College of EducationDr. Jill DrakeSpring 2012

Warm-up: Activating Prior KnowledgeCount the number of letters in your

first nameFor each letter in your name tell

your elbow partner something you remember about last week’s class.

Breaking Down the ChapterGet into groups of 5Count off in your group – 1 to 5

Breaking Down the ChapterEach group will be the “expert” of a section of the

chapterPerson 1 – Instruction in Mathematics and

AlgorithmsPerson 2 – Computational FluencyPerson 3 –Conceptual Learning and Procedural

Learning and Paper-and-Pencil Procedures Today Person 4 – Learning Misconceptions and Error

PatternsPerson 5 – Error Patterns in Computation

Breaking Down the ChapterYour group’s task is the determine the

important information for your assigned section.

Take notes on section and be prepare to make comments on the appropriate slide.

Instruction in MathematicsWe are not interested in students just doing

arithmetic in classrooms; we want to see the operations of the arithmetic applied in real-world contexts where students observe and organize data.

Instruction in mathematics is moving toward covering fewer topics but in greater depth and toward making connections between mathematical ideas.

Computational FluencyConceptual understanding- comprehension of mathematical

concepts, operations, and relations.Procedural fluency- skill in carrying out procedures flexibly,

accurately, efficiently, and appropriately.You should not focus on just paper and pencil work. We do

not want students to feel limited to computing only with paper and pencil. (& vise versa)

We need to integrate arithmetic and all of the mathematics we teach with the world of our students, including their experiences with other subject areas.

Students do not only need to know how to compute, but when to compute.

In order for our students to gain computational fluency, they need to be able to use different methods of computation in varied problem-solving situations.

The two methods of computation are approximation(mental estimation) and exact computation (mental computation, calculator, and paper and pencil)

AlgorithmsAn algorithm is a step-by-step procedure for

accomplishing a task, such as solving a problem.If some students have already learned different

algorithms for example, a different way to subtract learned in Mexico and Europe), remember that these students procedures are quite acceptable if they always provide the correct number

As we teach paper-and-pencil procedures, we need to remember that our students are learning and applying concepts as they are learning procedures

Conceptual Learning and Procedural Learning

Procedural learning must be based on concepts already learned.

Students need to understand the meaning of each operation is needed in particular situations.

Conceptual understanding is so important that some math educators stress the invention of algorithms by young students; they fear that early introduction of standard algorithms maybe detrimental and not lead to understanding important concepts.

Paper-and-Pencil Procedures TodayTeachers need paper and pencil to look for

misconceptions and error patterns in their written work.

Students learn from their errors by seeing their step-by-step written work, and with calculators this cannot always be seen.

The use of the Paper- and-Pencil procedure over time becomes more automatic.

The student employs increasingly less conceptual knowledge and more procedural knowledge( Proceduralization).

Learning Misconceptions and Error PatternsCareless mistakes and misconceptions Relate to diceStudents connect old concepts to new

informationYou don’t have to make a mistake to work hardAssess the students to figure out their

knowledgeOver generalizing- jumping to conclusionsStudents have a learned disability not a learning disability.

Error Patterns in ComputationErrors are seen as an opportunity to reflect and learnWe can use errors as problem solving situationsDifferent computing strategies, look for evidence of

how the students are thinking.Algorithms that incorporate error patterns are called

buggy algorithmsStudents who invent their own algorithms learn error

patternsRather than just scoring the paper we need to analyze

to see why their answer is incorrect and figure out the error that was made.

ReviewWhat is the difference between conceptual

learning and procedural learning?Which type of understanding would be more

beneficial to students? Why?

Vocabulary…Procedural Errors – involve errors in skills

and/or step-by-step procedures (algorithms) needed to solve mathematical problems

Conceptual Errors – errors that are caused by the misunderstanding of mathematical ideas such as place value, meaning of operations, and number sense.

Both Procedural and Conceptual Errors – errors that involve violations of an algorithm AND a misunderstanding of a mathematical idea.

Conceptual ErrorsConceptual understanding reflects a

student's ability to reason in settings involving the careful application of concept definitions, relations, or representations of either.

Such an ability is reflected by student performance that indicates the production of examples, common or unique representations, or communication indicating the ability to manipulate central ideas about the understanding of a concept in a variety of ways.

