Teachers who truly care about promoting inclusive relationships also promote mathematical thinking and reasoning. Research, however, documents many cases e. Subscribing to a proficiency agenda, teachers in this study believed that teaching should not offer students simplified tasks, but should challenge them and provide support for them to task risks.

Such an approach involves significantly more than developing a respectful, trusting and non- threatening climate for discussion and problem solving. It involves socialising students into a larger mathematical world that honours standards of reasoning and rules of practice Popkewitz, What was particularly effective was the way the teacher sustained the discussions. They shared and then transferred responsibility so that students could attain greater autonomy.

However, the specialized language of mathematics can be problematic for learners. Particular words, grammar, and vocabulary used in school mathematics can hinder access to the meaning sought and the objective for a given lesson.

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Words, phrases, and terms can take on completely different meanings from those that they have in the everyday context. In particular, mathematical language presents certain tensions in multilingual classrooms. Neville-Barton and Barton looked at these tensions as experienced by Chinese Mandarin-speaking students in New Zealand schools. Their investigation focused on the difficulties that could be attributable to limited proficiency with the English language.

It also sought to identify language features that might create difficulties for students. Two tests were administered, seven weeks apart. There was a noticeable difference in their performances on the two versions. What created problems for them was the syntax of mathematical discourse. In particular, prepositions, word order, and interpretation of difficulties arising out of the contexts. Vocabulary did not appear to disadvantage the students to the same extent. Importantly, Neville-Barton and Barton found that the teachers of the students in their study had not been aware of some of the student misunderstandings.

Word problems involving mathematical implication and logical structures such as conditionals and negation were a particular issue for students from senior mathematics classes. They also found technical vocabulary, rather than general vocabulary, to be problematic. Latu noted that English words are sometimes phonetically translated into Pasifika languages to express mathematical ideas when no suitable vocabulary is available in the home language.

The same point was made by Fasi in his study with Tongan students. Students, other than those from multilingual backgrounds, also have difficulties with mathematical language. Sullivan, Zevenbergen, and Mousley found that students with a familiarity of standard English usually students from middle-class homes had greater access to school mathematics.

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Competency with mathematical language involves more than technical vocabulary. It also encompasses the way it is used within mathematical argumentation. They point out that this instructional process depends upon the skilful orchestration of classroom discussion by the teacher. Yackel and Cobb reported from their research that building that understanding requires the teacher to first construct the norms for what constitutes a mathematically acceptable, different, sophisticated, efficient, or elegant explanation. Fraivillig and colleagues observed teachers who did more than sustain discussion—they moved conversations in mathematically enriching ways, they clarified mathematical conventions and they arbitrated between competing conjectures.

In short, they picked up on the critical moments in discursive interactions and took learning forward. In another study, Stein, Grover, and Henningsen report on the importance of a sustained press from the teacher for justifications, explanations, and meaning. In many cases, a press for understanding resulted in successive presentations that illustrated multiple ways of approaching a problem. The press for understanding is an aspect of quality mathematics pedagogical practice highlighted by many researchers. When does that make sense?

Franke and Kazemi make the important claim that an effective teacher tries to delve into the minds of students by noticing and listening carefully to what students have to say. Yackel, Cobb, and Wood provide evidence to substantiate the claim. Numerous studies of classroom discourse highlight the importance of teacher knowledge. The findings of studies undertaken by Ball and Bass and many others e. The teacher must make good sense of the mathematics involved to help move students towards more sophisticated and mathematically grounded Fraivillig et al.

The selection of quality instructional tasks is critical. Too often tasks can slip into being busy or fun type activities. Instances of displaced learning are also evidenced in studies of group activities—co-operative tasks or mathematical field trips for example—that have been insufficiently structured to engage students with mathematical ideas e. Teacher attempts to make mathematics interesting appeared to be at the expense of accuracy and meaning.

In contrast, students who engage in meaningful mathematical tasks are potentially able to treat tasks as problematic. In accord with findings about effective teaching in the primary context e. Large-scale empirical studies of educational change in the U. For example, Watters, English, and Mahoney demonstrated how use of extended modelling problems provided opportunities for learners to engage in a range of mathematical processes and develop mathematical understanding. Because the modelling activities in the study were designed for small-group work, they also provided opportunities for developing collaborative problem-solving skills and important metacognitive skills that enabled students to distinguish between personal and task knowledge and to know when and how to apply each during problem solution.

Other studies e. Situating tasks in contexts—be they real or imaginary settings—can provide a learning situation that is experientially real for students Gravemeijer, ; van den Heuvel- Panhuizen, When contextualizing tasks, however, researchers e. Researchers e. Quality tasks need to present suitable levels of challenge if the learner is to gain a sense of control and develop valuable mathematical learning and dispositions. Mathematical tasks that are problematic and offer an appropriate degree of challenge have high cognitive value. Houssart found that teachers of higher-streamed classes showed more enthusiasm for investigative tasks that encouraged creativity: There has to be an element of challenge about it Graham Challenge, especially with the top set Probably, had I had the lower set, the challenge bit would … be far lower down, until they got the basics in obviously.

