Coherence is primarily about how logically it is structured and ordered. A coherent project will have a beginning , middle and an end and the reader will always know where they are and where they are going as they are reading. This has implications for the planning (both planning your work to start with and planning how to write about it) Your work should tell a story, so to start with you need a story to tell and then you need to consider how the each part leads to the next. The story will follow a logical development and not appear disjointed. This is more about how one section links naturally to the next section helping to build the picture. You can lots of help with this from other people, let them read it, let them tell you what they understood and didn't.

Well organised - is often overlooked. Part of being coherent and well organised is about organising in to a logical structure. You will have an introduction that outlines the nature and aim of what you did and what the reader is about to see, then a body of work that constitutes the exploration and a conclusion that ties it up and suggests next possible steps. Then you need to think about the tools you can use to help you tell it. Heading structures, good use of text and diagrams and signposts as well as general appearance. Charts and graphs with annotations and interpretations nearby, and appendices for large data sets and extra references that might interfere with the flow in the main structure.

0 marks - This mark is going to be reserved for work that is largely incomplete. It is hard to imagine that many will get this. Clearly, if you haven;t really got anything to say or if what you submit seems to be a random collection of unconnected ideas then it is a possibility.

1 mark - You can get this if you have some coherence OR some organisation. Again this is only likely if you haven't got much to say or haven't taken any of the advice offered to you from your teacher.

2 marks - You should see this as a minimum requirement. Have you organised your work in to the different things you did AND then presented them in a logical order that tells the story? Any proof reader should be able to help you judge this. The descriptor talks about some coherence and some organisation. So we need to be able to identify the parts of the work that show those things.

3 marks - Clearly, to get this mark, you have to have the above already. The descriptor here describes the project as coherent and well organised, without the 'some'. As such, it would be expected that there aren't really any bits that don't seem to fit or that might leave a reader confused. It does depend on you having an idea that allows this to happen. If a reader feels that they don't really know what is going on in a given section or if a section rambles a bit and loses focus, then you might not get this mark. 

4 marks - This will be hard to get for everyone and will be reserved fro the explorations that are very well presented. The word 'concise' appears in the descriptors which implies that there is not really any wasted words and that the ideas are presented effectively so that the reader is not in any doubt about what is going on at any point.

The following is a list of the sort of the things you should pay attention to to help yourself with this criterion..

• Terminology - Use the correct mathematical vocabulary that goes with the techniques and mathematics you are writing about. Don't make it up, look it up and check that you have it right.
• Writing mathematics - It is highly recommended that you make use of an equation editor to write your mathematics and avoid at all costs an approach of 'That's the best I could do on a computer' All word processors have an equation editor built in and there are simply too many options for there to be any excuse not to. Accept that it takes bit more time (only in the beginning to learn how to use it) but that it is really worth it. In some cases it is appropriate to write mathematics by had either with a stylus or with pen and paper included as an image. The most important factor is that it is clear and correct. Too many explorations include poorly 'make do' attempts at writing mathematics that are not good. It is generally accepted that Times New Roman and italics is good for mathematics (for example)
• Computer notation - This should be easy to avoid, but you must recognise that things like '*' for multiplication and '/' for divide are part of computer coding notation and not appropriate in a written mathematical report. Number in standard form like '10e7' on a computer/spreadsheet must also be written formally and fully.
• Symbols - Find and use the correct symbol, do not use the character you think looks the best or closets, find the exact one. Insert object/symbol or equation editors will usually solve this. It matters. For a chi squared test, find the capital Greek letter Chi, not just a capital x. Find the divide and times symbols, not just an 'x' or '/'. Don't write 'xbar' for the mean, find the symbol. It is just important to convince yourself that it matters and be prepared to take the time.
• Rounding and Approximation - In general 3 significant figures is a wise choice but there are always exceptions. This is an area for consistency. You should use 3sf unless the situation demands something different in which case you should say so. For a particular calculation, you may choose to show the full calculator display before you round, but, for example in a table of expected frequencies, numbers with 8 decimal places and/or different degrees of accuracy look messy.
• Logical deduction/working out - When you are working through a calculation or proof, then make sure you lay out the steps of your work clearly, making sure you say what you are doing with each step. Typically we work vertically with mathematics with the equals symbol (when there is one) appearing underneath itself in each line. Again, there will be exceptions. Clear and correct is paramount.
• Diagrams, tables and charts - This a huge area and marks are often lost here.
  • Make sure all diagrams have titles/references that make it clear what they are so the reader does not have to play detective.
  • Make sure axes have clear labels including units where appropriate.
  • Make sure you have used the correct type of chart or diagram, eg histograms for continuous data and bar charts for discrete data. Pay attention to the difference between a scatter diagram and a line graph. In general, reflect carefully on the appropriateness of your choice.

