Aim: This assignment allows you the opportunity to research an area of

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Assignment Cover Sheet Teach First Programme 2019 – 2021 Complete all sections

Assignment Cover Sheet

Teach First Programme 2019 – 2021

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Drawing upon the breadth of your experience and development, work to transform your vision for your pupils into a reality.

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4521

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26-07-21

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26-07-21

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How does the approach of SSDD effect student’s problem-solving skills in the mathematics classroom?

Introduction

This essay continues on from the research of my module 4 assignment where I developed an intervention on mathematical problem solving. The focus of this assignment is to carry on with a problem-solving intervention, but to extend its impact and influence on other colleagues and key stakeholders. The current focus of my school’s mathematics department are difficult GCSE questions which require a good level of problem-solving skills. This is why my intervention was delivered to a group of 32 higher attaining year 9 pupils. This intervention will start at year 9 instead of year 11, which was the year group I delivered my previous intervention to, so that it can have a greater impact over time. After analysing the outcomes of my intervention, I present these to my KS3 coordinator and discuss my future recommendation for extending my work and influencing others. I conclude by discussing the merits, successes and limitations of my small research study. In my module 4 assignment, I focussed on low threshold, high ceiling problems (LTHC) with my lower attaining year 11 class. The reason I specifically chose this intervention for this class is because LTHC problems are designed so that they are accessible for all students. My previous assignment allowed students to think about mathematics differently How so? What were the main differences you observed? as they were approaching questions that were a different style to what they were used to and it was great in allowing students to be more creative. How did you observe this creativity in their problem solving? However, as this current assignment will focus on a group of higher attaining pupils, Why did you change your focus to higher attaining students? I have decided to change my focus to same surface, different depth (SSDD) problems, as these are usually designed for higher attaining pupils. A citation here would be useful for why these are better suited for higher attaining pupils. The goals of this small research study are to help students to choose the appropriate strategy for a problem All problems? Or a specific type? More clarity would be useful. and to improve students’ resilience when faced with unfamiliar questions in the hopes that they become better problem solvers. 

Literature review 

Problem solving and mathematical reasoning

Investigations by the government found that pupils were leaving school lacking problem-solving skills by way of former GCSE assessments. Citation? It was found that questions were predictable and often scaffolded for the pupils (House of Commons – From GCSEs to EBCs: the Government’s proposals for reform – Education Committee, 2021). Success in past assessments were dependent on recall and an application of standard procedures, with minimal opportunity for communicating, interpreting and representing mathematics (Sealey and Noyes, 2010). The new GCSE assessments are now designed so that they contain more problem-solving questions than ever before, therefore, emphasising the importance of teaching problem solving skills in the classroom.

Problem solving can be defined as a systematic means of addressing unfamiliar questions and discovering paths towards solutions (Koichu, 2014). George Pólya (year) identified four key steps to enhance problem solving skills in mathematics learning: 1) understanding the problem, (2) choose a strategy, (3) carry out the strategy and (4) reflect upon the problem. 

For the first step in understanding the problem, suggestions of questions to put forward are ‘what is the condition?’ and ‘what is the unknown?’. Citation? For the second step in finding a strategy, questions that could be asked are ‘have you seen it before?’ and ‘do you know a related problem?’. In the third step of carrying out the strategy, questions that can be helpful are ‘can you see that you have done the step correctly?’ And finally, for the fourth step, questions to be asked are ‘can you check the result?’ and ‘is there a different way to derive the answer?’ Making habits of thinking in this way are recommended for the teacher to incorporate into their lessons (Kang, 2012). Why is this the case? Therefore, these are the types of questions I used whilst delivering my intervention.

Blocked and interleaved practice You never actually defined interleaved practice, as you referred instead to it as mixed review, which is a slightly different concept.

