Many teachers in
America resort to the traditional algorithm when teaching number
concepts. Asking students to memorize procedures and rules before they
have developed a conceptual understanding of addition, subtraction,
multiplication, division, and fractions is putting students at risk. As
students rely on their memory of a series of rote steps, they are losing the
meaning for the concept, are not applying estimation or place value and are
working against their number sense. Many students develop misconceptions
because they either: confuse the algorithms, make computational mistakes within
the algorithm, overgeneralize what they know and apply some number concepts to
other examples, and/or have not fully grasped the concept in order to check if
their answers make sense.
Children need
many opportunities to talk about strategies for developing their number and
operations concepts, as well as, fractions, and decimals. They need to learn
how to apply number sense, estimation, or place value to their
calculations. Teachers need to strive for conceptual understanding that
connects the notation with the value that is being presented. The use of
manipulatives, such as craft sticks, rubber bands, connecting cubes, a hundreds
chart, number lines, counters and a cup, base – ten blocks on centimeter grid
paper, arrays should be used to tackle real – world problems facilitated by the
teacher. Students can use concrete models or draw pictures by partitioning
numbers and develop their own strategies or procedures that make sense to
them. Encountering many types of representations helps students focus on
the similarities among models, which strengthens their understanding of key
ideas.
Common Misconceptions Students Have:
Misconception
|
Examples
|
Ideas for
Instruction
|
| Some count two objects when saying “sev-en”
because of the double syllables in the word. Students skip over some words while returning to words that were already recited. |
·
Read Number
Books
·
Count out loud
often
·
Counting songs
and chants
|
|
| Thinking Addition Means “Join Together” and Subtraction Means “Take Away” |
Often, students confuse the operations when
the word more is used and sometimes think that take away means
to remove instead of finding the
difference. |
·
The teacher
needs to ensure that they use the words plus and add, as well as, minus
and subtract when referring to
subtraction symbol and reinforce this when students read their equations.
|
| Renaming and Regrouping when Adding and Subtracting Two - Digit Numbers |
Due to students’ difficulty understanding
place value, they are unable to compute correctly. |
·
Counting and
grouping objects by ones and tens.
·
Manipulatives
·
Hundreds Chart
|
| Misapplying Addition and Subtraction Strategies to Multiplication and Division |
Students confuse the commutative property of
addition with the commutative property of multiplication. When using estimation to multiply, students
forget to compensate at the end. |
·
Use the area
model to help students see another representation of their multiplication
expressions.
|
| Multiplying Two-Digit Factors by Two-Digit Factor |
When multiplying two digit factors by two
digit factors, students often forget to multiply by both digits. |
·
Model
cross-product problems using arrays that illustrate the commutative and the
distributive properties of multiplication.
|
| Understanding the Division Algorithm | Teachers come up with creative ways to teach the division algorithm, focusing on the
individual digits rather than the whole numeral.
Students are taught to ignore place value as they routinely work through a procedure they don’t necessarily understand.
|
·
Ask students to
share the strategies they used to get their answers and then discuss whether
these answers make sense.
· Divide by using Partial Products or
"Friendly Numbers."
|
| Understanding Fractions | Children often struggle with the concepts of
half and other equal fractional portions because of their prior experiences
of sharing or dividing things unequally. Students have difficulty dividing nontraditional shapes into parts such as thirds.  |
·
Explore the
concept of fair shares.
·
Present a
variety of shapes for students to divide so that students see that not all
shapes are divided in the same way.
·
Expose students
to more than just haves, thirds, and fourths
Video: Fourth - graders investigate how to share 12 cookies amongst 8 people Video: Understanding Fractions using Pattern Blocks |
| Adding and Subtracting Fractions | Students treat each digit in
a fraction as a whole number and add them separately. They are likely to add
the numerators together and the denominators together. |
·
Use
the number line to represent fractions.
·
Use
manipulatives, such as pattern blocks and fraction strips to count by fractions.
Video: Using Fraction Strips to Add Fractions |
|
|
Students have difficulty naming decimal fractions and are
unable to provide the correct word form.
They may not be able to show the relation of a decimal
expression to the whole.
Students have difficulty
comparing decimals and ordering them. |
· Teachers
need to name decimal fractions consistently and appropriately, thus reinforcing
place value.
· Use
tools such as base –ten blocks, a meter stick, Digi-blocks, and money to
effectively represent decimals.
|
Effective teachers need to be aware of the confusion
that students may face and provide engaging learning opportunities in order for
children to succeed. It is important for
students to grasp the foundational knowledge and concept of the skill that is
being taught before being introduced to a set of procedures provided by an
algorithm that they may not fully understand.
Questions to Ponder...
Should traditional algorithms be taught in the early elementary grades?
What are some common misconceptions your students have with number concepts?
How do you plan instruction to clear up any misconceptions that students have?
Here are some fun and interactive math games to help students practice their Basic Skills:
Addition
|
Subtraction
|
Multiplication
|
Division
|
Mixed Review
|
Fractions and Decimals
|
Bugabaloo:
Fruit Shoot
Penguin Party Addition
Addition Matching Game
Balloon Pop - Addition
Number Line Addition
Add Like Mad
Addition with missing numbers
Pac-Man Addition
Number Twins Addition
Kitten Match
Soccer Math: Adding 2 – Digit Numbers
Okta’s Rescue: Counting and Adding the number of Octapus’ rescued
|
Subtraction Harvest:
Fruit Shoot
Pearl Search
Subtraction Matching Game
Number Line Subtraction
Subtraction Action
Subtraction with Missing Numbers:
Pac - Man Subtraction
|
Stun attack
Multiplication Matching Game
Balloon Pop - Multiplication
Multiplication Station:
Multiplication with Missing Numbers:
Pac-Man Multiplication:
Shooting Fruits: Multiples
Pumpkin Multiples
Space Race
Grand Prix Multiplication
Penguin Jump Multiplication
|
Division Station:
Division with Missing Numbers
Pac-Man Division
Demolition Division
Drag Race Division
Division Derby
|
Interactive Hundreds chart
Math Lines
Monkey Drive
Catch the Falling Stars
Fruit Shoot: Number – Word Match
Fruit Shoot: Mixed Operation
Mixed Arithmetic/ Operations
Counting, Adding and Subtracting Bubbles under a Seashell
Make 24
Primary Krypto: Similar to Make 24, choose the operation (+,-,x,÷) to make the target number
http://illuminations.nctm.org/ActivityDetail.aspx?ID=173
Select from a list of Counting games, adding games, Multiplication or
Fraction Games
http://www.maths-games.org/
Timez Attack: As the player travels through the dungeon conquering monsters, they learn their basic facts in either multiplication, division, addition, or subtraction http://www.bigbrainz.com/ |
Comparing Fractions
Haunted Comparing 3 Sets of Fractions Game http://www.mathplayground.com/HauntedFractions/HFGameLoader.html
Equivalent Fractions
Adding Fractions
Jeopardy: Changing Fractions to Decimals to Percents
Puppy Chase Decimals
Hungry Puppies Decimals
Puppy Pull
Decimal
|
Resources
Math Misconceptions: Pre K - Grade 5: From Misunderstanding to Deep Understanding. Bamberger, Honi J; Oberdorf, Christine; Schultz - Farrell, Karren. Heinemann. Portsmouth, NH 2010.
