Tracing the Origins of Weak Learning of Spatial Measurementstemproj/presentations/STEM_MSUColloquium_2007.pdf8/3/2012 STEM Presentation, MEC, 9-07 14 STEM Project Sequence 1. Pick

Tracing the Origins of Weak

Learning of Spatial Measurement

Jack Smith & STEM Team*

•Leslie Dietiker, Gulcin Tan (METU), KoSze Lee, Hanna Figueras,

Aaron Mosier, Lorraine Males, Leo Chang, & Matt Pahl

“Strengthening Tomorrow’s Education in Measurement”

(NSF, REESE Program, 2 years & NSF via CSMC)

MSU Mathematics Education Colloquium, 9-19-07

8/3/2012 STEM Presentation, MEC, 9-07 2

Presentation Overview (major chunks)

The Problem: What is it and why address it?

Project Design: STEM structure & logic

“Results” #1: Our Curriculum Coding Scheme (in

development)

“Results” #2: Tough calls in deciding what is

measurement content

Wrap-Up and look ahead


The Problem (briefly)

Students don’t seem to understand spatial measurement very well (length, area, & volume)

• Highly practiced, routine problems OK

• Any sort of non-routine or multi-step problem not OK

• Poor written explanations

Not a problem of understanding the quantities and intuitively how to measure them (e.g., covering a surface)

Students “confuse” measures, in 2-D and 3-D situations

Length is poorly related to area and volume


NAEP Evidence (2003 Mathematics Assessment)

�Only about half of 8th graders solved the “broken ruler” problem correctly (L)

Less than half of 4th graders measured a segment with a metric ruler correctly (L)

Only 2% of 8th graders found a figure’s area on a geoboard and constructed another figure with the same area (A)

Only 39% of 8th graders found the length of a rectangle, given its perimeter and width (L)

Gap between poor & minority students and majority

students is greatest for measurement (4th & 8th)


Evidence from TIMSS

Overall (and consistently) our 4th graders perform pretty well and our 8th graders lag

The 8th grade lag was greatest for geometry and measurement

Textbook analysis also showed U.S. texts included less geometry & measurement content in grades 5 to 8

Lag is not made up in 12th grade


Evidence from Empirical

Research

Common finding: Confusion of area and perimeter for simple 2-D figures (Woodward & Byrd, 1983; Chappelle & Thompson, 1999)

Poor grasp of the relationship between length units and area units (e.g., inches and square inches) (Nunes, Light, & Mason, 1993; Kordaki & Portani, 1996)

Weak understanding of how length is related to area & volume in computational formulas (Battista, 2004)

Not all elementary students “see” rows and columns in a rectangular arrays (Battista 1998; Battista, et al. , 1998)


But… There is a Positive Side

Young children (e.g., 2nd grade) can learn to do and understand spatial measurement (Lehrer & Schauble’s work at U. Wisconsin)

• Carefully designed tasks

• Expert guidance from thoughtful researchers

• Teachers who understand & question children

Similar results for length from Stephan & Cobb (1st grade, 2005?)

Common element: “Problematize unit”; build a kids’ theory of measurement

Issue: What to do with these “existence proofs”?


Sharpening the Problem

What explains the problem of learning & teaching spatial measurement (length, area, & volume) in ordinary American classrooms?

Why is performance/understanding so low even with students’ extensive experience space and informal measurement outside of school?

Lots of evidence OF the problem but no explanation of its nature & genesis

Quandary for educators: What do we work on to help?

• Curriculum

• Pre-service teacher education

• Professional development


Six Possible Factors

1. Weaknesses in written curricula

2. Too little instructional time on measurement

3. The dominance of static representations of spatial quantities (esp. for area & volume)

4. Problems specific to talk about spatial quantities in classrooms (common everyday vocabulary, speakers talking past one another)

5. Instructional & assessment focus on numerical computation; numbers lose meaning as measures

6. Weaknesses in teachers’ knowledge


Other Factors?


Commentary on the Factors

These factors constitute a space of solutions

Cartesian analogy: Solution is a region in 6-space, with a range of values on each dimension

But… the dimensions are not independent; there are many relations of influence

Our approach (analogy to statistical models)

• Look for “main effects”

• Expect large (massive?) “interactions”

We start with Factor 1 (written curricula) because curricula are fundamental


End of Part I


STEM Project Overview

Goals: Assess impact of Factor 1 (quality of written

curricula) carefully and Factors 4 & 5 selectively

Focus exclusively on length, area, & volume, grades K–8

Develop an “objective” standard for evaluating the

measurement content of select written curricula

How much of the problem can be attributed to the content

of written curricula?

