I have a confession …. I bloody love a two-way table!! There! I’ve said it! …
Lots of you will see this as just a topic we teach and then move on from but for me, along with Venn diagrams and dare I say it … frequency trees (for me they aren’t a topic in their own right but that is a whole other blog post!) are tools for categorising and organising information that I use a lot in my teaching. I would probably add that I find myself saying “draw” a picture a lot – sometimes this will a stick picture to illustrate the story or a bar (typed bra then and made myself chuckle so badly I couldn’t “not” share!). They have the added benefit that when I’m using them in my modelling of other topics students automatically see some familiarity and are slightly more at ease. Let me explain.
Recently I’ve done a few sessions on problem solving and have come to the conclusion that one of the issues is that some teachers are looking for a panacea that they can teach students and then “Voila! Miraculously students will know how to solve problems!”. It doesn’t work like that! For me, there are lots of “tactics” (some of you will insist on calling them “strategies”, which, again is a whole other blog post but there is a massive difference between a strategy and a tactic!) that we use to tackle so-called “problems” and most of them you will already know. One of our issues is that when we use them we don’t verbalise that we are using (one of many!) problem-solving tactics and so students think that the only way to tackle a problem is using that specific method and don’t always see that it can be applied to other areas of maths.
As a topic, your teaching of two-way tables probably goes along the lines of the below where in the first instance you get them to complete a two-way table, extract information from it, use the information for (usually!) probability and then being able to design and use their own two-way table from words.
This is all well and good but they can be used for so much more!! For example, with reverse mean type questions such as this one:
With any two-way table I would encourage students to draw the table and GIVEN information in one colour (or pencil), ticking the information as they go along and then any calculations they’d done in another colour so that they could minimise the work in trying to unpick any mistakes if they needed to. It limits that frustration of “but Miss, its long!” when they get the final answer wrong! In terms of the above question I’d get them to lay it out as below …
Sometimes through exposure to a variety of reverse mean questions students will learn when to use the end “totals” column as in the below question one isn’t required but then it doesn’t make sense to use one either! Its all about the usefulness of summarising the information in a table format:
Another topic they could be used for are those horrible liquid A, liquid B and liquid C are combined to make liquid D type questions for density/mass/volume. Now is not the time for the discussion about how to teach the “actual maths” … no really, I’m not getting into the whole “Formula triangle/No Formula triangle” argument here! Again the thing I want to point out is how useful tables are for organising the information in the first place and for students to get their heads around the “story” of the question. The example below is actually about metals but you get the idea … in some contexts I’m not expecting students to get full marks on these type of questions but I know that by organising the information they are likely to get some way into the problem:
I feel the need to elaborate (don’t roll your eyes like that!!) … I’ve just written about some students getting someway into a question and I use the example to students/parents about wanting to get from here (wherever I’m sat at the time!) to the town centre. So the first thing I need to do is get to the end of the road and make a judgement call on which way to turn … oh there is lots of traffic going in that direction so I’ll go that way … then I get to the next road junction and have to make a similar decision. Sometimes I’ll get to a signpost that I recognise and then know what to do but sometimes I just have to keep making judgement calls etc etc. The fact is unless I make a start I will NEVER get anywhere and sometimes doing maths is like that … routes will open up that weren’t initially obvious at the outset but the first step is the most important one. Just try something!!
Another topic that students could do with some scaffolding for is speed/distance/time … again the way you teach the topic is up to you and what I’m really asking you to do is to think about how you extract the information from questions and then how you represent that same information … does it help your students? The below question is really quite straight forward but you can see that I’ve deployed several different tactics here: drawing a picture, ticking off the information and drawing a table of information.
In the below example you can see that its very similar to the density/mass/volume questions. Again pointing the similarities and the differences to other topics is a really effective thing to do … it helps make links for students and does ease their pain (as some of my students would say!) in learning new stuff!
This is by no means a definite list of where and how I’d use two-way tables to organise information but hopefully its given you food for thought.
As always … may or may not be useful!
*Most (if not all) of the questions are from Pearson/Edexcel … thanks!!! Don’t kick my ass for using them! Pretty please!