Year 6 and 7 Algebra and Patterns
Challenges
Solutions / Strategies
Justification and Origin
Transitioning from concrete patterning
to viewing patterns as functions
(Warren, 2005).

Origin: Often attributed to reasons such as students having never
developed the conceptual understanding of the target
mathematics concept/skill.
http://www.coedu.usf.edu/main/departments/sped/mathvids/in
dex.html
Conceptual understandings of a target mathematics concept/skill
is demonstrated by “the ability to recognize functional
relationships between known and unknown, independent and
dependent variables, and to discern between and interpret
different representations of the algebraic concepts. It is
exemplified by competency in reading, writing, and manipulating
both number symbols and algebraic symbols used in formulas,
expressions, equations, and inequalities.”
http://web.ebscohost.com.ezproxy.cqu.edu.au/ehost/detail
?sid=8d7c9040-f325-4cf0-a054ba8446eedede%40sessionmgr112&vid=2&hid=122&bdata=J
nNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=ehh&AN=6186365
3&anchor=AN0061863653-4
Identifying and articulating the missing
steps in an incomplete pattern. For
example the LM may provide students
with steps 1, 2 and 5. Students are
often unable to identify steps 3 and 4
(Choi, Huh, LaRue, 2010).
In this instance, gestures and manipulation of materials
add to the conversations, elements that are missing from
written responses.
http://emis.kaist.ac.kr/proceedings/PME29/PME29RRPape
rs/PME29Vol4Warren.pdf
Origin: LMs focus too heavily on growing patterns that are
consecutive rather than using patterns that require students to
find missing steps.
Is it possible to put in prediction and guessing into this???
As in they have to predict the 3rd step and could even work
out the 4th step from a guess and check strategy.
The use of concrete materials appeared to assist many children
ascertain the missing steps in the pattern. A number, when
completing the accompanying pen and paper worksheet,
recreated the pictorial pattern with the tiles and then used the
https://eee.uci.edu/wiki/index.php?title=Patterns_and_Alge
braic_Thinking&redirect=no
tiles to create the 5th and 10th step. They then drew a picture of
their solution on the worksheet.
Many children exhibited an ability to express the generalisations
orally, but such descriptions often lacked precision.
http://emis.kaist.ac.kr/proceedings/PME29/PME29RRPapers/PM
E29Vol4Warren.pdf
Fluency in the language of algebra demonstrated by
confident use of its vocabulary and meanings as well as
flexible operation upon its grammar rules (i.e. mathematical
properties and conventions) are also indicative of conceptual
understanding in algebra.
http://web.ebscohost.com.ezproxy.cqu.edu.au/ehost/detail?si
d=8d7c9040-f325-4cf0-a054ba8446eedede%40sessionmgr112&vid=2&hid=122&bdata=
JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=ehh&AN=6186
3653&anchor=AN0061863653-4
Converting a visual pattern to a table of
values so as to identify relationships
from within the table (Choi, Huh,
LaRue, 2010).
Origin: Concrete to abstract
Why is it important to be able to go from concrete to
abstract? Links to the curriculum? Links to real life
situations???
Patterns where the relationship between the pattern and position
were explicit. These types of patterns appeared to assist children
to verbally describe the relationship between the pattern and the
position, for example, it is twice the step number, it is the
same as the step number, it is one more than the step number.
http://emis.kaist.ac.kr/proceedings/PME29/PME29RRPaper
s/PME29Vol4Warren.pdf
Explicit questioning to link the position to the pattern- These
questions were of the form – What does the pattern look like?
How many rows? How many in each row? For the 3rd step, how
many on the bottom, how many on the top? The questions
explicitly related the position to the pattern’s visual components.
Generalising from the pattern in small position numbers, to large
position numbers- It was found that to articulate the relationship
between position number and the visual pattern in general terms,
children needed to discuss the relationship for increasingly
larger positions.
http://emis.kaist.ac.kr/proceedings/PME29/PME29RRPaper
s/PME29Vol4Warren.pdf
Students creating their own
patterns (Choi, Huh, LaRue, 2010).
Understanding of the equals sign and
how it is representative of quantitative
sameness (Vance, 1998).
using concrete materials to create patterns, specific
questioning to make explicit the relationship between the
pattern and its position, and specific questioning that assist
children to reach generalization with regard to unknown
positions.
http://emis.kaist.ac.kr/proceedings/PME29/PME29RRPape
rs/PME29Vol4Warren.pdf
 To reinforce the concept that a number can be
represented by many different expressions, learners
are asked to name a given number in several different
ways using two or more numbers and one or more
operations. For example, 9 can be named as 4 + 3 + 2
or 2 x 5 - 1.
 In a related activity, equations are to be completed so
that an expression has at least one operation on each
side of the equals sign. For example, given 7 + 5 as one
side of an equation, a student might write 7 + 5 = 2 X 6
or 7 + 5 = 10 + 5 – 3 (Vance, 1998).
http://www.learner.org/courses/learningmath/algebr
a/pdfs/AlgPerspective.pdf
Origin: Students too often complete LM provided pattern rather
than creating their own (Choi, Huh, LaRue, 2010).
Individual Mastery is important
Origin: This misconception is reinforced early on when they are
shown the vertical computational format for 3 + 2; the bar under
the lower number is a signal to find an answer. It is also true that
pressing 3 + 2 = on a calculator results in the standard form of the
number. Therefore, children need experiences in which they see
and write other types of number sentences, such as 5 = 3 + 2 and
3 + 2 = 4 + 1 (Vance, 1998).
Practical example is using a scale or see saw and showing
that the equal sign is an expression of two sides???
Variances in the standard number
sentence of a + b = c (Choi, Huh &
LaRue, 2010).
Bracket usage as a static signal telling
students which operation to perform
first in an algebraic equation (Gallardo,
1995).


Origin: Students who are used to seeing number sentences in the
form a + b = c do not see 9 = 3 + 6 as a "proper number sentence"
(Choi, Huh & LaRue, 2010).
Origin: Students tend to solve such expressions based on how the
items are listed, in a left-to-right fashion (in the case of English
speaking students), consistent with their cultural tradition of
reading and writing. Therefore, the rules underlying the order of
operations can actually contradict students’ natural ways of
thinking.
http://ehis.ebscohost.com.ezproxy.cqu.edu.au/eds/detail?vid=6&
hid=120&sid=56658b62-ac31-4c51-97e32878f9a574e3%40sessionmgr104&bdata=JnNpdGU9ZWRzLWxpd
mUmc2NvcGU9c2l0ZQ%3d%3d#db=ehh&AN=73959391 (CHECK)
INCLUDE ICT
References
Choi, S. Huh, L., & LaRue, C. (2010). Patterns and Algebraic Thinking. Retrieved April 2, 2012, from
https://eee.uci.edu/wiki/index.php?title=Patterns_and_Algebraic_Thinking&redirect=no
Vance, J. H. (1995). Number operations from an algebraic perspective. Teaching Children Mathematics, 4(5), 282.
Warren, E. (2005). Young Children's Ability to Generalise the Pattern Rule for Growing Patterns. Paper presented at the 29th Conference of the International
Group for the Psychology of Mathematics Education. Retrieved from
http://www.eric.ed.gov/PDFS/ED496965.pdf
Gallardo, A. (1995). Negative Numbers in the Teaching of Arithmetic. Repercussions in Elementary Algebra. Paper presented at the Annual
Meeting of the North American Group for the Psychology of Mathematics Education. Retrieved from
http://www.eric.ed.gov/PDFS/ED389549.pdf

Tessa’s Notes:
Maybe have examples of each challenge?