**Language: A Cultural Capital for Conceptualizing Mathematics Knowledge**

Nosisi Feza-Piyose

**Human Sciences Research Council- South Africa**

Science and Mathematics education in Africa is in crisis. Students continue to perform at a lower level

compared to other nations
including those with low GPD compared to them. Two factors have been highlighted
in research that impedes science and mathematics learning: teacher content knowledge and
irrelevant teaching strategies. This study contributes to this literature by
investigating five African (from a former White school) fifth grade students’
learning of length measurement with the aim of eliciting the students’ thinking
levels by using a length learning trajectory. Clinical interviews and teaching experiments
were employed for a comprehensive description of these students’ processes. The
findings reveal that students’ mother tongue is a psychological tool that enriches
their mathematics learning process, learning trajectory assisted in analyzing
students developmental processes with language and poor number development
impeded abstraction in learning of length measurement concepts.

**Keywords:**multilingualism, number development, internalization, learning trajectory.

**Introduction**

Mathematics performance of
African students in South Africa remains poor even after liberation from the
apartheid era. One of the contributing factors to this performance is quality
of instruction received by the majority of South African children. The majority
of students in the country lives below poverty level and school in poor
resourced schools. Mji and Makgato (2006), Adler (2001), and Howie (2003)
investigated factors that contributed to poor mathematics performance in these
schools and discovered that teaching strategies used do not cater for students’
needs. The second factor that affects the quality of mathematics education
received by these students is poor mathematics content knowledge demonstrated by
teachers (Mji & Makgatho, 2006; Van der Sandt & Niewoudt, 2003;
Wessels, 2008).

Addressing these challenges,
black parents of South Africa send their children to previously white schools
to seek quality education. However, little is known about how these students perform
mathematically, and if the multilingual research by Setati and Adler (2001) and
Setati (1998, 2005) applies in this new context.

This study aims to contribute to
the already growing literature on multilingualism and also to test the
relevancy of the learning trajectories in the South African context. Therefore,
this study describes how five fifth grade Black students internalize length
measurement concepts, and what mathematical experiences have they accumulated.
In achieving these objectives the study employed clinical interviews in
eliciting mathematical experiences about length and individual teaching
experiments in describing internalization of length measurement concepts.
Clements and Sarama’s (2009) learning trajectories was employed to analyze the
clinical interviews. The learning trajectories are used in this study because
they are informed by theory and research. The theories that influence these
learning trajectories are cognitive theories that have evolved over time to the
current research.

Length measurement is employed in
this study for its strength to bring together a variety of mathematical
concepts. Also, measurement creates opportunities for in-depth understanding of
space, number and algebra. Measuring involves spatial understanding, unit iteration
that involves continuous counting, conversion of units that uses multiplication
and decimal fractions. These concepts develop into pattern recognition that
develops algebraic thinking.

**Theoretical Framework**

If we wish to provide learning
opportunities for students, we must first reflect on what it means for the
student to “learn mathematics” and how the student goes about that task meaningfully
(Reynolds and Wheatley, 1996, p.564).

Reynolds and Wheatley explicitly
described the importance of giving students’ opportunities to show and tell us
how they learn. In learning how the students internalize length measurement
concepts, the literature on learning trajectories of length informs this study.
Also, addressing the psychological tools (the cultural capital) these students
bring into their learning, the literature on multilingualism in mathematics
classroom is employed.

**Learning Trajectories**

Learning trajectories
hypothesized by Sarama and Clements (2009) on learning of length measurement
concepts were designed from research-based theories that have been developed from
Piagetian and Vygotskian contributions of young children’s learning (Piaget,
1960, 1967); Vygotsky’s (1934, 1986) theory to the present research by (Sarama
and Clements 2009, Clements, 2010, Clements et al., 2011). The timeline covered
in this theory supports historical and continuous reforms in learning of
measurement concepts by young children.

This study employs this theory as
an analysis tool for investigating these students’ thinking processes. The
three parts that compose this theory are: mathematical goal, developmental path
in reaching the goal, and activities that are designed towards reaching the
mathematical goal.

**Mathematical goal**

Mathematical goals are the big
ideas of mathematics. Clements and Sarama (2009, 2004) describe the
mathematical goals as grouped concepts and skills that are mathematically essential
and logical, customary with children’s thinking and generative of future
learning. For example the length measurement is composed of these big ideas
that include understanding the distance that involves space, then how to
measure this distance such as deciding on the unit, aligning the measuring tool
correctly to the object of measure, using repeated units to measure, being able
to calculate overlaps or gaps using knowledge of units and likewise. All these
concepts form the cluster of the big ideas towards understanding length
measurement.

