Publications:
Whitten, D. (2012). Divergent learner pathways: Exploring the mathematical beliefs of young
adult learners. In A. Hector-Mason & D. Coben (Eds.), Synergy: Working together to
create more than the sum of the parts (proceedings of the 19th International conference of Adults Learning Mathematics – A research forum). Auckland. NCLANA.
Abstract: Maladaptive beliefs about mathematics have been described as those in which mathematics is characterised as facts, rules and procedures, thought to be learned primarily by rehearsal strategies, and perceived to reflect one’s intellectual abilities. These beliefs often develop during school years and may contribute to poor learner engagement in mathematics and subsequent poor mathematical achievement. Maladaptive beliefs may also be held by adults studying numeracy in a vocational context and contribute to negative attitudes and approaches to learning. The aim of this study was to explore the mathematical beliefs of young adult learners who had struggled with mathematics during their school years. Six young adults took part in individual interviews that explored their beliefs, attitudes and learning histories. The results indicate that the learners did hold maladaptive mathematical beliefs and that these beliefs influenced how they approached learning numeracy in the tertiary sector. Recommendations are made for an increased focus on improving learners’ beliefs as part of numeracy provision.
Whitten, D. (2013). The Influence of Mathematical Beliefs on Low-Achieving Adult Learners.
In V. Steinle, L. Ball & C. Bardini (Eds.), Mathematics Education: Yesterday, today and
tomorrow (Proceedings of the 36th annual conference of the Mathematics Education
Research Group of Australia). Melborne, VIC: MERGA.
Abstract: This paper explores how beliefs about mathematics may influence low-achieving adults’ re-engagement with mathematics in the tertiary sector. Adult learners who have problematic mathematical histories often hold negative beliefs about the nature of mathematics and how it is learned. In New Zealand these adults are often required to re-engage in mathematical provision in the tertiary sector to gain qualifications for employment. The beliefs they hold about mathematics may negatively influence their approach to learning mathematics and their affective response to it. This article explores several ways in which negative beliefs about mathematics may undermine adults’ success.
Whitten, D. (2016) Inside a math-for-work lesson on ratio. ICME (in press)
Abstract: This paper reports on divergent patterns of engagement of adult learners with problematic mathematical histories, as they attend a session on ratio of fuel mixes in a vocational programme geared to an agricultural context. Observational data were collected through multi-positional microphones and analysed thematically. The findings suggest that learners who judge the mathematical environment as potentially harmful to their social being constrain their engagement to such an extent that little mathematical understanding is developed. In contrast, learners who orient toward active and exploratory roles, and show no concern about social harm, engage in ways that may lead to success. This paper presents the situations, and explores the implications of the episode for mathematics education in and for work.
Some initial findings regarding the theme – Emotion and attitude
Having spent the last two months analysing and gradually synthesising the findings from the observation data I’m beginning to make some progress. Below is an very early overview of findings relating to the theme ‘emotion and attitude’.
Method
I coded and analysed ‘expressions of affect’ which include external expressions such as emotionally expressive words, interjections and exclamations, eye contact or lack thereof, facial expressions and multiple modes of body-language. Interestingly, the majority of emotive speech happened privately between two learners or as self-talk. Without the use of multiple microphones the entire range of emotional expressions would have been 'under-cover' of the layer of audible classroom dialogue.
The expressions can be broadly organised by the responses at consistent routines of the each lesson. Most lessons I observed were structured in three parts. These were: the initial introduction and demonstration of the skill/method/concept to be learned; learner application/practice - usually in groups; review, discussion and evaluation and marking of learner work.