Prepare for next steps (pursue a program of research on

this problem)


STEM Project Sequence

1. Pick a small number of representative elementary and middle school mathematics curricula

2. Locate the measurement content of these curricula

3. Develop an appropriate framework for evaluating the that content

• Mathematically accurate and deep

• Informed by existing research

4. Complete the evaluation

5. Report the evaluation, to the community & the authors

6. Examine some classroom “enactments” of specific measurement topics


Step 1: Choose the Curricula Elementary (K–6):

• Everyday Mathematics

• Scott Foresman-Addison Wesley’s Mathematics

• Saxon Mathematics

Middle School (6–8):

• Connected Mathematics Project

• Glencoe’s Mathematics: Concepts & Applications

• Saxon Mathematics

Criteria for choice

• Market-share

• Standards-based vs. publisher developed

• Saxon is different from both

Representativeness argument


Step 2: Find the Measurement Content

Should be easy, right? Just look for the “measurement”units

In fact, has not been so easy

We include “measurement” content, but also other content that looks like measurement to us

Units of text: units, lessons, problems

We include measurement lessons & problems (in non-measurement lessons)

Our criterion: Does this content very likely require reasoning with/about measures of length, area, or volume? If so, it is “in”


Step 3: Develop the Framework (henceforth, Curriculum Coding Scheme [CCS])

Quality of the analysis depends directly on the validity & applicability of the CCS

Core STEM question: Do students have sufficient opportunity to learn the mathematics of spatial measure?

Validity of the CCS depends on:

• Mathematical completeness & depth

• Learning from the empirical research literature

• Review by “experts”

Applicability of the CCS depends on:

• Match to textual types in written curricula

• Appropriate grain-size of measurement knowledge


Step 4: Code the Curricula (i.e., the spatial measurement content)

Our current state:

• Step 2: 90% complete, some thorny issues in middle school

• Step 3: Detailed CCS for length, 80% complete

• Have “test-driven” versions of the CCS

This semester: Code the elementary length content

• Some content explicitly involves multiple measures

• Still need to decide which of “length & area” and “length & volume” will included in the length analysis

Shape of the analysis (some options):

• Results for length, for area, and for volume OR

• Length, area, length & area, volume, length & volume, surface area & volume


Step 5: Explore Some Classroom

Enactments “Some enactments”: Limited time & resources

Want to extend the use of the CCS to classroom lessons

Same question: Do students in this classroom have sufficient opportunity to learn this measurement topic?

Our target lesson segments:

• Introduction to length

• Complex lengths

• Introduction to area

• Area & perimeter

• Surface area & volume

Videotape & analyze lessons; Not an evaluation of teachers

Focus: How do teachers who are using their curricula seriously transform it in their teaching? What effect on OTL?


End of Part II


Overview of Development Process (Curriculum Coding Scheme [CCS])

Initial focus was on conceptual knowledge, because research suggested doing without understanding

Identified elements of knowledge that holds for quantities in general (e.g., transitivity) before those that hold for spatial quantities specifically

Realization #1: Can’t just analyze the measurement knowledge; Need analysis of textual forms (e.g., statements vs. questions vs. demonstrations)

Realization #2: Can’t focus solely on conceptual knowledge

Realization #3: Need to attend to curricular voice, who speaks to students (teacher vs. text)


Curriculum Coding Scheme

“What happens to the

measure when the unit is

changed?”

Convert 9 ft. to yds.

3 feet = 1 yard

Conceptual Procedural Conventional

Statements

Questions

Problems

Demos

Games

One unit of length is

equivalent to some

number of a different

unit of length.

…multiply the given

length by a ratio of the

two length units.

Table of numerical

conversion ratios.

Text

Text

Text

Text

Teacher

Teacher

Teacher

By teacher

By others

Worked

Examples

Text

Teacher


Overview: Curriculum Coding Scheme

Textual Elements Conceptual

Knowledge

(40 elements)

Procedural

Knowledge

(25 elements)

Conventional

Knowledge

(9 elements)

Statements

Questions

Demonstrations ?

