Allow me attempt to generalise our current curriculum in a way that hopefully makes some sense and explains why Year 10 Quadratics (or parabolas for those on the other side of the Atlantic) is a topic that I can say, more than anything else, I teach differently to how I was taught.
Imagine solving a problem is a dish. Take graphing a quadratic/parabola and ravioli with pasta sauce. To graph it, requires a certain method, which can be followed by a recipe – not too big of a deal. Different levels of experience and confidence may result in consistency, time taken to prepare and cook, and probably quality – but by following the method, we should still be able to cook the dish. What is essential, however, is the ingredients. We’re going to need to go to the store to get the ingredients…
Now, imagine that the school year is like a conveyor belt at the grocery store, but unlike the ones at the registers, this stops for no one. Ingredients get scanned if they can be picked up, their barcode found or weight weighed in time. Once scanned, the item and it’s price is printed on the docket. What appears on the docket represents what students have been able to pick up and retain in their long term memory. If an item is not scanned (e.g. the belt was moving too fast for all things to be scanned in time), you walk out of the store without this ingredient. Now, let’s suppose that if you didn’t get something scanned at the register, you need ensure that you get it next time you go through the register. If the ingredients are the different concepts that make up the curriculum, then technically most students should get most items then pick them up the following year right? Hmmm… well, in addition to the items that weren’t scanned, you’ve got another list of ingredients to include, some of which are only useful if you have something else (imagine bread is angle properties and butter is trigonometric ratios). This compounds very quickly and soon enough you have a large amount of unscanned and now missing ingredients. In my experience, I have found that many students in my class come with a wide variety of ingredients in their basket with the same conveyor belt and running at the same speed. What does this result in? Students are trying to follow a recipe they have insufficient ingredients for. In the classroom, this is teaching content to students with massive gaps in their understanding – not even misconceptions, gaps.
So, what do we do? Well, have you ever been craving a certain meal only to realise that you don’t have the certain herbs and spices necessary to make that exact dish? One might know how to work around this, substituting certain ingredients for others to get a similar effect, like adding vinegar to milk in the absence of buttermilk for fluffy pancakes. This usually involves extra preparation or several other ingredients, like not knowing square numbers when simplifying fractional indices – there are ways to work around it but it’s much easier to know that 27 is also equal to 3^3. Readers of Craig Barton’s book, “How I Wish I’d Taught Maths”, would know that there is potential of Cognitive Overload if students don’t have all of the required ingredients on-hand and have to find work-arounds for most things on the list.
I’m not knowledgeable about substituting ingredients in the kitchen, so instead I would find a different recipe – one that either has minimal ingredients or is less intricate and more similar to the types of dishes I’ve made in the past. In addition, the dish needs to seem like a worthwhile exertion of time and energy and be achievable – I mean, who wants to spend hours trying to recreate the secret herbs and spices when you can just go to KFC and buy it. This is where I think a lot of work still needs to be done, as I wonder if we’re asking students to cook meals they don’t want to eat. Here’s how I try to get my students to cook meals that are still substantial:
- Whet their appetite using a Microwave Meal (Develop the need to learn)
- Use semi-prepared ingredients like pizza bases, dried spaghetti, beef stock etc. (understand the process in a simplified way)
- Learn how to make dishes from scratch or more complex and tastier versions (develop a deeper level of understanding)
Start with a Microwave Meal
This activity got students, who have literally no idea what a quadratic/parabola actually is, using them to create models to solve a problem – Dan Meyer’s Will It Hit The Hoop. I’ve always used this task but never as an introduction to Quadratics/Parabolas until now (I used the Desmos version after first playing one of the shots on the TV). As we all know, we can’t live on microwave meals, but they are quick and easy to cook, require no ingredients or preparation, and you get the essence of the meal. My students did the same with graphing Quadratics.
From here, I introduced a specific WODB problem (above) from Mary Bourassa to create the need for terminology about the different features of Quadratics (i.e. vertex, roots, concave, convex) and then backed this up with a Desmos Polygraph Activity to get the students to become more specific with the language they used to show them how terminology allows us to be more targeted and effectively descriptive. For our cooking analogy, this was like learning about the key ingredients of a dish, so that they know what they are when they see them written in a recipe.
The Quick’n’Easy Meal
Next, we looked closer at specific features like roots and the vertex and tried some ways of preparing these ingredients to create bigger meals. I did this by graphing out Quadratics in factored form and vertex form and hitting play on the slider which changed its position and then the slider which changed its shape.
5, 6, 7 lessons in, my students were able to graph Quadratics to pass though specific points and in different forms, without once putting pen to paper. Was I intentionally avoiding paper? No. Technology, here, allowed my students to engage with Quadratics and learn about them without any prerequisite knowledge. Like buying pasta sauce from aisle 2 and ravioli from the refrigerated section, my students had moved from microwave meals to cooking semi-prepared ingredients to make their dish. An appreciation of preparing these things was growing progressively.
Just like Nonna used to make
Next, we explored standard form and, what inevitably arose was expanding and factorising. Like learning how to prepare your own sauce and pasta, these things need some pretty essential ingredients on hand – any chef will tell you that the quality of these matters. Tomatoes for a great sauce are like times-tables for factorising and expanding. Many of my students would have been having pasta without sauce if I started here, as preparing the sauce would have been all too much – remember at this stage, they wouldn’t have yet realised that they were capable of cooking anything at all. It was important that they saw the value in making their own sauce – expanding and factorising equations. My students have been used to graphing equations (making the dish) and if they could get an equation from standard form into either vertex or factored form, they’d have their meal. From here, it was all about refining their skills to do more intricate recipes to create a wider variety of sauces (completing the square) and specific cooking techniques (quadratic formula).
This is all significantly different to how I was taught. Typically, Quadratics is quite an algebra-heavy unit, but starting from this point would have been like asking students to make a pasta dish from scratch when they haven’t even heated up a plate before. I retrospectively feel that this approach actually rings true to ideas surrounding teaching Novice learners (as I’ve assumed students know pretty much nothing to start with) while also maintaining an inquiry approach.
The reason I’ve written this is to show that there is a place in between. In society, we can easily find ourselves looking through a bias lens based on our perception of the author or speaker. Craig Barton, I feel through his book, How I Wish I’d Taught Maths, has allowed me to overcome any bias and freely agree or disagree with key points he raises, realising that research should inform what we do but not determine it. I have heard of some instances where teachers are being told which specific worked examples they are expected to show students, which I believe is an over-the-top interpretation of outcomes from researchers with little to no teaching experience. Craig, however, has outlined where he came from as a teacher and where he is at now – so at the very least, I can start from a place of agreement and be challenged from there, changing as needed but not unconditionally. I feel that this is really important to realise that teaching with an inquiry approach is not completely void of explicit instruction and vice-versa, and also that although processes and rationale may vary, neither one is completely superior to the other. I do, however, believe one on their own is lesser than somewhere between the two. I’ve used a bell shape curve below to try and show this in a simplistic way:
*my curve was far more skewed before reading Craig’s book, I’ll let you guess which direction.
So, in remembering that just because students don’t have all of the ingredients to make the most authentic dish from scratch, it doesn’t mean they still can’t cook. It definitely doesn’t mean that students should work on perfecting to dice tomatoes to make a delicious sauce (mastering times-tables to be able to factorise quadratic equations, if you haven’t caught onto the analogy by now). Also, we can’t slow down the conveyor belt or go back and re-scan all items that were missed, rather find a way that allows our students to still be chefs in their own right.