Findings
Anticipation of failure
Learners reacted strongly to being compelled to participate in numeracy lessons or activities. I’m struggling with the word ‘compelled’ but given the negative emotion expressed most adults would simply leave if they could. There are reasons why these learners don’t which have to do with PTE attendance procedures/rules and sociocultural pressures. When tutors introduced the topic learners become visibly anxious, exclaimed their hatred of maths or their poor historical performance with maths and often begin to withdraw their participation. I’ve linked these responses to the concept ‘anticipation of failure’. The anticipation of failure has links to learners’ prior experiences. Learners anticipate re-experiencing the worst aspects of their mathematics experiences. This answers one question: do adult numeracy environments activate mathematical beliefs? Without a doubt. I have been playing with the term ‘emotional inertia’ to describe how difficult it is for a tutor to positively engage learners in numeracy.
Emotional investment
The second part of the lessons (Learner application – group problem solving) evokes emotional responses also. But here they are a mix of positive and negative. On the positive side many learners emotionally invest into solving tasks, they work hard, persist and care deeply about the results. They appear motivated by the desire to demonstrate to others and themselves how good they are. They want to be the best and as such are performance oriented. These are the learners that challenge tutors, they are usually the best in the class and interact the most with the tutor during numeracy sessions. They care deeply about being right and will work hard to be so.
Learners also express negative emotions during problem solving phases of lessons. Often they become overwhelmed by the complexity of the task and disengage. These learners often are the hardest to engage as they withdraw at the first sign of difficulty. These learners have deep doubts about their ability and are motivated to avoid losing status. If the task looks like it could possibly damage their image they choose not engage. Other learners do engage and become frustrated at the lack of progress (they often resemble the positive emotion described above) but these learners are motivated by the desire to get the job done. They want to finish the task as quickly as possible. When it proves to be a challenge and they feel they are no longer moving toward completing the task they begin to get frustrated in a negative way.
Loading meaning onto correct/incorrect answers
At the third phase of the lesson (during which the tutor works through the answers) all learners who have completed the tasks are fixated on the tutor and answers. Many learners describe themselves as either ‘math people’ or 'non-math people’. Many adult learners use numeracy classes to either verify their existing identity or to reconstruct it. Hence many learners ‘load meaning’ onto their answers. When they have an answer it is not just a quantitative answer to a quantitative question rather it represents whether one is ‘smart’, ‘intelligent’, ‘good at maths’ or ‘bad at maths’. There were occasions in the observations where learners who got answers wrong had huge emotional responses and withdrew participation from that point on. For these learner’s getting the answer wrong confirmed their historical recollections and beliefs that they are not intelligent.
In summary, participating in numeracy is highly emotional for almost all learners whether they are ‘maths people’ or 'non-maths people’. Learners load meaning onto what happens in the class, how they learn, the answers they get correct, and the answers they get incorrect. Adult learners care about their performance with maths and numeracy. This is a huge opportunity for educationalists.
The question next is: Why do learners care? What do they believe about numeracy and mathematics that makes it so meaningful for them?
Mathematical discourse: Questioning
How learners ask and respond to questions reveals much about the type of information they deem important within a numeracy class. And what learners deem as important reveals much about the goals they set for themselves, and goals, be they implicitly or explicitly defined, reveal much about the beliefs learners hold.
And that is why this particular finding about adult learners' questioning practice is so troubling.
First let me explain the differences between good questioning and poor questioning using the model provided by Huffard-Ackels, Fuson & Sherin (2004) called the ‘math-talk community’. To my mind this model is the most appropriate for the adult sector and has been used everywhere from young school students to University mathematical classes.
Good questioning in a numeracy class:
Student to student talk is student initiated, not dependent on the teacher. Students ask questions and listen to responses . Many questions are ‘why?’ questions that require justification from the person answering. Students repeat their own or others questions until satisfied with the answers.
Poor questioning in a numeracy class:
Teacher is the only questioner. Short frequent questions function to keep students listening and paying attention to the teacher. Students give short answers and respond to the teacher only. No student to student math talk.
These two descriptions act as book-ends on a continuum. There are two intermediate descriptions also that demonstrate how learners may transition as they move from the poor description to the good description.
