Early Instruction in Mathematics: Laying the Foundation for Conceptual Understanding and Successful

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Early Instruction in Mathematics Laying
the Foundation for Conceptual Understanding and
Successful Achievement Ben Clarke, Ph.DPacific
Institutes for ResearchFebruary 24, 2005
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Contact Information

• Email
• clarkeb_at_uoregon.edu
• Phone
• [phone]
• Special Thanks to David Chard, Scott Baker,
  Russell Gersten, and Bethel School District

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ExerciseWhat are two ways to solve the following
problem?
Miss Spider is hosting a tea party for her 3
insect friends. If she wants each friend to have
two cookies with their tea, how many cookies will
she need to make?
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Possible Solution Strategies
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Discussion Point

• What ways did you solve the problem?
• What is the easiest way to solve the problem?
  The hardest?
• How would you differentiate instruction for
  students in your classroom?

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A primary goal of schools is the development of
students with skills in mathematics

• Mathematics is a language that is used to express
  relations between and among objects, events, and
  times. The language of mathematics employs a set
  of symbols and rules to express these relations.

(Howell, Fox, Morehead, 1993)
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Mathematical knowledge is fundamental to function
in society

• For people to participate fully in society, they
  must know basic mathematics. Citizens who cannot
  reason mathematically are cut off from whole
  realms of human endeavor. Innumeracy deprives
  them not only of opportunity but also of
  competence in everyday tasks. (Adding it Up,
  2001)

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Proficiency in mathematics is a vital skill in
todays changing global economy

• Many fields with the greatest rate of growth will
  require workers skilled in mathematics.
  (Bureau of Labor Statistics,1997)
• Companies place a premium on basic mathematics
  skill even in jobs not typically associated with
  mathematics.
• Individuals who are proficient in mathematics
  earn 38 more than individuals who are not.
  (Riley, 1997)

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Despite the efforts of educators many students
are not developing basic proficiency in
mathematics

• Only 21 of fourth grade students were classified
  as at or above proficiency in mathematics, while
  36 were classified as below basic. This pattern
  was repeated for 8th and 12th grade (NAEP, 1996).
• According to the TIMS (1998), US students perform
  poorly compared to students in other countries.
  United States 12th graders ranked 19th out 21
  countries.
• The result Students lack both the skills and
  desire to do well in mathematics (MLSC, 2001)

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Achievement stability over time

• The inability to identify mathematics problems
  early and use formative evaluation is problematic
  given the stability of academic performance.
• In reading, the probability of a poor reader in
  Grade 1 being a poor reader in Grade 4 is .88
  (Juel, 1988).
• The stability of reading achievement over time
  has led to the development of DIBELS.

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Trajectories The Predictions

• Students on a poor reading trajectory are at
  risk for poor academic and behavioral outcomes in
  school and beyond.
• Students who start out on the right track tend to
  stay on it.

(Good, Simmons, Smith, 1998)
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Developmental math research

• Acquisition of early mathematics serves as the
  foundation for later math acquisitions (Ginsburg
  Allardice, 1984)
• Success or failure in early mathematics can
  fundamentally alter a mathematics education
  (Jordan, 1985)

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Discussion Point Trajectories

• Do math trajectories and reading trajectories
  develop in the same way?
• How could they be similar?
• How could they be different?
• Are they the same for different types of learners
  (e.g. at-risk)?

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Math LD

• 5 to 8 percent of students
• Basic numerical competencies are intact but
  delayed
• Number id
• Magnitude comparison
• Difficulty in fact retrieval
• Proposed as a basis for RTI and LD diagnosis

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Math LD (cont.)

• Working memory deficits hypothesized to underlie
  fact retrieval difficulty
• Students use less efficient strategies in
  solving math problems due to memory deficits
• Procedural deficits often combine with conceptual
  misunderstanding to make solving more complex
  problems difficult.

