Exploring intermediate phase learners' use of mathematical language when learnin
What Are We Deciding?
These are the open questions. Nail these down and the proposal writes itself.
DECISION 1: Are teachers in or out?
Dr M asked this 3 times. This changes everything.
Our thinking: Teachers are OUT
- We are looking at learners' language, not teaching methods
- The study is about: what happens when a Grade 4-6 kid tries to explain fractions in their 2nd/3rd language?
- Teachers are part of the context (they set the tasks, they use certain language) but they are not the object of analysis
- Keeping scope tight = doable masters, not a PhD
But Objective 4 says "how do educators aid..." so either rewrite that objective or accept teachers are partially in scope.
DECISION 2: What exactly are we analysing?
What is the data? What do we collect? What do we look at?
- DATA Learner talk when they explain fractions (recorded, transcribed)
- DATA Written work where learners use maths vocabulary
- DATA The tasks themselves (what language do the questions use?)
- SCOPE Q Do we also look at what language the teacher uses when setting up the task? (context, not analysis)
DECISION 3: What are we looking FOR in the data?
When Innocent reads a transcript, what does he code/tag?
- LOOK FOR Vocabulary accuracy - do they say "numerator" correctly? Do they confuse terms?
- LOOK FOR Language switching - when do they drop into home language? What triggers it?
- LOOK FOR Misconceptions revealed by language - "the bigger number goes on top" shows a whole-number mindset
- LOOK FOR The gap between what the task asks (in English) and what the learner understood
DECISION 4: Why does this matter beyond school?
The "so what?" chain:
- Fractions are the gateway to algebra, ratios, percentages
- Fail fractions in Gr 4-6 = struggle in higher maths
- Struggle in maths = fewer career options
- If the barrier is language, not ability, that is fixable
- This study shows where language breaks down so teachers can intervene at the right point
Your proposal structure (simplified)
1.1 Intro = Language matters in maths. Here is why. 1.2 Background = SA context: CAPS, English as LoLT, rural Mpumalanga 1.3 Problem = Fractions are hard + language barrier + no research here 1.4 Rationale = Fractions affect life outcomes + language is fixable 1.5 Aims = Explore how learners USE maths language for fractions 1.6 Questions = 1 primary + 4 secondary (may need to drop Obj 4)
Exploring intermediate phase learners' use of mathematical language when learning
fractions in rural primary school in Mpumalanga province
By M.I Hadebe
Supervisors
Dr Jeanine Mwambakana and Prof Ogbonnaya
© University of Pretoria
Chapter 1: Introduction and background
1.1. Introduction
The language used in the mathematic class has a significant influence on learners’ communication of mathematical ideas and conceptual understanding of mathematics. Barwell (2016) added that mathematics is not only a subject of numbers and symbols, but a unique language utilized to convey mathematical reasoning and ideas (Prediger et al., 2016). Clarkson and Planas (2018) further noted that the evolution of mathematical language allows learners to explain and describe relationships and procedures, substantiate findings in the mathematical context (Plana, 2016). Barwell (2016) highlights that research in mathematics education the capability of a learner to engage with the mathematics language and discussions is fundamental to problem solving and conceptual understanding (Prediger et al., 2016).
Lamon (2012) noted that in the Intermediate phase (Grade 4-6) learners shift from basic whole numbers to more complicated mathematical concepts such as fractions (Siegler et al., 2016). Siegler et al. (2016) added that worldwide fractions are seen as one of the most demanding fields of mathematics for learners in primary schools because they need learners to comprehend relationships between quantities than whole numbers (Lamon 2012). Clarke et al. (2018) further added that to master fractions learners are required to understand and utilize mathematics language and terminology correctly, these terms involve ordinal numbers, mixed numbers, improper fractions, numerator and denominator (Charalambous, 2019).
of classrooms in South Africa, including those in Mpumalanga Province, mathematics is taught using English which is mostly the language of teaching and learning and often not the home language of the learners (Essien, 2021). Setati and Adler (2016) added that the language of teaching and learning can have a significant influence on learners’ discussions in the classroom and understanding of mathematical concepts (Essien, 2018).