**Developmental path**

Developmental path is more about
students’ levels of thinking (Clements & Sarama, 2004). These levels are
like a ladder as the next level become more complicated than the previous one.
Knowing these developmental levels allow the educator to plan instruction
tailored at the students’ level (Vygotsky, 1978). For example, from knowing
“big” a child begins to use gestures in explaining the meaning of big at a
different context. If it means tall, the child will show it by spreading
his/her harm high, whereas if it means wide two hands will be used parallel to
each other. This indicates that the understanding of “big” is getting richer
with more sophistication and requires more vocabulary to be expressed. Clements
and Sarama (2009) suggest that interpreting this development should be at the
child’s point of view. Learning trajectories recognize those innate abilities
and stretch them further through experiences to develop these measurement skills
and concepts more. The length measurement concepts themselves require

internalization as they are
meaningless as skills. This theory recognizes the power of mediation in
developing these concepts and therefore provides activities that enrich the learning
experiences.

**Instructional activities**

Measuring distance demands
reorganizing space that needs to be measured (Lehrer, 2003). Re-organization
leads to the development of units that need to be used in representing the
successive distance of multiple tiled units. These units need to be the same
and iterated. Therefore, once the student selects the unit of measure, s/he has
to put the unit end to end until s/he can cover the distance. Then the next
step is to count the units. This process reflects conceptualization of
iteration. Instructional activities should be directly focused at the students’
level of thinking. Developmentally appropriate activities need to be designed
to achieve this outcome. Lehrer et al. (1999), Clements (1999), and Barrett and
Clements (2003) divulged that the ability to iterate the same units and to use
the zero point correctly in measuring length typically develops with students’
age. In addition, Hiebert (1986) argued that there is a big difference between
conceptual knowledge and procedural knowledge. Conceptual knowledge refers to
the thoughts and intuitions about how mathematics works. Procedural knowledge
refers to skills, procedures, and formal symbols.

The major struggle in learning
measurement for students is to link procedures with understanding. For example,
the results of studies conducted by Bragg and Outhred (2000a, 2001) indicate
that most students could not demonstrate what a centimeter looks like in a ruler.
These results confirm that being able to use a ruler does not mean students
understand what they are measuring. Using a ruler to measure is a skill, and
counting the lines or spaces on the ruler while measuring is a procedure.
Conceptualization occurs internally when a mental ruler develops. This mental
ruler can be used anytime anywhere as a conceptual object.

Clements (1999), Lakoff and Nunez
(2000), Lehrer, Jaslow, and Curtis (2003), and Bragg and Outhred (2004) suggest
that teachers need to mediate the meaning of numbers on a scale and length as
movement on a scale from the point of origin, that is zero. Wilson and Rowland (1992)
and Van de Walle and Thompson (1985) highlighted the importance of building conceptual
understanding of zero as the point of origin and length as continuous not
discrete quantity.

Therefore, students need to make
a distinction between counting discrete objects and counting continuous length.
Also it is vital to provide instruction that will nurture students, enabling
them to rename zero when a line is not aligned with zero or when using a broken
ruler (Clements, 1999). Developing a strong conceptual understanding of length
will create opportunities for developing area and volume concepts effectively.
It will also attend to the misconceptions that are created when using
inappropriate units.

**Length Measurement Learning Trajectory.**

The theory suggests eight
hierarchical developmental progression levels for the length measurement trajectory.

**Pre- length quantity recognizer**(

**PLQR**). In pre-length quantity recognizer a student cannot identify length as an attribute. In this level students see everything as long regardless of the attribute.

**Length quantity recognizer**(

**LQR**). In this level the student identifies length as an attribute but cannot compare. This is the level when tall becomes a big thing. For example, “my mom is tall, my daddy is tall, and my brother is tall”. No comparisons at this level.

**Length direct comparer**(

**LDC**). In this level the student aligns objects physically to compare their lengths.

**Length indirect comparer**(

**LIC**). In this level the student uses a third unit to compare two objects’ lengths.

**Serial orderer to 6+**(

**SO+**). Sarama and Clements (2009) suggest that this level develops parallel with the next level, End-to-end measurer. Students order lengths in ascending or descending order from 1 to 6.

**End-to-end length measurer**(

**EELM**). Student lays units end-to-end when comparing or measuring length. In this level using the same unit repeatedly might not have developed yet.

**Length unit relater and repeater**(

**LURR**). Uses similar units repeatedly to compare and measure length. However, sometimes might still use a different unit in cases of a distance left that is shorter than the repeated unit. Students might change the unit to fit the distance. In other cases a student might be able to break the unit mentally into a fraction and finish up iteration.

**Length measurer**(

**LM**). The student understands the need for using the same unit for measuring length. He/she understands the relations between units. She recognizes and lays the object from zero when measuring.

**Conceptual ruler measurer**(

**CRM**). This is a level of abstraction. Units become mental units that a student can use anywhere anytime. Estimation skill is attained in this level.