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Big Ideas in Beginning Math Instruction

• Number Sense
• Informal to Formal Mathematics
• Instruction should be centered on enhancing
  student understanding of critical concepts that
  will be further developed in later grades
• Early Intervention and Prevention

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Discussion Point Number Sense

• What is number sense?
• What does number sense look like for the
  grade/students you work with?
• How does number sense change over time and what
  differentiates those with and without number
  sense over time?

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The Ghost in the MachineNumber Sense

• Number sense is difficult to define but easy to
  recognize
• (Case, 1998)
• Nonetheless he defined it!

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Case (1998) Definition

• Fluent, accurate estimation and judgment of
  magnitude comparisons.
• Flexibility when mentally computing.
• Ability to recognize unreasonable results.
• Ability to move among different representations
  and to use the most appropriate representation.

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Cases Definition (cont.)

• Regarding fluent estimation and judgment
• of magnitude (i.e. rate and accuracy).
• Recent empirical support for this insight
• Landerl, Bevan, Butterworth
• (2004), 3rd grade
• Passolunghi Siegel (2004),
• 5th grade

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Number Sense

• A key aspect of various definitions of number
  sense is flexibility.
• a childs fluidity and flexibility with numbers,
  the sense of what numbers mean, and an ability to
  perform mental mathematics and to look at the
  world and make comparisons
• Students with number sense can use numbers in
  multiple contexts in multiple ways to make
  multiple mathematics decisions.

(Gerston Chard, 1999)
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Big Idea More, Less, and Same

• The concept of more is developed before less and
  same
• Kids have the chance to develop a sense of more
• An understanding of less happens later
• Can be developed by tying it to more
• Have students name which group has more - than
  less

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Activity Find the Same Amount (Van de Walle 04)

• Give children a dot card and have them find the
  card with the same amount of dots
• Vary by having them find cards that have more or
  less dots

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Exercise

• What behaviors would you observe to note
  development of student skill?
• How would you differentiate the activity for
  students?
• What other activities could you use to teach the
  concepts of same, more, and less?

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Big Idea Counting

• Sequence words w/out reference to objects
• Students learn 1 through 12 Unstructured and
  learned through rote memorization
• Students learn 1-9 repeated structure (more
  difficult to apply to teen numbers 13-19)
• Students learn decade transitions

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Counting (cont.)

• Counting occurs when sequences words are assigned
  to objects on a one to one basis
• Counting first step in making quantitative
  judgments about the world exact

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Counting (cont.)

• 5 Principles of Counting
• One to one correspondence
• Stable order principle
• Cardinal principle (critical)
• Item indifference
• Order indifference
• (Gelman Gallistel, 1978)

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Cardinality

• Developed around the age of 4 (Ginsburg Russel,
  1981)
• All or nothing phenomena(Permangent, 1982)
• Can be taught and focus children on seeing
  individual items in terms or being part of a
  larger unit (Fuson Hall, 1983)

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Counting (cont.)

• Counting on and back are critical milestones in
  development because they allow new strategies in
  solving problems
• Counting on is considered a hallmark of early
  numeracy

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Activity Real Counting On (Van de Walle 04)

• Need cards with numbers 1-7, die, paper cup, and
  counters. First student takes card and places
  number of counters in cup (card is kept by the
  cup). Second student rolls the die and places
  the counter next to the cup. Students then
  figure out how many counters there are in total.

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Exercise

• What behaviors would you observe to note
  development of student skill?
• How would you differentiate the activity for
  students?
• What other activities could you use to teach
  counting on?

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Big Idea Number Symbols (Identification and
Production)

• Children learn about written symbol system for
  numerals before they enter school
• Children see numbers in multiple formats ordinal
  (floors), labels (phone), and measurement (dates)
• Most numbers are use for description not to
  represent cardinality (focus of formal
  mathematics)

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Activities

• Most activities require students to match a match
  a number to a set by writing or identifying the
  number (frequently used in reverse)
• After mastery, little gained from these types of
  activities
• Also common to have students trace numbers when
  first learning to write numbers

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BIG IDEA RELATIONSHIPS BETWEEN NUMBERS

• One and two more, one and two less
• Anchors/Benchmarks Five and Ten
• Part-Part-Whole Relationships

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One and two more, One and two less

• Not built when student rote or sequence count
• Builds on counting on counting back activities
• Activities Given a card make a set using
  counters that is one/two more or less.
• Extend by placing cards with number symbols next
  to sets
• Have students state number sentences Two more
  than four is six.

