in looking at all of the aspects of maths that are expected to be covered over the course of a year I feel that statistics offers some really great opportunities. It allows a really authentic way for students to get into some real maths with data that matters to them and to others. It is not that there is not authentic ways to engage with the other aspects of the course and get into some real maths, it just sometimes seems like it doesn't seem like an authentic situation for them, there is not the emotional buy in. With statistics you have the opportunity to introduce some very provocative data sets or ones that directly speak about them. You can have some really great discussions about how statistics are used to make decisions, how others use them to make decisions that effect you, to look at how you or other people can use them to build convincing arguments (even if the stats are quite deceiving).
I am at the very start of my unit on statistics, but I wanted to start it with them having a discussion about some data, their own diagnostic testing data. I chose to use this data for a because sometimes I feel they don't see the use of doing the testing, they don't always treat it properly and that skews our data. I also chose it as I feel that they do not always understand the results of the data, I wanted them to have much more awareness of what the data tells them and of how we use the data as eduators.
The last person to talk picks the next person to talk (from those with a counter out) by throwing them a ball or similar, only the one with the ball can talk. Once a person has used all of their counters they then cannot contribute any more to the discussion, they can sit and listen, but that is it. The teacher in this process is a facilitator, but tries to stay out of the discussion as much as possible. They provide the provocation to start off with, they pick the first person to talk and they will ask questions to stimulate the discussion if it dies off completely. In this role you need to avoid jumping in to help with that I tend to give myself the same number of talking stones that everyone else gets, if they only have three opportunities to add to the discussion then I only have three to stimulate it, it forces me to be strategic as well. This is hard when you completely disagree with what is being said, but you need to let students be the ones to respond them and to challenge that thinking.
I gave them two prompts to start their contribution with they could start with
When students think about presenting their solution to a mathematical problem visually I get very excited because I find those visual solutions hard to generate myself, I am getting better it, but I don't see the mathematics in that way when I am solving a problem generally, I get too wrapped up in the equations in the formal mathematical notation. The reason I get excited about those visual representations for a problem is that they often are much easier than the formal mathematics and they lead to some great mathematics.
The other day I did a task with my class that I got from Professor Peter Sullivan from Monash University who I had the pleasure of working with once per term for three previous years in my work as a numeracy coach. The question is as follow
I like this question on a range of levels
One of the groups working on this task came up with the solution shown below, it has been altered from the original (which was just dots showing the same information) but the same thinking was there. They first started by looking at the 0.7 and 0.6 and converting them to 70% and 60% which they then turned into 70/100 and 60/100 which simplified to 7/10 and 6/10. From this they reasoned that ten was the minimum number that could be on the train. From here they attacked the main part of the problem by first assuming that there was as much sharing as possible, making sure everyone with a computer also has a backpack. This gives six people having both items, this is shown by the image below on the left. They next thought about orgainsing the same but with as little sharing as possible, by making sure everyone who didn't have a backpack needed to have a computer. This scenario gave the image on the right below, an overlap of three. Hence they gave a range of answers as being between three and six out of ten people.
The animation below shows they range of answers with more clarity
What occurred to me as I was documenting their thinking on the board is how nicely this representation led into the intended purpose of the lesson, Venn diagrams and two-way tables. Simply by circling the laptops in one colour and circling the laptops in another I had generated a fairly accurate Venn diagram as shown below. You could clearly see in each case how many were in the intersection of the two sets, how many belonged to only one of the two sets and how many were outside of those two sets.
What I really liked about this is now easily it made the transition to Venn diagrams and two-way tables. from our initial starting point of probability of simple events.
Sometimes ideas for lessons get away from me and take me to some unexpected places, and sometimes the reaction of classes to those lessons are also unexpected. I was thinking about a lesson idea for circumference and I thought about a lesson I saw once from Andrew Stadel (lesson link here). Where he wanted students to determine how many rolls of tyre it would be for a tyre to hit a target, I liked to lesson but was not sure how my students would "buy in" to the task. I wanted to redesign the task a little to give students a greater reason for getting into the maths.
Like with many of my ideas I ran it past a teacher at the school that I often collaborate with on tasks such as this and we came up with what I think is an interesting alternative to that original task.
