EDMA310/360 Mathematics: Learning and Teaching Mathematics 2, 2014 – Assignment 1 – Template 1 of 3

Rational Number Assessment

[Jade Flanagan S00129143]

Australian Catholic University Teacher report on your student’s Rational Number Knowledge and any misconceptions (300 words) Grace has a sound understanding of moving from the whole to part fractions but struggles moving from the part fraction to a whole. This is largely due to her not recognising the amount that the part represents. For example, when shown 2/3 of a square and asked to draw the whole she drew 2 extra squares. While Grace understood that the square represented only part of the shape she was unaware she needed to only draw the other half of the shape, hence showing gaps in her proportional reasoning. Grace was able to use a benchmark to identify the larger fraction but was only comfortable with using ½ and was unfamiliar with ¼ or 1/3. This could be one of the reasons she was unable to convert bench mark fractions into decimals and percentages. Writing ¼ as 0.4 only recognising the denominator as the decimal equivalent. Grace was comfortable using circular area models but struggled when confronted with a rectangle not knowing if it was the whole and what it represented. In particular with a grid of 100, Grace struggled to recognise that when 6 out of 100 squares were coloured in it equates to 6/100 this again relays back to her lack of understanding around connecting decimals, fractions and percentages. Grace found the 100 squares overwhelming rather then viewing it as a percentage benchmark. Grace was incredibly comfortable with the addition of fractions and understood the use of the denominator and numerator and how to add them to create a fraction close to one. When comparing fractions, Grace also understood that 6ths were bigger then 8ths and but seemed unaware that a fraction is also division. Addition was the only operation she was comfortable with as she had the common misconception that multiplying fractions and decimals would always result in a greater number than division, showing she lacked knowledge of the structure of a fraction . Word count: 317 Critical evaluation of the usefulness of mathematics interviews for gaining knowledge about students’ current mathematical knowledge that can be used to plan future learning opportunities. Be sure to draw on relevant research literature to support your evaluation. (400 words) Mathematical interviews are an integral part of mathematical teaching. It allows for teachers to determine which students need challenging or scaffolding and arguably more importantly whether students are responding appropriately to teaching strategies (Gates 2002). Mathematical interviews allow teachers to gain insight into their student’s mathematical thinking, ability and their strengths and weaknesses. The information gathered from mathematical interviews can be used to guide and redirect future teaching instruction, methods and strategies. This can divulge common methods and misconceptions based on a holistic analysis of the class’s success and failures during the interview.

EDMA310/360 Mathematics: Learning and Teaching Mathematics 2, 2014 – Assignment 1 – Template 1 of 3

Teachers can however incorporate their informal judgements into their formal and summative assessments, it must be taken into consideration that not all students will perform to their greatest ability on the day of the assessment (Gates 2002) thus it is vital that teachers recognise the critical and analytical thinking behind students answers rather then just the answer itself (---). Mathematical interviews are constructed using higher order questions that require students to not only recall information and facts but also techniques and strategies to solve a problem (Way 2008). Assessment questions “allow the teacher to see how the children are thinking, what they understand and at what level they are operating” (Way 2008 pg 23). The very nature of a mathematical interviews allows the teacher to decipher whether students are working from “procedural knowledge or conceptual knowledge” ( Van De Walle 2004 pg90) and therefore if they have a sound, sophisticated understanding of the topic or are relying on inflexible process without understanding the method. This enhances professional judgement for subsequent instructional decisions ( pg 90 Van de Walle 2006) and permits both the student and teacher to reflect and assess their mathematical choices and techniques and processes. Mathematical Interviews post topic allows teachers to assess whether students have “accumulated debts of knowledge” (Daro, Mosher , Cororan 2011 p,48) This is possibly one of the most vital points that come from mathematical interviews, when students are left with holes in their maths foundational knowledge it becomes very difficult for them to form comprehensive understandings in other maths topics. Therefore the teacher is able to identify students that need more scaffolding in the area before moving on. While mathematical interviews have multiple positives it is important to note that their very nature doesn’t allow for ‘on the spot teaching’. When seeing a student struggle or misunderstand a concept a teacher’s instinct is to scaffold to the student during a perfect learning opportunity; interviews however prevent any teaching to occur and prime one on one learning opportunities are lost. It is very rare for teachers to get one on one time with students particularly for such an extended amount of time. This highlights how time consuming interviews are. They require the teacher to be away from the whole class for a significant amount of time and the time is spent learning about the student’s ability and not scaffolding or challenging them as they may require. It is, however, critical to remember that is it only through the interviews that teachers become aware of what scaffolding and challenges are required for students to become capable mathematicians. Words: 432 REFERENCES: Way, J. (2008). Using questioning to stimulate mathematical thinking. Australian Primary Mathematics Classroom. Gates, P. (2002). Issues in Mathematical Teaching. Australia: Taylor and Francis.

