Having decided to start blogging to help focus my professional reflection I suppose that I should articulate a little bit about what my teaching actually is. I’ve spoken in person to people about various parts of my perspective on teaching but have never gathered it all together in one place. So here goes…
My ambition is for the following outcomes for every pupil in my classes:
1. Able to apply algorithms to solve standard problems 2. Have confidence to attempt to apply knowledge to non-routine problems 3. Understand the concepts underpinning the topic 4. Have some sort of idea of why we are studying the topic 5. Enjoy the experience
I make no claims that all 5 outcomes are met, even most of the time for most of the pupils. I do, however, feel that what I do has a positive impact. My exam results have shown a positive trend, across various levels for a number of years. Pupil and parent feedback has been positive. Some might ask if all of the above is important. I don’t know if there is an absolute truth to it all. To me the above is important - that is why I aspire for it.
I don’t believe that all lessons should follow the widely accepted template of starter, middle, plenary. I am more interested in the progression over a number of lessons. In my opinion successful Maths teaching is far too complicated to fit into an identikit template. For that reason I can’t describe what a ‘typical’ lesson would look like in my class. It can vary a lot. Sometimes there is a starter task, sometimes not. There is rarely a mention of a learning intention(a future blog post). There is sometimes a textbook although, increasingly frequently not.
There is nothing revolutionary about my approach to teaching. I simply want the students to do most of the thinking and most of the work. I really do view myself as being there to nudge and guide them in the correct direction. I’m not an a strong advocate of any particular approach. Usually I will introduce a topic through:
Pupils draw conclusions by observing some pattern in a variety of cases
E.g. sketching straight lines from a table of values and developing their own comprehension of y = mx + c. This task might be done in pairs or individual. Similar approaches for transformation of trig, quadratic and log graphs etc.
Posing a number of problems pupils already understand while adding extension to introduce new topic
E.g. For teaching area of a sector I would ask pupils to find area of a circle, semi-circle and circles with angles of 90, 45,60 and 120 degrees indicated. The key question in the task would be the next question. Where the angle is, say, 17 degrees. Pupil discussion and then whole class discussion would pull together and solidify what the pupils where meant to conclude.
Direct instruction mixed with investigation/illustration
E.g. I introduce exponential graphs through various practical ideas. One of which is to ask pupils to count the number of sections created for each fold of a piece of paper. Pupils often come up with y = 2^x and are able to graph it themselves with minimal input from me. I might prompt that we create a table of value and suggest testing when x = 0, 0.5, -1 etc.
Engaging Real Life/Pseudo Real Life Scenario
E.g. I will discuss a real life scenario and demonstrate a new skill which ties in. Dan Meyer has a nice idea on Standard Deviation which I adapted and often use. It starts with a game on guessing ages of celebrities. It leads to a teacher led conversation (with a lot of planned questioning) which results in the concept of Standard Deviation. http://blog.mrmeyer.com/2007/how-old-is-tiger-woods/
The Ronseal approach - does what it says on the tin. ‘Here is some new material we have to learn’. Even with this approach I would try to bring it to life using Desmos/Autograph or Smartboard software.
Of course, a lot of the above applies when introducing a new idea. A lot of lessons are follow up lessons where we are consolidating or adding a slight extension. In essence though I do find that kids ‘get it’ more when they have participated through some form of directed activity rather than just been lectured to.
I’m not scared to take a little bit of time in class developing an idea and letting a concept secure itself at the expense of routine practice. Of course routine practice is important. For example for core skills such as adding fractions I will ask pupils to add a lot of fractions together! More interesting, though, is non routine practice.
Lower down the school, I often push the boundaries of what a pupil is capable of achieving with the knowledge they have at that point. I’ve always felt if I pushed a pupil slightly past what was required for ‘just now’ then the current content would seem more trivial to them. Eg: sub x = -1/2 into y = x^2 - 2x - 12 This is harder than a typical second year test question, but is a combination of second year skills. Later in the school, once a pupil has a fairly solid footing with a topic, I like to twist it and manipulate the questions as much as possible to ensure the pupil can not only handle the unexpected but also to ensure that the pupil actually understands the concepts underneath.
The preceding paragraph does seem like it is focused on more able pupils, however, I apply these same approaches, at different levels, with all pupils. I would never just teach National 4 to a class who were looking at Nat 5 in S5! That sounds like a future blog post in the making.