Students had to determine where on the ground they were going to put a chocolate frog so that it met two criteria:
The twist with this task was that they were only allowed to use 1 item from their bag to measure both the tyre and the distance on the ground.
In relation to this task I think there are three incredibly important aspects to it. Firstly is that they really have to think about the positives and negatives of anything that might be in there bag in relation to it's use as a measuring device, there was nothing in anyone's bag that could accurately measure the circumference of the circle in one measurement so they had to think about aspects such length, flexibility and having actual measurements on the device. Secondly was the idea that the criteria and the restrictions I provided made the task of achieving it quite difficult, Using something that was not ideal as a measuring device meant that there was going to be inherent errors in the way they could possibly measure the tyre. They had to be very accurate in what they used as the measuring errors could compound quickly as they measured both the tyre and the ground.Finally there was a high degree of ownership over the task as they seemed to really want to be able to not have a squashed frog. That little element of competition was very valuable in this task, That element of competition I have found to be very useful in a range of tasks I have done previously.
The Approaches and Challenges
The range of measuring devices people used to measure the tyre was quite interesting. Most chose their ruler from their bag to measure it. However since the ruler was only 30 cm and the tyre was significantly bigger than that both in circumference and diameter they needed to make lots of measurements and therefore compounding any measurement errors. Most were also using a rigid wooden ruler to measure a curved surface which made the measuring more difficult again, some had a more flexible ruler that they could wrap around, but some of the compounding errors remained. Many of these students decided to use a ruler as they could then stick within their comfort area of standard units of measurement.
Others chose to use headphone cables as measuring devices, they found them to be much more flexible and longer than the rulers, therefore they could accurately follow the contours of the tyre and it was long enough that the measurement errors were not as significant. However they seemed to encounter difficulty when dealing with lengths that were not quite a full headphone cable length. They were not sure how to figure out what fraction or proportion of the headphone cable was left over. When Introduced to the idea of folding the headphone cable to figure out how many parts were used they found this area. For example in the image below if they measured the tyre and got a total diameter of the blue part of the line below, they could fold it as shown to give an approximate length of just over 3/4.
The other part of this that students seemed to find challenging is using headphone cables as a unit of measurement in a mathematical formula. The seemed to have this impression that the formula for circumference of a circle would only work if you were working in standard units of measurement. They didn't realise that If you said that the diameter was 3/4 headphone cables then the diameter would be 3.14 x 3/4 = 2.36 headphone cables. This however was not unexpected, in mathematics we get so used to dealing with measuring devices with standard units of measurement that we do not often realise that sometimes the non-standard units of measurement can be just as valid and just as accurate.
Some of the students who engaged in the task a little more strategically came up with multiple ways to measure the tyre using the same device. The measured the circumference directly, but then they also measured the diameter and used to formula to determine the circumference, where there was significant differences between the two values they took measurements again to verify their results and if they were accurate then they had some discussion about which of their measurements they felt was more accurate. They realised that a discrepancy of 3 cm was not much but when you multiply it over 10 rolls you have just potentially lost a third of your target area.
The final challenge most of them faced was where to place the frog because that question is not as simple as it seems. If you put it too close to where you think the tyre will finish and your measurements are off then your frog will get squashed. Conversely if you put it close to 1 m away from where you think the tyre will finish and your measurements are off then you could be outside the target range. Most people put it closer to the tyre and a few put it closer to the end of the 1 m, very few considered putting it where they thought of the middle of the range may be which in my humble opinion is where I would be aiming.
Overall when we rolled the tyre the level of excitement was high, they were all keen to see how their calculations faired. Of the 8 groups that placed frogs down 1 group was more than 10 rolls away but outside the 1 m target range, four groups had their frog squashed (but by less than 15 cm) and 3 groups met both criteria. They seemed to really enjoy the activity and everyone got to eat the frogs at the end.
Below is a video of a lesson I have done for the last few years that is in relation to area. Students are required to determine both the number of post it notes that will fit on the board and the time it will take to complete the task.
In getting students to complete the task I first deprive them of the measurements I start with a set of estimation exercises. First they use the video of the first 20 as shown above as a way to estimate the time and the number for the whole board. They are then shown an image of 100 post-its on the board (shown above) and are asked to adjust their estimate and give reason for what change they made and why. This process is repeated with 200 and 300.
