The art of teaching Maths

My philosophy is that how to explain concepts an ideas is key to being a maths teacher. Interesting and fun activities are important but they come after the introduction. I will be writing about the best ways to introduce different concepts from my experience as a mathematics teacher and a mathematics graduate. Enjoy and share :).

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Many solutions

Quick post today to get kids thinking about different solutions to the same problem: Why does each row of Pascal's triangle add up to increasing powers of 2?

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Dividing with fractions

Dear Mathletes, This is a topic where teachers are just itching to say flip the fraction and multiply by the reciprocal. Wait!!! This is another great opportunity for students to discover this for themselves. Start off by getting students to divide by a half, a third, a quarter etc by analysing what it means to divide ie how many halves, thirds, quarters for in to so many whole ones. Now move on to dividing by two thirds or three quarters where the answer is a whole number. Here they can use diagrams to find how many two thirds for in to 6 whole ones for example. Now it's a case of asking probing questions for students to make connections and links. They can test out their theories and use a calculator to test it. Try it and promise yourself you will not mention any tricks! Happy teaching :)

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Solving equations

Dear Mathletes, To solve equations can start with two bits of advice so simple that you can give students nearly a whole lesson exploring solving different types of equations. This is opposed to introducing first with the variable on one side then on both sides then with brackets then with fractions and bla bla... 1. Whatever you do to one side of the equation you must do to the other. Never mention crossing the equals sign and becoming a negative, there is very little understanding from students in this. 2. Think 'what do I want to get rid of' and get rid of it in order to meet your objective of reaching x= That's it for the lesson until the end when it is important to allow students to discuss strategies and patterns in order to get rid of coefficients and constants. Try it, ten try following it up with a lesson of giving them ten tough equations to work on in groups to make it competitive. I always say I will give them a mark out of 10 but I'm not telling you which ones are wrong :) Happy teaching Mr Mathlete

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Good problem to start students thinking, analysing and making links between what they have learned. Great for when you are studying the binomial expansion.

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Introducing Differentiation

Dear Mathletes, Today I want to share with you some discovery maths. It is very tempting for teachers to just begin with the power rule of differentiation and express how simple this is to students. But actually this is robbing them of an opportunity to discover it for themselves and, in the process, develop their analytical skills. Using the graphing software of Geogebra, students can find the gradient at different points on the graph using the tangent line function. If students just focus on quadratic functions to begin with they can use their knowledge of linear sequences to generate the gradient functions. They can then use this in order to make predictions for cubic graphs and other types of non-linear functions. This is a great topic to allow students to explore and get them to develop not only their mathematics but also their learning skills. Please try! Mr Mathlete

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Introducing adding fractions

Dear Mathletes,

This is a topic close to my heart as one of my first experiences from teaching was to teach (or reinforce as I thought) adding fractions to 14 year olds. I presumed they knew the basics but sadly not and I was met by a sea of blank faces as I crashed. Clearly they had been taught but there was a lack of understanding.

The way I approached this with year 7 students was to make sure they understood the concept of equivalent fractions. We used diagrams and I started giving them the freedom of writing down whichever equivalent fractions they wanted. I then became a little strict and told them the denominator I wanted. This then progressed to adding fractions where they were already comfortable with making equivalent fractions so they just had to make the choice of the correct ones with the same denominator. It was also important to show through diagrams if we add 1/3 and 2/5 we get something we can’t describe using a fraction from looking at it whereas 2/13 added with 5/13 gave us a diagram with a fraction they could describe. This led them to the idea that denominators must be the same.

This took a few lessons but if we look at the bigger picture they won’t have to do the same thing again every year :)

Let me know what you think,

Me Mathlete

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Big Questions

On reflection, a lot of the posts I have done are to promote higher order thinking in students. I want to get away from questions where the one and only objective is to get an answer and the process has no learning outcome. So why not ask big questions to get students looking at the process. Here are some I have used: 1) when are two equations the same? Useful in year 11 when solving equations with graphs. Also with younger years to introduce balancing equations, you can get them to pair up equations that have started to be solved using the balancing method. 2) what is the general formula? 3) under what circumstances would you apply the binomial theorem? Can also replace binomial theorem with other methods. Get students to come up with their own questions to show. 4) y -x = 4, what does this give? Shared with me by my head of department, I can't wait to use it when introducing graphs to students. This type of thing needs to be slowly introduced to students. Try it at the end of a lesson and set out your expectations. Try 4 minutes on your own and 4 minutes sharing with a partner to get students reflecting on their own. Mrmathlete :)

