I watch a lot of sports anime (Japanese cartoons about basketball teams, volleyball teams, swimmers, figure skaters, you get the idea). In watching these cartoons, I have noticed that there is a distinct difference between the Japanese approach to exams, competitions and events, compared with the British or American approach.
The idea for this blog post hit me in the face when I was watching Free! Iwatobi Swim Club in Japanese with English subtitles, and I noticed what I thought was an odd translation. If you are familiar with anime you might have heard the Japanese expression "Ganbatte!" which literally means "try your best" and is used often before exams, competitions, and fighting evil pirates (OK that one wasn't a sports anime). However, in the anime I was watching, ganbatte was translated as "Good luck" in the subtitles which struck me as weird, because I knew the correct translation. That's when it hit me: before exams and competitions, British people say "good luck", whereas Japanese people say "try your best". Could this very simple habit be the reason Japanese students outperform British students in education?
Think about what "good luck" really means. It implies that the recipient can only do well in the exam by some fluke. It implies that their own knowledge and skills are not good enough, and that the only way for them to succeed is if they luckily manage to get easy questions, or they luckily guess how to answer the questions. They are implying that you don't really have any influence on your success in the exam, and it is all in the hands of fate. This is a very fixed mindset way of thinking.
Consider the Japanese mindset instead: by saying "ganbatte" you are simply encouraging that person to try their best, implying that the harder you try, the better your result will be and the greater your success. This is very growth mindset.
People of Britain: please stop saying "good luck" to students before exams, to performers before performances, and to teams before competitions. Let's start saying "try your best" instead. Maybe this subtle shift in focus is enough to encourage more of a growth mindset in this country.
And for more motivational tips, you really should check out some Japanese anime or manga. Assassination Classroom, in particular, is a good one for teachers wanting to inspire their students.
Emma x x x
Monday, 5 December 2016
Thursday, 1 December 2016
Why UCAS Grades Are Anti-Growth Mindset
I hate doing UCAS predicted grades.
In September, us year 13 teachers are asked to predict what grade we think the students will realistically achieve at the end of the year, and these are put on their university application form (the UCAS form). These have to be realistic and not inflated, because that's the only way that the system would be fair. As these predictions are made so early on in the year, we generally base them on their AS grade and maybe the first couple of assessments of year 13.
But here's the problem: many students are disappointed with their AS grade and aim to improve on it in year 13. They work twice as hard, now that they realise how difficult it is, and as a result many of them often do improve on their grade. But as a teacher, you can't necessarily predict this, and even if you think this might be the case, you can't really justify putting the UCAS grade higher if you have no evidence that they will improve.
I have a student who got a grade C in her AS Maths. I made her UCAS grade a C, because that seemed reasonable. But I think she underperformed last year and with a bit of extra dedication and a lot of support from me and my colleagues, she could get a grade B, or even higher. So when she asked me whether I could raise her UCAS grade for her so she would have a better chance of being accepted by universities, it was difficult for me to say no. If I said no, I would be telling her I don't think she can achieve above a grade C. Or at least, that's what she would think I was telling her. And this could affect her self-confidence, and could even become a self-fulfilling prophecy.
If we give a student a low UCAS grade (because evidence suggests that is the grade they are most likely to get if everything remains the same as last year), then we are sending the message that we don't believe they can improve. One of my ex-students came into school the other day and we were talking about how his UCAS grade was a D (which was perfectly reasonable, as he had only got an E at AS) but he ended up getting a grade B. He said, "I proved you wrong". This actually made me quite upset! He had obviously spent the whole year thinking I believed he could not achieve higher than a D. Now maybe this is what motivated him to go on to achieve a grade B, and hence it was a good thing his UCAS grade was so low. But it still upsets me to think that he thought I didn't believe in him!
Conversely, giving a student a high UCAS grade might give them a false sense of security. If they think they are very likely to get a grade A*, they might not push themselves as much. They might see their UCAS grade as the minimum grade they will achieve with the minimum amount of effort.
I believe that predictions can be very powerful in influencing outcomes. When I was revising for my third year exams at university, I made predictions of my percentage scores for every exam I took, wrote these on a piece of paper and stuck them on the wall above my desk. I made predictions before I started revising but these predictions were eerily accurate. Ever since then I have been convinced that what we believe will happen, will happen. I have actually made a vision board for this year's A2 results, and stuck this on the wall above my desk in the maths office. On it it has every year 13 student's name, and the grade I need them to achieve in order for us to get an ALPS grade 1. These grades are not the same as the students' UCAS grades, they are higher. So it doesn't make sense that I'm telling the students I believe one thing when I'm really aiming for another.