The findings
The findings of my classroom observations reveals that the greater part of the questioning discourse resembles the ‘poor’ description – disturbingly so. This is not to say there were no exceptions. There were some really good examples of questioning – but not many. Overwhelmingly this is what I found:
- Tutors asked almost exclusively polar questions while teaching
- Learners responded to these questions with single word answers
- Learners only responded when certain of their answer (no one risks being wrong in public)
- On occasions when tutors asked learner to explain their thinking the learners still responded with single word answers (even when prompted).
- Learners rarely asked questions and when they did the questions were requests for answers, not for explanations, clarifications or justifications.
- Learners very rarely initiated their own ‘why?’questions and when they did it was to the tutor not each other.
The pattern is simple: Tutor asks a closed question to the class, a learner who knows the answer answers, and the tutor carries on - over and over and over again. This is replicated in group work environments with no tutor present. This pattern is known as – Initiate, respond, feedback (IRF).
So, the discourse within adult numeracy classrooms resembles that of a traditional transmission-based classroom. Given the picture above it kind of makes sense how this idea may have sneaked into our psyche. A description of this pattern has been attributed to Friere: ‘The pedagogy of the question’. In other words a classroom culture in which questions and answers are used to control the structure, discourse and content of the class. But, Friere is suggesting it is the teachers ‘fault’ for implementing (or conforming) to the structure. But things are very different in this sector because the adult learners are now apply pressure to the tutor to conform to this model. And my thinking is that this is because adult learners believe that this is how maths classes should operate and therefore both exert influence for the pattern and conform easily to it.
My answer: No.
Do the adult learners themselves contribute to this pattern?
My answer: Yes.
My answer: Yes.
Can we as tutors change this?
My answer: I hope so but I'm not sure how.
My answer: I hope so but I'm not sure how.
Thoughts?
In essence, learners do not believe that their own mathematical ideas have any value and hence do exert energy in producing or evaluating them, nor do they exert energy in listening to other learners’ ideas. However, research strongly suggests this is an essential process in order to develop conceptual knowledge of mathematics. Learners must generate their own mathematical ideas. However, getting them to do so requires changes to occur at a multitude of levels. The first and not least, is to change the belief that mathematical knowledge is only produced by experts and cannot be self-discovered, a belief held by many tutors.
One primary aim of my research has been to identify how beliefs may act as constraints or affordances to learning. Affordances are things that help us to achieve things. For example, door handles help us to open doors and therefore are affordances to opening doors. It's a word that doesn't see much sunlight outside of acadamia.
The term 'constraints' on the other hand is pretty easy to grasp. Constraints can also mean items designed to reduce movement and behaviour, and my use of the word is much the same but related to mental and physical behaviour.
My research now clearly demonstrates that a wide range of beliefs operate to constrain adult engagement in mathematics. These include epistemic beliefs and beliefs about mathematics, oneself, and one's relationship with mathematics.
New findings
My most recent finding identify that many (almost all) lower-achieving NZ learners dichotimise 'understanding'. That is they view understanding as an either/or state, a bit like a switch. Either something is understood, or it is not. Secondly, they believe that understanding is something that will happen in a single instant. This is often expressed by them as 'getting it'.
"Sometimes I just get it, but when I don't get it I stop trying".
By 'getting' it, they mean sudden understanding. They don't necessarily mean they will understand something on the first exposure but they do expect the content to just 'click' at some point.
Thirdly, lower-achieving learners believe understanding is transferred to them by an expert, in a 'content packet'. If they don't understand the first time, then repeated explanations of the content are needed. My participants only had one responses to this question -"What should someone do if they haven't understood what is being taught?"
Answer: Ask the teacher to repeat it.
Classic non-agentic responses. Contrast it with higher achieving learners answers, "Oh, go and study it at home, go on Youtube, hit the teachers up, work through the problems till you start to see patterns".
The harm...