**Multilingualism in Mathematics Classrooms**

African students go through
different processes in conceptualizing mathematical concepts compared to their
peers whose mother tongue is English. It becomes a marginalizing treatment if
the educators treat them similarly (Khisty & Morales, 2004). Students from diverse
cultures bring more different backgrounds and experiences to the classroom than
their peers. They bring their initial cultures and languages that influence
their frame of reasoning (Bishop, 1985). Educators need to tap on this richness
to create opportunities for learning (Raborn, 1995). Recognizing individual
learning differences, educators can shape curricula and instruction for all
learners and give them opportunities to reach higher levels of mathematics.
Setati and Adler (2001) in their results of studying language practices in multilingual
mathematics classrooms of South Africa indicated that code switching between English
and the native language of a student enriched mathematical discussions. This mathematics
discussion becomes more conceptual and results in personalization of meaning (Vygotsky,
1978). Carignan et al. (2005) revealed that this code-switching was not allowed
in an ex-Model C school they studied. They observed that in this urban school
parents were required to speak English at home with their children instead of
their mother tongue to be enrolled in this school.

Research has proven that use of
native language in mathematics classroom enriches students’ understanding of
mathematical concepts (Nicol, 2005; Matang, 2006; Adler, 1998; Setati, 1998;
2002; 2005; Setati and Adler, 2001; Setati and Barwell, 2006). Therefore, instead
of native languages becoming stigmatized languages in the classroom they should
be treated as cultural capital students’ bring and be acknowledged as their
point of reference.

Literature on multilingualism
reveals that mother tongue is used for sense making, understanding of new ideas
and conceptual discourses (Setati, 1998, 2006; Setati and Adler, 2001). The
three uses of native language indicated in the studies are fundamental for
learning of mathematics. Sense making integrates known to the unknown that lead
to internalization of ideas. When ideas are internalized they become personalized
and applied in different contexts (Vygotsky, 1978). Understanding new ideas is
a process that requires cultural tools. The cultural tool in this case is
language that students attained socially and use in learning new ideas. Denying
them this opportunity of using their cultural tools then denies them access in learning
of new ideas. These ideas need to be internalized and become mental structures
that can be used in solving problems. Justifications, explanations and
argumentations are elements of conceptual discourse. Language is a tool for
conceptual discourse, students need language they are confident to justify,
explain and argue for their reasoning.

**Methodology****Research Design**

This study reports data from my
dissertation supervised by Prof Julie Sarama. A case study design was employed
to focus on in-depth understanding of how black fifth grade students learn
length measurement. The case study focused on having an in-depth understanding
of students’ conceptualizing processes, learning tools they use in learning measurement.
Classroom observations were used to understand the kind of instruction received
by these students in the former White school, followed by clinical interviews
of five selected students to determine their thinking levels on measurement
concepts (length, perimeter, area and volume). The analysis of the clinical
interviews was used to inform the creation of teaching experiments. There were
four main teaching experiments for each measurement concept with episodes for each
teaching experiment. The number of episodes was determined by the needs of each
student for each teaching experiment. This paper reports only the pre and post
clinical interviews and teaching experiments of the length concept. The total
number of length episodes were 8, however not all five learners needed all of
the eight episodes depending on their progression the episodes were more for
those who were taking longer route to understand the concept. The instructional
episodes for the length teaching experiment included, practical activities
measuring lengths using different measuring instruments, using broken rulers
measuring lines, reading of calibrated measuring instruments and meaning of
numbers in a ruler, activities involving conversion of units of length, conversion
activities, number development, basic operations, and fraction activities. Each
length episode session lasted 20 minutes totalling to 160 minutes per students.
As mentioned previously this 160 minutes was used differently for different
students. These individual length teaching episodes were conducted on daily
basis over 8 school days. After the 8 days of teaching post-clinical interviews
were conducted to support the claims that emerged from the analysis of the
teaching experiment weather students’ conceptual understanding of length progressed,
regressed or remained the same.

**Participants**

This study collected data from 5
black fifth grade students. The selection of participants focused on different
levels of performance with relevant gender representation. Out of two fifth
grade classes, 16 students represented gender and mathematical performance
based on students’ grades among all fifth graders in the school. From the 16
students they were grouped into three performance levels, high, average and
low. The students’ grades ranged from 74-84% were 7 girls only, 52-58 % was 2
boys and 2 girls, and 37-48 % was 2 girls and 3 boys. The 5 participants
selected were selected from the 16 students using the performance and gender
representation. The representation was 2 girls from the high performance, 1 boy
and 1girl from the average group and 1 boy from the low performing group.

**Research Site**

The site used to collect this
study’s data was a former White school in the Eastern Cape Province, South
Africa. The majority of students in this school are African with 3% White students.
The teacher - student ratio in this school is 1:55. The majority of teachers
are White Afrikaans speaking with 3 African teachers. The medium of instruction
is English.

**Data Collection**

Clinical interviews. Structured
clinical interviews were administered to elicit the levels of thinking attained
by the students in their prior learning experiences. These structured interviews
were administered before teaching experiments were conducted to inform teaching
experiments. At the end of the teaching experiments the same structured
interviews were administered as an assessment tool to support the claims that
come out of the teaching experiments and to measure students’ developmental
progressions. Measurement studies that are influenced by Piaget and
Inhelder’s cognitive development theory fall short in informing the literature
about the mediation needed to develop understanding (Lehrer, 2003), because their
focus has been on using clinical interviews over a period of time to describe developmental
epistemology. Therefore, conducting clinical interviews solely denies researchers’
access to pedagogical needs of students. Hence, this study employs clinical interviews
in combination with teaching experiments.

