2019 Learning Guide HMMA032 PDF

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Summary

This is a learning guide for method of mathematics in general for varsity students....

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UNIVERSITY OF LIMPOPO Turfloop Campus School of Education Department of Mathematics, Science and Technology Education Name of Programme: Bachelor of Education – Senior Phase and FET (BEdSPF) National Qualification Level (NQF Level): 6 Module Name: Method Mathematics for Educators 3B Module Code: HMMA032 Credits: 6 Contact Dates: July - November 2019

Facilitators:

Mutodi P, Muthelo DJ & Maphutha BK

Ba c h e l o ro fEd u c a t i o nSe ni o rPha s ea n dF u r t h e rEd u c a t i o na n dT r a i n i n g( B. Ed. SPF ) LEVEL 3 Mo d ul eNa me : Mo d ul eCo de : Cr e d i t s : Pr e r e q u i s i t e s : Pe r i o d sp e rwe e k Du r a t i o n: L e c t u r e spe rwe e k : Pr a c t i c a l p e rmo d ul e : T u t o r i a l sp e rmo d ul e :

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MET HODMA THEMA T I CSF OREDUCA T ORS3 B HMMA0 3 2 6 HMMA0 2 1 , HMMA0 2 2 4 1 6 4p e r i o d sx4 5mi n u t e s NI L NI L Co n t e n t( s e me s t e r2 )

F a c i l i t a t i o na n dAs s e s s me n t i nma t h e ma t i c s( t h r o u g ha c t i v i t yb a s e dl e a r n i n g ) Po r t f o l i oa saf o r mo f a s s e s s me n t . Cr i t e r i o na n dn o r mr e f e r e n c e di n t e r p r e t a t i o no f s c o r e s .

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Pr e p a r a t i o no f t e s t i t e msa n dme mo r a n d u m. I n t r o d u c t i o nt oe d u c a t i o n a l r e s e a r c h( r e fl e c t i v ep r a c t i c ea n da c t i o nr e s e a r c h ) . Co mp u t e r sa n do t h e rl e a r n i n ga i d si nma t h e ma t i c s( o t h e ri n s t r u c t i o n a la i d swh i c ha r e u s e f u li nt h ec l a s s r o o mi n c l u d eo v e r h e a dp r o j e c t o r ,VCR a n dmo n i t o r ,s l i d ep r o j e c t o r , s c i e n t i fi cc a l c u l a t o r s ) . As s e s s me n t

As s e s s me n t wi l l b ed o n eu s i n gb o t hc o n t i n u o u sa n ds u mma t i v ea s s e s s me n t ( fi n a l e x a mi n a t i o n ) . Co n t i n u o u sa s s e s s me n t ma r kwi l l b eo b t a i n e dt h r o u g ho r a l p r e s e n t a t i o n ss u c ha sd e b a t e so r i n d i v i d u a l p r e s e n t a t i o n , a s s i g n me n t s , p r o j e c t s , q u i z z e s , e t c e t e r a . Av a r i e t yo f a s s e s s me n t s t r a t e g i e s( g r o u p , p e e ra n ds e l f )wi l l b eu s e d . T h efi n a l ma r kf o rt h emo d u l ewi l l b eo b t a i n e df r o ma c o mp r e h e n s i v ewr i t t e ne x a mi n a t i o n( 4 0 %) a n dt h ec o n t i n u o u sa s s e s s me n t ( 6 0 %) . Pr e s e n t e r s :DrPMu t o d i , MrDMu t h e l o&Mr sBKMa p h ut h a

ACTIVITY 1: ACTION RESEARCH

1.

Read through Bryan Meyer’s proposal (attached) to answer the questions that follow.

1.1.

Suggest a title for Meyer’s proposed study.

1.2.

Briefly discuss what prompted Meyer’s proposed study – the rationale.

1.3.

Identify the three sentences that Meyer uses to highlight the problem statement.

1.4.

Write the purpose statement for Meyer’s proposed study.

1.5.

What is/are the research question/s in the proposed study?

1.6.

Suggest the significance of Meyer’s proposed study.

1.7.

