Each summer I find I have the time to reflect on what is currently working well in my teaching and, importantly, what improvements I might be able to make in the next school year. Of course these improvements should be about making things better for my students, but if they save me time and stress in the long-run then that is also a worthy goal!
With reference to the latest (research-backed) buzzwords that are flying around these days, I’m keen to build in more ‘retrieval practice’. In particular, for my Further Maths classes, I quite often use warm-up questions that bring to the forefront any prior knowledge that the current topic depends on. In my usual style, these were always made up on the fly, but now I am thinking about tightening up on my planning in this area.
Here is that same screenshot, with additional annotations to help with my explanations below:
Warm-up with a purpose
The questions numbered with Roman numerals are designed to be tackled by the students independently at the very start of a section of study. The intent is that they recap prior knowledge, but with a focus on specific skills that will be called upon in the current section. In my example above, questions (i) and (ii) are clearly designed to remind students about solving ‘hidden quadratics’ of the form that will arise in the subsequent exercise set. Question (iii) brings the concept of a long-term value to the surface and that feeds in to question (iv). The real purpose of these is to understand the behaviour of tanh x in questions (3) and (4) and, indeed, to help us later with graph sketching. Question (*v) is marked with an asterisk, indicating that I don’t expect all students to attempt or fully complete it. (It is there for those who are quick to complete (i)-(iv) and is still useful for those who do tackle it.) In general, questions marked with a * are either optional or more challenging than necessary! In this case, a student who completes question (*v) should have no trouble sketching y = tanh x independently.
I wanted a way to highlight the most fundamental knowledge for each section of study and opted to do this via the blue ‘key knowledge’ boxes. This information is only presented once and, for example, later warm-up questions ask students to write the exponential definitions down themselves, or sketch the graphs etc – more of that lovely retrieval practice! (My hope is also that these boxes will reduce those annoying, recurring questions in class: “what does the graph of sinh look like again?…” The students will either have a better memory or at least an easy way to look things up.)
I am not intending these sheets to completely replace the textbook. Ultimately, I find books helpful as a repository of practice questions and indeed a number of my ‘questions’ on the sheet are simply directions to complete specific work from the textbook. However, there are particular skills that I want to highlight and also certain, subtle types of scaffolding that I want to put in place. Thus the questions with Arabic numerals will be a significant part of classwork and I might even choose some to use as examples with the class.
In the screenshot above, I’ve grouped questions 8-12 together as an example of the scaffolded/ramped practice. I phrased question 8 in particular to provide a strong guide to the method of its solution. This scaffolding is then taken away (although, for example, question 9 deliberately says “each solution” to suggest there is more than one). By question 12, students are working at the standard required for this particular skill. Questions like 12 will then pop up in the “warm-up” questions for later sections in this same chapter – more retrieval!
You will likely notice the intention of questions 14-16, too, foreshadowing the next section of work that will look at the inverse hyperbolic functions.
It’s my intention to end each section with at least one extension question of some sort, although I have not always been able to think of appropriate questions for all sections yet. These will typically come from STEP papers (the STEP database makes it relatively easy to search for problems on a given topic).
Even with this very first page, I already notice at least one small change that I think would be worthwhile: in the warm-up section I feel it would be good to have a question asking students to simplify e^ln(3) and e^-ln(3) as preparation for questions 5-7.
One of the perennial questions I’m asked (by both students and teachers alike) is “So, where are the answers?”. And I shrug. My own plan is to make much greater use of OneNote next year so that a lot of notes, examples and solutions from class will all be collated in one place. If I make a neat solutions page one day then I’ll be sure to share it, but please don’t hold your breath!
Absolutely! I’m a strong believer that homework has value when students are set work within their capability but that includes ‘desirable difficulty’ and, again, a large part of my homework next year will be… retrieval! Our students complete A-level Maths in year 12 before moving on to the FM content in Year 13. I am currently thinking of the following structure for a 10 questions-per-week homework sheet:
- 4 questions (2 pure; 1 mech; 1 stats) of Maths revision
- 3 questions of FM revision (that is, testing topics what were studied 3-4 weeks or more previously*)
- 3 questions on the current FM topic of study
* our students actually start some FM topics at the end of year 12, so those will form the FM revision section for the early weeks of September
Absolutely! Our school policy is actually to have a test every week. In the past I have always focussed these very specifically on the most recent topic of study. However, next year I think they will be structured in a similar vein (and probably corresponding topics) to the weekly homework sheet.
So, what next?
Watch this space, essentially. Over the course of the summer I will be chipping away at the ‘Core Pure 2’ content and tidying up these class sheets. Drafts for three of them are already online but these will be revised again before the summer is out, so don’t rush to download and save them! Also, I’ll be sure to blog again once term has started to reflect on how well (or not) things are going…