One structure that mathematics lessons can take is that each taught topic is followed by a set of practice problems, a ‘practice set’, and this is usually arranged in one of two ways. Blocked practice consists of pupils studying a topic and then straight after practising questions on that specific content (Rohrer, 2009). For example, a lesson on the Pythagorean theorem would be followed by several questions on this topic. The blocked problems could also include a combination of procedural problems and worded problems but the underlying topic would be the same for each question. In contrast, mixed review, are groups of problems drawn from different lessons. The ordering of the problems is also presented in such a way that the same skill or concept does not appear in a consecutive pattern. The mixed review problems can come in one of two ways. Firstly, problems on a specific topic are spaced across several practice sets, for instance, presenting pupils with a trigonometry question every few lessons. Alternatively, problems related to different topics can be mixed within each practice set. An example of this would be a question that combines several topics like trigonometry and the Pythagorean theorem. Spacing problems allows for a review of the topic which improves long term retention and using different topics for practice sets improves the student’s ability to pair a problem with the appropriate concept or procedure (Rohrer, 2009).

The skill of choosing an appropriate strategy can be difficult as the superficial features of a problem may not always point to an obvious strategy. An example of this type of question would be: 

A bug flies 3m east and 4m north. How far is the bug from where it started? 

The solution to this problem would be to use the Pythagorean theorem. However, this problem does not explicitly mention the Pythagorean theorem or even a triangle. Although students must learn to choose an appropriate strategy, they are denied this opportunity if they are only learning through blocked practice problems. There is a lot of reflective writing here, but this needs to be supported by the literature, too. More references to support your claims, such as the preceding one, would be necessary to strengthen your literature review. In an alternative approach, the problems in each assignment can be drawn from previous lessons so that no two consecutive questions require the same approach. With this, students must choose an appropriate strategy. Problems that seem similar at first glance can be termed ‘superficially similar problems’ and the similarity between these types of questions may hinder students’ ability to distinguish between different kinds of problems. Several studies Which ones? Where is the citation and what are the benefits you refer to toward the end of this sentence? have required students to learn and distinguish between problems that are nearly identical in appearance, and the benefit this has on test performance. In contrast, (Rohrer, Dedrick and Burgess, 2014), showed that this benefit can also be true for problems that do not look alike. Despite all of the benefits you have referred to so far regarding interleaved practice, you have not highlighted one of the primary ones, which is to allow students to practice identifying the strategy rather than automatically assume the procedure that will be applied based off of what was done most recently in class.

Mayfield and Chase (year) experimented to compare the use of blocked practice and mixed review in the classroom. In the study, a group of college students attended sessions where they were taught the laws of indices (e.g. am x an = am+n) in the same manner. However, the students were split into a blocked practice group and a mixed review group. In the blocked practice group, each session consisted of a single law of indices. In the mixed review group, each session included interleaving of problems for each of the previously learnt rules. Two tests were taken after the sessions, the first test was 1-2 days after the sessions, and the second was 4-12 weeks after the sessions. The mixed group outscored the blocked practice group on both tests. The mixed group’s average performance on the first test was 97% which was more than 10% higher than that of the blocked practice group. In addition to this, on the final test, the mixed group had an average percentage score of 82.6% correct, in comparison to the blocked practice group who achieved 56%. To add to this, the mixed review group were also able to solve problems faster than the blocked practice group. These results suggest that mixed review practice of skill is an effective method in training pupils to become better problem solvers (Mayfield and Chase, 2002). There is no need here to summarize the whole study, but rather consider synthesizing instead the major finding along with others presented above and below.

In addition to this, in an experiment by Rohrer and Taylor (year), college students worked on sets of practice problems that were either blocked practice problems or mixed review problems. College students were taught how to find the volume of several 3d shapes in three sessions one week apart. Each session included a practice session and the final session included a test. Both groups worked on the same practice problems but they were either blocked – e.g. having four questions on finding the volume of one shape consecutively, or they were mixed problems. The distinct feature of mixed problems in comparison to blocked problems is that in the mixed group, students are required to pair a problem with its appropriate procedure before solving. In the first practice session, the blocked group had a higher average than the mixed group that was statistically significant. In the second practice session, the blocked practice group scored only slightly higher. In the final session where the college students took a test, the mean test performance was far greater for the mixed group than that of the blocked group. An analysis of the tests was done and the number of times a student provided the correct formula but not the correct answer was counted. This was only found on two occasions – one by a student in the blocked group and one by a student in the mixed group. This shows that if the correct formula for each question was correctly identified, the correct answer was almost always found. This means that the majority of college students (regardless of group) knew how to correctly solve each kind of problem given that they could recall the formula. And subsequently, the poor performance by the blocked group in the test session was due to their inability to correctly recall the formula for each problem. Inability to recall a formula or identify which one is necessary to solve the problem? There is an important distinction here. Therefore, it appears that when the students receive the necessary training What is this training? Be more specific. to be able to choose an appropriate strategy, their test performance was better. Several caveats can apply to the findings of this study. First of all, the students were in college and these effects may not be seen in younger students. So where are the studies that are more relevant to your age group? Secondly, the questions found on the tests were problems exactly like those found in the practice sessions, and it is not known whether these findings would be transferable to tests where the problems were unfamiliar (Rohrer and Taylor, 2007).