In your own words explain what Meyer possibly means by mathematical agency?

1.8.

Identify and explain the research approach that Meyer proposes to use in his study.

1.9.

Identify and explain the research design that Meyer proposes to use in his study.

1.10.

What was the proposed intention of the data collection methods for this study?

1.11.

Give examples of questions used in the survey.

1.12.

Why did Meyer propose to use blogs?

1.13.

Use Meyer’s writing to explain what focus group interviews are.

1.14.

Explain how sampling for focus group interviews was proposed.

ACTION RESEARCH PROPORSAL BY BRYAN MEYER I will never forget my first week of teaching. Fresh out of my credentialing program, I was placed as an 11th Grade Mathematics teacher. I was determined to carry out the complex instruction that had intrigued me in my program. I envisioned students working collaboratively, creatively solving problems, and engaging in critical thinking. On the third day of class, I gave the students the following prompt: “In the NFL, teams score 7 points for a touchdown and 3 points for a field goal. Given only these two opportunities for points, are there any final game scores that are impossible to obtain? If so, is there a highest impossible score?” A couple minutes later about a third of the class started experimenting, but the others…still nothing. I walked over to a few students who were sitting idly and asked, “What are you thinking about?” They replied, “We don’t know what to do. “I was shocked. The only skill necessary to “access” this problem is the ability to add numbers together, which all of the 11th graders in my class could do. Yet, still, they sat waiting for someone (probably me) to tell them exactly what to do and precisely how to think. People are born with an innate curiosity and a desire/need to make sense of their world through their experiences and interactions with it (Glasersfeld, 1995). We naturally experiment, conjecture, visualize, look for patterns, and attempt to communicate; in short, we are all born mathematical and there is significant evidence to suggest that young children develop mathematical structures even before attending school (Resnick,1987). The typical transmission of this “mathematics” usually follows a similar process. The teacher stands at the front of the classroom and covers some new idea or topic for the day while the students take notes. Then the class does a few problems together with the instructor to make sure that the students have successfully adopted the teacher’s procedure for “solving” these types of problems. After that has been accomplished, the instructor assigns a set of problems, all relatively identical, in which students practice the same procedure over and over until they have reached mastery. It should not be surprising, then, that students are puzzled when explicit instructions about how to proceed have not been provided. As Alan Schoenfeld describes, “When mathematics is taught as received knowledge rather than as something that (a) should fit together meaningfully, and (b) should be shared, students neither try to use it for sense-making nor develop a means of communicating with it. They have little idea, much less confidence, that they can serve as arbiters of mathematical correctness, either individually or collectively. Indeed, for most students, arguments (or proposed solutions) are merely proposed by themselves. Those arguments are then judged by experts who determine their correctness. Authority and the means of implementing it are external to the students” (1994, p. 57). Students have learned that they are observers to, rather than participants in, mathematics and that experts must be consulted about how to solve a problem and how to ratify its solution. The vision of a student sitting helpless as they face an uncertain problem has surfaced many times since that third day of my career in the teaching profession. I knew then, and still know now, that this is a problem in our educational

