With only two students in the room, it was not possible for the presence of the observer to be discrete. However, I do not feel the flow or atmosphere of the class was any different to normal because of this. It felt like a fairly typical session.
This is in contrast to the session my mentor observed, which felt tense and this tension seemed to vanish with a big sigh when the mentor left.
Thursday, 24 April 2008
Week 10: Integration as Anti-Differentiation (also peer observation)
This week's material was introduction to Integration by treating it as the reverse process to Differentiation. I was observed by another member of my PGCHE group. This was the first lecture since the coursework was handed in.
Two students attended. One is a mature student returning after a period in employment to complete a second degree who has covered the material before in that degree but does not remember it well. Call her A. The other is straight from school and has not covered the material before, having only GCSE mathematics. Call her B. They obtained two of the top marks in the coursework. They are highly capable but lack confidence.
In what follows I will give a run down of the class while it is fresh in my mind.
The observer and I arrived 5 minutes before the hour and one student had already arrived. Shortly afterwards the other arrived. I introduced the observer but didn't go through the whole explanation since I had explained the observation idea and my taking the PGCHE to both students on prior occasions.
I handed out the coursework. Both students were pleased with their marks and one noticed that means they have passed the module already. I said this was not the way to look at it!
A few minutes later we decided no more students were likely to arrive and started. I began with an explanation of Xeno's paradox of Achillies and the tortoise to introduce the concept of an infinite sum. This will be more applicable next week when we do Integration as area under a curve, but was intended to introduce the concept of infinite sums totalling finite amounts and to provide a warm up bit of thinking.
I followed this with a reminder that speed is the derivative of distance, and we differentiate it again to get acceleration. Then we can reverse this process, anti-differentiating speed to get distance.
Following this I went through the deduction of the integral of x^2 using logical reasoning.
Finally I showed the rules for integrating polynomials, natural logarithm and simple exponential functions. The students attempted exercises.
Student A got on with the exercises fairly confidently. She asked a couple of questions and was capable. She noticed an error in the solutions.
Student B was much less confident. She said she didn't understand the material several times even though she did, she just didn't have the confidence in what she was doing. After a few exercises she grew more in confidence.
The students sat much further apart than usual, which meant I could follow one student or the other, but not both. I moved between the two and this seemed to work fairly well.
Student A finished the exercises much earlier than student B. I said she could read through next week's material or go early. I didn't want to go through the next week's material on the board because Student B was not ready, though I did not say so of course.
Student A read through next week's material, said it was familiar and she would wait for next week, and left early.
Student B continued through the exercises and by the end was completing them at an acceptable pace. She lacks confidence. I told her she did understand the material but just hadn't practised enough yet, as she was with differentiation.
Just after the session the observer told Student B she said "I don't understand that" too much and that she clearly was capable and needed to be more confident. I agreed. She also said the student would be fine in the end test. I agree with this also, but am nervous of setting a sense of complacency, particularly as the student has already passed the module on the coursework alone but could achieve an impressive mark. The student, unprompted, said she would be sure to do plenty of revision and I agreed this was a good idea.
Two students attended. One is a mature student returning after a period in employment to complete a second degree who has covered the material before in that degree but does not remember it well. Call her A. The other is straight from school and has not covered the material before, having only GCSE mathematics. Call her B. They obtained two of the top marks in the coursework. They are highly capable but lack confidence.
In what follows I will give a run down of the class while it is fresh in my mind.
The observer and I arrived 5 minutes before the hour and one student had already arrived. Shortly afterwards the other arrived. I introduced the observer but didn't go through the whole explanation since I had explained the observation idea and my taking the PGCHE to both students on prior occasions.
I handed out the coursework. Both students were pleased with their marks and one noticed that means they have passed the module already. I said this was not the way to look at it!
A few minutes later we decided no more students were likely to arrive and started. I began with an explanation of Xeno's paradox of Achillies and the tortoise to introduce the concept of an infinite sum. This will be more applicable next week when we do Integration as area under a curve, but was intended to introduce the concept of infinite sums totalling finite amounts and to provide a warm up bit of thinking.
I followed this with a reminder that speed is the derivative of distance, and we differentiate it again to get acceleration. Then we can reverse this process, anti-differentiating speed to get distance.
Following this I went through the deduction of the integral of x^2 using logical reasoning.
Finally I showed the rules for integrating polynomials, natural logarithm and simple exponential functions. The students attempted exercises.
Student A got on with the exercises fairly confidently. She asked a couple of questions and was capable. She noticed an error in the solutions.
Student B was much less confident. She said she didn't understand the material several times even though she did, she just didn't have the confidence in what she was doing. After a few exercises she grew more in confidence.
The students sat much further apart than usual, which meant I could follow one student or the other, but not both. I moved between the two and this seemed to work fairly well.
Student A finished the exercises much earlier than student B. I said she could read through next week's material or go early. I didn't want to go through the next week's material on the board because Student B was not ready, though I did not say so of course.
Student A read through next week's material, said it was familiar and she would wait for next week, and left early.
Student B continued through the exercises and by the end was completing them at an acceptable pace. She lacks confidence. I told her she did understand the material but just hadn't practised enough yet, as she was with differentiation.
Just after the session the observer told Student B she said "I don't understand that" too much and that she clearly was capable and needed to be more confident. I agreed. She also said the student would be fine in the end test. I agree with this also, but am nervous of setting a sense of complacency, particularly as the student has already passed the module on the coursework alone but could achieve an impressive mark. The student, unprompted, said she would be sure to do plenty of revision and I agreed this was a good idea.
Pre-observation
In today's class I will be observed by another member of my PGCHE group.
Today's class is the first on integration, treating it as the reverse process to differentiation. I have been thinking a lot about this session since the introduction to differentiation class, in which I feel I failed quite dramatically to get over the deep concepts and the students went away with a toolbox they don't understand. I do not know if this is a problem. People use computers but have no idea how they work; similarly cars. Does it matter if I give someone a mathematical tool to apply to scientific problems in their area and they don't understand how that tool is derived? From the notes I have been given to teach from there are several areas where a rule or technique is derived and I have generally skipped over this as they do not need to know the mathematical proof in order to apply the technique. But we feel Calculus is an area where the application requires some understanding of the deep meaning.
So I will be trying to get across concepts of infinity. I will do this using Xeno's paradox of Achilles and the tortoise and a bouncing ball, which I have been carrying in my pocket and occasionally bouncing for just this purpose for the last few weeks.
Today's class is the first on integration, treating it as the reverse process to differentiation. I have been thinking a lot about this session since the introduction to differentiation class, in which I feel I failed quite dramatically to get over the deep concepts and the students went away with a toolbox they don't understand. I do not know if this is a problem. People use computers but have no idea how they work; similarly cars. Does it matter if I give someone a mathematical tool to apply to scientific problems in their area and they don't understand how that tool is derived? From the notes I have been given to teach from there are several areas where a rule or technique is derived and I have generally skipped over this as they do not need to know the mathematical proof in order to apply the technique. But we feel Calculus is an area where the application requires some understanding of the deep meaning.
So I will be trying to get across concepts of infinity. I will do this using Xeno's paradox of Achilles and the tortoise and a bouncing ball, which I have been carrying in my pocket and occasionally bouncing for just this purpose for the last few weeks.
Wednesday, 23 April 2008
PGCHE: Finalise micro-teaching sessions
I attended a PGCHE session today. This outlined what was needed for the rest of Module 1 (this semester). In the next session we will be doing 15 minutes "micro-teaching" so I must pick a topic relevant to my teaching but accessible by the class.
Tuesday, 22 April 2008
Developing mathematical reasoning
I attended an interesting research seminar today given by Dr. Chris Sangwin of the University of Birmingham. This was on a software package called Geogebra which is a tool for dynamic geometry.