• Defining variables - Probably the most overlooked idea. For example, a student might carry put a Pearson's correlation coefficient using the variables 'x' and 'y' without ever telling us what they are in the context of the exploration. You need define variables explicitly like 'Let be 'x'  the Human Development index of a country....'. This is true for all of the processes you will use, particularly where we use conventional symbols in a calculation. For example, expected and observed frequencies, the values of A, B and C in a trigonometry calculation and many more. Wherever you use a letter or symbol in a calculation or statement you should make sure you have explicitly stated what it represents in the context of your exploration.

• Consistency - This can be hard to pay attention to. We have already talked about being consistent with rounding. The same is true for the variables you defined above. Once defined then try to use the same symbols throughout or explain when and why you have changed. You might be using a graph plotter that labels the x' and 'y' axis by default. If your variables are 'S' for speed and 't' for time, then either change what is on the plotter or define x' and 'y' in the context of that graph.

0 marks - rare and reserved for explorations that show none of the below.

1 mark - For this one you need 'some relevant mathematical communication' that is 'partially appropriate'. This leaves a fair bit of wiggle room. This should be a given for any piece of work that student actually finishes and submits.

2 marks - Here were take away the 'Partially appropriate' which implies that your mathematics needs to be 'the right thing in the right place'. This should not be difficult to achieve and I would see this as a minimum to aim for students who pay attention.

3 marks - This mark requires your mathematical communication to be 'relevant, appropriate and mostly consistent'. This is a pretty high standard and there will not be much room for error here. Some errors with consistency are common and will be allowed and this will be harder for really simple projects that don't offer much opportunity to show this.

4 marks - This one requires consistency throughout and real attention to detail. This is reserved for students who have understood the important elements of clear mathematical communication. (some minor errors can be overlooked if they don't have a detrimental effect on the message)

A busy IBDP student has a lot on their plate and original ideas are tough. For that reason you can imagine that a lot of explorations might look a little formulaic. It might come across that students have just gone through the motions with a string of processes that think they are supposed to be doing. This is not a necessarily a bad thing and does not mean that your work is not good, it will just mean that you don't score highly in this criterion. It is for that reason I suggest that it might be the most difficult. If you have struggled to find an idea and struggled to know what to do and relied heavily on suggestions and guidance from your teacher, it will probably show here. To do well, you have to take control, ask good questions and make decisions based on your research. It needs to be clear that you have reacted independently and productively from the off and to the things that happen along the way.

Evidence - A key word here is that your teacher needs to be able to point to evidence in your written work that can be used here. If your personal engagement cant be referenced in this way then you will not get credit for it, even if you were engaged! Your teacher can't just say - 'Yes, the student showed lots of personal engagement'. The examples below give an idea of the sorts of things you can do and write about to help get the credit. It is not an exhaustive list, but should help to understand the idea.

• Demonstrate that you thought independently - This just means that you took decisions about what to do at any given point by your self based on your knowledge and understanding of the situation you were in.  'In this scatter graph, I noticed that although the data was monotonic with what appeared to be a strong correlation, it is most certainly not linear. As such, I decided to carry out a Spearman's Rank analysis to test the strength of the relationship between the two variables.'
• Demonstrate that you thought creatively - When you face a particular problem, did you find a creative solution and have you explained it?
• Presenting mathematical ideas in your own way - did you choose or devise a representation that you felt was better suited to the specifics of your exploration. This might be as simple as orientation (I put these box plots all on the same access and annotated them to show who the outliers are so we could make important comparative judgements) and don't forget that you have to tell us about these decisions so we can understand what you wre thinking.
• Look at the topic from different perspectives - did you show us that you considered different perspectives and convince us why this is important to the outcomes? (Hear the ToK bell ringing!!!)
• Make and test predictions - Did you tell us a out what you were expecting to happen and why? Did you then test the outcomes to reflect back on those predictions?
• Research - Did you do some research on to the topic and have you demonstrated how the outcomes of this research have informed the decisions you made? Maybe you have contacts in an field/industry that you can interview. Perhaps you choices can be informed by previous similar work.
• Unfamiliar mathematics - Have you had to research, learn, understand and use unfamiliar bits of mathematics to help make sense of your work?

If your exploration is disjointed and misses obvious opportunities to go in a particular direction then evidence for the above is harder to find. If you have used 'textbook style' explanations of your mathematics that don't adapt to the context of your exploration, then evidence will be harder to find.