system that needs attention. As teachers, we should be instilling in our students powers of inquiry, confidence and creativity in problem solving, and a love of learning. Certainly, there needs to be a shift in how we define mathematics and in our vision of what the teaching and learning of that mathematics looks like in schools. It led me to wonder, “How does a mathematics classroom centered in habits of mind support students’ mathematical agency?” I want my students to know that we are all mathematical. I want them to recognize their own mathematical thinking and habits. I want them to view themselves as mathematicians, people who view the world with a mathematical lens and make sense of problems in the best way they can. I want them to have confidence in their own thinking. A Note About My Research Methods I have chosen to take a very qualitative approach to my action research methods and data collection. Most of my research has involved me looking at my classroom and teaching practice through the lens of student identity and agency and in looking for “critical observations” where moments of agency (or lack thereof) seem to somehow be occasioned by the conditions of the mathematics classroom. Of course, my perspective influences my interpretation of this data, but even the analysis of quantitative data could be no other way. I should also note that I taught two different “sections” during this research period – three classes of 10 th Grade Integrated Mathematics and one class of 12th Grade Mathematical Thinking. The tenth grade course had the expectation of being rooted in standards-based mathematics while the twelfth grade course was open to almost any form of mathematical experimentation and play. While my intent at the beginning of the research period was to look solely at the tenth grade class, it turned out that the twelfth grade course provided a unique and important opportunity to investigate the way identity and agency is impacted in the classroom. As a result, I have drawn student quotes, work samples, and reflections from both classes as necessary to fully investigate agency and mathematical habits of mind. WHOLE CLASS METHODS The methods described in this section were not necessarily intended to document changes in attitude, identity, and agency that were occasioned or caused by my work with students. Although some of that is present, my methods were mostly to provide a broad sample of student opinion and reflection. Student Survey Students enter our classrooms each year with preconceived notions about definitions of mathematics and about their identity as mathematicians. It was important to capture their thoughts about these things at the beginning of the year and at the end of the research period so that I could see how their thoughts were changing over time. I conducted a beginning of the year survey (see Appendix A) that asked students questions about the nature of mathematics and about how they perceive themselves as mathematical thinkers. Questions such as the ones below are indicative of the overall theme of the survey:

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It is important to memorize things in mathematics.

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I can create mathematical ideas, formulas and rules.

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I have to know a procedure for a problem before I can try it on my own

Students were asked to rate their agreement with particular statements on a discrete sliding scale (strongly agree, agree, disagree, strongly disagree). I summarized their responses from the beginning of the year survey using bar graphs to visualize responses. I gave the exact same survey near the end of the research period. By summarizing

responses in the same way as before, I was able to identify areas of significant change. With respect to those areas, I asked students specific questions to try and understand more about the elements of their experience that were related to their shift in thinking. Habits of a Mathematician Blog Towards the beginning of the year, my classes worked on a variety of rich, open-ended tasks that allowed them to engage in the activity of “doing mathematics.” Upon reflecting on their work together with those tasks, they identified some key moves that had allowed them engage more deeply with the task. With some guidance, they constructed the “habits of a mathematician.” Shortly after that, each student set up a blog that they would use to complete weekly reflections regarding evidence of the habits in their work. Each week, students picked one piece of mathematics that they were most proud of and completed a blog post in which they identified a specific habit that was crucial to their work. They answered the questions: 

Describe the task/activity that you were working on.2.

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How are the “habits of a mathematician” present in your work?

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Give specific examples.3.

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Were the habits helpful in making sense of the task? If yes, how?

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If not, how would you revise your use of the habits so that they were helpful?

These blogs allowed me to capture lots of student work and to read their thinking about how these habits were helping them identify their own mathematical thinking. Over time, they also began to blog about things that happened outside of our class in which they found themselves using the same mathematical thinking. FOCUS GROUP METHODS Much of my data and analysis came from focus group discussions with students from my classes. Although the focus groups did not always involve the same students, the groups were chosen to represent variety in student ethnicity, social status, traditional mathematics achievement level, and gender. Group Interviews Most of the focus group interviews that I conducted involved between 5 and 15 students. Most often, I would witness a powerful moment in the classroom surrounding students’ mathematical agency, identity, and authority and I would assemble a focus group to learn more about that moment or theme. Over time, I conducted focus groups on:

task design-



unit design-



habits of a mathematician-



grading and assessment-



collaborative group work

I annotated as students talked amongst each other and recorded their conversations so that I could go back and pull out direct quotes. Mostly, I would look for quotes that I believed spoke to their definitions of mathematics, their own

identity in relation to mathematics, times when they clearly saw their own thinking in the mathematics of the classroom, and social/group work issues. All of these themes I felt had significance to an investigation of mathematical agency.

Work Samples As an on-going part of my data collection and analysis, I would collect student work that illustrated some of the themes that emerged from our focus group discussions. Mostly, I would look for examples of student work that demonstrated their authoring of mathematics, work that showed them pursuing their own ways of thinking, and work that students felt particularly proud of. The analysis was not to prove causation or correlation, but rather to document moments of student agency and authorship in action and to find out more from them about how they experienced the task.