**Teaching experiments.**

With the results from
pre-structured clinical interviews, teaching experiments were developed to
instruct at the students’ actual level and stretching them to the next level.
Teaching experiments allow access to prior conceptions and how they are used to
make meaning of new ideas (Thompson, 1979). Teaching experiments give access to
both students’ mathematical reasoning and mathematical learning (Steffe &
Thompson, 2000).

**Data coding.**

Data analysis was inseparable
with data collection in this study to inform continuous data collection and to
engage with data. Analytical memos assisted the researcher in making dialogue
and making sense of the data (Ely et al., 1991). After conducting the pre-structured
clinical interviews, analysis assisted in determining the actual levels of
thinking the students demonstrated. Each student’s transcripts from the video
recorder with non verbal cues was typed and grouped according to the relevant
developmental progression level of the length learning trajectory (Clements and
Sarama, 2009), then the audio data from the tape recorder was also grouped
using the developmental progression, and the researcher’s notes were grouped
too using the developmental progressions. A table with three columns of audio, video
and field notes were created. Under each column developmental progressions were
tabled and color-coded according to similarities and differences. The visual
picture assisted the researcher in creating a common teaching experiment with
different levels of instruction determined by individual student’s performance
on the interviews. This triangulation gave credibility and trustworthiness to
the claims about students’ thinking levels. The similar analysis for
post-structured interviews occurred with a different focus. That analysis supported
the teaching experiments and indicated the amount of growth students attained after
the teaching experiments.

Teaching experiments were
analyzed differently. After each episode of the teaching experiment, an
analysis was conducted to inform the next episode. The video, audio and field notes
were all typed and annotated. The annotations were tabled for each source under
the source column. These annotations were color-coded and analytical memos were
used to make sense of the visual display. The triangulation assisted in
preparing for the next episode. When the tables did not give a clear picture,
other visual displays were used to display relationships and differences. A
total of 14 Descriptive codes emerged from the teaching experiments of all students.
Four themes emerged from the 14 descriptive codes.

**Findings**

The findings of this study are
reported in three parts: the pre-clinical interview results, teaching
experiments and the post-clinical interviews. In pre-interviews the findings
report the thinking levels of the five black fifth grade students before they
were involved in teaching experiments. It is important to note that South
African schools have quarterly terms and this data was collected during the
last two terms of the year.

**Pre-structured Interview Results**

All five students demonstrated
that measuring any length requires alignment of the measured object accurately
from the Zero origin. That is, on all occasions all started from zero with accurate
alignment of the ruler. However, four students when required to measure in
millimetres measured accurately but erroneously in centimetres, with only one
of the students measuring accurately in millimetres. Thus, all but one of the students
measure with limited conceptual understanding of units. For example, one of the
students was measuring a distance between the shelf and the beginning of the
window in the library room the researcher was using. The following episode
presents the dialogue:

**R:**Can you measure the distance from the shelf to the beginning of the window.

**Lulama:**(takes the tape-measure. Measure the wall from zero to the end of the distance).

138

**R:**138 what?

**Lulama:**138 millimeters, no kilometers.

According to the Learning
Trajectories that are hypothesized by Sarama and Clements (2009) these students
are not conceptual measurers. They do not have mental images of millimeters nor
centimetres. Lulama here mentions “kilometres” that is longer than the length
of a classroom. He does not show mental understanding of how long is a
kilometre. He is reading numbers only in the measuring tool but cannot measure
length. The learning trajectories place these students on End-to-end length
measurer as they are able to align the measured object with the measuring
instrument from the zero. These students are able to use the ruler when measuring
but do not recognize units of measuring (Sarama and Clements 2009).

**Teaching Experiment Themes**

Curry et al (2006) suggest five
principles that should be applied in measurement instruction. These were used
to plan these teaching experiments. These principles include: “The need for
repeated units, the appropriateness of a selected unit, the need for the same unit
to be used to compare two or more objects, the relationship between the size of
the unit and the number required to measure, and the structure of the repeated
units” (p. 377).

Figure 1 indicates the
developmental paths the five students took from end-to-end length measurer to
the conceptual ruler. This figure presents the overall diagram of the five
students that presents the two themes of number development and mother tongue
use.

However, individual diagrams for
the five students will present individual paths taken by each of the five
students. The path is similar for all five students but the end points are
different and tools used to get to the next level are unique to each student’s
level of thinking about number and use of language.

Figure 1. Summarized learning
paths of the five students.

In teaching experiment one,
episode 1 focused on the appropriate selection of a unit and the need for a
repeated unit. However, each student had unique needs and therefore the teaching
experiment was adapted to address such needs. Episode 2 of teaching experiment one
focused on differentiating between millimeters and centimeters and creating an understanding
of the relationships between the two units.