Methods

In this section, I outline the methodology for my intervention. New GCSE assessments contain more problem-solving questions than ever before. As a result, my current departmental focus are the more difficult GCSE questions to improve students’ problem-solving skills. For that reason, I decided to focus my intervention on SSDD problems to improve students’ problem-solving skills. 

My intervention explores the approach of same surface, different depth (SSDD) problems and their effect on problem skills in the mathematics classroom. SSDD problems are a special set of problems that look the same at first glance, but each question requires a different mathematical approach to solve (SSDD Problems, 2021). An example of one of the tasks I used is shown in Figure 1. The study was carried out over six weeks to a group of high attaining year 9 pupils in a school in North London. Why high attaining? My current teaching practice for the start of my lessons include a retrieval starter activity, usually with 4-5 questions, as a point of recall, where the style of questions is familiar to the students. My change in practice will involve replacing the current starter activity with SSDD problems. The SSDD questions are on the topic of geometry, such as, similar shapes, Pythagoras and trigonometry as these are the current taught topics on the year 9 curriculum at my school. Each of the SSDD tasks will comprise of four problem style questions. These have been retrieved from SSDD problems from Craig Barton. How is this now related to the research on blocked/interleaved practice you dedicated time to exploring? These connections have not been made explicit.

Figure 1 – an example of an SSDD task using in my intervention 

The intervention comprised of allowing pupils to attempt these problems, while asking them questions that are important in the problem-solving process according to George Polya. For example, ‘do you understand what you are being asked to do?’, ‘can you restate the problems in your own words?’ and ‘can you check your result?’. This is then followed by coaching the pupils through the modelling process of these problems. How will you do this? I will collect quantitative and qualitative data to analyse the outcome of my intervention. I intend to give pupils a pre-test and a post test. Measuring what? Over 6 weeks, students will do two tests to quantitatively assess the outcome of my intervention. The qualitative data will be assessed using observations Observations of what exactly? that I make during the SSDD tasks. 

The ultimate aims of my intervention are to explore the effects of SSDD tasks on problem solving skills in the hope that my students become better problem solvers and also, to improve resilience when met with unfamiliar problems. You refer to resilience but it is not presented as a construct in your literature review and it is unclear how you plan to measure it.

Practitioner Enquiry 

Practitioner enquiry (PE), defined by Menter et al (2011) is an investigation with a rationale and approach that can be explained or defined. These findings are then shared with others so that it is more than a reflection. As practitioners become “agents of their professional learning”, PE can play a part in challenging existing thoughts and can help teachers “let go, unlearn, innovate and re-skill in cycles of professional learning throughout their career in response to changing circumstances” (Baumfield, Hall, Wall and Baumfield, 2013). Page numbers required for direct quotes; additionally, the formatting of this in-text citation is different than the rest of the paper.

I plan to extend the impact of my study beyond my classroom. On completion of my intervention, I will present my findings to my KS3 coordinator to discuss how other colleagues in my department can implement my intervention. If effective, my intervention has the potential to impact the pupils in KS3, leading up to KS4 where pupils will face more problem-solving questions in the lead up to their GCSEs.  

Ethical considerations

Another consideration to make when involved in practitioner enquiry research is that the sharing of data and outcomes must consider the ethical implications when research involves human participants. The British Educational Research Association (BERA)’s guidance on ethical responsibilities states that consent, transparency and privacy are important when undertaking this type of study, where participants are giving consent where necessary (Ethical Guidelines for Educational Research, fourth edition (2018), 2021). It is important to be completely transparent with the key stakeholder and the participants involved.