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Be Vague

Hi all, Students think Maths is all about getting the right answer, wrong! Giving vague instructions can help students to see that they can go further and they can take control of the topic and answer the questions they are curious about. I gave the instruction to some students to sketch the graphs of some functions and label on the important points of the graph. The first thing I was asked was 'what are the important points?'. I told them to make their own judgement and this caused them to really think how to draw a graph but also what happens to these 'important points' if the conditions change. They took it further without even me asking them. The graphs were reciprocal graphs which is upper school stuff but I will definitely be trying this with all the students I teach in a lot of topics! Mr Mathlete

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Exploring mathematical representation

This is something I feel passionately about... It is the relationship between graphs and functions/equations/expressions. Students sometimes like the algebra but make no connection with the graph and I think teachers can accentuate this problem by seeing an obvious connection. I let my top set year 11 explore the relationship between a quadratic function and it's graph. They used completing the square, the quadratic formula and factorising into double brackets to explain how the graph was behaving in relation to the function. They could then move onto making predictions of a graph just by looking at the function. For kids in KS3 this should be done with straight line graphs, dedicating a few lessons to the discovery of gradient and y-intercept. Give students a sheet with a few guided questions and bring the discussion together later. A great way to do this is to have groups of four and then get two of each group to be nomads so at ten minute intervals they move to other groups so they can spread their views while also listening to others'. Also packages such as geogebra will help students to explore and explain. To just go through a step by step process of drawing a table then entering x-values and finding y-values then finding coordinates and blah blah blah helps no one and doesn't forge any connections. There is so much to do on this topic and expand into! Please let me know if you have any good ideas to do this! mrmathlete

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National holiday

No Maths teaching today as it is a national holiday so I will share with you this website which is quite popular... Diagnosticquestions.com Get a question from this up, let everyone vote and if everyone gets it right then it's time to move on. If some get it wrong it is a great discussion point as incorrect answers are misconceptions. So you can get students to back up their answers. If students finish early then get them to figure out how students can get the other wrong answers, they seem to enjoy finding the mistakes! Don't tell them the right answer straight away, get them to debate and you will find everyone will realise the correct answer eventually. If you've used this a different way let me know! Some teachers use it to check knowledge, I like to use it as a way to introduce a topic, see if students can figure it out themselves... Day off and still managed to talk so much! Happy teaching

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New School Year, one new focus

Dear reader, New year, time for some fresh ideas mixed with the successful tried and tested ones... I had an epiphany that good mathematicians are risk takers, they are not afraid to get things wrong and in doing so they get closer to the right method as they narrow them down. I realised that sometimes when a student gets something wrong we provide the write method too quickly or hints too quickly. Here is an a question for your risk taker students to have a go at - this would be for high ability students as I was trying to get students to be more creative, especially those bright students who just want to learn method after method and reproduce it on a test! What whole integer values of n will make the following an integer: (n - 9)/n^2 - 1) They should try and do something with the fraction and I think you will find they will get there in the end. Also I won't tell you the answer to bring out the risk taker in you! Happy risk taking, Mr Mathlete

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Day 8 - Ideas for pre starter

Dear all, apologies for delay in this post, I have just got back from holiday and am refreshed for a new term!

Before the holiday I expressed the importance of a pre starter, today I will start with some ideas.

The first idea will help you to link from the previous lesson. At the end of a lesson I like to set a challenging question so kids are still thinking as they leave the class. They do this question on a piece of paper by themselves (because this skill of learning by themselves is needed as well as team work) and then hand it to me as they leave. An exit slip with a twist. For the next lesson I then distribute these around the class to mix up the pairs and students discuss their answers and will mostly all come to the correct answer without you having to give them it. Some will also have been thinking about the question outside of class which is just what you want! 

This idea has so many benefits not least that students are making the most of every lesson and have something they know they will be doing right from the start and can get started on their own while you deal with any small tasks!

Try it!