The following themes emerged from
the teaching episodes: mother tongue use and number development. Episodes of
each student will be presented under the two themes. Gloria’s developmental
path starts like all others at the end-to-end length measurer because she has
no conceptual understanding of measuring units, could not measure with a tool
that is not calibrated, however was able to use the skill of aligning the ruler
correctly with measured object. Figure 2 presents Gloria’s developmental path
in conceptualizing length measurement.

Figure 2. Gloria’s developmental
path.

The following episode present the
process of development presented by the diagram when Gloria decided to convert
centimeters into millimeters and multiplied 158 by 10 and got 1048.

**R:**Does the 8 become just 8? What is 0x8

**Gloria:**8

**R:**You are multiplying not adding. Do you understand what it means?

0x8 means 0 + 0 + 0 + 0 + 0 + 0 +
0 + 0 =

0 + 0 is

**Gloria:**1

**R:**Ukuba ndingadibanisa into engekhoyo kwengekhoyo. (If you want to add nothing

to nothing) What do you get?
(Exact language used in the teaching experiment)

**Gloria:**Nothing.

**R:**0x5

**Gloria:**0

**R:**So when we multiply by 0 what do we get?

**Gloria:**0

**R:**So to get millimeters what do we do?

**Gloria:**No juffrou we must multiply by 10. (Calculating the following)

158

×10

000

1580

1580 mm

In this selected episode Gloria
was challenged by properties of zero in both addition and multiplication.
Conceptual meaning of zero was not realized and therefore her mother tongue pushed
her very fast to realizing zero. When she had to apply the new attained
knowledge it became easy for her to convert “14cm 8mm” to “148 mm.”

Figure 3 presents Simphiwe’s
developmental path indicating his unique needs during mediation. Simphiwe is
one of the students who did not demand or require mother tongue instruction
during teaching episodes and his path does not include mother tongue as
presented in Figure 3. Siphiwe was able to convert using counting as the
conversion tool even for fractions. Below is the episode that Simphiwe and
researcher experienced when Simphiwe measured a 3½ cm strip with his ruler
correctly.

**R:**Ok let’s see, how many millimeters in the 3½ centimeters?

**Simphiwe:**(counts) 10, 20, 30, and 5 mm.

**R:**If we want to present 137 cm you measured in millimeters what should we do?

**Simphiwe:**10, 20, 30, 40, 50, 60, 70, -------1370mm (counting in tens throughout)

**R:**How long is the length of this pencil?

**Simphiwe:**(measures the pencil accurately) 14½ cm.

**R:**What will it be in millimeters?

**Simphiwe:**140 mm ½ cm

Figure 3. Simphiwe’s
developmental path.

Figure 4 presents Siziwe’s
developmental path that was not easily observable. In her processes Siziwe did
not demonstrate any of the strategies she used in converting units. Instead she
converted units efficiently including fraction units. Siziwe used mother tongue
for expressing her reasoning compared to her peers.

Figure 4. Siziwe’ s developmental
path.

The episode on Siziwe supports
Figure 4 in drawing the picture of her path in attaining length measurement.
When Siziwe was measuring the length of the classroom she explained her actions
using mother tongue.

**Siziwe:**(picks the tape-measure)

**R:**Why are you taking that one?

**Siziwe:**Inde iklasi nayo inde (it is long and the classroom is also long)

**R:**Ok what about the strip?

**Siziwe:**You can use it for into ezincinci (you can use it for small things)

**R:**How long is this pencil?

Sinesipho

**:**14 ½ cm.**R:**Can you tell me how long is it in mm?

**Siziwe:**yi 145 mm.

In her transition to English, she
started by mixing both languages in a sentence and then continued with English
only. Her mother tongue dialogue was about the length concepts like “ubude bale
klasi” meaning the length of the classroom, “inde iklasi nayo inde” meaning the
meter stick is long as the classroom and “into ezincinci” meaning for small
things. These “isiXhosa” phrases give clear meaning of her conversation
conceptually. Then “into ezincinci” refer to something smaller than short to
her. Meaning she wanted to say ‘tiny” in English. Thus, Siziwe’s use of mother
tongue assisted her in clearly expressing her thinking.

Figure 5 presents Lulama’s path
that is not progressive like his peers. He could not move from his level to the
next level.

Figure 5. Lulama’s developmental
path.

**R:**From here to here how many 10’s (0-----2cm).

**Lulama:**20.

**R:**From here to here how many 10’s (0------3cm).

**Lulama:**40.

**R:**Can you count in tens for me?

**Lulama:**10, 20, 40.

**R:**If you change 36 cm to mm what will that be?

**Lulama:**1, 2, 3, 4.

**R:**Does it really get to 4. How many mm in 1 cm?

**Lulama:**10.

**R:**If there are 2cm how many will be those in millimeters?

**Lulama:**19.

Lulama demonstrated understanding
that 1-centimeter is equal to 10 millimeters.