I’m not anti-textbook, but sometimes I’ve felt that they can cover up for a lack of planning on my own behalf. My new department has pretty much no class sets of textbooks, they use a lot of these legendary (at least in Scottish Maths teacher circles) worksheets: http://www.knightswoodsecondary.org.uk/personal/Resources/Hillhead/Resources_hillhead.htm. As they were written by one of our colleagues. I know of good exercises in various books that I can rely on, but a lot of the time there is nothing that has been published commercially that does the job. A classic example is for logs and exponential at higher. To give the kids a good amount of practice of the various ins and outs I’ve often felt that a combination of Maths In Action, Heinemann, other worksheets and past paper questions is required. Neither of the books on it’s own is sufficient (nothing has changed from what I see in the latest updates for CfE).
Rigour I am by no means into ‘trendy’ educational ideas. True, I do engage with the likes of John Hattie and the work he has done on the relative effect sizes of various L&T approaches. However, I am very traditional in the sense that I like my senior classes to copy down notes from the board - where I model (what I hope are) mathematically rigorous examples of how I would like them to attempt their own workings. I have a real dislike of ‘tricks’. I don't think that we should be trying to make maths easier, as such. Instead I think that we should helping the kids become equipped with the skills required to engage fully with the subject. For example, I never teach ‘change side, change sign’ for solving linear equations. This isn’t for any pretentious reason such as it not being mathematically rigorous (which it isn’t) but because it doesn’t fully equip pupils for later and, in fact, can lead to a lot of misconceptions.
4x = 2 x = 2/-4
I would rarely, if ever see this from a kid taught the balancing method for equations. Where as, from my experience of marking with SQA, tutoring kids from other schools etc, I see this is a common sort of mistake. The culprit is almost always that they have this idea of solving equations. Why not just persevere in S1/2 with balancing. ‘Whatever you to do one side you do the other’, ‘ what is the inverse of addition/multiplication?’ etc. Pupils do find balancing harder initially, no doubt. But by giving up on them and going with ‘change side, change sign’ I feel that a dis-service is being done. The same applies to cross multiplication - whatever the hell that is! Consider solving an exponential equation. To eradicate the exponential we take logs of both sides. Everything becomes seamless and fluid. If you start with a rule such as ‘change side, change sign’ then you are building on an unstable base to begin with and the further you go, the harder it is to say that the rule always works. Because it doesn’t. I know this might be controversial - but that’s where I’m at!
I can make similar arguments for novelty ideas of adding fractions etc. None of it makes any real sense and we aren’t doing the kids any favours, particularly if they are going to take their maths forward. These tricks and shortcuts work for very specific cases but aren’t sustainable. So why waste time on them?
I like to do one large homework task per week. Occasionally a question or two to do that evening, but not much more. Pupils have a lot of other subjects to focus on too. Every senior class (in my place that is really S3 up) get an ‘ink exercise’ every week. I will mark two out of every three of them. The one I don’t mark will be marked by the pupil from a solutions sheet I issue in class. I issue forms which pupils complete indicating how many marks they got for each question. I feed the info from these sheets into a spreadsheet. Even though I know the marking isn’t always accurate I still get a picture from the class of where the difficulties lay and can then plan lesson starters/future homework questions accordingly. The homework the pupil self mark is never new content. It is always a revision exercise of work previously covered. I’d love to mark for every class every week, but it’s infeasible. Being PT, having a heavy timetable, being a dad of twin toddlers and generally wanting to have a wee bit of a life I find that two sets of jotters is what I can manage, particularly if Higher/Advanced Higher is one of those sets. In first/second year I rely much more heavily on peer/self marking from a set of solutions. I run round the class to check homework is done and have and talk to pupils about questions they have got wrong or are unsure about. I try to mark a handful of formal exercises for each S1/2 during a term. However, I just can’t seem to manage to incorporate it into my work schedule with any sort of regularity. To this end, S1/2 exercises are often drill exercises designed to reinforce key skills. A lot of my feedback and knowledge of S1/2 pupils comes form AiFl approaches in class such as using mini-whiteboards etc. I do think this is an area where I need to improve. This leads nicely to…
I’d say that my strength lies with Higher and Advanced Higher. I have rarely had many/any fails at either of those levels and tend to help get the kids better grades in maths than they get elsewhere. I’m don't feel that I add as much value for the kids who struggle most with maths. I’m aiming to improve on this over the next couple of years. I’ll be focusing a lot of time on lower ability S1/2 classes. Because I’m now PT I’m investigating ideas such as the mastery curriculum models which seem to be having some success in England. I’m looking to improve the quality of my explanations for these pupils too and also find better resources than I’ve used previously. I feel I’ve done OK with these classes in the past, but could do better.
All of the above is, of course, generalisation. It is how I think today. It wasn’t how I thought five or ten years ago and I assume my opinions will continue to change over time. I like to think I can be adaptive and if that means compromising my principles occasionally if I genuinely think it will be best for a particular child/group of children then I will do that.