Tomorrow more ideas to make the most of your lesson,

Mr Mathlete

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Day 7 - the importance of a pre starter

Again, like yesterday, a tale from my qualification days. I was in a lesson to observe and pick up ideas, it was a year 11 lesson and I sat waiting while the kids came in. It took about 10-15 minutes before any work started while the teacher chatted to students and waited for everyone to get ready, what a waste! I also just had a learning objective and starter on the board while students came in. Some finished in a couple of minutes while some took close to 10 minutes. While they were working it was still a waste. I now have a pre starter students can do without my help before we start the lesson. It can be some number or algebra questions or it can be a problem solving question. Another type is a question that gets them hooked, one that will be answered using today's skills so students can get involved more in the learning. For example I got my year 12s to derive the double angle formulae and adapt it in different ways asking them what it was used for. They didn't know so we explored the usefulness, each time wondering if we would need it. It was a great lesson, lots of energy in the room. Eventually we didn't use it and it only increased their interest, they still want to know. One day they will find out I told them, one unassuming day I will slip in a question... I am such a big fan of the pre starter I will be sharing some of my best over the next few posts, Keep reading! Mr Mathlete

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Day 6 - Rank them

When I was just starting as a Maths teacher I observed a class on simultaneous equations. It was with a C/D borderline class and they were obviously struggling with it. The teacher gave them one type of question and they solved it together and then students practiced but as soon as they came to a question with a slight difference such as negatives they were stuck and the class had to stop again and go through this type of question. The class proceeded like this with no real understanding going on. I have since faced simultaneous equations with a weaker group and they understood the basic concept. To progress them I gave them a group of scattered questions on the board and asked them to rank them from hardest to easiest and to describe the reason why they had ranked them. This caused discussion amongst students about the difference between the work they could do and these slight variations. This meant all the problems had been solved by them to turn the questions into ones they could solve and they could practice. I have used this method time and again and it does get students to focus on one particular aspect of a questions rather than seeing the whole thing as one big scary worksheet. It also removes the fear as I didn't expect them to solve these questions although if they were feeling really brave they could after they had ranked them.

Until tomorrow,

Mr Mathlete

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Day 5 - Create your own

We spend our teaching careers dishing out questions for students to solve. It would be, however, much simpler for students to create their own questions. This is not an innovative thought but it is important that students can explain through their writing about how their question is a little harder than the ones we have already done. If they create multiple questions then how their questions have developed with difficulty. This gets students to consider the misconceptions themselves while also dealing with a solution path. It is much more likely to stick this way as they are putting more deep thought into their work. I am a strong believer that writing about their work is an important attribute to have and also helps all students to understand more if they are able to explain their working. Students understanding is the upmost importance to me, I repeatedly tell them that there are no 'tricks' taught in my class. 

Hope you are having a good start to the week :)

Mr Mathlete

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Day 4 - What's the point?

Students spend their education being fed facts and how to apply them. Have we ever stopped to ask students why are they being taught this?

For example, while teaching a year 12 class, students discovered a few identities. I then posed the question of why are they learning this? I gave them the task that they must try to come up with some questions where this is applicable. This worked extremely well with students who came up with the exact type of questions that they would solve. This exercise made students practice a skill we rarely deal with in class, the skill to identify what techniques to use to solve questions. This is only addressed when students actually take the tests themselves. In a lesson students deal with one main concept, they therefore know what concept to use to solve questions in that lesson. However this does not prepare them for a test when they have all the concepts they possibly know to choose from. Understand?

To further explain, my year 9 class were studying pythagoras' theorem, to solve a question of this nature you need a right angled triangle which they know and will always search for but when it came to a test most didn't spot when to use this concept as it was relatively new to them...

Question to ask: 'What type of questions can we apply this to?'

Until next week,

Mr Mathlete

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Day 3 - Extension ideas

We can fall into a trap as teachers to just give harder examples in order to extend students. It makes sense but students will reach a point where they will study this harder example. Why not teach a student to be analytical of problems not just teach them a step by step method. At some point we want students to be set free so they can form their own methods. I tried a very simple extension activity today.

The activity:

After solving questions I asked students to group the questions whichever way they wanted. Instead of just going through the answers we discussed the way some students would group them (I got them to write about it to practice explaining their ideas and I find this produces more deep thinking). This produced great discussions about problem solving and we discussed all the skills needed and the possible steps that could be taken to solve the questions so after the discussion they didn't need the answers, they could already see their mistakes and how to move forward.

Here is my theory:

We are not just teaching students to solve equations, add fractions etc, we are teaching them to be problem solvers to look for solutions, not just in Maths but in everything they do. By presenting students with a topic and giving them a point by point solution to copy we are robbing them off learning this skill. As a Maths graduate, the way I look at complex problems (in my spare time because I do genuinely love maths) is to look for a path from the information I have to the solution. I don't always have a path to begin with, sometimes just a few steps but with those steps a longer path forms and I can get to the solution. I always encourage students to try things, if they get it wrong nothing bad will happen. This analysing technique should not be ignored, it should be taught and taught well if we are to produce independent life long learners. Try it, when it gets to a point where problem solving is a must it will be too late for you to start.

Until tomorrow

Mr Mathlete...