However, his counting skills were
becoming a barrier for progress. He struggled to count in tens beyond 20. He
also made an obvious mistake like adding 10 and 10 and getting 19. In addition
Lulama showed that he skip counts when counting in tens as the researcher
engaged with him. He responded:

**R:**If they are 3cm. How many mm?

**Lulama:**40.

**R:**Can you count in tens for me?

**Lindisipho:**10, 20, 40.

Figure 6. Mimi’s developmental
path.

Mimi’s number sense nurtures her
length concepts. For example her knowledge of fractions, and multiplication
accelerated her understanding of length measuring units. The following present
Mimi’s learning process:

**R:**It’s 140 what?

**Mimi:**140 + 5 mm=145 mm.

**R:**Can you change it into cm?

**Mimi:**14cm and 5mm.

**R:**That 5mm is what to a cm?

**Mimi:**1/2 cm.

**R:**Can I call this 14½ cm?

**Mimi:**Yes, it is 14½ cm.

Mimi easily converted 140 mm to
14 cm without a struggle because of her division and multiplication knowledge.
She went further to covert numbers to fractions successfully and vice-versa.

**Post Clinical Interviews**

All five students were able to
accurately draw 10 centimetres, 12 millimetres, and a 29 centimetres line
segment. When they also were required to draw a straight path of 29 centimetres
with two turns; all five students drew an accurate path of 29 centimetres but without
two turns as instructed. Teaching experiments were able to mediate conceptual understanding
of units of measuring length and were able to mediate the relationships between
these units. According to the Revised National Curriculum Statement (2002)
fifth grade students are expected to be able to use appropriate measuring
units, instruments and formulae in a variety of context. Figure 6 presents the
development of each learner at different times of the learning process.

Figure 7. Eight length development progressions.

**Discussion**

The findings of this study reveal
that (1) number development of these students lacked foundation and created a
barrier for abstraction. Gloria was challenged by the properties of zero and just
needed conceptual meaning of zero as a number and once she got that through mother
tongue explanation she progressed very fast. Only one student was not
challenged by number concepts and that allowed her to reach the abstract
developmental progression of length measurement; (2) mother tongue of these
students play a significant role to some learners in

**conceptualizing**number and length concept. While one student used mother tongue to express her conceptual thinking. (3) Mediation that was on the students actual thinking levels nurtured students’ levels of thinking about length (Setati and Adler, 2001; Vygotsky, 1978); (4) both procedures and understanding are needed during mediation to develop higher order thinking levels, length learning trajectory guided understanding of conceptual development of length. All four students lacked understanding of skip counting, reading numbers, and properties of 0. This reflects to their foundation phase experiences on number that did not develop conceptually. One of the students demonstrated that he has not attained cardinality after counting. He could not determine how many centimetres the length of the pencil was. Lulama did not demonstrate development or regression. His teaching episodes demonstrated his lack of number sense. He could count in tens further than 20, could not add 10+10. The teaching experiments were at a higher level for him and measurement concept was far ahead of him. All he could do during the study was to align the ruler from 0 to the end with no understanding. In fact he began as a direct comparer and ended as a direct comparer. His number development closed the possibilities for observable growth.
South African
studies on mathematical performance of students report performance at lower levels
than expected (Mji & Makgatho, 2006; Van der Sandt & Niewoudt, 2003 and
Wessels, 2008). This study revealed an additional component to teacher lack of
content knowledge, and poor teaching strategies. This component is the lack of
strong mathematical foundation of the students in the beginning years of
schooling before Grade 5. South African early childhood in mathematics begins
in grade R leaving behind the developmental stages of children when they are
young, hence one of them still struggles with cardinality that they should
start developing before Grade R. Research has proven that exposing young
children as early as 3 years to quality mathematical experiences predict
success in literacy and mathematics learning in their elementary years
(Barnett, 1995; Cross et.al., 2009; Sarama and Clements, 2004). These results
confirm the call for formalisation of early childhood learning to account for
gaps of knowledge children enter schools with. South African research needs to
inform policy on quality early childhood mathematics curriculum and practice.
This study proved that Setati’s research on multilingualism has implications
for all multilingual students regardless of their learning environment.

### This research is perfectly in harmony with the revelation I received from God in 2012 about why Africa is backward and poor that I posted on this blog. Read details of the revelation at http://www.africason.com/2014/06/african-school-of-grassroots-science.html

### This research applies to conceptualizing all sciences not only mathematics. I can be of help to the government of South Africa, I know exactly the initial problems you'll face; which is creating contents in your native language. Simply contact me, I'm ready to come to your country and let you know how to do it. You need just a model for creating terminologies in your native language. Only then shall African children learn naturally as it should be.

### I made a song about this topic after receiving the revelation from God. Please spread this message all over Africa through your radios, TVs, mouth to mouth, anyhow you can. The song is called African school and could be found on iTunes, artiste name: Africason

**References**

Adler, J. (1998). A language of teaching dilemmas:
Unlocking the complex multilingual

secondary mathematics classroom. For the Learning of
Mathematics, 18, 24–33.