I don't want to make the students' UCAS grades too high because it seems against the rules of UCAS, and I want to do things fairly. But am I being silly? Do all the other schools inflate their grades on little evidence? Should I just predict them all A*s and then work my socks off to make sure they get those grades?
Anyone else in a similar situation? What's your policy for making UCAS grades?
Emma x x x
In September, us year 13 teachers are asked to predict what grade we think the students will realistically achieve at the end of the year, and these are put on their university application form (the UCAS form). These have to be realistic and not inflated, because that's the only way that the system would be fair. As these predictions are made so early on in the year, we generally base them on their AS grade and maybe the first couple of assessments of year 13.
But here's the problem: many students are disappointed with their AS grade and aim to improve on it in year 13. They work twice as hard, now that they realise how difficult it is, and as a result many of them often do improve on their grade. But as a teacher, you can't necessarily predict this, and even if you think this might be the case, you can't really justify putting the UCAS grade higher if you have no evidence that they will improve.
I have a student who got a grade C in her AS Maths. I made her UCAS grade a C, because that seemed reasonable. But I think she underperformed last year and with a bit of extra dedication and a lot of support from me and my colleagues, she could get a grade B, or even higher. So when she asked me whether I could raise her UCAS grade for her so she would have a better chance of being accepted by universities, it was difficult for me to say no. If I said no, I would be telling her I don't think she can achieve above a grade C. Or at least, that's what she would think I was telling her. And this could affect her self-confidence, and could even become a self-fulfilling prophecy.
If we give a student a low UCAS grade (because evidence suggests that is the grade they are most likely to get if everything remains the same as last year), then we are sending the message that we don't believe they can improve. One of my ex-students came into school the other day and we were talking about how his UCAS grade was a D (which was perfectly reasonable, as he had only got an E at AS) but he ended up getting a grade B. He said, "I proved you wrong". This actually made me quite upset! He had obviously spent the whole year thinking I believed he could not achieve higher than a D. Now maybe this is what motivated him to go on to achieve a grade B, and hence it was a good thing his UCAS grade was so low. But it still upsets me to think that he thought I didn't believe in him!
Conversely, giving a student a high UCAS grade might give them a false sense of security. If they think they are very likely to get a grade A*, they might not push themselves as much. They might see their UCAS grade as the minimum grade they will achieve with the minimum amount of effort.
I believe that predictions can be very powerful in influencing outcomes. When I was revising for my third year exams at university, I made predictions of my percentage scores for every exam I took, wrote these on a piece of paper and stuck them on the wall above my desk. I made predictions before I started revising but these predictions were eerily accurate. Ever since then I have been convinced that what we believe will happen, will happen. I have actually made a vision board for this year's A2 results, and stuck this on the wall above my desk in the maths office. On it it has every year 13 student's name, and the grade I need them to achieve in order for us to get an ALPS grade 1. These grades are not the same as the students' UCAS grades, they are higher. So it doesn't make sense that I'm telling the students I believe one thing when I'm really aiming for another.
I don't want to make the students' UCAS grades too high because it seems against the rules of UCAS, and I want to do things fairly. But am I being silly? Do all the other schools inflate their grades on little evidence? Should I just predict them all A*s and then work my socks off to make sure they get those grades?
Anyone else in a similar situation? What's your policy for making UCAS grades?
Emma x x x
Thursday, 17 November 2016
Draw Your Brain: a Growth Mindset Activity
Today I shared an activity with my colleagues and they seemed to like it so I thought I'd share it here too.
I told my colleagues (Maths, Science and Learning Support teachers) that they needed to get into the mindset of 15 year olds so that they could get the most out of the demonstration. Of course then one of my colleagues got a little bit too in character and drew an item of male anatomy on his piece of paper. Actually, knowing this particular colleague's personality, he would have done this even if he wasn't pretending to be a fifteen year old.(Typical Physics teacher).
