Number lines: Building conceptual and procedural skill in the elementary and middle school classroom Verónica Ocampo Milwaukee Public Schools Greenfield Bilingual School
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Goals for this session Learning Intentions ● Present some features of different number line models and show they are used in elementary and middle school classrooms. ● Share examples of issues and ideas for strong applications of number lines in classrooms. ● Discuss ideas for the design and evaluation of number line student work.
Success Criteria ● If you gathered one or two strategies or concepts you can take to your classroom. ● Recognized some strong applications of the number lines in your daily practice. ● Identify one idea for stronger evaluation of student work on the number line.
Mental number line Consensus: -From a very young age, humans have an internal leftto-right number line. -This number line resembles left-right coordinates of space or ordinal aspects of number. -This is especially true for people culturally raised in the Western countries, with a leftto-right writing system.
Santens, S. & Gevers, W. (2008). The SNARC effect does not imply a mental number line. Cognition, 108(1), 253-270. Stott, D. (2013) Number line image generator —A website review. Learning & Teaching Mathematics, 14, 22-25.
Number line task (Middle school) Location Death Valley, CA New York, NY Milwaukee, WI Desert Shores, CA New Orleans, LA Madison, WI
Altitude (ft above sea level) -264 34 614 -200 -7 874
Death Valley, CA is higher than Desert Shores, CA. New York, NY is farther from 0 than New Orleans, LA. Milwaukee, WI is lower than Madison, WI F Sketch a number line to answer these statements.
T T
F F T
The ENL in the CCSSM -The ENL was slowly introduced in Great Britain. In the US, it started to appear as spare exercises in curriculums like Investigations or CMP2. -In 2010, the CCSSM adopted the ENL model. It has been integrated in several domains across all grade levels. The number line is the most influential model in mathematical cognition research. (Dehane, 2003).
The ENL as conceptual model -The ENL has gradually taken the place of place-value blocks, hundred charts, and other area models. -This substitution is because the ENL “constitutes a unifying and coherent representation for the different sets of numbers which other models cannot do.” -A special strength of the ENL is the very close resemblance it has with continuous quantities. -Area model: more suitable to represent discrete quantitities or fractions less than the whole.
Structured number lines -Structured, traditional number lines had been around for some time. -Structured number lines have marked line segments (tick marks). Each mark represents a length. -Structured number lines have distinct start and endpoints. -Structured number lines provide students a model to structure their work. (Stott, p. 1)
Empty Number Line (ENL) -Number line fever started in the Freudenthal Institute (Utrecht University, Netherlands). -TIMSS reports showed a remarkable good performance of Dutch children. -This mainstreamed the Realistic Mathematics Education (RME) movement, which uses the empty number line (ENL) as a model to create fluency in mental addition and subtraction up to 100 in second and third grade. -This model also stresses interactive discussion of problems.
Beishuizen, M. Two types of mental arithmetic and the empty number line. Retrieved from http://bsrlm.org.uk/ IPs/ip17-12/BSRLM-IP-17-12-4.pdf
Solve with a number line… Sketch a number line to solve the following problem. Compare your number line with that of your table partners. What is similar in your number lines? What is different? Could you identify what kind of number line model you used? (Use the chart on the next slide.)
Problem: There are 32 students in Mr. Marlin’s homeroom. Of the students, ⅜ are boys. How many girls are in his homeroom?
Types of number lines ● ● ● ● ●
Empty (no beginning or end points) Closed (with beginning and end points) Open (with a beginning but no end point or viceversa) Structured (with well-defined partitions and labeling) Semi-structured (with some partitions and labeling missing) ● Proper fraction number lines (1) Applies to decimals. ● Positive and negative integers number lines
The ENL Revolution -The Empty Number Line has no numbers or markers. -Students display their own thought processes on the ENL, making them visible to themselves and others. -The ENL encourages students to develop flexible mental strategies for problem solving. -The ENL allows students to visualize and trace the steps they have completed in a given problem.
The Empty Number Line: The Challenge and the Hope -The ENL stimulates distinct mental moves which result in mental math activities, which is the desired outcome. Discussion of such mental moves is a great opportunity to construct learning as a social experience. -However, the ENL is “a rather difficult model for weaker pupils.”
(Beishuizen, p. 18)
-It might also be difficult for teachers, more familiar with the traditional number line or ingrained algorithms, who need to build conceptual understanding before applying the ENL to the standards.
Why closed number lines are not necessarily bad Online number line generator
A good closed number line practice might higher level counting / estimating / locating tasks. A well-designed warm-up can be conducted as a number talk task.
Large whole numbers
Positive and negative integers
Why closed number lines are not necessarily bad Number line tasks What is the number missing on this number line?
What strategies are necessary to identify the number? What strategies did students use? Can you identify a pattern of student misunderstandings?
Why closed number lines are not necessarily bad Plot 29/3 on this number line.
What strategies are necessary to identify this number? What strategies did students use? Can you identify a pattern of student misunderstandings?
Learning Mathematics Through Representations Curriculum (LMR) University of Berkeley The LMR curriculum proposes an integrated number theory. It is a research-based intervention approach that integrates fractions and positive and negative integers. LMR uses the number line as the main representational model. Different mathematical representations and communication within the classroom are essential to learning.
LMR Curriculum
The LMR curriculum is free for noncommercial use.