Adler, J. (2001). Teaching mathematics in multilingual
classrooms. Dodrecht: Kluwer.

Barnett, W. S. (1995). Long-term effects of early
childhood programs on cognitive and

school outcomes. The Future of Children, 5(3), 25-50.
doi:10.2307/1602366

Barrett, J. E., & Clements, D. H. (2003).
Quantifying path length: Fourth-grade children's

developing abstractions for linear measurement.
Cognition and Instruction, 21(4), 475-

520. doi:10.1207/s1532690xci2104_4

Bishop, A. J. (1985). A cultural perspective on
mathematics education. Keynote address to

the National Seminar on Teaching Mathematics to
Aboriginal Children, Alice Springs,

Australia.

Bragg, P., & Outhred, L. (2000a). What is taught
versus what is learnt: The case of linear

measurement. In J. Bana and A. Chapman (Eds.),
Mathematics Education Beyond 2000-

Proceedings of the 23rd Annual Conference of the
Mathematics Education Research

Group of Australia (Vol. 1, pp. 112-18). Sydney:
MERGA.

Bragg, P., & Outhred, L. (2001). So that’s what a
centimetre looks like: Students’

understanding of linear units. In Marja van den
Heuvel-Panhuizen (Eds.), Proceedings of

the 25th Conference of the International Group for the
Psychology of Mathematics

Education (Vol.2, pp.209-216). Utrecht, The
Netherlands: Freudenthal Institute, Utrecht

University.

Bragg, P., & Outhred, L. (2004). A measure of
rulers - The importance of units in a measure.

Proceedings of the 28th Conference of the
International Group for Psychology of

Mathematics Education (Vol.2, pp.159-166).

Carignan, C., Pourdavood, R. G., King, L. C., &
Feza, N. (2005). Social representations of

diversity: Multi/intercultural education in a South
African urban school. Intercultural

Education, 16(4), 381–393.
doi:10.1080/14675980500304371

Clements, D. H. (1999). Teaching length measurement:
Research challenges. School Science

and Mathematics, 99(1), 5-11.
doi:10.1111/j.1949-8594.1999.tb17440.x77 N. Feza-Piyose

Clements, D. H., & Sarama, J. (Eds.) (2004).
Hypothetical learning trajectories [Special

Issue]. Mathematical Thinking and Learning, 6(2).

Clements, D. H., & Sarama, J. (2007a). Early
childhood mathematics learning. In F. K.

Lester, Jr. (Ed.), Second Handbook of Research on
Mathematics Teaching and Learning

(pp. 461-555). New York: Information Age Publishing.

Clements, D. H., & Sarama, J. (2009). Learning and
teaching early math: The learning

trajectories approach. New York: Routledge.

Clements, D. H. (2010). Tools, technologies, and
trajectories. In Z. Usiskin, K. Andersen &

N. Zotto (Eds.), Future curricular trends in school
algebra and geometry (pp. 259-266).

Charlotte, NC: Information Age Publishing, Inc.

Clements, D. H., Sarama, J., & Wolfe, C. B.
(2011). TEAM—Tools for early assessment in

mathematics. Columbus, OH: McGraw-Hill Education

Cross, T. C., Woods, T. A., & Schweingruber, H.
(2009). Mathematics learning in early

childhood: Paths toward excellent and equity.
Washington: The National Academic

Press.

Curry, M., Mitchelmore, M., & Outhred, L. (2006).
Development of children’s understanding

of length, area, and volume principles. In J. Novotná,
H. Moraová, M. Krátká, & N.

Stehlíková (Eds.), Proceedings of the 30th Conference
of the International Group for the

Psychology of Mathematics Education (Vol. 2, pp.
377–384). Prague: PME.

Ely, M., Anzul, M., Friedman, T., Garner, D., &
Steinmetz, A. M. (1991). Doing qualitative

research circles within circles. Bristol, PA: The
Falmer Press, Taylor & Francis Inc.

Hiebert, J. (1986). Conceptual and procedural
knowledge: The case of mathematics.

Hillsdale, NJ: Lawrence Erlbaurn Associates, Inc.

Howie, S. J. (2003). Language and other background
factors affecting secondary pupils'

performance in mathematics in South Africa. African
Journal of Research in Mathematics

Science and Technology Education, 7, 1-20.

Khisty, L. L., & Morales, H. Jr. (2004). Discourse
matters: Equity, access, and Latinos'

learning mathematics. Retrieved March 21, 2007, from

http://www.icmeorganisers.dk/tsg25/subgroups/khisty.doc.

Lakoff, G., & Nunez, R. E. (2000). Where
mathematics comes from: How the embodied mind

brings mathematics into being. New York: Basic Books.

Lehrer, R., Lee, M., & Jeong, A. (1999).
Reflective teaching of Logo. Journal of the

Learning Sciences, 8, 245-288. doi:10.1207/s15327809jls0802_3

Lehrer, R., Jaslow, L., & Curtis, C. (2003).
Developing an understanding of measurement in

the elementary grades. In D. H. Clements, & G.
Bright (Eds.), Learning and teaching

measurement (pp. 100–121). Reston, VA: National
Council of Teachers of Mathematics.