2. Read through Lisa’s proposal (attached) to answer the questions that follow. 2.1.

What is action research?

2.2.

What are the stages of action research process?

2.3.

Why is it important for teachers to investigate their work?

2.4.

What is the problem that made Lisa’s proposal necessary?

2.5.

What is the purpose of the proposed action research by Lisa?

2.6.

Which methods were employed by researchers cited in Lisa’s proposal to improve students’ mathematical skills?

2.7.

How will Lisa’s proposed study be useful to student?

2.8.

What findings did Lisa make from the literature review?

2.9.

Who was proposed to participate in Lisa’s action research?

2.10.

Which research question(s) did Lisa propose to answer through her action research?

2.11.

Which data collection methods did Lisa propose to use?

2.12.

What procedures/plans were to be followed by Lisa for each of these methods?

2.13.

How will the proposed study help Lisa improve the basic math skills of her students?

Action Research Proposal - Lisa Booth http://lisabooth.weebly.com/uploads/1/1/9/7/1197525/action_research_proposal.pdf

The purpose of this action research is to improve the basic math skills of algebra I students at North Panola High School. One of the fundamental pieces of algebra is taking a word problem and translating it to an algebraic expression. In order to successfully complete this, students must be able to recognize and use basic mathematical terms (Maccini, 2000). In addition to writing algebraic expressions, students must be able to solve these problems. This requires knowledge of how to perform basic mathematical operations (Maccini, 2000). I have found that many of my students are lacking in the ability to perform basic mathematical operations. They struggle with simple multiplication facts, as well as working with positive and negative numbers. These are necessary skills to have in order to succeed in any higher level math class, including algebra I. Beginning teachers, myself included, are faced with the struggle of teaching students with varying levels of ability all at the same time. The biggest challenge is helping those students of lower ability master grade level content knowledge when they continue to lack basic skills. To do this, teachers must fill in whatever gaps students have in their basic knowledge. The teacher must fill in those fundamental pieces of algebra that each student is lacking. Much research has been done on methods to improve, or compensate for, a lack of basic skills (Carroll, 1994; Maccini & Hughes, 2000; Mevarech & Kramarski, 2003; Sweller & Cooper, 1985; Sweller, et al, 2001; Ward & Sweller, 1990). These methods include using worked examples, (Carroll, 1994; Sweller & Cooper, 1985; Ward & Sweller, 1990; Sweller, et al, 2001) problem-solving strategies, (Maccini & Hughes, 2000; Mevarech & Kramarski, 2003; Sweller, et al, 2001) and peer tutoring (Allsopp, 1997). Finding a way to improve the basic skills of students will allow students to succeed in algebra specifically, as well as all math courses. While much research has been done, there is a lack of research in rural, high poverty areas. This study is relevant for stakeholders in my individual classroom, including students, parents, and myself, as well as other algebra teachers who are struggling with a lack of basic skills in their students. The first study (Sweller & Cooper, 1985) investigated the use of worked examples as a way to develop mental schema which are required to categorize and more easily solve problems. The study used year 9 students at a Sydney high school. All students were given two worked example problems to study and were then allowed to ask questions. The students were divided into two groups, one group worked on conventional practice problems, while the second group was given the same worksheet with every other problem worked for them. The results showed that students in the worked example group spent less time in the acquisition phase and made fewer errors than the conventional group. Carroll (1994) took Sweller and Cooper's (1985) findings and applied it to a Midwestern, United States city. Students were given a pretest to pair them based on ability and then were divided into a conventional practice group and a worked example group. After instruction and practice (either conventional or worked examples), a post test was given. The results found that the worked example group did significantly better on the post-test compared to the conventional practice group. These studies raised the question, how do you write a beneficial worked example? Sweller and Ward (1990) performed a study about what makes a worked example effective. In this study, two year 10 mathematics classes in Sydney were used. The students were taught two topics, one using conventional practice and the other using worked examples. The results showed that worked examples are most effective when students do not have to split their attention between and integrate two or more sources of information. Sweller and Ward ...