The interesting part, beyond the actual demonstration of the tool, was the way Chris had used Geogebra. He had been teaching a course on mathematical reasoning using the tool and teaching this using Moore method teaching. This is a method developed by Robert Moore in which the teacher does not show the students how to do the mathematics but rather gives problems and the students solve these using their own reasoning. Students then present their solutions to the class. In this case, the group got through 31 problems in a 10 week course, which Chris regarded as slow going in terms of content but he felt very strongly that the course had helped his students develop their mathematical reasoning skills.
The problems were geometry problems. An example is the following: "A ladder leaning against a wall slides to the ground. What path does a cat sitting halfway down the ladder make at it falls?" The students were not given the solution, nor any clues.
Chris reported the use of Geogebra as a reasoning/visualisation tool and as a presentation tool by some students (some preferred pen and paper). He said some students would go down lines of reasoning that were not getting anywhere, and not all students solved all problems before seeing a solution from another student. But he felt that this exploratory reasoning procedure has made them better mathematicians.
The point here is that when I wrote about small group teaching I felt that having a small group discuss the material rather than be shown it by the teacher would not work for my course. What interests me here is that this is exactly the sort of thing Chris is doing with the Moore method. However, it should be pointed out that he is teaching mathematicians to reason, whereas I am teaching scientists and business students a tool they can apply. I do not believe this method is appropriate for building a mathematical toolkit, though it would be interesting to give it a go with some students who are interested in learning mathematics for its own sake.
Because I am showing the students how to do problems then having them follow me, they are not necessarily developing an understanding of the deeper complexity that might help them see how the mathematics should be applied, or help them remember the lessons better. But am I just thinking this because I am a mathematician?
The interesting part, beyond the actual demonstration of the tool, was the way Chris had used Geogebra. He had been teaching a course on mathematical reasoning using the tool and teaching this using Moore method teaching. This is a method developed by Robert Moore in which the teacher does not show the students how to do the mathematics but rather gives problems and the students solve these using their own reasoning. Students then present their solutions to the class. In this case, the group got through 31 problems in a 10 week course, which Chris regarded as slow going in terms of content but he felt very strongly that the course had helped his students develop their mathematical reasoning skills.
The problems were geometry problems. An example is the following: "A ladder leaning against a wall slides to the ground. What path does a cat sitting halfway down the ladder make at it falls?" The students were not given the solution, nor any clues.
Chris reported the use of Geogebra as a reasoning/visualisation tool and as a presentation tool by some students (some preferred pen and paper). He said some students would go down lines of reasoning that were not getting anywhere, and not all students solved all problems before seeing a solution from another student. But he felt that this exploratory reasoning procedure has made them better mathematicians.
The point here is that when I wrote about small group teaching I felt that having a small group discuss the material rather than be shown it by the teacher would not work for my course. What interests me here is that this is exactly the sort of thing Chris is doing with the Moore method. However, it should be pointed out that he is teaching mathematicians to reason, whereas I am teaching scientists and business students a tool they can apply. I do not believe this method is appropriate for building a mathematical toolkit, though it would be interesting to give it a go with some students who are interested in learning mathematics for its own sake.
Because I am showing the students how to do problems then having them follow me, they are not necessarily developing an understanding of the deeper complexity that might help them see how the mathematics should be applied, or help them remember the lessons better. But am I just thinking this because I am a mathematician?
Friday, 18 April 2008
Coursework submissions
I collected the coursework today. 16 were submitted and 7 were not. The lady on the submissions desk said she wouldn't be surprised if most of the 7 came in on Monday. I must find out what happens to late submissions. Otherwise, it is a worry that 7 students have ruled themselves out of 50% of the marks. A mark of 80% is needed on the end test to pass the module if the coursework has attracted a zero mark.
I now need to find some time for marking.
I now need to find some time for marking.
Thursday, 17 April 2008
Missing the difficult one
1 student turned up today! She hadn't been to the previous Differentiation classes so we went through all 3 weeks of Differentiation. She had done most of it before years ago so it was a refresher rather than new material, which was more managable.
I am concerned about all the students who have not turned up for this, which I think is the most difficult class so far. Next week on to Integration...
I am concerned about all the students who have not turned up for this, which I think is the most difficult class so far. Next week on to Integration...
Do they even know the coursework has been set? Redux
It looks like of the three students I had emailed who had not yet downloaded the coursework last week, one has since done so. None of the three have replied to my email to say they have or don't need the coursework. That means there are still 2 students for whom I have no evidence they have seen the coursework one day before the deadline. But I have emailed those 2 five times about the coursework (initially to say it was released, once before Easter I emailed those who hadn't yet downloaded it and after Easter I emailed all students as a reminder and also those who still hadn't downloaded it again separately) and I don't know they haven't got it from another source/stopped doing the module/etc., so I feel I have done all I can.
Being calculating
The more calculating students, having studied last year's coursework and end test (given as mocks) may have noticed that 100% of the marks on last years, and this years, coursework and 70% of the marks in the end test are on the first 6 weeks of lectures. This accounts for 85% of the mark, which is a good first. Furthermore, these are all questions for which the answer can be used to generate the question for checking purposes.
This means that the (arguably more difficult, more useful) material on calculus at the end of the course is only actually accounting for 15% of the mark, yet it covers 5 of 11 weeks of teaching.
I will, of course, not be telling my students this. But I wonder if any have realised. I could change the end test; there is still time as I haven't written it yet. But whenever for the past 3 months students have asked what will be on the end test I have said "look at the mock" so this may be unfair. I will have to have a think.
The students I am seeing in class seem keen to learn. Whether they don't realise this or realise it and know they might need to know the material for future courses (i.e. are not purely assessment-driven), I am not sure. But a couple of them have commented on examples in this course that they have seen in other science modules.
This means that the (arguably more difficult, more useful) material on calculus at the end of the course is only actually accounting for 15% of the mark, yet it covers 5 of 11 weeks of teaching.
I will, of course, not be telling my students this. But I wonder if any have realised. I could change the end test; there is still time as I haven't written it yet. But whenever for the past 3 months students have asked what will be on the end test I have said "look at the mock" so this may be unfair. I will have to have a think.
The students I am seeing in class seem keen to learn. Whether they don't realise this or realise it and know they might need to know the material for future courses (i.e. are not purely assessment-driven), I am not sure. But a couple of them have commented on examples in this course that they have seen in other science modules.
The difficult one
I am a little anxious in advance of today's class. The topics today are, in my opinion, a step above everything else on the course so far in terms of difficultly. Furthermore, tomorrow is the coursework deadline so the distraction I experienced last week with students preferring to work on the coursework may be more acute.
Thursday, 10 April 2008
Week 8: Differentiation of standard functions
3 students attended. One had attended last time and so had seen intro to differentiation. Two of them wanted to work through the coursework though I made it clear I couldn't answer specific questions.
I ran through quickly what was done last time and the students did some problems. This was really good because it meant that they refreshed the work done before the Easter break although it meant I didn't do very much for the observation.
Then I went through a couple of bits on the board and they did more problems.
I think they had a reasonable grasp of the topic and hopefully are well prepared for the more advanced differentiation topics next week.
I ran through quickly what was done last time and the students did some problems. This was really good because it meant that they refreshed the work done before the Easter break although it meant I didn't do very much for the observation.
Then I went through a couple of bits on the board and they did more problems.
I think they had a reasonable grasp of the topic and hopefully are well prepared for the more advanced differentiation topics next week.
Observation by mentor
Today I will be observed in class by Prof. Tony Sackfield, who has agreed to be my mentor. Tony is my PhD supervisor. I met with him today and gave him the pre-observation form and discussed the group and the lesson.
Catching up on my correspondance
I am concerned about the upcoming coursework deadline.
I have been to the coursework submission desk and there are no more submissions. This is not really a problem since there is over a week until the deadline, but a couple of students had told me they would submit early to avoid conflict with other deadlines.
On the VLP, 3 students have not yet downloaded the coursework. It is possible they have got a copy from someone else on the course, or that they are no longer taking the module and yet are still in the system. It is also possible they have simply not looked at the coursework yet.