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Mental Number Line

• Hypothesized to underlie early math development
• Developed by most students when they enter school
  through the number 9
• Robust in making quantity comparisons and in
  beginning addition and subtraction

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Quantity Comparison (cont.)

• Hypothesized that mental number line is packed
  more tightly at higher numbers and not as tight
  at lower levels. Thus 2 and 3 are conceptualized
  as farther apart than 92 and 93.
• (Siegler and Robinson, 1982)

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Quantity Comparison

• Number of studies investigated reaction times for
  solving quantity comparison problems to
  hypothesize about number lines
• Magnitude of the discrepancy Symbolic distance
  effect (e.g. 4 and 8 compared more quickly than 4
  and 5)
• Smaller the size of the smaller stimulus the
  faster the comparison Min effect (e.g. 3 and 5
  compared more quickly than 5 and 7)

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Anchoring numbers to 5 and 10

• Create sense of number that is made up of
  critical parts (e.g. 7 is 5 and 2, 13 is 10 and
  3)
• Most common model is the ten frame (Wirtz 1974)

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Model Ten Frame Making 4 (one less than 5)
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Model Ten Frame Making 8 (5 and 3 more)
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Extending the Ten Frame

• Builds an initial understanding of the base 10
  system that will form a conceptual basis for
  later mathematics
• Can be used to demonstrate parts and wholes of
  numbers

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Discussion Point Ten Frame

• How would you introduce and fade ten frame
  models?
• How could you use the ten frame to introduce the
  concept of base 10 and regrouping for later
  mathematics?

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Part-Part-Whole Relationships

• Not built by counting activities
• Allows greater understanding of the flexibility
  of number
• Can begin by building wholes and advance to
  finding missing parts
• Useful in solving a variety of basic word
  problems (Jitteranda 05)

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Activity Covered Parts (Van de Walle)

• Have a set of counters equal to the whole you
  want to examine (e.g. 7). Have one student place
  all under a tub and then pull out some number.
  The other student must identify the missing part
  under the tub.

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Exercise

• What behaviors would you observe to note
  development of student skill?
• How would you differentiate the activity for
  students?
• What other activities could you use to teach
  missing parts?

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Big Idea Informal to Formal Mathematics

• Early math concepts are linked to informal
  knowledge that a student brings to school
  (Jordan, 1995)
• Linking informal to formal math knowledge has
  been a persistent theme in the mathematics
  literature (Baroody, 1987)

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From counting to addition

• Addition makes counting abstract
• Addition is counting sets
• 2 apples and 3 apples

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The link to addition

• Count All starting with First addend (CAF)
• Count All starting with Larger addend (CAL)
• Count On from First addend (COF)
• Count on from Larger addend (COL)

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Development of early addition (cont.)

• Students first use CAF and COF supporting a
  uniary view of addition (e.g. changing one
  number)
• CAL and COL supports a binary (I.e. combining two
  number) view of addition
• Based on principle of commutativity
• Students who understand communtativity can use
  the COL strategy

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Addition Strategies

• COL strategy has been termed the Min strategy
  because it requires the minimal amount of
  counting steps to solve a problem
• Recognized as the most cognitively efficient

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Addition (cont.)

