Joan Moss
Ruth Beatty
Susan London McNab
Ontario Institute for
Studies in Education
Janet Eisenband
Teachers College
Columbia
The Potential of Geometric Sequences
to Foster Young Students’ Ability to
Generalize in Mathematics
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Understanding Patterns in Mathematics: A Study on Fourth Grade Students, Slides of Algebra
The ways in which fourth grade students understand and engage with patterns in mathematics. It discusses their challenges and difficulties, as well as the role of geometric and numeric patterns in their learning. The document also presents findings from a study that used Knowledge Forum to help students conceptualize and communicate about mathematical ideas.
What you will learn
- How can geometric and numeric patterns be integrated in mathematics instruction for fourth grade students?
- How do children understand patterns in mathematics at the fourth grade level?
- What are some common challenges students face when working with patterns in mathematics?
- How can teachers support students in developing a deeper understanding of patterns and functions in mathematics?
- What role did Knowledge Forum play in helping students understand patterns and functions in mathematics?
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The Potential of Geometric Sequencesto Foster Young Students’ Ability to Joan Moss Ruth Beatty Susan London McNab^ Ontario Institute for^ Studies in Education Janet Eisenband^ Teachers College^ Columbia
Generalize in Mathematics
Abstract
-^ NCTM has proposed that patterning may promote algebraicreasoning skills; however, persuasive evidence suggests thatpattern work may not support a smooth transition to algebraicthinking. While students tend to be adept at extending bothrepeating and growing patterns to “next” positions, they displaydifficulty making predictions far down the sequence of growingpatterns. Employing Case’s theory of mathematical development,we have been designing, implementing and assessingexperimental patterning curricula in second and fourth gradeclassrooms in diverse urban settings. An important feature of thisinstructional design is the focus on the integration ofvisual/geometric and numeric patterns. Our goal is to explicitlylink ordinal pattern positions with the number of elements in thatposition, and to bridge the learning gap between scalar sequenceand functional relations. The components of the curricula includegeometric pattern building with position cards, function machinesand t charts, and activities that integrate these activities.
Abstract
Furthermore, in both research projects the rate of change frompre- to post-test was consistent over high, medium, and lowerachievers. Qualitative analyses of the grade 2 studies revealed anumber of interesting and unanticipated results that will bepresented. in particular, we found that the experimental groupwho had not had formal instruction in multiplication either beforeor during the intervention, invented a series of informal strategiesthat allowed them to find correct answers to multiplicativefunction problems. In addition, these students in the experimentalgroup were more able to apply multiplication than the students inthe control group all of whom had received formal instruction inmultiplication. • Analyses of the fourth grade study indicated that these studentswere able to generalize not only in terms of specific data sets, butalso across structurally similar problems. Further, the KnowledgeForum database revealed a trend in the students’ increasingability to generalize, to negotiate multiple rules, to use variousmodes of representation. Finally the data revealed students’developing use of mathematical symbolic language andincreasing sophistication for offering proof and justifications fortheir conjectures.
Can you a build a^ “Times 2 plus 1”^ pattern?(video)
Agenda
-^ Early Algebra and Patterning – AnIntroduction affordances and difficulties •^ Overview of theory and patterning research •^ Grade 2 study •^ Grade 4 study •^ Knowledge Forum discussions •^ Overall results and relation to Strands ofMathematical Proficiency •^ Discussion and implications^ – Focus and priority on spatial representations ofpatterns^ – Algebrafied arithmetic?
Why Study Patterns?
-^ “Patterns, Relations & Functions” part of the mostrecent preK-12 Algebra recommendations(NCTM 2000)•^ Patterns can constitute one route towardgeneralizing (e.g. Ferrini-Mundy et al.; Kenny &Silver; Schliemann et al.; Smith; Kaput)•^ Patterns can develop understanding of thedependent relations among quantities thatunderlie mathematical functions•^ Young children naturally interested (Seo andGinsburg, 2004) and older students find patternscompelling•^ Spans ability levels
How Do Children Do? • Limited idea of patterns only as
repeating
-^ “A pattern is something that goes on forever that never stops.Like if you said big little big little from this day until you wereabout 10 that would be a pattern.” •^ Children can
extend
patterns, but have
trouble
describing & generalizing -
finding
elements far down the sequence
Challenges
-^ Tendency to find and use the
scalar,
or^ recursive
,
rules^
Position
Blocks 1
1 2 3
x 3+
Whole Object:
Tables & Chairs (video)
Difficulties with Patterns • Route from perceiving patterns to finding useful rulesand algebraic representations is very difficult (Kieran,Noss et al, 1997) • Patterning problems become “arithmetic exercises”with tables: little understanding of relationshipsimplied in underlying structure • Difficulty with geometric representations of patterns • Students lack rigour and commitment to justifications – “Once students selected a rule for a pattern, they persisted intheir claims even when finding a counter example to theirhypotheses”. (Cooper and Sakane 1986)
In a recent survey that weconducted with students fromgrades 2-5 we found thatstudents had a great deal ofdifficulty with all kinds ofpatterns….
Normative Patterning Study • 97 interviews with students in grades 2-5 (53 male, 44 female)• Each interview consisted of five patterning questions includingdifferent mathematical functions: 3x+1, 2x+2, 3x+1, 5x+1 and 3x• Each function was presented in one of five formats: A B 1 4 3 104 136 199 41 n 4,7,10, 13, __, __, …
ndst (^21) rd^3 th^4
st^ nd 12
rd^3 th 4
Jane really wants a brand new scooter. She has only $1.00 saved up. Shedecides she’s going to save money she earns from babysitting. She gets a jobbabysitting for an hour each day. After babysitting on the first day, she putsthe money she earned from babysitting in her piggy bank along with the $1.00she already had saved. Now, altogether, she has $4.00 in her piggy bank.After babysitting for a second day, she now has $7.00 altogether in her piggybank. After the third day of babysitting, Jane now has $10.00 altogether in herpiggybank.
Ley 2005
Overall (all grades combined) frequencyof strategy use in each of the five formats
Ley, 2005
Our Pilot Research
Goals:To have children engage with geometric andnumeric patterns in ways that could •^ Forge connections between different patternrepresentations •^ Illuminate mathematical structures underlying patterns •^ Provide natural situations for mathematical problemsolving, discussion, reflection and knowledge building •^ Our study:
Design, implement and assess pattern activities in grades 2 and 4 classrooms