Lehrer, R. (2003). Developing understanding of
measurement. In J. Kilpatrick, W. G. Martin,

& D. Schifter (Eds.), A research companion to
principle and standards for school

mathematics (pp. 179–193). Reston, VA: National
Council of Teachers of Mathematics.

Matang, R. A. (2006). Linking ethnomathematics,
situated cognition, social constructivism

and mathematics education: An example from Papua New
Guinea. ICME-3 Conference

Paper, New Zealand.

Mji, A., & Makgato. M. (2006). Factors associated
with high school learners’ performance: A

spotlight on mathematics and physical science. South
African Journal of Education,

26(2), 253-266.

Nicol, C. (2005). Exploring mathematics in imaginative
places: Rethinking what counts as

meaningful contents for learning mathematics. School
Science and Mathematics, 105(5),

240. doi:10.1111/j.1949-8594.2005.tb18164.x

Piaget, J. (1960). The psychology of intelligence.
Oxford: England.

Piaget, J. (1967). The child’s conception of the
world. London: Routledge & Kegan Paul.

Raborn, D. T. (1995). Mathematics for students with
learning disabilities from language-minority backgrounds: Recommendations for
teaching. New York State Association for

Bilingual Education Journal, 10, 25-33.

Reynold, A., & Wheatley, G. H. (1996). Elementary
students’ construction and coordination

of units in an area setting. Journal for Research in
Mathematics Education, 27(5), 564–

581.

Sarama, J., & Clements, D. H. (2004). Building
Blocks for early childhood mathematics.

Early Childhood Research Quartely, 19, 181-189.
doi:10.1016/j.ecresq.2004.01.014

Sarama, J., & Clements, D. H. (2009). Early
childhood mathematics education research:

Learning trajectories for young children. New York:
Routledge.

Sarama, J., & Clements, D. H. (2009). Teaching
math in the primary grades: The learning

trajectories approach. Young Children, 64(2), 63-65.

Setati, M. (1998). Code-switching in senior primary
class of second language learners. For

Learning of mathematics, 18(2), 114–160.

Setati, M., & Adler, J. (2001). Between languages
and discourses: Language practices in

primary multilingual mathematics classrooms in South
Africa. Educational Studies in

Mathematics, 43, 243–269.

Setati, M. (2002). Researching mathematics education
and language in multilingual South

Africa. The Mathematics Educator, 12(2), 6–20.

Setati, M. (2005). Mathematics education and language:
Policy, research and practice in

multilingual South Africa. In R. Vithal, J. Adler,
& C. Keifel (Eds.), Researching

mathematics education in South Africa: Perspectives,
practices and possibilities.

Capetown, South Africa: HSRC Press.

Setati, M., & Barwell, R. (2006). Discursive
practices in two multilingual mathematics

classrooms: An international comparison. African
Journal for Research in Mathematics,

Science and Technology Education. 10(2), 27-38.

Steffe, L. P., & Thompson, P. W. (2000). Teaching
experiment methodology: Underlying

principles and essential elements. In R. Lesh & A.
E. Kelly (Eds.), Research design in

mathematics and science education (pp.267-307).
Hillsdale, NJ: Erlbaum.

Thompson, P. (1979). The constructivist teaching
experiment in mathematics education

research. Paper presented at the Research Reporting
Session, Annual Meeting of the

National Council of Teachers of Mathematics, Boston.

Van der Sandt, S., & Niewoudt, H. (2003). Grade 7
teachers’ and prospective teachers’

content knowledge of geometry. South African Journal
of Education, 23(3), 199-205.

Van de Walle, J., & Thompson, C. (1985). Let’s do
it - Estimate how much. Arithmetic

Teacher, 32(8), 8-12.

Wilson, P. & Rowland, R. (1992). Teaching
measurement. In P. Jenner, (Ed), Research Ideas

for the Classroom - Early Childhood Mathematics (pp.171-191).
New York: Macmillan.

Vygotsky, L. S. (1978). Mind in society: The
development of higher psychological process.

Cambridge, MA: Harvard University Press.

Wessels, H. (2008). Statistics in the South African
school curriculum: Content, assessment

and teacher training. In C. Batanero, G. Burril, C.
Reading & A. Rossman (Eds.), Joint

ICMI.IASE Study: Teaching Statistics in School
Mathematics. Challenges for Teaching

and Teacher Education. Proceedings of ICMI Study 18
and 2008 LASE Round Table

Conference.

Authors

Nosisi Feza-Piyose, Postdoctoral fellow, Education and
Skills Development

Research Program, Human Sciences Research Council,
South Africa;

Source: http://www.iejme.com/022012/d2.pdf

Africason is a Musician/independent recording artiste and a die-hard believer in Africa.

Twitter: @african_school

Web: www.africason.com

Email: info(AT)africason.com

Find my songs on iTunes, artiste name: Africason

## No comments:

## Post a Comment