But anyway, I had given everyone a piece of A5 paper and a felt tip. I told them to pretend to write on their piece of paper, and pretend they were answering a really difficult maths problem. They had to pretend they had worked on it for ages and they were really struggling. Finally, they come to an answer. However, a minute later, their teacher tells them their answer is wrong. I asked my colleagues to think about the emotions they would experience then. They gave me some examples: anger, frustration, embarrassment, disappointment. Then I told them to take all of those emotions, and channel those emotions into crumpling their piece of paper up into a ball. I told them to really make sure that all of their anger, frustration, and feelings of failure were screwed up into that ball of paper. And then I told them to throw their piece of paper as hard as they could, and with it release all of those negative emotions.
Then I asked them to retrieve their piece of paper and uncrumple it, and smooth it out so it's nice and flat. I told them that the piece of paper represented their brains. All of the tiny crease marks on the paper are the synapses, or pathways, inside their brain. I got them to draw over the crease marks with a felt tip pen, and as they did that, I asked them to think about the knowledge flowing through their brains thanks to these pathways. Every time you make a mistake, your brain gains an extra synapse. The only way to gain extra synapses is through making mistakes. So all of the connections and pathways in your brain are due to making mistakes. The piece of paper representing their brain would not have any synapses at all if it hadn't been crumpled up. Those pathways are only there because they made a mistake.
With students, I then tell them to keep this picture of their brain in their folder, and every time they make a mistake in Maths, they should look at it and think about how their brain has just gained another synapse. They can even draw on another synapse each time they make a mistake, and by the end of the year they will be able to see how much their brain has grown, and how much progress they've made.
This is a great activity to do with a class that lacks confidence or is stuck in a fixed mindset. I did this with my year 11 intervention class last year, and I think it was a real turning point for them.
Please try this with a class and let me know what impact it has!
Emma x x x
PS I got this idea, plus many others, from the book Mathematical Mindsets by Jo Boaler.
I told my colleagues (Maths, Science and Learning Support teachers) that they needed to get into the mindset of 15 year olds so that they could get the most out of the demonstration. Of course then one of my colleagues got a little bit too in character and drew an item of male anatomy on his piece of paper. Actually, knowing this particular colleague's personality, he would have done this even if he wasn't pretending to be a fifteen year old.
But anyway, I had given everyone a piece of A5 paper and a felt tip. I told them to pretend to write on their piece of paper, and pretend they were answering a really difficult maths problem. They had to pretend they had worked on it for ages and they were really struggling. Finally, they come to an answer. However, a minute later, their teacher tells them their answer is wrong. I asked my colleagues to think about the emotions they would experience then. They gave me some examples: anger, frustration, embarrassment, disappointment. Then I told them to take all of those emotions, and channel those emotions into crumpling their piece of paper up into a ball. I told them to really make sure that all of their anger, frustration, and feelings of failure were screwed up into that ball of paper. And then I told them to throw their piece of paper as hard as they could, and with it release all of those negative emotions.
Then I asked them to retrieve their piece of paper and uncrumple it, and smooth it out so it's nice and flat. I told them that the piece of paper represented their brains. All of the tiny crease marks on the paper are the synapses, or pathways, inside their brain. I got them to draw over the crease marks with a felt tip pen, and as they did that, I asked them to think about the knowledge flowing through their brains thanks to these pathways. Every time you make a mistake, your brain gains an extra synapse. The only way to gain extra synapses is through making mistakes. So all of the connections and pathways in your brain are due to making mistakes. The piece of paper representing their brain would not have any synapses at all if it hadn't been crumpled up. Those pathways are only there because they made a mistake.
With students, I then tell them to keep this picture of their brain in their folder, and every time they make a mistake in Maths, they should look at it and think about how their brain has just gained another synapse. They can even draw on another synapse each time they make a mistake, and by the end of the year they will be able to see how much their brain has grown, and how much progress they've made.
This is a great activity to do with a class that lacks confidence or is stuck in a fixed mindset. I did this with my year 11 intervention class last year, and I think it was a real turning point for them.
Please try this with a class and let me know what impact it has!
Emma x x x
PS I got this idea, plus many others, from the book Mathematical Mindsets by Jo Boaler.
Monday, 15 August 2016
Using Circle Geometry to Hunt Pokemon
If you know me in real life, it will come as no surprise to you that I have spent most of my summer holiday so far playing Pokemon Go. Those of you who don't know me in real life, if you see a pink-haired twenty-something with a phone precariously attached to a Pikachu lanyard wearing dungarees (optimal number of pockets) and Go Walk Sketchers (optimal walking shoe) walking at as close to 7km per hour as possible (optimal PoGo speed) then that's me.