I emailed all students to remind them of the coursework deadline. I emailed the 3 (BCC so they don't know who's who) and said they haven't downloaded a copy. I asked them to email me and let me know if they have seen a copy or are not for some reason or download it if not.
I also took the opportunity to email the student who is not attending deliberately and ask how it is going. They have downloaded the coursework so that is a good sign.
I have been to the coursework submission desk and there are no more submissions. This is not really a problem since there is over a week until the deadline, but a couple of students had told me they would submit early to avoid conflict with other deadlines.
On the VLP, 3 students have not yet downloaded the coursework. It is possible they have got a copy from someone else on the course, or that they are no longer taking the module and yet are still in the system. It is also possible they have simply not looked at the coursework yet.
I emailed all students to remind them of the coursework deadline. I emailed the 3 (BCC so they don't know who's who) and said they haven't downloaded a copy. I asked them to email me and let me know if they have seen a copy or are not for some reason or download it if not.
I also took the opportunity to email the student who is not attending deliberately and ask how it is going. They have downloaded the coursework so that is a good sign.
PGCHE: Student learning
I attended the PGCHE session yesterday on "Student learning: an overview of behaviourism, cognitive psychology and humanistic theories".
This was interesting but I'm not sure I followed it all. I regognised a lot of behaviourism in the way I treat my cats, but I'm not sure about my students. I think my teaching fits best with cognitive methods, in that each week builds on previous work and there is a real thinking / doing to learn approach.
This was interesting but I'm not sure I followed it all. I regognised a lot of behaviourism in the way I treat my cats, but I'm not sure about my students. I think my teaching fits best with cognitive methods, in that each week builds on previous work and there is a real thinking / doing to learn approach.
Sunday, 6 April 2008
A post about a poster
My PhD is, in part, looking at methods for dynamically generating mathematics assessment. I have been compelled to submit a poster abstract for the School of Science and Technology internal research conference in May. I wrote this today and it is highly informed by my experience of and reflection on teaching Scientific Mathematics, so I reproduce it here.
"Dynamic generation of problems in formative assessment of mathematical subjects"
Mathematics is a subject you cannot learn without having a go at some problems. Indeed, it may be necessary for students to repeat a technique many times before understanding and confidence is achieved. However, this can be time consuming and tedious for the tutor, who must set many similar problems, and variety in the activity may be necessary to engage the interest of all students.
A problem is made up of a question and a context. Part of the beauty of mathematics is its application to many different fields: a question on one technique in mathematics might be written into a problem in one of several diverse topics in science or engineering, or even finance or other subject areas. To a student, these may appear to be different (varied) problems even though the same technique is being learned. The core mathematics of a question can be written and rules be applied to form this into different problems for different fields.
Once the format has been set many similar problems can be written which teach the same lesson simply by altering the numbers in the question. There is no need for the tutor to manually make this repetition; this process can be automated. Then a great many problems can be generated based on a single manually written question. In some cases it might be necessary to include a diagram or simulation which varies dynamically also.
"Dynamic generation of problems in formative assessment of mathematical subjects"
Mathematics is a subject you cannot learn without having a go at some problems. Indeed, it may be necessary for students to repeat a technique many times before understanding and confidence is achieved. However, this can be time consuming and tedious for the tutor, who must set many similar problems, and variety in the activity may be necessary to engage the interest of all students.
A problem is made up of a question and a context. Part of the beauty of mathematics is its application to many different fields: a question on one technique in mathematics might be written into a problem in one of several diverse topics in science or engineering, or even finance or other subject areas. To a student, these may appear to be different (varied) problems even though the same technique is being learned. The core mathematics of a question can be written and rules be applied to form this into different problems for different fields.
Once the format has been set many similar problems can be written which teach the same lesson simply by altering the numbers in the question. There is no need for the tutor to manually make this repetition; this process can be automated. Then a great many problems can be generated based on a single manually written question. In some cases it might be necessary to include a diagram or simulation which varies dynamically also.
Learning without interaction
I attended the Annual Learning and Teaching Conference 2008, NTU's internal learning and teaching conference.
I attended a session on engagement and interaction with e-learning methods. Within this there seemed to be an assumption that the students who were engaging (in this case, posting to an online forum) were fine, were learning and we should worry about the ones who are not interacting. The seminar focussed in part on how to encourage interaction between students and with the tutor.
I do not think that because a student is not interacting it means they are not learning.
In my last two classes I have had two students listening to me when I lecture. One is outwardly keen and talkative and operates the calculator and helps the process along. The other shrinks into his chair when I talk to him. Afterwards, they both work through the exercises at a similar rate and do similar things right and wrong. They have both learned the technique (and in mathematics I have the luxury of instant feedback when they get the questions right or wrong). I am not concerned at all about the student who doesn't talk back to me or ask questions "in group". He is obviously benefiting from observing the interaction between myself and the other student.
I understand that if both students were asking informative questions I would feel more confident they are following what I am saying, that in a larger group that would become more acute and that in a subject without the instant feedback of a problems class this might be all the feedback I am going to get from them, but I would not be comfortable putting this student in a position he isn't comfortable with and potentially running over his preferred learning style just to satisfy my sense of whether the students are understanding me.
It reminds me of myself in school. I was not one of the students who would naturally talk in front of the class. When a question was asked I would generally not put up my hand (unless no one else did). Generally, at my first parents evening with a new teacher in maths or science I would be reported as "too quiet" and told that it was vital I speak up more in class or I risked falling behind. I never took this advice. At the second parents evening with that teacher (some time later; usually after some summative assessment) it would be reported that I was a little quiet but obviously a strong student destined to do well. After the teacher had some test results, it seems to me, they were content with my progress. In the absence of such results all they had is whether I was speaking up in lessons, so this is what they based their opinion of my academic abilities on. The one is not necessarily a good indicator of the other and though I can see some cases where it might be the best available it should not be relied on.
I attended a session on engagement and interaction with e-learning methods. Within this there seemed to be an assumption that the students who were engaging (in this case, posting to an online forum) were fine, were learning and we should worry about the ones who are not interacting. The seminar focussed in part on how to encourage interaction between students and with the tutor.
I do not think that because a student is not interacting it means they are not learning.
In my last two classes I have had two students listening to me when I lecture. One is outwardly keen and talkative and operates the calculator and helps the process along. The other shrinks into his chair when I talk to him. Afterwards, they both work through the exercises at a similar rate and do similar things right and wrong. They have both learned the technique (and in mathematics I have the luxury of instant feedback when they get the questions right or wrong). I am not concerned at all about the student who doesn't talk back to me or ask questions "in group". He is obviously benefiting from observing the interaction between myself and the other student.
I understand that if both students were asking informative questions I would feel more confident they are following what I am saying, that in a larger group that would become more acute and that in a subject without the instant feedback of a problems class this might be all the feedback I am going to get from them, but I would not be comfortable putting this student in a position he isn't comfortable with and potentially running over his preferred learning style just to satisfy my sense of whether the students are understanding me.
It reminds me of myself in school. I was not one of the students who would naturally talk in front of the class. When a question was asked I would generally not put up my hand (unless no one else did). Generally, at my first parents evening with a new teacher in maths or science I would be reported as "too quiet" and told that it was vital I speak up more in class or I risked falling behind. I never took this advice. At the second parents evening with that teacher (some time later; usually after some summative assessment) it would be reported that I was a little quiet but obviously a strong student destined to do well. After the teacher had some test results, it seems to me, they were content with my progress. In the absence of such results all they had is whether I was speaking up in lessons, so this is what they based their opinion of my academic abilities on. The one is not necessarily a good indicator of the other and though I can see some cases where it might be the best available it should not be relied on.
The silent majority
I am concerned about the students who are not attending. Not necessarily that they are not attending, just that I have nothing on which to base their abilities on yet.