What was interesting about the process of this activity this year is the strong desire of the students for the measurements initially. The just wanted to apply a formula they knew to the problem, they placed absolutely no value on the process of estimating both the number of post its and the time, they did not see it as a valid way to approach the problem.
However in reality estimation is an incredibly important skill for students to become comfortable as we use it all the time, in fact we use it a lot more than we use an exact calculation. In an article I read a few years ago (I can't remember the source of the article now) it stated that of all the maths we doing in our lives 80% of it is estimation, only 20% requires an exact calculation. If this is the case then the estimates I was getting them to do is arguably a more important process than getting them to calculate the exact numbers. However I also feel that little emphasis is place on estimation within mathematics classes, our desire as teachers often is to apply the formula and to deal with exactness of the solution, we don want to look at the messy and imprecise nature of an estimate.
Students also seem to have a fundamental level of misunderstanding of how to estimate. In some of the diagnostic testing we do this misunderstanding of estimation also becomes evident. When asked how they would estimate the value of 18 x 79 the most common response is to calculate 18 x 79 and then give the answer. Other common answers have students performing the calculation and then rounding. The number of students who calculate 20 x 80 instead as a good estimate is very small but is the most appropriate way to approach the task. It makes me think that estimation needs to be a much larger section of my program than it currently is and I know that one very good source of questions requiring estimation is Estimation 180. It is a great website that potentially has tho ability to provide students with 1 estimation question for each lessonn of the year as a warm- activity.I am also thinking more about how I can get students to estimate before they calculate more often over the course of their year.
I am so proud of the kids I teach today. Since the start of the year we have had a much stronger focus on patient problem solving than they might have been used to in the past and until today there has been a lot of resistance to attempt some of the tasks. But today they were superstars, it is the first time where almost all of them had persisted in solving a problem, even when they found it difficult. There had been pockets of this in the past, but to see most of the class doing it today was fantastic.
The lesson that we decided to look at was based on an ad produced by the motor accident commission in South Australia. The idea behind the lesson, which they came up with at the start, was to find out how long they were actually in the back of that taxi. For the sake of trying something a bit different we combined three maths classes with three maths teachers to see how it would go.
Students were then asked to generate some ideas about what is the sort of information that they might need to know to answer the question and they came up with
From here they need to have some information in order to be able to progress forward firstly was how they calculate the taxi fares. Students were given the formula below, but were not shown how to use it at all.
Cost of the fare = Distance cost + Waiting Time Cost + Flagfall
It was great to see how students had made the decision that they needed to compile information from several sources realising that they needed to use the time of departure to find out what tariff they would use and then using the tariff information to relate back to the distance traveled. This continuous movement between different sources of information was really great to see. The path they took from this point was pretty different with each group. some were able to work through it themselves and some needed some further prompts, but very few actually gave up on the problem. Even when they did not have much success initially they really took on the prompt and tried to incorporate it into their thinking and that is why I am so proud of them today.
It is always a little nice when I still find some of the answers I get for questions I give to students surprising. It is not that the answers are unexpected, it is more that sometimes I get excited by the solutions.
A local supermarket chain on and off for the last few years has been running a bit of a promotion. The promotion involves a series of 108 animal cards. For every $20 spent at their store you get a pack of 4 animal cards. I have a three year old daughter who loves these cards. She loves collecting them and seeing which ones we have missing. I know that over the course of the promotion we have collected a lot of cards and have got a lot of doubles, it got me wondering how many cards I would need to collect (doubles and all) to collect the full set of 108.
and sometimes less gaps, but the 20 represents the average number of cards missing. At 400 cards the average missing drops to about 2, a substantial gain in cards. However at this point you have spent $2000 dollars at their store. In running the multiple simulations of the 400 cards there was only one or two occasions out of the dozens of times tested that I managed to get a full set of 108 out of the 400 cards.
Raising this again to 600 cards there is still a gap. about a third of the time there was a full set cards, but in two thirds of cases there was an average of one card missing, it was only when taking the number of cards to 800 that we began to consistently got a full set of cards. At 800 cards you have spent $4000 in their stores.