Worked Examples ? Ø

Problems

Games ? Ø


Focus on Length First in CCS

Length is fundamental spatially

Length gets lots of curricular attention (e.g.,

measured in sheer number of pages & problems)

Introduced early in elementary grades, still part of

the middle school curriculum


Common Length Topics (by grade band)

Grades K-2 Grades 3-5 Grades 6-8

Estimate & Measure

objects; non-stan. units

Measure with rulers Perimeter formulas

Estimate & Measure

objects; standard units

English & metric systems

& unit conversions

Scaling & similarity

Draw segments of given

length

Find perimeters of

polygons

Pythagorean Theorem

Find perimeters of

polygons

Estimate lengths Slope


Conceptual Knowledge (length)

General truths about length & the measurement of length

Some examples of “deep” conceptual knowledge:

Transitivity: “The comparison of lengths is transitive. If length A > length B, length B > length C, then length A > length C.”

Unit-measure compensation: “Larger units of length produce smaller measures of length.”

Additive composition: “The sum of two lengths is another length.”

Multiplicative composition: “The product of a length with any other quantity is not a length.”


More Conceptual Knowledge Examples (length)

Midpoint Definition: The midpoint of a segment is the

point that divides the segment into two equal lengths.

Pythagorean Theorem: “In right triangles, the area of

the square on the hypotenuse is equal to the sum of

the areas of the squares on other two sides.”

Circumference to radius: “The circumference of any

circle is proportional to the length of the radius (or

diameter).”


Procedural Knowledge (length)

General processes for determining measures

Broad interpretation of “process”

• Visual, e.g., comparison, estimation

• Physical, e.g., using a ruler

• Numerical, e.g., computations with measures

Visual as well as physical & numerical processes

Generally, elements of PK are not procedural images of CK

Two instances were the match is close:

• Perimeter: Meaning/definition (CK); How to compute (PK)

• Pythagorean Theorem: The relationship (CK); How to compute missing sides (PK)


Some Procedural Knowledge Examples (length)

Visual Estimation: Use imagined unit of length, standard or non-standard, to estimate the length of a segment, object, or distance.

Draw Segment of X units with Ruler: Draw a line segment from zero to X on the ruler.

Unit Conversion: To convert a length measure from one unit to another, multiply the given length by a ratio of the two length units.


Conventional Knowledge (length)

Cultural conventions of representing

measures; devoid of conceptual content

Notations, features of tools (e.g., marks on

rulers), numerical ratios in English system


End of Part III


Back to Step 2: Is it Measurement?

Recall criterion: “…very likely involves measurement reasoning”

Process: Team discussion toward consensus; time intensive

Face validity of the process: We have excluded nothing that curriculum authors present as measurement

Four coding categories:

• ** “very likely measurement reasoning required”

• ?? “measurement reasoning possible”

• P “pre-measurement”

• No code

Only ** content will be analyzed

Want to show/discuss some surprising results & problematic choices


Interesting Result: Fractions via Partitioned Regions

Many ways to introduce fractions: Positions on the number line, parts of sets, parts of 2-D shapes

The last may be the most common

• Construct an equal partition of a shape (“a whole”)

• Quantify a subset of the resulting parts

Initial view: Fractions is a number/operations topic

But when criterion is applied, we include some partitioning and some fraction problems b/c they entail measurement reasoning (i.e., visual comparison of areas)

Consider two examples


Problematic Topic: Ratios of Lengths

Ratio and related topics are important measurement content in middle school

Lengths can be arguments in ratios; numbers (quantities) to be related/compared

Lengths can also be found by reasoning with ratios (similarity, trig)

Our struggle: In a variety of ratio contexts, when is measurement reasoning very likely?

Consider two examples


Sum Up: Is it Measurement?

Much of the K–8 spatial measurement content is

unproblematic to identify & include

But some has not been; Surprises & problems

Must get this step right; consequences of mistakes

at this step are large and negative

If not coded as measurement, will not be analyzed


Conclusion

We hope that we have convinced you of the importance of the problem

In search of an explanation, we must explore a complex space (main effects & interactions)

We hope to be back next year with real results

But we have miles to go before we rest

Thank you!

Documents

Tracing the Origins of Weak Learning of Spatial Measurementstemproj/presentations/STEM_MSUColloquium_2007.pdf8/3/2012 STEM Presentation, MEC, 9-07 14 STEM Project Sequence 1. Pick