• Some problems were solved quicker than expected
• Based on patterns such as doubles, tens
• Indicate development of number sense
• (Groen Parkman, 1972)
• Siegler (1982) hypothesized that use of the min
  effect was the critical variable in 1st grade
  math and failure to do so was predictive of later
  failure in mathematics

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Recommendations for teaching mathematics

• Efforts to improve students' mathematics learning
  should be informed by scientific evidence and
  their effectiveness should be evaluated
  systematically. Such efforts should be
  coordinated, continual, and cumulative.
• Additional research on the nature, development,
  and assessment of mathematical proficiency

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Moving to Instruction

• Children enter school with a base of math
  knowledge and the ability to interact with number
  and quantity. Very context dependent. (6)
• Instruction in math is based on the interactions
  between student, teacher, and content. Students
  must link informal knowledge with formal often
  abstract knowledge.(9)

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Numbers are abstractions

• To criticize mathematics for its abstraction is
  to miss the point entirely. Abstraction is what
  makes mathematics work. If you concentrate too
  closely on too limited an application of a
  mathematical idea, you rob the mathematician of
  his or her most important tools analogy,
  generality, and simplicity (Stewart, 1989, p.
  291)
• The difficulty in teaching math is to make an
  abstract idea concrete but not to make the
  concrete interpretation the only understanding
  the child has (i.e. generalization must be
  incorporated).

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Abstraction and Culture

• Mathematics is at the same time inside and beyond
  culture it is both timely and timeless (xiv)

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The Number 7

• Could be used to describe
• Time
• Temperature
• Length
• Count/Quantity
• Position
• Versatility makes number fundamental to how we
  interact with the world

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Five Strands of Mathematical Proficiency

• 1. Conceptual Understanding-comprehension of
  mathematical concepts, operation, and relations
• 2. Procedural Fluency-skill in carrying out
  procedures flexibly, accurately, efficiently, and
  appropriately
• 3. Strategic Competence-ability to formulate,
  represent, and solve mathematical problems

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Five Strands (cont.)

• 4. Adaptive Reasoning-capacity for logical
  thought, reflection, explanation, and
  justification
• 5. Productive Disposition-habitual inclination to
  see mathematics as sensible, useful, and
  worthwhile, coupled with a belief in diligence
  and one's own efficacy.

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Instruction What do we know?
The knowledge base on documented effective
instructional practices in mathematics
is less developed than reading.
Mathematics instruction has been a concern to
U.S. educators since the 1950s, however,
research has lagged behind that of
reading.
Efforts to study mathematics and mathematics
disabilities has enjoyed increased interests
recently.
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Purpose

• To analyze findings from experimental research
  that was conducted in school settings to improve
  mathematics achievement for students with
  learning disabilities.

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Identifying High Quality Instructional Research
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Method

• Included only studies using experimental or
  quasi-experimental group designs.
• Included only studies with LD or LD/ADHD samples
  OR studies where LD was analyzed separately.
• Only 26 studies met the criteria in a 20-year
  period (Through 1998).

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RESULTSFeedback to Teachers on Student
Performance

• Seems much more effective for special educators
  than general educators, though there is less
  research for general educators.
• May be that general education curriculum is often
  too hard.

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Feedback to Teachers on Student Performance
(cont.)

• 3. Always better to provide data and
  suggestions rather than only data profiles (e.g.,
  textbook pages, examples, packets, ideas on
  alternate strategies).

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Feedback to Students on Their Math Performance

• Just telling students they are right or wrong
  without follow-up strategy is ineffective (2
  studies).
• Item-by-item feedback had a small effect (1
  study).
• Feedback on effort expended while students do
  hard work (e.g., I notice how hard you are
  working on this mathematics) has a moderate
  effect on student performance.

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Goal Setting

• Studies that have used goal setting as an
  independent variable, however, show effects that
  have not been promising.
• Fear of failure?
• Requires too much organizational skill?

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Peer Assisted Learning

• Largest effects for well-trained, older students
  providing mathematics instruction to younger
  students
• Modest effect sizes (.12 and .29) were documented
  for LD kids in PALS studies. These were
  implemented by a wide range of elementary
  teachers (general ed) with peer tutors who didnt
  receive any specialized training (More recent
  PALS data not included)

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Curriculum and Instruction

• Explicit teacher modeling, often accompanied by
  student verbal rehearsal of steps and
  applications
• -moderately large effects
• Teaching students how to use visual
  representations for problem solving
• -moderate effects

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Key Aspects of Curriculum Findings

• The research, to date, shows that these
  techniques work whether students do a lot of
  independent generation of the think alouds or the
  graphics or whether students are explicitly
  taught specific strategies

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Overview of Findings

• Teacher modeling and student verbal rehearsal
  remains phenomenally promising and tends to be
  effective.
• Feedback on effort is underutilized and the
  effects are underestimated.
• Cross-age tutoring seems to hold a lot of promise
  as long as tutors are well trained.
• Teaching students how to use visuals to solve
  problems is beneficial.
• Suggesting multiple representations would be
  good.