0 marks - This should be rare and reserved for explorations where there is little or no tangible evidence of personal engagement.

1 mark - When there is some evidence of some personal engagement. It is hard to be specific here about 'some' but If there are 1 or preferable 2 clear references to evidence your teacher can point to then you can get this mark.  I see this as a minimum.

2 marks - This descriptor talks about evidence of 'Significant personal engagement' - and 'A few occasions'  when clear evidence can be found and referenced. The evidence wil show that your engagement with the exploration 'drive its direction' and that this really helps the reader to understand what you were going for.

3 marks - Here we talk about 'Outstanding personal engagement' - and 'Numerous instances and high quality evidence'. As with 2 marks, the evidence needs to show that your engagement 'drove' the direction of the investigation 'creatively' and that your  approach/engagement has led you to a thorough understanding of the context/topic. This needs to be well conveyed in your writing.

The following routine should help you to be reflective and then demonstrate it by writing about it. After every stage  of your work - or as often as you appropriate...

• State the outcome of the process
• Consider carefully and then discuss what it means in context with specific reference to the research questions you started with and the goal of the exploration.
• Consider and discuss the limitations of any conclusions that could be made about the results. How objective are the outcomes and how might they look from different perspectives? To what degree can they be interpreted differently? Essentially - how substantial are the conclusions and what is their significance.
• Do/should the outcomes of what you have done influence what you are going to do next?How and why?

0 marks - As will the others, this is rare and reserved for those explorations that don't meet the following levels.

1 mark - For this mark, we are looking for evidence of limited reflection. This is a basic stating/describing of the results of what you did without any elaboration or depth of discussion. This is a minimum, but everyone can reasonably aim for 2 and that should be achievable.

2 marks - At this level the key description is 'Meaningful reflection'. For your commentary to be meaningful, you will have to related it clearly back to the questions you asked that gave rise to the exploration. What does it mean in context? What do you learn from the result? What limitations are there on the result?

3 marks - This is a big step up in that the descriptor talks about 'Substantial evidence' and 'Critical reflection'. This kind of reflection involves high levels of insight that show you have thought deeply about the mathematics used and the result, its meaning and its limitations in the context of your work. Perhaps the reflection is decisive in the direction your exploration takes and has an obvious impact on your understanding. Critical reflection will involve seeing the strengths and weakness of different approaches that consider different perspectives and make valuable contributions to the story you are telling. For this to be considered substantial evidence, it will need to be present throughout the work.

Focus on your questions and remember that 'Less may be more' - We have all seen lots of long rambly projects where students have tried to shoe horn every bit mathematics they can in to the project. These projects do not do well. The mathematics has to be used in response to the questions you are trying to answer and the goals of the exploration. Simple projects wth well chosen, well executed processes that are used to make progress with the stated aims of the exploration are often the best ones. Clearly you can use lots of mathematics - great! - but it is important to establish which part of the story you are telling each time and make sure that it all links together. Large amounts of mathematics that are not actually used or end up being repetitive and that, as such, don't really contribute can work against you. The pattern should be - what mathematics do I need to answer this question as opposed to how can I bet an independence test in to my exploration.

There are a few key terms that define this criterion. The different levels are then different and increasing combinations of them. As such, they are hierarchical in that each one assumes the one before it.

Relevant - This is very easily overlooked if you don't spend enough time making sure that process you are using is the appropriate one for your circumstances. Students will use graph generators on spreadsheets with thought thinking or checking if it is appropriate - the spreadsheet doesn't care, it just crunches the numbers you give it to crunch. Sometimes, this will mean avoiding a complex process where a simple one wil do. Check, reflect, ask..

Commensurate - Clearly, the ideal is that yo are using mathematics that you have learned during the course. If you never go beyond simple mathematics that came before then it wont be commensurate. Make sure you pay attention to this IA - What Mathematics? page to give you an idea! You might explore some unfamiliar mathematics and you might equally use some simple ideas from before IB, but it makes sense to be sure that the bulk of your mathematics comes from the syllabus and is there fore 'commensurate'

Demonstrating understanding - This is difficult - if pressed, most people can talk about mathematics they have used without truly understanding. There are contexts in which tis OK, but this is a mathematics course and the whole point is that you use mathematics and then demonstrate that you know and understand what it means in context. To demonstrate your understanding, you will have use words, notation and reason to convey that understanding to the reader. You will need to offer examples and explanations of how it applies in context. At least some if this is required for everything above a level one. Remember too, that even if you do understand, your teacher has to be able to point to parts of your exploration that demonstrate it as described in this paragraph.