How might these beliefs hurt low achieving learners? First let's quickly discuss how understanding really develops. There are multiple theories, but they all basically say that understanding is created by the individual as they struggle to make sense of and incorporate new information into existing understandings. New understandings are also thought to result from receiving information you do not understand, that challenges your existing understanding, causing you to have to reconfigure your existing ideas to accommodate the new ones (long sentence!).
In other words, new understandings require you to 'make sense' of information that you do not understand. It requires that you struggle and ultimately reconfigure your existing understandings. This takes time and effort. It is an entirely different process to hearing something and understanding it.
Do you see the problem?
This set of beliefs leads students to expect to understand information when they hear it. If they don't understand they think there is something wrong. Then they think they are dumb, and that others think they are dumb, so they disengage. If they are persistent and motivated they ask the teacher for help. By 'help', they mean repeating the information, again and again. If they still don't understand then they are out of options (often blaming the teacher or themselves for not being smart enough).
By expecting understanding to occur in response to an explanation learners take a very passive role in their education, expecting the packet of content to match their existing understandings.
The truth is that understanding is a process, not a state. It takes time, and it is constantly evolving. There is no 'either I understand' or 'I don't understand', there is only coming to a further understanding.
Summary
Beliefs are the foundation of the mind. They determine how you interpret the world and what you deem important, and as such are the most essential factor in all education.
Mathematical discourse
We were made to talk
When thinking about education I can't avoid working from this basic premise: We were made to talk and laugh and have fun. And learning was meant to be joyful and easy. Unfortunately things in the educational world don't always turn out this way.
I have spent the last few months analysing discourse patterns that occur between adults (learners and tutor) within numeracy lessons. The findings shed light on whether learners are "talking and having fun" in mathematically productive ways. In my mind, the findings have huge implications for the tertiary sector.
The word ‘discourse’ has several meanings relating to slightly different contexts and purposes. My use of the word really just means the type of talking/communication that occurs between students and the tutor in classrooms. I have analysed particular features that have emerged as important.
Setting the scene
There is strong evidence that the quality of learner discussions has huge implications for how learners engage in, and consequently develop, mathematical knowledge.
In other words: The type of discourse that occurs in numeracy classrooms COUNTS.
So, in adult numeracy classes do the discourse patterns enhance learning, or constrain and inhibit learning? Well, unfortunately the discourse patterns as they relate to mathematics are particularly poor. There are four areas of specific concern. I’ll post the first of the four below with the other three to follow in a further post.
The source of mathematical ideas
The learners in my study believe that the tutor, and associated resources, is the only valid source of mathematical ideas. They view the tutor as the expert who possesses the right type of mathematical knowledge. That is, the math knowledge that is generated, owned and passed down by established academic institutions (and official mathematicians). Because of this belief, learners see little value in generating their own ideas, sharing their ideas with others, or listening to other learners ideas. Learners do not see group problem solving as a learning opportunity and require the tutor to verify any and all answers before they celebrate success (note that whether an answer is correct or incorrect should be self-evident if learners understand the problem and answer). Keep in mind that learners may be motivated to group problem solve due to the social aspect and a sense of fun. They will still engage but not really believe that they will learn what they need from the experience.
The learners in my study believe that the tutor, and associated resources, is the only valid source of mathematical ideas. They view the tutor as the expert who possesses the right type of mathematical knowledge. That is, the math knowledge that is generated, owned and passed down by established academic institutions (and official mathematicians). Because of this belief, learners see little value in generating their own ideas, sharing their ideas with others, or listening to other learners ideas. Learners do not see group problem solving as a learning opportunity and require the tutor to verify any and all answers before they celebrate success (note that whether an answer is correct or incorrect should be self-evident if learners understand the problem and answer). Keep in mind that learners may be motivated to group problem solve due to the social aspect and a sense of fun. They will still engage but not really believe that they will learn what they need from the experience.