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Program Development What not to do

• US curriculum covers more topics but more
  superficially than textbooks in other countries
• Students, especially those who struggle, have
  difficulty in connecting understanding across
  topics

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Program Development What to do

• Three year grant to develop and refine
  Kindergarten math curriculum
• Y1 Intervention development and refinement
  Measurement refinement and validation
• Y2 Intervention efficacy Implementation
  analysis and hypothesis development
• Y3 Hypothesis testing re differential
  effectiveness of intervention

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Curriculum content

• Scope and sequence based on 4 integrate strands
• Numbers and operation
• Geometry
• Measurement
• Vocabulary

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Curriculum Content (cont.)

• Key goals
• Building conceptual understanding to abstract
  reasoning via mathematical models
• Building math related vocabulary
• Procedural fluency/automaticity
• Building competence in problem solving

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Sample Teacher Language Concept Introduction

• Introduce activity patterning
• Tell children that you will start a pattern and
  that when they have figured out the pattern, they
  can join in. Start the AB pattern clap, slap
  your legs clap, slap your legs clap, slap your
  legs. Reinforce children as they join in. Ask a
  child to describe the pattern. Reinforce the
  pattern by saying, Yes, the pattern was clap,
  slap clap, slap. What should come next in our
  pattern? Reinforce appropriate responses by
  extending the pattern as the children tell you
  what the next activities should be.
• Tell children that you will start a new pattern
  and that when they have figured out the pattern,
  they can join in. Start the AAB pattern touch
  your head, touch your head, touch your nose
  touch your head, touch your head, touch your
  nose. Reinforce children as they join in.
• Ask a child to describe the pattern. Reinforce
  the pattern by saying, Yes, I touched my head,
  touched my head, and touched my nose. I touched
  my head, touched my head, and touched my nose.
  What should come next in our pattern? Reinforce
  appropriate responses by extending the pattern as
  the children tell you what the next activities
  should be.

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Student Materials
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Automaticity Builder Software (IntellItools, 2004)
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Structure

• Lessons sequenced in sets of 5
• Designed for whole class delivery in 20 minutes
• Culminates with group problem solving activity
  which integrates math discourse with strands
  taught during the previous 4 lessons

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Assessment

• Assessment goals
• Continued refinement of screening and progress
  monitoring measures
• Development of authentic measures to examine
  specific areas of mathematics instruction and
  skill development
• Determine relationship between screening,
  progress monitoring, authentic, and high stakes
  measures of mathematics.

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Math Assessment

• Critical pre-math skills may be centered around
  the concept of number sense.
• Number sense has been defined as
  a childs fluidity and flexibility with
  numbers, the sense of what numbers mean, and an
  ability to perform mental mathematics and to look
  at the world and make comparisons

(Gerston Chard, 1999)
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Two Tier System

• Screening Measures
• Quick and easy to administer
• Predictive of later math achievement
• Diagnostic Measures
• Used with select students as identified by
  screening measures
• In-depth analysis of critical early math concepts

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Measures

• Screening Measures
• EN-CBM Oral Counting measure
• Students orally count for one minute. No student
  materials.
• EN-CBM Number Identification measure

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Measures (cont.)

• EN-CBM Quantity Discrimination measure
• EN-CBM Missing Number measure
• Diagnostic Measure
• Number Knowledge Test

2 3
4 1
5 10
9 4
2 __ 4 6 7 __ __
4 5
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Sample Number Knowledge Test Items (Levels 0,1)
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Sample Number Knowledge Test Items (Levels 2 3)