When I set the mock coursework I deliberately didn't release the answers. Partly this will stop the "answers in the back of the book" mentality and partly this was to make sure they come to me in class and I get a sense of how well the group are able to cope prior to the actual coursework. Not one student has approached me to mark their mock. A couple have asked me questions on the mock coursework questions, but those are the ones who are attending class and I know anyway.
Prior to the Easter break only one student submitted the coursework. I have looked at this and every question is answered correctly, although there are a couple of rounding errors and slight notational problems that may knock the mark off 100% it will be pretty close. This means this student has passed Scientific Mathematics, since 40% is a pass and the coursework accounts for 50% of the assessment. This is marvellous news, though of course I can't tell the student until the coursework deadline has passed, which is a pain as I know she will ask me in this week's class.
My worry is that the students who aren't attending will not do the coursework correctly. Still, they know when the classes are and can attend if they find they have problems. And there is a later opportunity for correction since the end test is worth 50% of the assessment and 40% is a pass. So I will try not to worry.
When I set the mock coursework I deliberately didn't release the answers. Partly this will stop the "answers in the back of the book" mentality and partly this was to make sure they come to me in class and I get a sense of how well the group are able to cope prior to the actual coursework. Not one student has approached me to mark their mock. A couple have asked me questions on the mock coursework questions, but those are the ones who are attending class and I know anyway.
Prior to the Easter break only one student submitted the coursework. I have looked at this and every question is answered correctly, although there are a couple of rounding errors and slight notational problems that may knock the mark off 100% it will be pretty close. This means this student has passed Scientific Mathematics, since 40% is a pass and the coursework accounts for 50% of the assessment. This is marvellous news, though of course I can't tell the student until the coursework deadline has passed, which is a pain as I know she will ask me in this week's class.
My worry is that the students who aren't attending will not do the coursework correctly. Still, they know when the classes are and can attend if they find they have problems. And there is a later opportunity for correction since the end test is worth 50% of the assessment and 40% is a pass. So I will try not to worry.
Learning from the master
During small group teaching, one of the group discussions was a discussion of how to begin a session with a new group. It interested me greatly that the other members of my group felt they needed to introduce themselves and their credentials to the students at the start of the first session. I didn't. In particular they felt the need to explain to the students why they were qualified to teach the course.
One aspect of this of course is that for me to stand up at the start of the first class and say "I'm a PhD student and they couldn't find anyone to teach this." would be demoralising and cause a lack of confidence.
However I think there is more to it than that. I could have easily told them of my mathematics degree from Nottingham, my membership of the Institute of Mathematics and its Applications, the professional body for mathematics, etc., and shown I was more than capable of teaching the course. The thought that I might didn't cross my mind.
If I went to a lesson on, say, writing a successful grant application, I might reasonably expect the tutor to explain why they are qualified to teach this lesson; why do they know more than I or the man on the street on this topic? That sort of topic is subjective and I would want to know: if I disagree with something the tutor says, why should I revise my viewpoint? If they are an expert and have a record of experience in writing successful grant applications, their opinion on subjective issues may become of more interest.
Mathematics, at the level I am teaching, is not subjective in this way. What was true for Napier on logarithms almost 400 years ago and Newton on Calculus later that century is still true today. The authority, then, comes from the body of knowledge I am teaching. I wonder if this is why I didn't feel my credentials were relevant? I know this content; if I didn't I wouldn't have been employed to teach it. And the mathematics in the course is built up over thousands of years. I see myself as a guide to this world, rather than a master of the topic with insight to learn from.
One aspect of this of course is that for me to stand up at the start of the first class and say "I'm a PhD student and they couldn't find anyone to teach this." would be demoralising and cause a lack of confidence.
However I think there is more to it than that. I could have easily told them of my mathematics degree from Nottingham, my membership of the Institute of Mathematics and its Applications, the professional body for mathematics, etc., and shown I was more than capable of teaching the course. The thought that I might didn't cross my mind.
If I went to a lesson on, say, writing a successful grant application, I might reasonably expect the tutor to explain why they are qualified to teach this lesson; why do they know more than I or the man on the street on this topic? That sort of topic is subjective and I would want to know: if I disagree with something the tutor says, why should I revise my viewpoint? If they are an expert and have a record of experience in writing successful grant applications, their opinion on subjective issues may become of more interest.
Mathematics, at the level I am teaching, is not subjective in this way. What was true for Napier on logarithms almost 400 years ago and Newton on Calculus later that century is still true today. The authority, then, comes from the body of knowledge I am teaching. I wonder if this is why I didn't feel my credentials were relevant? I know this content; if I didn't I wouldn't have been employed to teach it. And the mathematics in the course is built up over thousands of years. I see myself as a guide to this world, rather than a master of the topic with insight to learn from.
The spice of life
My classes don't last their full two hours. I had not been too worried about this. They generally last more than one hour and the students are covering the material at a perfectly adequate pace. That they get to go early is an indication that I am not boring them at the start with too many examples, and I imagine they leave feeling good that they are able to leave early.
In Small Group Teaching a discussion was held on operating a variety of activities in order to maintain engagement and interest. It occurred to me that my last class, in which I broke up the "doing problems" period with some further explanation, was the longest yet. It is difficult to do many different activities : sitting and working through problems is very much the point. Wondering which mathematical models to use and how they relate to the real world are areas for the home course. I wonder if breaking the class up into lecturing, problems, lecturing, problems, is enough to maintain interest. Of course, it may seem more likely that as the material is getting more difficult and the assessment more imminent the enthusiasm to get the most of out classes is increasing.
In Small Group Teaching a discussion was held on operating a variety of activities in order to maintain engagement and interest. It occurred to me that my last class, in which I broke up the "doing problems" period with some further explanation, was the longest yet. It is difficult to do many different activities : sitting and working through problems is very much the point. Wondering which mathematical models to use and how they relate to the real world are areas for the home course. I wonder if breaking the class up into lecturing, problems, lecturing, problems, is enough to maintain interest. Of course, it may seem more likely that as the material is getting more difficult and the assessment more imminent the enthusiasm to get the most of out classes is increasing.
Remember last week
In Small Group Teaching, the tutor recommended refreshing last week's material at the start of a lesson and gave this process a name which I have not noted down. There was also a discussion of learning by rote without understanding.
In mathematics it is very necessary you will learn the material by doing it, so learning by rote without understanding is absent (though you may not understand the deep concepts). Also, this weeks material will very likely build on last weeks, since we are building concept on concept on axiom. So refreshing last weeks material is usually a given.
In mathematics it is very necessary you will learn the material by doing it, so learning by rote without understanding is absent (though you may not understand the deep concepts). Also, this weeks material will very likely build on last weeks, since we are building concept on concept on axiom. So refreshing last weeks material is usually a given.
Personal responsibility
It is important that my students attend classes and learn the techniques needed. In order that they continue to do so, I mustn't make attending too unattractive a prospect.
On attendance: They do not have to attend. No one is marching them to lectures. Then, if they want to leave early that is up to them. If I try to say they can't leave early they may well not come back next week.
Equally, if they want to not attend and work at their own pace, that is also their choice. I cannot march them to lectures, nor would I want to.
They are not children, they are responsible for their own education at this point.
On attendance: They do not have to attend. No one is marching them to lectures. Then, if they want to leave early that is up to them. If I try to say they can't leave early they may well not come back next week.
Equally, if they want to not attend and work at their own pace, that is also their choice. I cannot march them to lectures, nor would I want to.
They are not children, they are responsible for their own education at this point.
Tabula rasa
In my PGCHE class are represented a variety of different subject areas. I have noticed that some people are teaching subjects where they will be the first person ever to teach that subject to that student. A tertiary education mathematics teacher will not have this experience. All my students have done some mathematics before and they all have preconceptions about it.
I am reminded of my first driving lesson. My instructor asked if I had driven before. I said I had not. He said this was good; that a number of his students had already done some driving with a parent and his first job was to wade through the preconceptions and bad habits already formed to get back to a clean slate.
I am not saying I would like to teach addition to 18 year olds! But it is interesting to wonder about how this affects my teaching: I am always building on a body of knowledge.