In terms of data security, I ensured that any data collected was kept in a password protected place, which I would only have access to and is also anonymised to protect the participants. I ensured that I was transparent with all of the participants and stakeholders involved in my small research task. It was essential that I clearly explained all of the details of the research and how I intended to use the information they provided me. 

Analysis and reflection

In this section, I evaluate the outcome of my small study, reflecting upon the successes, merits and limitations of my intervention. Additionally, using the data obtained and my professional judgement, I will reflect on the potential impact of my study. Finally, I conclude with the recommendations for extending the impact of my intervention and influencing others beyond my classroom. 

Domain specific knowledge, as defined by Tricot and Sweller (2013), is “memorised information that can lead to action permitting specified task completion over indefinite periods”. Page number required. For example, to be an expert in the Pythagorean theorem, learners must not only learn the theorem itself, but also the various ways in which the Pythagorean theorem can be applied, therefore, developing domain-specific knowledge. It can be argued that pupils cannot succeed in problem solving without domain-specific knowledge stored in their long-term memory (Tricot and Sweller, 2013). This can also be seen as blocked practice, which was mentioned in the literature review, where pupils study a topic and then straight after practice questions on this topic (Rohrer, 2009). For this reason, before implementing my intervention, pupils were taught in detail some of the topics of geometry. These topics included similar shapes (extending to area and volume scale factors), how to use the Pythagorean theorem and trigonometry on right angled triangles. I used maths genie and Corbett maths worksheets on these topics to provide pupils with an in-depth knowledge of these topics and also allowed pupils to apply their knowledge. This was intended to give the pupils the best foundation possible before attempting the SSDD tasks. But how did you ensure they knew the information? Remember that exposure does not guarantee understanding. To add to that, as the literature suggested, I used mixed review questions in the form of retrieval starter activities where no two questions were the same for the reason of reinforcing knowledge and allowing pupils to retain this knowledge in their long term memory (Rohrer, 2009). This was done at the start of every lesson for the three weeks that I was teaching domain-specific knowledge.

Following these lessons, a pre-test was completed. What was the pre-test measuring? IF you are interested in problem solving and whether or not SSDD problems are associated with better problem-solving skills, the pre-test should have been an opportunity to assess students’ problem-solving skills. As the pre-test was created by the KS3 coordinator, there were several topics in the pre-test as a point of interleaving. These topics included the area of a circle, compound measures and similar shapes. There were three questions on similar shapes on the pre-test. Two of these questions were straight forward for the pupils So how were they problem-solving questions? as they were the questions pupils were familiar with. The first question was answered correctly by 86% of the pupils, and the second question was answered correctly by 76%. The reason for this could be that the pupils were explicitly taught these types of questions and so were familiar with these questions, hence why they were answered correctly. However, the third question on similar shapes was only answered correctly by 6% of the pupils. A reason for why the pupils may have struggled with the final similar shapes question may be that it was an unfamiliar type of question and required an element of problem solving. To add to this, around 40% of pupils had not even attempted the question. Therefore, using SSDD tasks, I aimed to improve the ability of pupils to improve problem-solving skills and increase student resilience when faced with unfamiliar problems. This would mean increasing the percentage of pupils getting the problem-solving questions correct and increasing the percentage of pupils attempting the questions. Only one question on the pre-test even measured problem-solving skills, so how can you use this to determine the effectiveness of your intervention?

The first SSDD problem I used was on the topic of similar shapes presented in Figure 1. These problem-solving questions were given to pupils, and at first glance, all the questions looked identical, but each problem required a different approach. I gave pupils as much time as they required which amounted to 15 minutes in the first task. As I was circulating, a student was confused, he raised his hand and asked “did we do this last lesson?”. This suggests that the student had noticed that this is an unfamiliar set of questions to him, however, I prompted him and the rest of the class by asking “what is the question asking me to do?” and “what do you know about the question?”. This allowed the pupil to then recognise that we have done a similar question to this, it just looked slightly different to what he was used to. He then flourished with the other questions and was able to correctly choose the correct strategy for each question. All in all, the work from the class was not received well as about 45% of pupils had attempted all the questions. 

