerals and numbers (which are
essentially the same) and digits.
For example,
73 is a numeral that represents the number value 73 and has two digits – 7 and
3.
• Old habit to eliminate: “Addition makes
things get bigger.”
The
problem: When negative numbers are introduced, the old habit has to be
debugged.
New habit
to adopt: Addition is about combining.
• Old habit to eliminate: “Subtraction makes
things get smaller.”
The
problem: As with addition, negative numbers make this wrong.
New habit
to adopt: Subtraction is about difference.
• Old habit to eliminate: When borrowing, saying,
“We don’t have enough ones so we need to go to the next place.”
The
problem: Students don’t understand that in the number 10, there are ten ones,
but in the decimal system, we don’t “see” them.
New habit
to adopt: “We can’t see the ones we need, and we need to find those ones.”
• Old habit to eliminate: “You can’t take a
big number from a little number.”
The
problem: The statement is intended to help elementary students deal with
borrowing, but it’s mathematically inaccurate and leads to problems later on.
New habit
to adopt: “We could take a larger number from a smaller number, but we would
get a negative number. You will learn about these later, but right now we will
learn to solve this problem using all positive numbers.”
• Old habit to eliminate: “Let’s ‘borrow’ from
the tens place.”
The
problem: This doesn’t prepare students for more-difficult borrowing and
fractions.
New habit
to adopt: Use “regrouping,” “trading,” or “decomposing” instead.
• Old habit to eliminate: Multiplication
“makes things bigger.”
The
problem: This is true only when using positive whole numbers and will confuse
students later on.
New habit
to adopt: Teach the three structures of multiplication: repeated addition;
finding how many unique possibilities there are when matching one set with
another; and finding a total amount or area when two sides are known.
• Old habit to eliminate: Division “makes
things smaller.”
The
problem: As with multiplication, this is not true a lot of the time.
New habit
to adopt: Teach the different structures of division: repeated subtraction of
groups; answering the question “how many for each one?”; and finding a side
when an area and another side are known.
• Old habit to eliminate: “Doesn’t go into”
(for example, 7 doesn’t go into 3).
The
problem: Even elementary school children understand intuitively that sometimes
cookies need to be split up when they don’t divide up exactly.
New habit
to adopt: Prepare students for later learning by using accurate mathematical
language. A teacher might say, “We could divide 3 by 7, but the result won’t be
a whole number. When you begin working with fractions, you will solve problems
like this regularly. Here we want to consider numbers that divide into other
numbers without creating fractional parts or leftover pieces.”
• Old habit to eliminate: Saying “and” means
decimal point.
The
problem: In common parlance and math parlance, “and” generally means to
combine, add to, or augment. Insisting on using “and” only when there’s a
decimal buries the opportunity to have a discussion that focuses on considering
unit sizes and different ways to form a number.
New habit
to adopt: Don’t create false rules for language. In other words, it’s not a big
deal to call 145 “one hundred and forty-five.”
• Old habit to eliminate: Canceling out – for
example, “These eights cancel out.”
The
problem: Students don’t notice how often properties are used and how important
they are.
New habit
to adopt: Explicitly use and discuss the idea behind simplifying. A teacher
might say, “Here I have an 8 divided by an 8, and we know that anything divided
by itself equals 1. So you can see here that we have simplified this expression
without changing its value.”
• Old habit to eliminate: Referring to “the
answer.”
The
problem: If the goal is to find answers, there’s a tendency to forget the most
important part: How did we do that? Why did we do that? How did you know that?
New habit
to adopt: Use “the model” or “the relationships” or “the structure” or “justify
your answer.”
• Old habit to eliminate: Guess-and-check as a
strategy.
The
problem: While this sometimes involves using number sense, it’s not logical or
mathematical and doesn’t prepare students for more difficult challenges.
New habit
to adopt: Teach systematic math representations – bar models, for example – to
teach students to think like mathematicians.
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