I am reminded of my first driving lesson. My instructor asked if I had driven before. I said I had not. He said this was good; that a number of his students had already done some driving with a parent and his first job was to wade through the preconceptions and bad habits already formed to get back to a clean slate.
I am not saying I would like to teach addition to 18 year olds! But it is interesting to wonder about how this affects my teaching: I am always building on a body of knowledge.
PGCHE: Small Group Teaching
I attended a PGCHE class on small group teaching. I found this stimulating and useful, even though it transpires that what I teach is not "small group" (despite involving few students) but instead is "problem class" teaching, which the tutor said he could do a whole session on alone!
What follows are several reflections jotted down during and after this session, in no particular order.
What follows are several reflections jotted down during and after this session, in no particular order.
Easter break
Now comes the Easter break. The coursework is due around one week after they return.
Just before the Easter break I started the Postgraduate Certificate in Higher Education (PGCHE) course. Consequently the Easter break is characterised as a period of reflection. All that has been written in this journal so far has been written after this point in the schedule. From this point on, the journal is properly reflective.
Just before the Easter break I started the Postgraduate Certificate in Higher Education (PGCHE) course. Consequently the Easter break is characterised as a period of reflection. All that has been written in this journal so far has been written after this point in the schedule. From this point on, the journal is properly reflective.
Do they even know the coursework has been set?
Since I set the coursework I have only seen 3 students. I was anxious whether the students were aware the coursework was out and what the deadline was. I asked the students in week 7 whether they thought their classmates were aware of the coursework. They said they weren't sure and one student suggested I send an email through the Outlook system that required the student to answer "Yes" or "No" to whether they have downloaded the coursework. This seemed a good idea; those who do not answer can be more carefully targeted.
After the class I realised the VLP does this for me. It contains a document tracking system so I checked and all but 6 students had downloaded the coursework or been in a class where it was handed out. If they have downloaded it I am happy they are at least aware of it. There are several issues to bear in mind with this sort of automated tracking: a student could very well have printed more than one copy of the coursework or photocopied a friend's coursework. A colleague pointed out that students frequently change modules without deregistering properly and so are left as ghosts in the system. I have noticed a slight variety in students as I have gone on. In my register from the first week I notice that I have a couple of students who are no longer on the VLP register and that in the first few weeks I met all but three of the students yet now there are six in the system who I have not met. So there is clearly movement taking place.
After the class I realised the VLP does this for me. It contains a document tracking system so I checked and all but 6 students had downloaded the coursework or been in a class where it was handed out. If they have downloaded it I am happy they are at least aware of it. There are several issues to bear in mind with this sort of automated tracking: a student could very well have printed more than one copy of the coursework or photocopied a friend's coursework. A colleague pointed out that students frequently change modules without deregistering properly and so are left as ghosts in the system. I have noticed a slight variety in students as I have gone on. In my register from the first week I notice that I have a couple of students who are no longer on the VLP register and that in the first few weeks I met all but three of the students yet now there are six in the system who I have not met. So there is clearly movement taking place.
Week 7: Introduction to Differentiation
Mindful of the fact I had released the coursework I was prepared for the fact that attendance might jump this week and fill the room with students wanting to ask questions about the coursework material. However, if we don't introduce the concept of differentiation this week we will be running to catch up all through Calculus, which is a terrible place to lose them. My strategy was to go through this weeks material and field questions on the coursework afterwards.
3 students attended. The same 3 as last week.
Anyway, I had to get across to them what is going on when we differentiate a function. This is not a simple concept and involves ideas of infinitely small distances, etc. It is further complicated by the fact that knowledge of what happens in differentiation is not needed at all to get through the exam. We like to imagine that it is important for the students to have a look under the hood, as it were, and get some feel for what is really going on when they apply this ubiquitous technique. However, in the same way that we don't feel compelled to take a computer apart before we use one students outside of pure mathematics are not particularly keen to do so. Still, I know it is important to de-mystify the mathematics; understanding makes a technique better remembered. Also a sense of how differentiation works and what it does will be useful when we learn integration in a few weeks time.
The reality of the situation was that I was pretty ill and kept losing my place and having to refer to my notes. I found the situation fairly frustrating as I know this explanation quite well and had revised it prior to class. I tried to break down the derivation into manageable parts and take the students through it slowly and carefully. At the end I showed them the formula that is needed for differentiating any polynomial function. I realised up until then that they were thinking they would need to derive differentiation as a concept to answer every question, as this was what I had done so far. When they realised all they needed to do what apply a simple formula they became much more comfortable with the technique. Polynomial functions draw on the material from previous weeks and was back in the comfort zone. I really feel I could have done better here, and a clear head would have gone a fair way towards this.
I explained for a bit, they did some questions, then I took them through the next part on the board and they finished with some more questions. They left early, but the closest to 2 hours we have been so far.
3 students attended. The same 3 as last week.
Anyway, I had to get across to them what is going on when we differentiate a function. This is not a simple concept and involves ideas of infinitely small distances, etc. It is further complicated by the fact that knowledge of what happens in differentiation is not needed at all to get through the exam. We like to imagine that it is important for the students to have a look under the hood, as it were, and get some feel for what is really going on when they apply this ubiquitous technique. However, in the same way that we don't feel compelled to take a computer apart before we use one students outside of pure mathematics are not particularly keen to do so. Still, I know it is important to de-mystify the mathematics; understanding makes a technique better remembered. Also a sense of how differentiation works and what it does will be useful when we learn integration in a few weeks time.
The reality of the situation was that I was pretty ill and kept losing my place and having to refer to my notes. I found the situation fairly frustrating as I know this explanation quite well and had revised it prior to class. I tried to break down the derivation into manageable parts and take the students through it slowly and carefully. At the end I showed them the formula that is needed for differentiating any polynomial function. I realised up until then that they were thinking they would need to derive differentiation as a concept to answer every question, as this was what I had done so far. When they realised all they needed to do what apply a simple formula they became much more comfortable with the technique. Polynomial functions draw on the material from previous weeks and was back in the comfort zone. I really feel I could have done better here, and a clear head would have gone a fair way towards this.
I explained for a bit, they did some questions, then I took them through the next part on the board and they finished with some more questions. They left early, but the closest to 2 hours we have been so far.
Ideal Easter break
In an ideal world I would now report that the Easter break happened. The students have covered the first 6 of 11 lessons and have learned the topics needed to complete the coursework. The remaining 5 lessons are on calculus, which has some tricky concepts.
However, it was not to be. The Easter break is positioned to separate "Introduction to Differentiation" from "Differentiation of standard functions" by 4 weeks. Oh well, perhaps the distribution over time will force something other than short term memory to kick in.
However, it was not to be. The Easter break is positioned to separate "Introduction to Differentiation" from "Differentiation of standard functions" by 4 weeks. Oh well, perhaps the distribution over time will force something other than short term memory to kick in.
Releasing the coursework
After week 6 all the material has been covered to allow the students to complete the coursework. I handed it out to the few students who attended and put it on the VLP for those who were not. I placed an announcement of this on the VLP and sent an email to all students saying so.
I knew it couldn't be correct
In class in week 6 a student answered a question on logarithmic form. The question related to pollution levels in a lake and gave the values of pollution at time t=1 and t=2. Part of the question asked to predict the value at time t=3. One student had converted to logarithmic form, solved the simultaneous equations and built a negative exponential function to predict the value. Her process seemed fine and the answer seemed reasonable (right order of magnitude, etc.). But she had noticed it was wrong; the value she had at t=3 was between the values given at t=1 and t=2. The value at t=3 must be less than that at t=2, since the equation describes a decay. That the student had deduced this really shows an important deep understanding of how the formulae and numbers relate to the real situation.
There can be a tendency in mathematics for the student to believe what the calculator tells them. It is really important when teaching mathematics to those who will be applying it to real situations that they are able to relate the numbers back to the situation they are modelling. It is no good training a plumber who can't tell what's wrong when the calculator tells him to measure out kms of pipe for a central heating system!