McDonough A, Clarke D (2002). DESCRIBING THE PRACTICE OF EFFECTIVE TEACHERS OF MATHEMATICS IN THE EARLY YEARS. Australian Catholic University. Daro, P Mosher, F, Cororan T (2011).Learning tragectories in mathematics. A foundation for standards, curriculum assessment instruction. Philadelphia, PA: Consortium for policy research in education. John. A Van De Walle Karen D Karp Jennifer M, Bay- Williams (2004). Elementary and middle school Mathematics. Teaching Mathematically.

EDMA310/360 Mathematics: Learning and Teaching Mathematics 2, 2014 – Assignment 1 – Template 1 of 3

Critical evaluation of the usefulness of Open Tasks with Rubrics for gaining knowledge about students’ current mathematical knowledge that can be used to plan future learning opportunities. Be sure to draw on relevant research literature to support your evaluation. (400 words) Open tasks allow students to showcase their knowledge and mathematical proficiency regardless of their ability level thus all students feel proficient in completing the task. This allows for critiquing the way students approach and complete the task rather then which tasks they are able to complete. Open tasks allow students to demonstrate their understanding without the constraints of peer comparison as there are multiple ways of solving the problems. (Ferguson 2008). Consequently open task rubrics often focus largely on process and understanding of the task rather than necessarily getting the correct answer. It is because of this, open tasks require a customised rubric outlining the task and teacher’s expectations of the students. Teachers are therefore able to put more weight on certain dimensions of the task they deem to be more imperative to the topic. (Maggie, McGatha , Darcy 2010) Rubrics also allow students to learn what teacher’s value and show students what to strive for (Ferguson 2008). This can be argued that students work towards the teacher’s expectations rather then explore their own mathematical ideas. This, however , can also guide students towards placing more emphasis on the process rather than the outcome.

Mathematics often has a common misconception that the right answer is more important than thinking progressions. Hence open task rubrics allow for a ‘grey area’ that grading does not. (Bush,Williams S. Leinwand 2000). Open task rubrics acknowledge a student’s process, mathematical thinking and use of strategies regardless of whether the correct response was achieved (Olsen 2004). Teachers therefore are able to better monitor student’s understanding of the process and strategies required for a certain topic, if an entire class scored low in an open task rubric this data can be analyzed and the teacher can re-address the topic and choice of teaching strategies. Similarly if all students score highly in all elements of the rubric the teaching can be guided towards a more challenging concept.

“Students and teachers both benefit from the use of rubrics; the use of rubrics has a beneficial effect on teachers by helping them to clarify their assessment criteria and score fairly” (Maggie, McGatha , Darcy 2010) and thus supports the consistency of assessment.” However handing out a rubric does not in itself guarantee a learning impact on students” (Maggie, McGatha , Darcy 2010) They are incredibly time consuming to create and don’t always have the intended impact. Student created rubrics, particularly in open tasks, not only allows students to understand the expectations but also outlines what they need to strive for and what the critical elements of the task are. Students are much more likely so succeed when they are explicitly aware of the expectations. This also allows the teacher to recognise what the students value and centre task towards their wants as well as their needs In particular, students perform better when they find the task purposeful and relevant (Panadero Romero 2014).

REFERENCES''

Panadero E, Romero E (2014). To rubric or not to rubric?. The effects of self-assessment on self-regulation, performance and self-efficacy.

EDMA310/360 Mathematics: Learning and Teaching Mathematics 2, 2014 – Assignment 1 – Template 1 of 3

Bush,Williams S. Leinwand,S (2000). Mathematic Asssessment :Practical Handbook for Grades 6-8. Reston,

Olson, J. & Barrett, J. (2004). Coaching teachers to implement mathematics reform recommendations. Mathematics Teacher Education and Development, 6, 63–78.

Ferguson, S. (2008). Same Task, Different Paths: Catering for Student Diversity in the Mathematics Classroom. Australian Catholic University. –

Maggie B. McGatha Peg Darcy (2010). Rubrics at play. National Council of Teachers of Mathematics.