Example: Circle the triangles

Procedural ErrorsStudents demonstrate procedural

knowledge in mathematics when they select and apply appropriate procedures correctly; verify or justify the correctness of a procedure using concrete models or symbolic methods; or extend or modify procedures to deal with factors inherent in problem settings.

Example: Write the numbersOne: _________

Two: __________

Three:_________

Four: __________

Both Procedural Errors and Conceptual Errors

Case Study ReviewBasic Facts Sequence:

1. Interview and administer the Pre-test2. Discover if they know the concept

Teach the concept if it is not know

3. Teach Number strategies4. Develop Automaticity (80% Accuracy)5. Administer the Post-test

Updated Fall 2009 21

Diagnosing Basic Facts…Use a Sample: Basic Facts Cards

Can use as a pre-test AND a post-test.Procedure:

Create flash cards of ALL times tables. One table at a time, jumble cards for that table, flash one

card at a time, and count three seconds in mind. Based on response to each flash card, create three piles

of cards: Know Do NOT know Do not know WITHIN 3-second limit

Use piles to complete basic facts card (A legend/code/key is needed to distinguish the meaning of the three piles)

Key: ☺- Answered CorrectlyX- Answered Correctly After 3 SecondsX- Answered Incorrectly

Multiplication Pretest

Updated Fall 2009 23

Correcting Basic Fact ErrorsTeach the meaning of the operation.

Teach number relationship strategies.Addition/Subtraction – Van de Walle, pp. 159-168Multiplication/Division – Van de Walle, pp. 168-174

Provide ongoing drill to develop automaticity (3 second answers).See Van de Walle, 2004, p. 175 (Activities 11.21 and

11.22).Games, individual speed tests, competitions, oral

drills, written drills, repetitive writing of facts, and so on that are designed to achieve fast answers to unknown facts.

Draw me a picture that shows what 3x7 mean?

One group of 7. A second

group of 7. A third group of 7.

Three groups with 7 things in each.The x means “of” – three groups of 7.

Three groups of 7 equals 21.

Number Strategy: Commutative Property

List of Basic Facts I worked on:

7x37x47x57x6

Matching flash cards to practice strategy:3 x 7 = 7 x 3 =

214 x 7= 7 x 4 =

285 x 7= 7 x 5 =

356 x 7= 7 x 6 =

42

I put each of these on flash cards. First, we practiced these basic facts and our number strategy by matching flash cards. Then, we practiced by playing concentration. The last time we practiced, we played a grab relay: who could match the most sets of cards. I end this part of the session by asking the answer of each fact and having the student to tell me how he knew the answer. I expected him to use the commutative property (“I knew 3x7=21 so, 7x3=21”).

Automaticity:

9/19/09: We meet for 22 minutes and I did 7s, 8s, and 9s flash cards with him. I ended the session by having him write and re-write each of these facts he missed until he had them memorized.

9/21/09: I had him do a Mad Minutes Timed Test. I took him 4 minutes to finish the test. I ended the session by having him write and re-write each of these facts he missed until he had them memorized.

Basic Facts Multiplication Number Strategieshttp://www.kathimitchell.com/mathfolder/

mathpage.htmZeros and onesDoubles, and Double AgainSkip Counting (twos, fives)Commutative PropertyNifty NinesIf tens, then nine.Make 9 with one lessHelping Facts (adding up or subtracting down)

Interviewing: Data Source #2Goals of an interview

Gain insight into thinking, attitudes, and strengths and weaknesses of student and his/her environment

Verify pre-diagnosis of error by ruling out reasons for errors

How to begin an interview?Insights – use list of questions from readingVerification – say something like…

“Show me how to solve this problem.” “Solve this problem and talk out loud as you solve it.”

Organize data into Solve Sheet #1.

Final Thoughts…Remember:

Diagnosing and Correcting Mathematical Errors involve four processes: the Diagnosing Process, the Correcting Process, the Evaluating Process, and the Reflection Process.

Use your diagnosis to guide your instruction. Interviewing students and observation of students are

important tools for diagnosticians. They can help rule out and/or reveal reasons for errors.

When correcting errors related to basic facts, refer to last week’s lecture notes.

When diagnosing errors related to whole numbers, refer to the list of errors related to whole number operations.

Many procedural errors occur because students do not have an adequate understanding of the concepts that underlie these procedures. For this reason “procedural knowledge must be tied to conceptual knowledge” (Ashlock, 2006, p. 47) when correcting errors.