Adler (2019) notes that research done in South Africa reveals that the diversity in language often create challenges for learners when explaining their thoughts, reasoning and when interpreting mathematical problems (Venkat & Adler, 2017).
Setati-Phakeng (2020) added that in a language diverse classroom, learners may have challenges understanding mathematics terminology and explain their reasoning using the language of learning and teaching (Essien, 2019).
Furthermore, research conducted by Clarke et al. 2018 suggests that learners often rely memorizing formulas instead of understanding mathematical concepts such as fractions (Siegler & Thompson, 2019). Hence, comprehending how learners use mathematical language when learning fractions is significant for enhancing learning and teaching methods.
This study aims to explore how learners in the intermediate phase (grade 4-6) in rural areas of Mpumalanga use mathematical language when learning fractions.
1.2. Background
The Curriculum and Assessment Policy Statement (CAPS) which was designed by the Department of Basic Education (DBE) guides mathematics education in South Africa. Hence, according to the (DBE, 2011) the CAPS curriculum in mathematics encourages conceptual understanding and procedural fluency and fosters learners to explain their reasoning using the correct mathematical language (DBE, 2011).
However, Planas (2016) noted that in the multilingual classrooms of South Africa, learners from Grade 4 going forward are mandated to learn mathematics in English even though English is not their home language (Prediger et al., 2016). Research conducted by Setati and Adler (2016) emphasizes that the language of learning and teaching have a significant influence on learners’ discussions in the classroom and understanding of mathematical concepts (Essien, 2018).
Research concentrating on linguistically diverse mathematics classrooms in South Africa have shown that language of teaching and learning can either challenge or support learners understanding of mathematics (Setati-Phakeng, 2020). For example, study conducted by Venkat and Adler (2017) shows that language used in South African plays a huge role in shaping learners mathematical thinking (Venkat & Adler, 2017). Similarly, research conducted by Setati and Adler (2016) highlights that language practices in classrooms have a significant effect on how learners participate in mathematical discussions and in translating mathematical activities (Setati & Adler, 2016).
Furthermore, studies conducted by Essien (2018) further highlight the significance of recognizing language as a resource for learning mathematics in linguistically diverse classrooms as opposed to it being a barrier (Essien, 2019). Such research shows that when a learner’s home language is integrated strategically by an educator in a mathematics classroom it can support conceptual understanding (Essien, 2021).
Alternative studies conducted in South African mathematics education by Robertson et al. (2021) demonstrates that learners in rural-underprivileged schools often encounters obstacles such lack of resources, poverty, and language diversity (Essien, 2024). Sepeng (2018) agrees that such background factors can have a significant effect on how learners conceptualize and engage with the mathematics language (Venkat, 2023).
As an educator who had the privilege to teach in the intermediate phase (grade 4-6), I have noticed that learners are experiencing challenges when dealing with fractions because they include abstract relationships between quantities and learners to translate symbolic and linguistic notations at the same time (Cramer, 2020). Ni and Zhou (2017) added that learners must understand mathematical vocabulary such as numerator, denominator and ordinal numbers to be able to explain mathematical problems (Ni & Zhou, 2017).
Regardless of the trend in international study on language in mathematics education, there is still lack of evidence-based study concentrating particularly on learners in the intermediate phase in rural Mpumalanga use the mathematics language when engaging fractions. Hence, exploring this challenge may present a significant outlook into how language impacts learners’ mathematical discourse and conceptualiation.
1.3. Problem statement
As Siegler et al. (2016) highlights that fractions create an essential element of the intermediate phase curriculum and as a bedrock for mathematical topics such as algebra and ratios in higher grades (Siegler et al., 2016). Nonetheless, Cramer (2020) emphasizes that studies demonstrate that majority of learners are still experiencing challenges when it comes to conceptualiing mathematics (Venkat, 2020).
Prediger et al. (2020) noted that a conceivable cause that influences these challenges is learners lack of use and understanding of mathematical language (Essien, 2019).
There are specialized mathematics terminology and symbolic notations which are used to communicate mathematical ideas (Essien, 2022) and learners must be able to translate and use this language to understand fractions (Prediger et al., 2020).