The discourse patterns during group work were dominated by learners who ‘knew’ the correct method (taught to them by previous teachers and therefore authentic) and simply told the group how to solve the problem. The group did not discuss ideas such as solution options, alternate interpretations or alternate solution strategies. Also, once the tutor began to work through the problems the learners simply used the tutor’s ideas. Learners very rarely spoke up and gave a new or unique method of solving or thinking about a task. The one exception was ‘numeracy experts’, that is, those learners who had high numeracy skills and wished to show their proficiency to the class.
New findings regarding the relationship between beliefs and adult engagement in mathematics/numeracy
One primary aim of my research has been to identify how beliefs may act as constraints or affordances to learning. Affordances are things that help us to achieve things. For example, door handles help us to open doors and therefore are affordances to opening doors. It's a word that doesn't see much sunlight outside of acadamia.The term 'constraints' on the other hand is pretty easy to grasp. Constraints can also mean items designed to reduce movement and behaviour, and my use of the word is much the same but related to mental and physical behaviour.
My research now clearly demonstrates that a wide range of beliefs operate to constrain adult engagement in mathematics. These include epistemic beliefs and beliefs about mathematics, oneself, and one's relationship with mathematics.
New findings
My most recent finding identify that many (almost all) lower-achieving NZ learners dichotimise 'understanding'. That is they view understanding as an either/or state, a bit like a switch. Either something is understood, or it is not. Secondly, they believe that understanding is something that will happen in a single instant. This is often expressed by them as 'getting it'.
"Sometimes I just get it, but when I don't get it I stop trying".
By 'getting' it, they mean sudden understanding. They don't necessarily mean they will understand something on the first exposure but they do expect the content to just 'click' at some point.
Thirdly, lower-achieving learners believe understanding is transferred to them by an expert, in a 'content packet'. If they don't understand the first time, then repeated explanations of the content are needed. My participants only had one responses to this question -"What should someone do if they haven't understood what is being taught?"
Answer: Ask the teacher to repeat it.
Classic non-agentic responses. Contrast it with higher achieving learners answers, "Oh, go and study it at home, go on Youtube, hit the teachers up, work through the problems till you start to see patterns".
The harm...
How might these beliefs hurt low achieving learners? First let's quickly discuss how understanding really develops. There are multiple theories, but they all basically say that understanding is created by the individual as they struggle to make sense of and incorporate new information into existing understandings. New understandings are also thought to result from receiving information you do not understand, that challenges your existing understanding, causing you to have to reconfigure your existing ideas to accommodate the new ones (long sentence!).
In other words, new understandings require you to 'make sense' of information that you do not understand. It requires that you struggle and ultimately reconfigure your existing understandings. This takes time and effort. It is an entirely different process to hearing something and understanding it.
Do you see the problem?
This set of beliefs leads students to expect to understand information when they hear it. If they don't understand they think there is something wrong. Then they think they are dumb, and that others think they are dumb, so they disengage. If they are persistent and motivated they ask the teacher for help. By 'help', they mean repeating the information, again and again. If they still don't understand then they are out of options (often blaming the teacher or themselves for not being smart enough).
By expecting understanding to occur in response to an explanation learners take a very passive role in their education, expecting the packet of content to match their existing understandings.
The truth is that understanding is a process, not a state. It takes time, and it is constantly evolving. There is no 'either I understand' or 'I don't understand', there is only coming to a further understanding.
Summary
You are in trouble if you believe any of the following:
- Learning is about understanding what you hear at the time, and then applying it.
- Not understanding what you hear, might mean you are not as smart as others who can.
- If you don't understand the content you must get the teacher to explain it to you again so that you can understand.
If you answered yes to more than one, then you have either been very lucky, or have some issues around education and learning. Yes, the first one is what our 'training' sector is based on but has quite different goals than developing understanding.
All of my low-achieving participants believe these three, and all my high-achieving participants don't. That should get people thinking.