The SSDD task attempted by the students was then followed by an explanation of each question, accompanied by my thought process. The main strategy I used to improve problem solving skills were Polya’s four steps in problem solving. Therefore, questions that I asked myself out loud while working through the problems included “what is the question asking me to do?”, “what do I already know about the question?”, “is there a way to check your result?” and so on (Polya, 2014). I emphasised the importance of using these steps in providing a structure to problem solve. I also told pupils to write these questions down so that they could use it for the next SSDD task. This intervention allowed pupils as much time as they needed to consider and understand what was being asked of them as there was no time limit on the SSDD tasks. 

The next SSDD task is shown in Figure 2. Again, by the nature of SSDD tasks, all of the problems looked identical but required a different approach to solve them. As students had previously written the steps to think about when solving a problem, I told them to refer to those when answering these questions. The response to the questions is much better this time round, and about 55% of the pupils are attempting all the questions. Samples of work done by a pupil for this task is shown in Figure 3 and Figure 4. The sample shown in Figure 3 shows that the pupil had attempted one question and copied out the working out for the other 3 questions. In contrast, Figure 4 shows a pupil who has attempted 3 out of the 4 questions. In the next two lessons, I presented pupils with two more SSDD tasks. These were both well received in the class and the first had a 65% completion rate and the final one had a 70% completion rate. The evidence suggests that pupils are becoming more resilient as they are answering more questions. As you never defined or presented resilience as a construct in your paper, it is difficult to now integrate it in your study, particularly as there are many reasons beyond resilience why more students may have completed the questions as you moved on.

Figure 2 – the second SSDD task used 

Figure 3 – student sample

Figure 4 – student sample work

After completing the four SSDD tasks, students completed a post-test a week later. Again, this post-test was created by the KS3 coordinator, and therefore, had topics outside of geometry. There was one question on compound measures, but the majority of the questions were on the topics of similar shapes, Pythagoras and trigonometry. There was one problem solving question which combined ratio and trigonometry. There was an improvement from the pre-test in the percentage of pupils who got this question correct as 30% of pupils answered this question correctly. This is an increase from the 6% of pupils who correctly answered the problem-solving question in the pre-test. 

Merits and limitations of my study 

On reflection, regarding the merits and limitations of my study, I feel that my intervention successfully targeted my desired outcomes. The ultimate aims of my intervention were to improve the pupils’ problem-solving skills by helping them to choose the appropriate strategy for a problem and to increase pupil resilience when faced with problems which are unfamiliar to them. By using SSDD tasks, I was able to see my aims come into fruition. Pupils were answering far more questions than they were at the start of the intervention and the percentage of pupils getting the problem-solving percentage correct increased from 6% to 30%. A higher percentage of pupils had also attempted the problem-solving question the second time round (from 60% to 82%). This suggests an increase in resilience and willingness to complete questions which are unfamiliar to them. Also, I feel as though my pupils achieved an increase in attainment and an increase in confidence when it came to answering the questions. How did you assess this? To add to this, in working to improve pupils’ problem-solving skills, specifically in the area of geometry, I applied Polya’s four steps as described in the book How To Solve It. Using Polyas steps is regarded very highly and can be seen as a success of my small study. The study also provided me with evidence that I could use with little ambiguity. It was also low cost, effective and had no ethical concerns.

Limitations of the study would be that the pre-test and post-test were created by the KS3 coordinator. Although there were problem-solving questions in both tests to assess the pupils’ understanding, the tests did not fully cater to the needs of the intervention. It would have been better had I created the tests myself. It is also hard to tell whether the problem-solving questions in both tests were of the same difficulty. Perhaps, a reason why the percentage of pupils getting the question correct increase was due to second problem solving question being easier. Additionally, the sample size of 32 pupils was small and the duration of the intervention was 6 weeks which is short. In future, I would increase the sample size and try this type of intervention on other classes and would do it for a longer period of time to see what the effect would be over a longer period of time. While this short study focussed on the topics of geometry, I believe the structure of using Polya’s four steps could easily be applied to other areas of mathematics.