After a few weeks of aimlessly walking around Coventry city centre or sitting on the steps of the bank where some kind stranger will always drop a lure, I decided that I had caught enough Drowzee and would like to make a bit of progress towards catching 'em all. My pokedex was hovering around the 80 mark for a long time, despite making trips to Leeds, Manchester and Birmingham in the hopes of finding exotic local pokemon and hatching eggs like Bernard Matthews. So I decided to give pokemon tracking a try.
In the bottom right-hand corner of the screen when you're in map view, there is a window you can expand that says "sightings" and lists some pokemon that have spawned nearby. Sometimes I would see a rare pokemon on this window, but learned helplessness has taught me that the rare ones never pop up when you want them to. By "pop up" I mean appear on your map as a tappable, catchable pokemon. So I decided that instead of wandering aimlessly and hoping for the rare ones to pop up, I would hunt them down strategically using geometry.
A pokemon will appear in "sightings" when you are within 200m of it. The pokemon will be catchable when you are within 70m of it. So when a rare pokemon, let's say a Charizard, appears on your sightings, you should be picturing the following diagram:
Of course, you could be anywhere in the purple circle (but not in the pink circle, or the Charizard would already be catchable). The fact that the pokemon just appeared in your sightings could either mean that you have just stepped inside the purple circle, or the pokemon has just spawned. Pokemon disappear after 15 minutes, so what you do next needs to be efficient and at a bit of a jog if possible. If you have a buddy with you, this is much easier, as you'll see in a minute.
Let's call the point you're at point A. What you need to do is identify a straight path that you can walk along that goes in both directions from point A. This can be very difficult, depending where you are. I have found that it is much easier in a park than in the city centre. Now, walk along that path, remembering where you started. Ideally, you would count your paces as you walk. Keep walking until the Charizard disappears from your sightings. The point where that happens we'll call point B. This will be a point on the circumference of the circle in your head. Of course, you could get lucky and walk right into the pink circle, in which case, get the razz berries and ultra balls ready! But let's assume the unluckiest situation.
Next, you need to turn around and walk in the exact opposite direction, back to point A and beyond it until the Charizard disappears from your sightings again. Call this point C. If you have a buddy with you, they can do this bit whilst you are doing step one, to save time. Again, it would be good if you could count your paces.
Now that you have identified two points on the circumference of the circle, and have walked a chord of the circle, this is where the geometry comes in. The perpendicular bisector of any chord of a circle always passes through the centre of the circle.
If you don't believe me, think about this: take a random chord of a circle and join up its end points to the centre as in this diagram:
You should be able to see that this makes an isosceles triangle, because two of the sides are radii.
This means that this triangle has a line of symmetry and if you cut it down this line, you get two right-angled triangles:
And clearly this line of symmetry goes through the centre of the circle.
So, back to our hunt. We're at point C, and we need to find the perpendicular bisector of the line segment BC which is the path we have just walked. The first thing we need to do is find the midpoint of B and C. In some places it is easy to do this by eye, if you have a good map or if you're in a very flat area. But if you have counted your paces, you will be able to find the midpoint much more accurately. I personally don't bother with counting. Because we're only trying to get inside the pink circle, not get to the exact centre, we don't have to be that accurate. So, walk to this midpoint, which we'll call D.
Now, turn ninety degrees and walk. But Emma! (I hear you cry) There's two ways of turning ninety degrees! Yes, you're right. At this point, you have not uniquely defined the purple circle. If you draw two random dots on a page, there are always exactly two different circles with a given radius that pass through those two points. You don't know which circle it is, so you have to guess. So turn ninety degrees in any direction and start walking (or running!) until one of two things happen: the Charizard pops up and you catch it, or the Charizard disappears from your sightings, in which case you do a 180 degree turn and run for it! If you have a buddy with you, you can take one direction each and invent some kind of signal for "I found it!" (smoke signal? A whistle? Make a sound like a dying giraffe?)
You have to do all this in the space of 15 minutes which can be tricky, and sometimes involves running and looking like a loon. But me and my husband went out to Coombe Abbey country park and Coventry's War Memorial Park last week and managed to use this method successfully several times. The handy thing is, pokemon can't spawn just anywhere, there are a set number of spawn points in a given area. So once we identified some of the spawn points, we didn't even have to use the method, we could run to the nearby spawn point we found earlier. There are still enough different spawn points around to keep the game challenging though.