Watson and Watson and De Geest provide evidence of enhanced instructional practices that support the mathematical thinking of students previously identified as low attainers. Based on their belief that these students were entitled to access mathematics, teachers in the Improving Attainment in Mathematics Project chose not to simplify mathematical activities. Working with students identified as learning disabled, Behrend and Thornton, Langrall, and Jones found that, given the opportunity, these students successfully engaged with rich and meaningful problem tasks.

Task challenge is also crucial for academically gifted students. Diezmann and Watters report on teachers who successfully increased challenge by way of task problematisation.

Without changing the mathematical focus, a task can be problematised by methods such as inserting obstacles to the solutions, removing some information, or requiring students to use particular representations or develop generalisations. Diezmann and Watters found that problematizing, adapting, and enriching regular curriculum tasks provided underachieving gifted students with the opportunity to oscillate between regular activities and more challenging activities according to their capability, confidence, and motivation. Designing quality tasks is not the end of the matter. Without effective pedagogy we know that high quality tasks can fail to achieve their desired purpose.

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Stein et al. Frequently teachers took the student through the chain of reasoning and students merely filled in the gaps with the arithmetical answer, or low-level recall of facts.

This factor was frequently accompanied by the modelling of competent performance by the teacher or by a capable student—often in the format of a class presentation of a solution. Used effectively, tools and representations—artefacts—offer spaces to help organise mathematical thinking Askew, Numerous studies e. In mathematics the opportunity to access non-linguistic representation is particularly important; inscriptions in the form of notations, graphical, pictorial, tabular, and geometric representations abound.

Ball and Lampert , among others, have found that effective teachers select and construct artefacts that their students can relate to and have the intellectual resources to make sense of. In challenging an over-reliance on adult contrived equipment researchers contend that representational contexts need to be real or at least imaginable; be varied; relate to real problems to solve; be sensitive to cultural, gender and racial norms and not exclude any group of students; and allow the making of models Sullivan et al.

Authentic situations that use artefacts to provide a bridge between the mathematics and the situation can occasion effective learning experiences. Students were observed to extend, adapt, and revise mathematical ideas.

## Extending the conception of mathematics teaching

They readily established their own sense of authenticity by aligning the problem with their personal experiences and understandings. Whilst research see Thornton et al. In three of the five classrooms, manipulatives became the focus rather than a means for thinking about mathematical ideas. A distinctive feature of instruction for those teachers who engaged target students in mathematical thinking was the way they used a variety of representations of a concept prior to the use of the manipulative specified in the curriculum. For example, in a geometry lesson, parallel lines were represented by a range of arm movements, lengths of string were used to create angles, calculators were used, and finally representations were transferred to geoboards.

The mathematical textbook, together with the worked example, are examples of often taken-for-granted tools Goos, A group of research studies point to ways that effective teachers make use of these tools. Discursive practices of mathematical inquiry, we have seen earlier, are a hallmark of effective pedagogic practice. Tools provide an effective way for students to communicate their thinking. For example, Hatano and Inagaki describe an instructional episode involving first grade children.

Students familiar with join—separate problems were presented with the problem: There are 12 boys and 8 girls. How many more boys than girls are there? Most of the children answered correctly, but one child insisted that subtraction could not be used because it was impossible to subtract girls from boys. None of the students who had answered correctly was able to argue persuasively against this assertion.

It was only after the students physically modelled the situation that they realised that finding the difference was a matter of subtracting the 8 boys who could hold hands with girls from the 12 boys. In recent times, there has been intensive interest in research that links learner outcomes with pedagogies that utilize new technologies. Studies have shown that technological tools, like other conceptual mediators, can act as catalysts for classroom collaboration, independent enquiry, shared knowledge, and mathematical engagement. Likewise, Goos, Renshaw, Galbraith, and Geiger provided evidence that the graphics calculator can be a catalyst for discursive interactions focused on mathematical thinking that simultaneously support personal small-group and public whole-class knowledge production.

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Providing opportunities for mathematical exploration, technological tools can also serve to increase the relevance and accessibility of mathematical practices for learners. Nason, Woodruff, and Lesh report on a study in which groups of students developed spreadsheet models to record quality of life in a number of Canadian cities. As a result of public and critical scrutiny of their ideas, the students learned about mathematical efficiency and organising information for presentation.

The computer became a mediator not only for building personal knowledge but also for the development of learning at the interpersonal level. It did this by occasioning interactions within and between student groups in the classroom. Yelland also noted the impact of ICT on the community of young learners. For example, Pierce, Herbert, and Giri found that where teachers continued to privilege the high value of done-by-hand algebraic manipulations, students more likely perceived that CAS offered insufficient advantages over a graphics calculator to warrant the time and cognitive effort required to become effective users of this new technology.

Ball and Stacey suggest that teachers should share decision making about technology-based approaches with their classes and have the students monitor their own underuse or overuse of technology. These researchers argued that the use of CAS technology needs to be accompanied by the development of algebraic insight.

When students see an algebraic expression, they should think about what they already know about the symbols used, the structure and key features of the expression, and possibly its graph before they move further into the question.