There can be a tendency in mathematics for the student to believe what the calculator tells them. It is really important when teaching mathematics to those who will be applying it to real situations that they are able to relate the numbers back to the situation they are modelling. It is no good training a plumber who can't tell what's wrong when the calculator tells him to measure out kms of pipe for a central heating system!
Week 6: Reduction of laws to linear form
3 students attended. One was the student from previous weeks who has been following the material capably in advance of the teaching schedule but feels the need to have me nearby in case she has a question. She was taking the mock coursework; this week we are learning the technique applied there. Also in attendance was the student from the visa email.
Many natural processes follow an exponential form and can be reduced to more manageable linear form using logarithms, so that can apply more straightforward techniques to them. This was today's lesson. This takes the students to the point where they can now do the coursework and so it was set in this class. I had intended to withhold this until the end of the class, so they paid attention to me solving problems on the board and didn't spend their time reading the coursework questions. However my excitement to express how important today's lesson was meant that I let slip that the coursework was in my bag and they students became agitated until I showed it to them. There was no great detrimental effect and it perhaps helped to encourage enthusiasm in the material to see it so closely linked with the assessment.
I learned an interesting trick in today's class. I forgot my calculator and had to ask one of the students to make the calculations as I went through the examples on the board. This really engaged that student in the process and her confidence grew as her answers agreed with those I had scribbled in my notebook. I suppose with a larger group I could ask them to take turns making calculations.
By the end of the class, the students were solving with ease the types of questions that had scared them at the start of week 5. The one student who had been in attendance did not really seem to appreciate this and I remain unsure whether it was a good idea. It pleased me greatly as a marker of progress but then that is a selfish reason for pursuing a particular strategy.
The students left early, but by no means as early as in previous weeks. I wonder if this had something to do with the coursework being set.
Many natural processes follow an exponential form and can be reduced to more manageable linear form using logarithms, so that can apply more straightforward techniques to them. This was today's lesson. This takes the students to the point where they can now do the coursework and so it was set in this class. I had intended to withhold this until the end of the class, so they paid attention to me solving problems on the board and didn't spend their time reading the coursework questions. However my excitement to express how important today's lesson was meant that I let slip that the coursework was in my bag and they students became agitated until I showed it to them. There was no great detrimental effect and it perhaps helped to encourage enthusiasm in the material to see it so closely linked with the assessment.
I learned an interesting trick in today's class. I forgot my calculator and had to ask one of the students to make the calculations as I went through the examples on the board. This really engaged that student in the process and her confidence grew as her answers agreed with those I had scribbled in my notebook. I suppose with a larger group I could ask them to take turns making calculations.
By the end of the class, the students were solving with ease the types of questions that had scared them at the start of week 5. The one student who had been in attendance did not really seem to appreciate this and I remain unsure whether it was a good idea. It pleased me greatly as a marker of progress but then that is a selfish reason for pursuing a particular strategy.
The students left early, but by no means as early as in previous weeks. I wonder if this had something to do with the coursework being set.
A work visa
I got an email in the week asking about the progress of one of the students on my module. Apparently this student required a progress report to continue his visa. Unfortunately he was not a student I had met; he had never signed my attendance register. I could do nothing but reply to that effect.
Setting the coursework
I had a very pleasant morning sat in my conservatory in the spring sunshine scribbling on a notebook writing the coursework problems. I based this heavily on last years coursework. In mathematics it is common to use the same questions but change the numbers and words slightly. So the question where you have to draw a linear graph of a logarithmic dataset is changed from a "decline in pollution levels in a lake" problem to a "increase in infections during a pandemic" problem, and so on. Knowing I am a new lecturer, and that the students are above expectations, I wanted the assessment to be obviously of similar difficulty to previous years.
An interesting feature of the coursework for Scientific Mathematics is that 6 out of 8 questions are nearly identical and reduce to the techniques used in the other 2. So really it is just the same question over and over. The problems are dressed with different contexts, different numbers obviously and slightly different focus means that the questions appear different, though fundamentally they are all the same technique.
The following contexts are used to dress the same question, reduction of natural law to linear form, into several different problems:
An interesting feature of the coursework for Scientific Mathematics is that 6 out of 8 questions are nearly identical and reduce to the techniques used in the other 2. So really it is just the same question over and over. The problems are dressed with different contexts, different numbers obviously and slightly different focus means that the questions appear different, though fundamentally they are all the same technique.
The following contexts are used to dress the same question, reduction of natural law to linear form, into several different problems:
- Decay of pollution levels in a lake over time;
- Increase in number of people infected in the early stages of a pandemic;
- Decrease in temperature of a body;
- Radioactive decay;
- Compound interest.
Week 5: Linear functions; exponential functions
3 students attended; the three who were working at the speed of the teaching schedule last week.
Weeks 5 and 6 work in tandem: this week we learn some rules of linear functions and logarithms and next week we apply them to solving 'real' problems, which build on the work of previous weeks and take the students to the point where they can now do the coursework.
I decided in advance of the class to demonstrate this process by showing them a problem of the sort used in last years coursework and solving it quickly on the board. There is a trick whereby exponential functions, which are easy to demonstrate as nasty pieces of work, are reduced to linear functions, which we know how to deal with. My intention was to create a little healthy confusion and then tell them that by the end of next week they would be solving problems like this no trouble. On reflection I think I just created confusion and scared them a little. They seemed much more relieved back in the comfort zone of working through the material rather than listening to this madman at the front.
They left early.
Weeks 5 and 6 work in tandem: this week we learn some rules of linear functions and logarithms and next week we apply them to solving 'real' problems, which build on the work of previous weeks and take the students to the point where they can now do the coursework.
I decided in advance of the class to demonstrate this process by showing them a problem of the sort used in last years coursework and solving it quickly on the board. There is a trick whereby exponential functions, which are easy to demonstrate as nasty pieces of work, are reduced to linear functions, which we know how to deal with. My intention was to create a little healthy confusion and then tell them that by the end of next week they would be solving problems like this no trouble. On reflection I think I just created confusion and scared them a little. They seemed much more relieved back in the comfort zone of working through the material rather than listening to this madman at the front.
They left early.
Week 4: Simultaneous and quadratic equations
Week 4 was the first week where I felt the material went beyond revision for a significant number of those attending. In advance of the class I prepared a set of simultaneous equations and 2 quadratic equations (one that is easy to solve by eye and the other which required the formula). In a previous week I had noticed a student solving simultaneous equations by a quite laborious method and I had performed a magician's trick showing a small group an easier way to solve them. I intended to perform the same magic here. The second topic of the day was quadratic equations. These are a traditional scapegoat in mathematics, often cited when people express their hatred for the subject. I intended to present them as methodical to solve and not worthy of attention; certainly not hatred.
So this was my first time writing on a white board in a class since tutorials at university. Nerve-wracking!
4 students attended. One was well in advance of the lesson plan and content to work through the material and ask questions if needed. The other three knew each other and formed a nice little group I talked to collectively.
On simultaneous equations, they copied down my method and seemed pleased with it if not astonished.
They had bad memories of quadratic equations. I showed them the "by eye" one and explained this is often a problem point for teaching this subject. The lecturer will say "and we see obviously that.." and the students are lost if they can't see how to complete the square. I told them not to worry if they didn't see how to do it. I then showed them the second method. I showed them the second method would solve the "by eye" quadratic equations too and that it was not incorrect, simply overkill, to use the method where a solution was obvious. So I said don't worry if you can't see the trick, the second method will work.
Then they had a go at some problems themselves. They seemed capable and confident at applying these half-remembered techniques and all got on well.
They left early.
So this was my first time writing on a white board in a class since tutorials at university. Nerve-wracking!
4 students attended. One was well in advance of the lesson plan and content to work through the material and ask questions if needed. The other three knew each other and formed a nice little group I talked to collectively.