Number line tasks (Elementary) Elementary: Many elementary teachers have explored and found the potential of ENL, developing great flexibility themselves. These are some tasks: -Memorization and practice of counting and creating patterns with with cardinal and ordinal numbers. -Operations with whole numbers -Representation and understanding of benchmark fractions -Comparing and ordering fractions, and equivalent fractions -Representing/identifying fractions as magnitudes on a number line -Operations with fractions
http://mathcats.org/ideabank/fractionsites.html
Number line tasks (Middle school) Middle school: Number lines are powerful tools to make the transition from arithmetic to algebra -Extend representation of numbers to decimal numbers, positive and negative integers, and square and cubic roots -Comparing and ordering rational and irrational numbers -Operations with all rational numbers -Solving and graphing inequalities -Demonstrate density of fractions and decimals -Locate extreme values in a number line -Determine best scales to represent different magnitudes 8th grade routine problem on a number line
Areas of improvement... When teaching, it is important to avoid these approaches ● Emphasis on area models ● Low-level tasks on a closed number line ● Limited range of whole numbers (numbers no larger than five) ● Treating positive and negative integers as another category of numbers, rather than as part of the rational numbers system ● Treating negative fractions as another category of numbers, rather than as part of the rational numbers system
To evaluate student work… When evaluating Create a strong rubric before observing and evaluating student work. Make a preliminary list of possible answers and misunderstandings. Solve your tasks before students and generate a list of the mental strategies and mistakes that you might possible encounter when analyzing student work.
Area models Before the CCSSM, area models were important in the conceptual understanding of fractions. Number lines were very sparsely used. CMP2 has several number line exercises in Bits and Pieces I but, as it often happens, those were the ones teachers might skip in order to finish the investigations.
Task 1 (CMP2, Bits and Pieces 1)
It is tempting to think that students will be able to translate naturally from one representation to the other. In this case, my students needed to transfer their knowledge from area models to this closed number line model. They couldn’t. I revised my teaching and started working intensely with number lines.
From area models to numerical, number line, and measurement models
Next instructional years were centered on making this transfer happen. A new non-routine number line task showed that the transference did not take place. What limitation can you observe in this sample of fraction representation?
Analysis of student work
Huinker, D. (2015). Pathways Fraction Task 6-8.
What kind of task is this? Counting, Estimating, Operating, Comparing What strategies are necessary to identify this number? What strategies did students use? Can you identify a pattern of student misunderstandings?
Count, Partition, Estimate, Measure, Operate????? Some suggestions to analyze student work Identify type of task Identify type of strategies needed to complete the tasks ● ● ● ● ●
Partitioning Solving Estimating Measuring Counting
Identify strategies students used to complete the tasks. Identify patterns of strengths and misunderstandings.
http://mathdifficulties.blogspot.com/2010/12/discussion-here-isfocused-on-math.html
Begin with an end in mind...
Lack of preparation to analyze student work, makes the assessment of this task a very difficult job. Revision of student work needs to be planned and methodical. Following is a list of suggestions to make this task more efficient and productive.
Analysis of student work: Fractions and decimals (N/P) What are the solution strategies? What are the misunderstandings? Is there a pattern to the misunderstandings? Is this pattern applicable to other students? What are the strengths and weaknesses of these students’ work?
Analysis of student work: Inequalities (drawing vs. solving)
Drawing
Solving
Analysis of student work: Integers (counting vs. solving) Counting
Solving
Analysis of student work: Large numbers (counting vs. solving) Counting
Solving
Number line pitfalls Logarithmic vs. linear representation of numbers
This model responds to Siegler’s and others developmental theory of number representation. -”A lack of accuracy when dividing segments also results in the incorrect location of fractions” (Wong, 2008, p. 598).
Improving teaching with number lines ● Introduce different kinds of tasks. ● Use different kinds of numbers (whole numbers, fractions, decimals, positive and negative integers, negative fractions, square and cubic roots, irrational numbers, 𝝅, and big and small numbers) ● Depending on the problem and types of numbers, share your strategies with students. ● Elicit discussion on why you or other students chose a number line type over others. ● Analyze student work, preserving anonymity. ● Have students work in pairs and groups to solve challenging problems. ● Model flexibility and encourage students to also show flexibility in their approaches, design, and problem solving with number lines.
If things like this start to happen... No matter what you do in your classroom with the number line, if students measure and compare nail lengths during their breaks… It is a LIKE to the number lines.
Online resources Online resources Learning Mathematics Through Representations Curriculum (LMR) – University of Berkeley http://www.culturecognition.com/lmr/ Number line generator http://www.senteacher.org/worksheet/104/NumberLineMaker.html Number line image generator http://www.oliverboorman.biz/projects/tools/number_lines.php Printable number lines and quizzes http://www.helpingwithmath.com/resources/oth_number_lines.htm Measurement of number sense and Approximate Number System (ANS) h"p://panamath.org/
Online number line generator http://www.oliverboorman.biz/projects/tools/number_lines.php
Number Line Image Generator - reviewed in Research Gate - generates different types of number lines for different purposes - labels measurement units - includes integers, decimals, and fractional distances - download and save images as pdf - suitable for whole class discussions BUT… - cannot create fraction labels (Use Paint to correct.)
Online resources SEN Teacher (with difficulty levels) Printable number lines Online number line generator Online quiz: Fractions on a number line
Panamath
Measurement of number sense and approximate number system (ANS)