Correct - You would think this goes with out saying. Clearly correct ness matters, but the criteria are written so that you can still get credit for good attempts at mathematics that contain some errors.

0 marks - As usual - rare and reserved for those explorations that do not meet the criteria below. Watch out though, if the mathematics you have used is arguably not relevant then you might well get 0.

1 mark - Some relevant mathematics

2 marks - As above with limited understanding demonstrated

3 marks - As above but with the requirement that the mathematics is commensurate with the level of the course.

4 marks - Relevant, commensurate mathematics. Partially correct with some knowledge and understanding demonstrated.

5 marks - Relevant, commensurate mathematics. Mostly correct with good knowledge and understanding demonstrated.

6 marks - Relevant, commensurate mathematics. Correct with thorough knowledge and understanding demonstrated.

HL students - not surprisingly, there is a greater expectation of an HL student's ability to use, know, understand and be sophisticated, rigorous and precise with mathematics and this is reflected in the criteria. As such, these criteria are more demanding in general as well as the fact that the expression 'commensurate with the level of the course' means commensurate with the HL course.

Focus on your questions and remember that 'Less may be more' - We have all seen lots of long rambly projects where students have tried to shoe horn every bit mathematics they can in to the project. These projects do not do well. The mathematics has to be used in response to the questions you are trying to answer and the goals of the exploration. Simple projects wth well chosen, well executed processes that are used to make progress with the stated aims of the exploration are often the best ones. Clearly you can use lots of mathematics - great! - but it is important to establish which part of the story you are telling each time and make sure that it all links together. Large amounts of mathematics that are not actually used or end up being repetitive and that, as such, don't really contribute can work against you. The pattern should be - what mathematics do I need to answer this question as opposed to how can I bet an independence test in to my exploration.

There are a few key terms that define this criterion. The different levels are then different and increasing combinations of them. As such, they are hierarchical in that each one assumes the one before it.

Relevant - This is very easily overlooked if you don't spend enough time making sure that process you are using is the appropriate one for your circumstances. Students will use graph generators on spreadsheets with thought thinking or checking if it is appropriate - the spreadsheet doesn't care, it just crunches the numbers you give it to crunch. Sometimes, this will mean avoiding a complex process where a simple one wil do. Check, reflect, ask..

Commensurate - Clearly, the ideal is that yo are using mathematics that you have learned during the course. If you never go beyond simple mathematics that came before then it wont be commensurate. Make sure you pay attention to this IA - What Mathematics? page to give you an idea! You might explore some unfamiliar mathematics and you might equally use some simple ideas from before IB, but it makes sense to be sure that the bulk of your mathematics comes from the syllabus and is there fore 'commensurate'

Demonstrating understanding - This is difficult - if pressed, most people can talk about mathematics they have used without truly understanding. There are contexts in which tis OK, but this is a mathematics course and the whole point is that you use mathematics and then demonstrate that you know and understand what it means in context. To demonstrate your understanding, you will have use words, notation and reason to convey that understanding to the reader. You will need to offer examples and explanations of how it applies in context. At least some if this is required for everything above a level one. Remember too, that even if you do understand, your teacher has to be able to point to parts of your exploration that demonstrate it as described in this paragraph.

Sophistication -  For this to be achieved, we are looking for mathematical ideas and concepts that go beyond the SL requirements. It may be an SL idea that uis used in more complex way and it may be something specifically from the HL syllabus amd to distinguish form simply commensurate, it will be used in a way that is particularly incisive in the context of the exploration that exposes underlying structures.

Rigour - This is really about clarity of language and logic that covers all the bases and means that work and claims are thoroughly abd clearly justified and pursued to cover all the eventualities.

Precision - This word is to reflect a step up from correct. Precise leaves no room for error with accuracy.

Correct - You would think this goes with out saying. Clearly correct ness matters, but the criteria are written so that you can still get credit for good attempts at mathematics that contain some errors.

0 marks - As usual - rare and reserved for those explorations that do not meet the criteria below. Watch out though, if the mathematics you have used is arguably not relevant then you might well get 0.

1 mark - Some relevant mathematics with limited understanding demonstrated

2 marks - As above and partially correct with some knowledge and understanding demonstrated.

3 marks - Relevant, commensurate mathematics. Correct with some knowledge and understanding demonstrated.

4 marks - Relevant, commensurate mathematics. Correct with good knowledge and understanding demonstrated.

5 marks - As above with -  demonstrates 'sophistication' and 'rigour' and thorough knowledge and understanding demonstrated.

6 marks - As above with 'precise'