On simultaneous equations, they copied down my method and seemed pleased with it if not astonished.
They had bad memories of quadratic equations. I showed them the "by eye" one and explained this is often a problem point for teaching this subject. The lecturer will say "and we see obviously that.." and the students are lost if they can't see how to complete the square. I told them not to worry if they didn't see how to do it. I then showed them the second method. I showed them the second method would solve the "by eye" quadratic equations too and that it was not incorrect, simply overkill, to use the method where a solution was obvious. So I said don't worry if you can't see the trick, the second method will work.
Then they had a go at some problems themselves. They seemed capable and confident at applying these half-remembered techniques and all got on well.
They left early.
Week 3: Rearrangement of formula
8 attended, including 3 who had not attended before. Again the students included a group who were much more advanced through the material than the lesson plan. There was one student who had completed the entire course. He went round, as I did, helping those who needed help. He told me he found it a useful educational experience explaining the material to others. I told him that was good, knowing that my experience of school was very similar. I kept an eye out to check he was giving correct advice, though. Again they left early.
Talking with the target student
One student emailed me prior to week 3. I knew him from class. At the end of week 1 he had not been as advanced as the lesson plan but said he would work on the material in between classes. The module specification allows 3 hours work at home for each 2 hour session, so this seemed fine. The next week he had caught up to the lesson plan. He told me he visited the Maths Support Centre in between weeks 1 and 2 and I had told him that was a good idea since this is a friendly atmosphere where one-to-one tuition from a very capable tutor was available. As he left in week 2 he was again behind the lesson plan and said he would work on it in his own time.
In the email he said he was finding the material difficult and the sessions weren't useful for him as his fellow classmates just wanted to chat and not do the work while he was struggling. He said it takes him a long time to get to university and he could be working. He complimented me as a tutor, said I was likable and had a "fantastic knowledge". He said he wasn't going to send this message to any one else as he didn't want me getting into trouble but he felt more comfortable carrying on at his own pace alone. He added at the end that he might be a little embarrassed "struggling so far behind the others."
I was pretty upset by this; he was at that point the only student sticking to the lesson plan! My reply is below.
I am sorry to hear you are not finding the lessons useful but I sympathise with the points you make. Of course you understand I can’t recommend non-attendance as a course of action, rather it is your choice.
I must confess I am surprised by the number of students who have sufficient mathematical knowledge already to take the course and I am a bit put off by people leaving half way through but it is their choice. I had been told to expect students who were perfectly capable but rusty (i.e. hadn’t done maths for a while) and I think you fit that description well, and that the material would be largely unseen by the students before. I think from talking to you last week that you are moving along at the speed of the lesson plan I was given that was used in previous years so you have nothing to worry about on that respect.
I also think that as the course goes on more students will encounter unfamiliar material and the need for class teaching might be more acute. For example, this week I have prepared myself to work through a set of simultaneous equations and a couple of quadratic equations as I imagine more students will struggle with these concepts. I worry a little how well students will cope unaided with the differentiation and integration concepts towards the end of the course.
One suggestion might be if you would like to come and stay for the second hour, which people mostly seem to be leaving for, and I can help you more intensely. Otherwise I know you have been to the Maths Support drop in centre and I think that is really good as Sarah is an excellent tutor. I imagine you know but she runs sessions in the City as well as Clifton so that might be less of a journey for you? In case you haven’t seen, the information is on the VLP at:
https://elearn.ntu.ac.uk/mle/system/resources/maths.asp?section=skills
If you aren’t going to be in classes it would be good if you could keep me up to date on how you are progressing through the course notes. Via email to this address is fine. Also if you have any questions.
They have finally got me onto the course on the VLP so I will put up last years coursework and test for students to take as a mock. You might find this useful to see the kind of thing that will be on this years tests.
All the best, and I hope to see you in class sometime
I didn't hear back.
In the email he said he was finding the material difficult and the sessions weren't useful for him as his fellow classmates just wanted to chat and not do the work while he was struggling. He said it takes him a long time to get to university and he could be working. He complimented me as a tutor, said I was likable and had a "fantastic knowledge". He said he wasn't going to send this message to any one else as he didn't want me getting into trouble but he felt more comfortable carrying on at his own pace alone. He added at the end that he might be a little embarrassed "struggling so far behind the others."
I was pretty upset by this; he was at that point the only student sticking to the lesson plan! My reply is below.
I am sorry to hear you are not finding the lessons useful but I sympathise with the points you make. Of course you understand I can’t recommend non-attendance as a course of action, rather it is your choice.
I must confess I am surprised by the number of students who have sufficient mathematical knowledge already to take the course and I am a bit put off by people leaving half way through but it is their choice. I had been told to expect students who were perfectly capable but rusty (i.e. hadn’t done maths for a while) and I think you fit that description well, and that the material would be largely unseen by the students before. I think from talking to you last week that you are moving along at the speed of the lesson plan I was given that was used in previous years so you have nothing to worry about on that respect.
I also think that as the course goes on more students will encounter unfamiliar material and the need for class teaching might be more acute. For example, this week I have prepared myself to work through a set of simultaneous equations and a couple of quadratic equations as I imagine more students will struggle with these concepts. I worry a little how well students will cope unaided with the differentiation and integration concepts towards the end of the course.
One suggestion might be if you would like to come and stay for the second hour, which people mostly seem to be leaving for, and I can help you more intensely. Otherwise I know you have been to the Maths Support drop in centre and I think that is really good as Sarah is an excellent tutor. I imagine you know but she runs sessions in the City as well as Clifton so that might be less of a journey for you? In case you haven’t seen, the information is on the VLP at:
https://elearn.ntu.ac.uk/mle/system/resources/maths.asp?section=skills
If you aren’t going to be in classes it would be good if you could keep me up to date on how you are progressing through the course notes. Via email to this address is fine. Also if you have any questions.
They have finally got me onto the course on the VLP so I will put up last years coursework and test for students to take as a mock. You might find this useful to see the kind of thing that will be on this years tests.
All the best, and I hope to see you in class sometime
I didn't hear back.
Week 2: Revision of Algebra 2
The second week of Scientific Mathematics saw attendance drop to six. Two of these had not attended last week. It was clear that most of the students had convinced themselves they were sufficiently adept in the course materials that they need not attend. The students who attended were at various places during the module, mostly not within the week 2 material.
I did not lecture again, as no two students were at the same point in the material after last week and work completed at home. However, for the most part these were the "weaker" students and they were closer to the front of the module. I toured the room answering questions as they arose. I was required to answer questions on any part of the material, which was a little daunting and lucky I know this material fairly well; in a more advanced module I would have struggled without having revised!
Again the students left early. One student who was fairly advanced decided to call it a day and miss the traffic and the others followed.
I did not lecture again, as no two students were at the same point in the material after last week and work completed at home. However, for the most part these were the "weaker" students and they were closer to the front of the module. I toured the room answering questions as they arose. I was required to answer questions on any part of the material, which was a little daunting and lucky I know this material fairly well; in a more advanced module I would have struggled without having revised!
Again the students left early. One student who was fairly advanced decided to call it a day and miss the traffic and the others followed.
Pigmaei gigantum humeris impositi plusquam ipsi gigantes vident
Each week uses the techniques learned in last week and builds on them a little further. Then, Scientific Mathematics is a gradual progression.
The basic structure of course, derived from the lesson plan given to me with my copy of the notes, is:
Week 1:
Arithmetic: ratios, fractions; their addition, subtraction, multiplication, division
Algebra: the same techniques applied to letters instead of numbers
Week 2:
Rearrangement of formula: the same techniques as last week only making sure we don't change the value of the formula as we rearrange
Week 3:
Equations: the same techniques as last week only with an equals sign; keeping both sides equal as techniques are applied
Week 4:
Simultaneous equations: same technique as last week only with 2 equations that have the same solution
Quadratic equations: equations with a term squared; equation used with techniques from previous weeks to solve these
Week 5:
Functions: extending concept of equations from previous weeks, which hold for one or more specific values of the independent variable to functions, which are the same in format but which hold for all values of the independent variable. Linear functions are just like the equations and formulae handled so far in form.