Having to spend $4000 to in a sense guarantee a full set of cards that probably cost them less than $5 to produce is not necessarily something that sits well with me. Part of the money generated is being used to support a few zoo's, so it is not so bad. In reality people will swap cards with other people to fill the gaps, and they are likely to spend this money any way, they are not doing more shopping just to get cards. However if the parents of these children used to shop at another supermarket chain, then the introduction of these cards. if only for a short time, has no doubt swung quite a bit of business their way. These were the sort of discussions that were so valuable in conducting this task with the class.
the other way we try and make the work relevant is by linking the work to their interests using things such as sport to calculate statistics and optimal angles, but due to many students dislike of mathematics quite often what comes back is "why are you trying to ruin football for me my making it into a maths lesson". They enjoy the sport and playing it, but don't want to know about the maths involved.
The argument about relevance also becomes about who it is relevant to. If you make the lesson about football, then it is not personally relevant to anyone who does not like that sport, this could easily be more than half of your class.
The idea of relevance can be used very successfully for engagement, there have been some highly relevant lessons that I have taught, that students have also found highly engaging. But I have also had some highly relevant lesson fail miserably. Over the last few years I have found that embracing the idea of presenting students with ridiculous problems has increased student engagement more than anything else.
So how do I define an ridiculous problem
Some of the questions of this type I have had success with in the past are as follows
All of these questions immediately gained the students interest, the looked at me like I was insane, but also had that look of "I want to know the answer to this". As can be seen they are all very much related to something they have seen before, money, burgers, post-it's nuggets but it takes how they have thought about these things far outside the realm of what we consider "normal". It is this deviation from the normal that students find really interesting and the work they have produced on these sort of tasks has been quite stunning. In my observations of students working on these type of tasks it is clear that they are discussing the work more with others, coming up with stronger, well reasoned responses and also add value to the task for looking at ways they can extend it such as looking at how many calories in the 100 x 100 burger or the cost of a 1000 x 1000 burger. In particular the chicken nuggets question still continued to have students talking about it 8 weeks after the task had finished, students even went to McDonalds and tried to order that number of nuggets to see what they would say.
Ok, before I get too many hateful thoughts from people who may read this, I don't have anything against literacy, I think it is an incredibly important skill that all people must develop and it is an incredibly important part of being able to talk about mathematics. My issue is not with literacy itself, but more with the way the word literacy is being used, particularly in relation to terms such as statistical literacy and financial literacy. In trying to make my point I have taken the following definitions for statistical and financial literacy from Wikipedia
Statistical literacy is the ability to understand statistics. Statistical literacy is necessary for citizens to understand material presented in publications such as newspapers, television, and the Internet. Numeracy is a prerequisite to being statistically literate. Being statistically literate is sometimes taken to include having both the ability to critically evaluate statistical material and to appreciate the relevance of statistically-based approaches to all aspects of life in general (http://en.wikipedia.org/wiki/Statistical_literacy)
Financial literacy is the ability to understand how money works in the world: how someone manages to earn or make it, how that person manages it, how he/she invests it (turn it into more) and how that person donates it to help others. More specifically, it refers to the set of skills and knowledge that allows an individual to make informed and effective decisions with all of their financial resources. (http://en.wikipedia.org/wiki/Financial_literacy)
In looking at those definitions it should be clear that these are not only very important, but it should also be clear that these are not literacy skills, they are numeracy skills. For many it would be "so what, it is just a label" but it is much more than that. As it is, numeracy is not a term that is respected as much as it should by the general public, and people cannot generally identify the aspects of their lives that deal with numeracy. By taking two very large aspects of numeracy and labeling them with a literacy tag, people start to associate them less with numeracy and more with literacy.
It seems like the term literacy is being associated with every aspect of learning considered an essential life skill. Apart from the ones already mentioned there are terms such as technological or computer literacy, emotional literacy and physical literacy, this was just with a quick Google search. Although not necessarily numeracy skills, they are also not literacy skills, just terms given the literacy tag again to hopefully gain support for their importance.