Presentation to key stakeholder

The outcomes and recommendations of my intervention were presented to my key stage 3 coordinator at my school in a 10-minute presentation. I explained the rationale for my intervention and the key reasons which underpinned the design of my intervention. I emphasised the importance of building pupils’ problem-solving skills at an early stage and the importance of being resilient when faced with unfamiliar problems, especially considering the way that the GCSE specification has reformed to include more problem-solving questions. Finally, I presented my data which showed a small improvement in test scores. After my presentation to my KS3 coordinator, I was commended on my intervention, but we discussed that the small sample and the short duration of the intervention meant that my findings may not have had the intended outcome. It is too soon to say whether pupils are better problem solvers as more time is needed and more SSDD tasks should be used to see a more accurate outcome. However, my intervention will be used to support the department and several SSDD tasks have now been added to the central planning area for other colleagues to use as Do Nows if they wish to. 

Conclusion 

The focus of my small assignment was to see whether SSDD problems would affect the problem-solving ability of the pupils in my higher attaining year 9 class. As the new GCSE assessments have now changed to include an increase in problem solving questions, pupils must acquire these skills to perform well in their maths GCSE. My ultimate aim was to improve pupils’ problem-solving skills and to increase the resilience of these pupils when they are faced with questions which look unfamiliar to them. I feel as though I was successful in my desired outcomes as students were attempting more questions and getting more questions correct.

My key finding from this study is that domain-specific knowledge is a pre-requisite in problem solving. It is also important that pupils understand the thought process required to problem solve and that they were continuously working their way through the steps suggested by George Pólya (Polya, 2014). The percentage of pupil getting the problem-solving question increased from 6% (in the pre-test) to 30% (in the post-test). It is hard to tell whether this increase was due to the problem-solving question being easier or whether the SSDD tasks along with the four steps in problem solving helped with the increase.

This intervention deepened my understanding of practitioner enquiry and the process behind creating, implementing, analysing and sharing my interventions and particularly how the intervention can be used to impact and influence others beyond my classroom. I intend to include SSDD tasks into my teaching practice and regularly use George Polya’s four steps to enhance problem solving with my future classes.

References

Baumfield, V., Hall, E., Wall, K. and Baumfield, V., 2013. Action research in education. Los Angeles, Calif.: SAGE.

Bera.ac.uk. 2021. Ethical Guidelines for Educational Research, fourth edition (2018). [online] Available at: [Accessed 26 July 2021].

Kang, W., 2012. Implications from Polya and Krutetskii. 12th International Congress on Mathematical Education,

Koichu, B., 2014. Reflections on Problem-Solving. Mathematics & Mathematics Education: Searching for Common Ground, pp.113-135.

Mayfield, K. and Chase, P., 2002. THE EFFECTS OF CUMULATIVE PRACTICE ON MATHEMATICS PROBLEM SOLVING. Journal of Applied Behavior Analysis, 35(2), pp.105-123.

Polya, G., 2014. How to Solve It. Princeton University Press.

Publications.parliament.uk. 2021. House of Commons – From GCSEs to EBCs: the Government’s proposals for reform – Education Committee. [online] Available at: [Accessed 9 July 2021].

Rohrer, D., 2009. The effects of spacing and mixing practice problems. Journal for Research in Mathematics Education, 40, pp.4-17.

Rohrer, D., Dedrick, R. and Burgess, K., 2014. The benefit of interleaved mathematics practice is not limited to superficially similar kinds of problems. Psychonomic Bulletin & Review, 21(5), pp.1323-1330.

Rohrer, D. and Taylor, K., 2007. The shuffling of mathematics problems improves learning. Instructional Science, 35(6), pp.481-498.

Sealey, P. and Noyes, A., 2010. On the relevance of the mathematics curriculum to young people. The Curriculum Journal, 21(3), pp.239-253.

SSDD Problems. 2021. SSDD Problems. [online] Available at: [Accessed 26 July 2021].

Tricot, A. and Sweller, J., 2013. Domain-Specific Knowledge and Why Teaching Generic Skills Does Not Work. Educational Psychology Review, 26(2), pp.265-283.[supanova_question]