Should Algorithms Be Taught in School?

                 According to the 2009 scores from the Program for International Student Assessment (PISA) report, which compares the knowledge and skills of 15-year-olds in 70 countries around the world, the United States performed below –average in mathematics and was ranked 25th out of 34 OECD countries.  In addition, “The United States spends more per student, on average, than other countries. The report notes that countries like Estonia and Poland perform at about the same level as the United States, while spending less than half the amount per student (USA Today,December 7, 2010).”  Does this information tell you anything about the way mathematics is taught in the United States?  In my opinion, these results signify a dire need for change in the education system of the United States.

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
Counting with Word Numbers	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 ·       Counting games Video: Creative Ways to Teach Your Child How to Count at Home
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
Representing, Ordering and Adding /Subtracting Decimals	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: http://sheppardsoftware.com/mathgames/earlymath/bugabalooShoes.htm   Fruit Shoot http://sheppardsoftware.com/mathgames/popup/popup_addition.htm     Penguin Party Addition http://sheppardsoftware.com/mathgames/popup/popup_addition.htm       Addition Matching Game http://sheppardsoftware.com/mathgames/matching/matching_addition.htm   Balloon Pop - Addition http://sheppardsoftware.com/mathgames/numberballoons/NumberBalloons_add_level1.htm     Number Line Addition http://sheppardsoftware.com/mathgames/earlymath/fruit_shoot_NumberLine.htm   Add Like Mad   http://sheppardsoftware.com/mathgames/Add%20Like%20Mad%20Math/AddLikeMad_easy.htm   Addition with missing numbers http://sheppardsoftware.com/mathgames/matching/AdditionX.htm   Pac-Man Addition http://sheppardsoftware.com/mathgames/mathman/mathman_additon.htm   Number Twins Addition http://sheppardsoftware.com/mathgames/numbertwins/numbertwins_add_10.htm           Kitten Match   http://www.mathplayground.com/ASB_KittenMatch.html   Soccer Math: Adding 2 – Digit Numbers http://www.math-play.com/soccer-math-adding-two-digit-whole-numbers/adding-two-digit-numbers.html   Okta’s Rescue: Counting and Adding the number of Octapus’ rescued http://illuminations.nctm.org/ActivityDetail.aspx?ID=219     Make 10 Frame http://illuminations.nctm.org/ActivityDetail.aspx?ID=75   Five Frame http://illuminations.nctm.org/ActivityDetail.aspx?ID=74	Subtraction Harvest: http://sheppardsoftware.com/mathgames/earlymath/subHarvest.htm     Fruit Shoot http://sheppardsoftware.com/mathgames/fruitshoot/fruitshoot_subtraction.htm      Pearl Search http://sheppardsoftware.com/mathgames/popup/popup_subtraction.htm     Subtraction Matching Game http://sheppardsoftware.com/mathgames/matching/matching_subtraction.htm   Number Line Subtraction http://sheppardsoftware.com/mathgames/earlymath/FS_NumberLine_minus.htm   Subtraction Action http://sheppardsoftware.com/mathgames/Add%20Like%20Mad%20Math/SubtractionAction_easy.htm   Subtraction with Missing Numbers: http://sheppardsoftware.com/mathgames/matching/SubtractionX.htm   Pac - Man Subtraction http://sheppardsoftware.com/mathgames/mathman/mathman_subtraction.htm	Stun attack http://sheppardsoftware.com/mathgames/popup/popup_multiplication.htm   Multiplication Matching Game http://sheppardsoftware.com/mathgames/matching/matching_multiplication.htm   Balloon Pop - Multiplication http://sheppardsoftware.com/mathgames/numberballoons/NumberBalloons_times_level1.htm     Multiplication Station: http://sheppardsoftware.com/mathgames/Add%20Like%20Mad%20Math/multiplcationstation_easy.htm   Multiplication with Missing Numbers: http://sheppardsoftware.com/mathgames/matching/MultiplicationX.htm   Pac-Man Multiplication: http://sheppardsoftware.com/mathgames/mathman/mathman_multiplication.htm   Shooting Fruits: Multiples http://sheppardsoftware.com/mathgames/multiple/multiple_frenzy.htm Pumpkin Multiples http://www.mathplayground.com/multiples.html     Meteor Multiplication http://www.mathplayground.com/ASB_MeteorMultiplication.html   Space Race http://www.mathplayground.com/ASB_SpaceRaceMultiplication.html   Grand Prix Multiplication http://www.mathplayground.com/ASB_GrandPrixMultiplication.html     Penguin Jump Multiplication http://www.mathplayground.com/ASB_PenguinJumpMultiplication.html   Product Game: 2-players – Get Four in  a Row http://illuminations.nctm.org/ActivityDetail.aspx?ID=29   Interactive Multiplication Table http://illuminations.nctm.org/ActivityDetail.aspx?ID=155	Croc Doc http://sheppardsoftware.com/mathgames/popup/popup_division.htm   Division Matching Game http://sheppardsoftware.com/mathgames/matching/matching_division.htm   Division Station: http://sheppardsoftware.com/mathgames/Add%20Like%20Mad%20Math/divisionstation_easy.htm   Division with Missing Numbers http://sheppardsoftware.com/mathgames/matching/DivisionX.htm   Pac-Man Division http://sheppardsoftware.com/mathgames/mathman/mathman_division.htm   Demolition Division http://www.mathplayground.com/ASB_DemolitionDivision.html   Drag Race Division http://www.mathplayground.com/ASB_DragRaceDivision.html   Division Derby http://www.mathplayground.com/ASB_Division_Derby.html     The Quotient Café http://illuminations.nctm.org/ActivityDetail.aspx?ID=224	Interactive Hundreds chart http://www.apples4theteacher.com/math/games/100-number-chart-one.html   Math Lines http://sheppardsoftware.com/mathgames/mathlines/mathlinesmenu.htm   Monkey Drive http://sheppardsoftware.com/mathgames/monkeydrive/monkeymath.htm     Catch the Falling Stars http://sheppardsoftware.com/mathgames/catchthestars/catchthestarsmenu.htm     Fruit Shoot: Number – Word Match http://www.sheppardsoftware.com/mathgames/earlymath/fruitShootNumbersWords.htm Fruit Shoot: Mixed Operation http://sheppardsoftware.com/mathgames/fruitshoot/FS_Mixed_Operations.htm   Mixed Arithmetic/ Operations http://sheppardsoftware.com/mathgames/arithmetic/arithmetic.htm   Counting, Adding and Subtracting Bubbles under a Seashell http://illuminations.nctm.org/ActivityDetail.aspx?ID=198   Make 24 http://www.mathplayground.com/make_24.html   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/	Fraction Models http://illuminations.nctm.org/ActivityDetail.aspx?ID=11     Comparing Fractions http://www.mathplayground.com/ASB_TugTeamFractions.html   Haunted Comparing 3 Sets of Fractions Game http://www.mathplayground.com/HauntedFractions/HFGameLoader.html   Equivalent Fractions http://www.mathplayground.com/Triplets/Triplets.html   Bridge Builders Fraction Game http://www.mathplayground.com/FractionGame/FractionGame.html   Adding Fractions http://illuminations.nctm.org/ActivityDetail.aspx?ID=18     Jeopardy: Changing Fractions to Decimals to Percents http://www.math-play.com/Fractions-Decimals-Percents-Jeopardy/fractions-decimals-percents-jeopardy.html           Puppy Chase Decimals http://www.mathplayground.com/ASB_Puppy_Chase_Decimals.html   Hungry Puppies Decimals http://www.mathplayground.com/ASB_Hungry_Puppies_Decimals.html     Puppy Pull Decimal http://www.mathplayground.com/ASB_Puppy_Pull_Decimals.html

   

Resources

Math Misconceptions: Pre K - Grade 5:  From Misunderstanding to Deep Understanding.  Bamberger, Honi J; Oberdorf, Christine; Schultz - Farrell, Karren.  Heinemann.  Portsmouth, NH 2010.