Adler (2019) supports this that in rural Mpumalanga classrooms, where learners are learning mathematics using a different language from their home language (Essien, 2019), issues such pertaining mathematical language may have a great impact on how learners understand fractions (Essien, 2021). A study conducted by Venkat (2023) in South African mathematics education adds to this that language used in the mathematics classroom when teaching fractions have a significant influence on how learners explain mathematical tasks and how they reason mathematically (Adler, 2019).
Furthermore, Robertson et al. (2021) also noted that learners may have challenges explaining their reasoning mathematically, misrepresent mathematical vocabulary and depend on fixed methods without conceptually understanding the operations of fractions (Cramer, 2020). Nqabeni et al. (2023) added that these obstacles can be seen in rural schools where learners who are speaking different languages and coming from different socio-economic backgrounds are found (Nqabeni et al., 2023).
Nevertheless, studies have investigated mathematical misconceptions about fractions, however there is still lack of empirical research on how learners’ use of mathematical language impacts how they comprehend fraction ideas in rural Mpumalanga classrooms. Lacking comprehension of this relationship may make intervention aimed at improving learner’s performance on fraction ineffective.
1.4. Rationale
This research is significant for numerous factors:
Firstly, as Siegler et al. (2016) notes that fractions are an important foundation for learning mathematics such as ratios and algebra in the higher grades. Studies have demonstrated that having challenges with fractions in the lower grades may hinder success in the higher grades (Siegler et al, 2016).
Secondly, as Barwell (2016) notes that mathematical language has a significant influence on learners’ conceptual understanding of fractions, and their ability to explain mathematical concepts (Prediger et al., 2020). Learners with a good grasp of mathematical terminologies are most likely to conceptually understand the operations of fractions (Charalambous, 2019).
Thirdly, as Essien (2022) notes that linguistically diverse mathematical classrooms offer different chances and obstacles for teaching and learning. This agrees with Setati-Phakeng (2020) who added that it is important to understand learners’ language needs and aid the development of mathematical classroom discussions (SetatiPhakeng, 2020).
Fourthly, as Robertson et al. (2021) states that most schools in the rural areas are under privileged with issues such as lack of resources and socio-economic factors. Challenges as such may have an impact on how learners participate during mathematics teaching and learning.
Lastly, the outcome of this research may have a significant contribution in enhancing mathematics teaching by giving an outlook into how educators can aid the advancement of mathematical language when teaching fractions in rural Mpumalanga. This research may also help in designing professional teacher training frameworks that can enhance mathematics learning and teaching in rural Mpumalanga.
1.5. Aims and objectives
The purpose of this study is to explore how intermediate phase learners in rural primary schools in Mpumalanga use mathematical language when learning fractions.
1.5.1. Objectives
To explore how accurately do learners use fractions vocabulary such numerator, denominator and ordinal number.
To investigate what language misconceptions, occur when learners explain fractions
To explore how does the instructional language impact learners’ explanation of fractions ideas.
To explore how do educators aid the advancement of mathematical language during lessons of fractions.
1.6. Research questions
1.6.1. Primary research questions
How do intermediate phase learners in rural primary schools in Mpumalanga use mathematical language when learning fractions?
1.6.2. Secondary research questions
How accurately do learners use fractions vocabulary such numerator, denominator and ordinal numbers?
What language misconceptions occur when learners explain fractions?
How does the instructional language impact learners’ explanation of fractions ideas?
How do educators aid the advancement of mathematical language during lessons of fractions?
Top 5 Grammar Fixes
These are the patterns Dr Mwambakana flagged most. Fix these five things and 80% of the comments go away.
1. "They" with no person attached
They shift from whole numbers to fractions
Learners shift from whole numbers to fractions
Every "they/them/their" must point back to a name or group in the same sentence or the one before it. If you cannot find who "they" is in 2 seconds, replace it with the actual group name.
2. Missing small words (prepositions and conjunctions)
learners often rely memorizing formulas
learners often rely on memorizing formulas
issues such pertaining
issues such as those pertaining
Read each sentence out loud. If it sounds like you skipped a word, you did. The small words (on, in, of, as, that, to) are the glue that holds sentences together.