Indices and logarithms: functions involving x^a and log(x)
Week 6:
Reduction of laws to linear form: Methods for applying logarithms (encountered last week) to exponential functions (encountered last week) to reduce them to linear functions (encountered last week). Then all techniques learned so far can be applied to exponential functions.
Coursework set
Week 7:
Differentiation: building on the work on linear functions and their gradients, a method for finding the gradient of any polynomial function.
Easter break
Week 8:
Further differentiation: differentiation of polynomial functions was discovered last week; here are some rules for differentiating some standard functions encountered earlier
Maxima and minima: converting derivatives (last week and this) into equations and solving these to find stationary points (using techniques learned in weeks 3-4).
Week 9:
Applications of differentiation: Applying the techniques of differentiation (last two weeks) to problems like those in real life (or the exam).
Week 10:
Integration as anti-differentiation: Applying the technique of differentiation (last 3 weeks) in reverse to define integration.
Week 11:
The Integral as area under a curve: Using the integral with limits to define an equation (first 3 weeks) and evaluating this to find the area under a curve.
Revision
End test
The basic structure of course, derived from the lesson plan given to me with my copy of the notes, is:
Week 1:
Arithmetic: ratios, fractions; their addition, subtraction, multiplication, division
Algebra: the same techniques applied to letters instead of numbers
Week 2:
Rearrangement of formula: the same techniques as last week only making sure we don't change the value of the formula as we rearrange
Week 3:
Equations: the same techniques as last week only with an equals sign; keeping both sides equal as techniques are applied
Week 4:
Simultaneous equations: same technique as last week only with 2 equations that have the same solution
Quadratic equations: equations with a term squared; equation used with techniques from previous weeks to solve these
Week 5:
Functions: extending concept of equations from previous weeks, which hold for one or more specific values of the independent variable to functions, which are the same in format but which hold for all values of the independent variable. Linear functions are just like the equations and formulae handled so far in form.
Indices and logarithms: functions involving x^a and log(x)
Week 6:
Reduction of laws to linear form: Methods for applying logarithms (encountered last week) to exponential functions (encountered last week) to reduce them to linear functions (encountered last week). Then all techniques learned so far can be applied to exponential functions.
Coursework set
Week 7:
Differentiation: building on the work on linear functions and their gradients, a method for finding the gradient of any polynomial function.
Easter break
Week 8:
Further differentiation: differentiation of polynomial functions was discovered last week; here are some rules for differentiating some standard functions encountered earlier
Maxima and minima: converting derivatives (last week and this) into equations and solving these to find stationary points (using techniques learned in weeks 3-4).
Week 9:
Applications of differentiation: Applying the techniques of differentiation (last two weeks) to problems like those in real life (or the exam).
Week 10:
Integration as anti-differentiation: Applying the technique of differentiation (last 3 weeks) in reverse to define integration.
Week 11:
The Integral as area under a curve: Using the integral with limits to define an equation (first 3 weeks) and evaluating this to find the area under a curve.
Revision
End test
Saturday, 5 April 2008
Now turn to the answers
In the course material I have been given to teach from, and in mathematics more generally, the answers appear on the next page, or sometimes even the same page, as the questions. This can present a real problem. Students will wrangle the calculator in ways they don't quite understand, but if they can get it to display the answer that is given they move on to the next question. They may not have understood the topic; the point of the problems is to understand through doing, so if that understanding is not full the student should repeat or try a similar problem.
For example: I answer a question and look in the answers. I have not got the answer given. I try the question a slightly different way and still get an incorrect answer#. I try a third time and finally I get the same answer as in the book. Have I really understood the technique I am applying? Which application will I use in the exam? Still, I have the correct answer, so I move on happily.
Here is an example, the type of question most commonly given on the coursework. I am given a formula:
y = A * exp(k * t)
I am told the values of y at two values of t (e.g. "when t=1, y=63.2") and asked to find the values of the constants A and k. I convert to logarithmic form and make a pair of simultaneous equations. Solving these give the values of A and k (let's say 40 and 0.33). Now the formula is
y = 40 * exp(0.33t)
For the second part of the question I am asked to predict the value of y when t is some greater value; t=7, say. To do this, I put t=7 into the equation.
The crucial point to notice here is that I can check whether my values for A and k are correct by putting the values I have found into the formula and asking what is the value when t=1. Since I have been told the value at t=1 is y=63.2, say, I can calculate my value at t=1 and see if they match. If they match for both the points given in the question, I have the correct values for A and k. Using my values to predict the value at t=7, say, I am now completely confident I have the answer correct. There is no need to check the answers in the back. In this mathematics the truth is absolute.
I have tried to stress to the students the power of working the answer back into the question like this. I did this in my A-Level Mathematics and came out of the exam confident I had an A because I knew I had answered so many of the questions correctly. (In Further Mathematics and at degree level it is not so absolute, but at this level of mathematics as a toolbox it is the case). Given the question we can find the answer; given the answer we can find the question; etc. After all, this is how I set the questions in the first place, picking an answer arbitrarily and working back from there to form the question.
Two points come from this line of thought:
And of course, this process of proving the answer correct is far more realistic. There are no answers in the real world.
# of course, there is always the possibility that the answer given is not correct and this has happened in the course notes. It is most gratifying when a student notices an answer that is incorrect, since this shows real deep understanding and confidence (and represents a valuable life lesson: we are all only human and prone to err), but it can really destroy the confidence of a student who has not yet fully built up their understanding.
For example: I answer a question and look in the answers. I have not got the answer given. I try the question a slightly different way and still get an incorrect answer#. I try a third time and finally I get the same answer as in the book. Have I really understood the technique I am applying? Which application will I use in the exam? Still, I have the correct answer, so I move on happily.
Here is an example, the type of question most commonly given on the coursework. I am given a formula:
y = A * exp(k * t)
I am told the values of y at two values of t (e.g. "when t=1, y=63.2") and asked to find the values of the constants A and k. I convert to logarithmic form and make a pair of simultaneous equations. Solving these give the values of A and k (let's say 40 and 0.33). Now the formula is
y = 40 * exp(0.33t)
For the second part of the question I am asked to predict the value of y when t is some greater value; t=7, say. To do this, I put t=7 into the equation.
The crucial point to notice here is that I can check whether my values for A and k are correct by putting the values I have found into the formula and asking what is the value when t=1. Since I have been told the value at t=1 is y=63.2, say, I can calculate my value at t=1 and see if they match. If they match for both the points given in the question, I have the correct values for A and k. Using my values to predict the value at t=7, say, I am now completely confident I have the answer correct. There is no need to check the answers in the back. In this mathematics the truth is absolute.
I have tried to stress to the students the power of working the answer back into the question like this. I did this in my A-Level Mathematics and came out of the exam confident I had an A because I knew I had answered so many of the questions correctly. (In Further Mathematics and at degree level it is not so absolute, but at this level of mathematics as a toolbox it is the case). Given the question we can find the answer; given the answer we can find the question; etc. After all, this is how I set the questions in the first place, picking an answer arbitrarily and working back from there to form the question.
Two points come from this line of thought:
- There might be benefit in getting students to set their own questions, since they will see the symbiosis of question and answer and this could help them in solving this sort of problem.
- It might be interesting to set questions which are back to front. In some ways we do this already, the second part of the question above (predicting the value at some future time t) is really just finding values to set up the question with. However, to bring this home the answer could be given and we ask "what was the question?"
And of course, this process of proving the answer correct is far more realistic. There are no answers in the real world.
# of course, there is always the possibility that the answer given is not correct and this has happened in the course notes. It is most gratifying when a student notices an answer that is incorrect, since this shows real deep understanding and confidence (and represents a valuable life lesson: we are all only human and prone to err), but it can really destroy the confidence of a student who has not yet fully built up their understanding.