It is interesting that people are starting to believe that these things are really literacy skills, and not numeracy skills. Having spoken to some teachers from other schools, teachers who work in English faculties, they believe skills such as reading and constructing graphs are truly literacy skills, but I also doubt that these skills have ever been taught in an English class, but they are taught every year in Mathematics classrooms. Just because there is a fact, figure, graph or table as part of a written text, it doesn't make it a literacy skill, simply it is just a part of the text that needs some numeracy to fully comprehend it.
The argument has been made for years that you can't be numerate without being literate, and with that I do agree, at least to some degree. Many mathematical problems require the comprehension and decoding of written texts or problems and being able to transcribe them into a mathematical construct. However I would also argue the reverse is true, that you can't been literate without also being numerate, it is a reciprocal relationship not a one way relationship. Often texts include numbers that students may not recognize the size of, graphs and tables that they find it difficult to interpret and map, flowcharts and other diagrams requiring spatial reasoning that they may not be able to fully interpret
If we are to have any significant impact on both numeracy and literacy levels in students we need to acknowledge that they are separate skill sets that are connected by the idea that you cannot fully have one without the other. Central to doing this is acknowledging the numeracy that is present and not forcing the literacy tag onto everything.
The title of this post is a question I hear almost daily in my Maths classroom. I could retire a happy man if I was given a dollar for every time I have heard it "When am I ever going to use this?". The question bugs me a bit, because they don't ask the question anywhere near as much, if ever, in Science and I am sure it doesn't happen very often in other subjects either. There is something about Maths that makes people believe that everything they do, every lesson will be a mathematical concept that they can take out and immediately apply to their everyday life, this just doesn't happen and to be honest none of my answers in the past have necessarily hit the mark for the students.
Take for example solving quadratic equations. More than 95% of all students who walk through the door will only need to solve quadratic equations for the purposes of year 10 Mathematics they will never need to apply it to any job they have or any task they will need to complete as part of their lives, at least not solving quadratic equations in the manner they occur in class. Even if it is something they will need for their job, in most cases they will have a piece of software that will solve it for them. So how do you justify studying it to students who will only ever use this concept for 5 weeks of their lives or will have a computer do it for them?
My new thinking on how to tackle this question has two trains of thought. Firstly is to show them how they are already unconsciously using the Maths that we are studying. For example lets take the situation of a person kicking a football to a running player. This is an incredibly complicated situation as essentially you are kicking the ball into empty space with the hope that the person that the person will be there to catch it by the time it lands. As the person kicking the ball you have to pick a spot to kick to, you then need to judge how fast the person is running and use this to estimate how long it will take them to reach that spot. You then need to determine how hard to kick the ball and at what angle to make sure the time of flight of the ball is exactly the same as the amount of time for the running person to reach that spot from their current position
Complicated huh? it involves estimation, extrapolation, solving linear equations, solving quadratic equations, probably a bit of physics and trigonometry. The maths in this sort of problem would be too much for most high school maths classrooms, but yet our mind does it almost seamlessly through a mixture of experience and practice. Part of addressing this question of where will I ever use this is to show them how they are unconsciously already using this maths, the lesson then becomes about putting a formal mathematical construct to what their brain is doing automatically. Therefore the question then is answered by showing them how much maths actually permeates their lives, whether they like it or not, the universe the live in, is written in the language of mathematics.
The video below illustrates this point well. What I like most about it however is that it was made by high school students for a video contest called Math-O-Vision. Therefore it shows they maths that they see in day to day life.
I think the second part of addressing the idea of "when am I ever going to use this?" is in getting students to rethink the purpose of maths education, what it really should be about. For me mathematics is about learning to think in a logical way and it is also learning to think in a completely different way to most other subjects. Ideally a maths class presents you with a complex, messy, complicated problem and it requires you to look at what you have, what you need, and then to proceed with solution that follows a clear line of logic, it is not about the continual accumulation of a range of skills. In approaching maths in this kind of way, it is about trying things, making mistakes, learning from them and trying again, it is not about following the carefully structured approach that the teacher has given you. Therefore I think if students see mathematics is a way of looking at problems they may encounter in a completely different way, if they realise that mathematics is about teaching them how to think, then to some extent the question may also go away.
Senior Leader of Pedagogical Innovation and Mathematics Coordinator in Regional South Australia.
Opinions in this blog are my own and do not necessarily represent the views of my employer.