3. One paragraph, one idea
Dr M flagged this three times. Multiple themes crammed into single paragraphs.
Write the main idea of your paragraph in the margin. If you wrote two ideas, split it into two paragraphs.
Test: can you summarise your paragraph in one sentence? If you need two sentences, you need two paragraphs.
4. Finish the thought after "supports" / "added" / "noted"
Adler (2019) supports this that...
Adler (2019) supports the argument that...
After words like "supports", "agrees", "noted", always say WHAT they support or agree with. Never leave the reader guessing.
5. Link your paragraphs together
Dr M asked: "how does this connect with the previous paragraph?"
First sentence of every new paragraph should have a bridge:
- "Building on this..."
- "In contrast..."
- "A related challenge is..."
- "While X found Y, Z found..."
The bridge tells the reader WHY you are moving to a new point and how it connects to what came before.
4 Papers. Read Only These.
Each paper answers a specific question you need for the proposal. Read with those questions in your head.
1. Essien (2022) DEFINES YOUR LENS
Language and multilingualism in mathematics education
Read this to answer: What IS mathematical language? Is it just vocab words, or something bigger?
- Does Essien treat language as a BARRIER (problem to fix) or RESOURCE (tool to use)? Your proposal must pick one.
- What does "mathematical language" actually include? List the categories. This becomes your coding framework.
- When learners switch to home language during maths, is that a failure or a strategy? What does Essien say?
After reading, you should be able to say: "In my study, mathematical language means X, Y, and Z. I am treating it as [barrier/resource/both]."
2. Prediger, Clarkson & Bose (2016) TELLS YOU WHAT TO LOOK FOR
Purposefully relating multilingual registers
Read this to answer: What are the different "registers" and how do you spot them in data?
- 3 registers: everyday ("sharing pizza"), school ("equal parts"), technical ("numerator"). Can you hear the difference?
- When a learner says fractions wrong, is it a MATHS misconception or a LANGUAGE problem? How do you tell?
- What data collection method would let you capture register-switching? (This shapes your methodology.)
After reading, you should be able to say: "I will analyse learner language across [X] registers. I will collect data by [method]."
3. Siegler, Thompson & Schneider (2016) PROVES WHY IT MATTERS
Integrated theory of whole number and fractions development
Read this to answer: Why are fractions the make-or-break moment? What goes wrong in kids' heads?
- The "whole number bias": kids think 1/3 > 1/2 because 3 > 2. How would this show up in what they SAY?
- At what stage (age/grade) do fraction misconceptions solidify? Does Grade 4-6 catch them in time?
- Fail fractions = fail algebra = fewer life options. Can you trace this chain with their evidence?
After reading, you should be able to say: "Fractions matter because [X]. The common misconceptions are [Y]. My study catches learners at the critical window."
4. Robertson, Lovemore & Graven (2021) YOUR CONTEXT
Mathematics learning in contexts of poverty
Read this to answer: What is different about rural SA classrooms? Why does your study need to happen HERE?
- What obstacles do rural Mpumalanga schools face that make the language problem WORSE than in cities?
- Did they study language specifically, or just general maths performance? (If general, that is your gap.)
- What was their sample? How did they collect data? Can you adapt their method?
After reading, you should be able to say: "My study fills the gap because Robertson et al. looked at [X] but not [Y]. I am looking at [Y]."
Audio Deep Dives
Three episodes covering all 4 papers. Start simple, build to expert-level insights. Listen while commuting, walking, or just thinking.
Episode 1: Why Fractions Matter
Siegler + Essien foundations
From "what is a fraction" through whole number bias, to why fractions predict life outcomes and how language shapes conceptual understanding.
Episode 2: Language as the Hidden Variable
Prediger's registers + Essien on language-as-thinking
Everyday vs school vs technical registers. Code-switching in multilingual classrooms. What to listen for when you observe teachers and learners.
Episode 3: Designing Your Study
Robertson + Graven context, then methodology synthesis
Rural SA reality. Case study design. What data to collect, how to code it, how to write the proposal as a coherent story. Walk away ready to write.
Study Notes
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