In case you were wondering, my Pokedex is now up to 94. I promised my tutor group that I would have caught 'em all before term starts again in September, which is looking very unlikely. Then again, my tutor group promised me they wouldn't fail their AS levels, so I might have a bit of leverage there...
I hope this method helps you hunt down some rare pokemon and maybe understand the relevance of circle geometry a bit more. (Oh my gosh, did I just come up with a "real life" application of geometry??? Noooooo! Keep Maths pure, people!)
Emma x x x
After a few weeks of aimlessly walking around Coventry city centre or sitting on the steps of the bank where some kind stranger will always drop a lure, I decided that I had caught enough Drowzee and would like to make a bit of progress towards catching 'em all. My pokedex was hovering around the 80 mark for a long time, despite making trips to Leeds, Manchester and Birmingham in the hopes of finding exotic local pokemon and hatching eggs like Bernard Matthews. So I decided to give pokemon tracking a try.
In the bottom right-hand corner of the screen when you're in map view, there is a window you can expand that says "sightings" and lists some pokemon that have spawned nearby. Sometimes I would see a rare pokemon on this window, but learned helplessness has taught me that the rare ones never pop up when you want them to. By "pop up" I mean appear on your map as a tappable, catchable pokemon. So I decided that instead of wandering aimlessly and hoping for the rare ones to pop up, I would hunt them down strategically using geometry.
A pokemon will appear in "sightings" when you are within 200m of it. The pokemon will be catchable when you are within 70m of it. So when a rare pokemon, let's say a Charizard, appears on your sightings, you should be picturing the following diagram:
Of course, you could be anywhere in the purple circle (but not in the pink circle, or the Charizard would already be catchable). The fact that the pokemon just appeared in your sightings could either mean that you have just stepped inside the purple circle, or the pokemon has just spawned. Pokemon disappear after 15 minutes, so what you do next needs to be efficient and at a bit of a jog if possible. If you have a buddy with you, this is much easier, as you'll see in a minute.
Let's call the point you're at point A. What you need to do is identify a straight path that you can walk along that goes in both directions from point A. This can be very difficult, depending where you are. I have found that it is much easier in a park than in the city centre. Now, walk along that path, remembering where you started. Ideally, you would count your paces as you walk. Keep walking until the Charizard disappears from your sightings. The point where that happens we'll call point B. This will be a point on the circumference of the circle in your head. Of course, you could get lucky and walk right into the pink circle, in which case, get the razz berries and ultra balls ready! But let's assume the unluckiest situation.
Next, you need to turn around and walk in the exact opposite direction, back to point A and beyond it until the Charizard disappears from your sightings again. Call this point C. If you have a buddy with you, they can do this bit whilst you are doing step one, to save time. Again, it would be good if you could count your paces.
Now that you have identified two points on the circumference of the circle, and have walked a chord of the circle, this is where the geometry comes in. The perpendicular bisector of any chord of a circle always passes through the centre of the circle.
If you don't believe me, think about this: take a random chord of a circle and join up its end points to the centre as in this diagram:
You should be able to see that this makes an isosceles triangle, because two of the sides are radii.
This means that this triangle has a line of symmetry and if you cut it down this line, you get two right-angled triangles:
And clearly this line of symmetry goes through the centre of the circle.
So, back to our hunt. We're at point C, and we need to find the perpendicular bisector of the line segment BC which is the path we have just walked. The first thing we need to do is find the midpoint of B and C. In some places it is easy to do this by eye, if you have a good map or if you're in a very flat area. But if you have counted your paces, you will be able to find the midpoint much more accurately. I personally don't bother with counting. Because we're only trying to get inside the pink circle, not get to the exact centre, we don't have to be that accurate. So, walk to this midpoint, which we'll call D.
Now, turn ninety degrees and walk. But Emma! (I hear you cry) There's two ways of turning ninety degrees! Yes, you're right. At this point, you have not uniquely defined the purple circle. If you draw two random dots on a page, there are always exactly two different circles with a given radius that pass through those two points. You don't know which circle it is, so you have to guess. So turn ninety degrees in any direction and start walking (or running!) until one of two things happen: the Charizard pops up and you catch it, or the Charizard disappears from your sightings, in which case you do a 180 degree turn and run for it! If you have a buddy with you, you can take one direction each and invent some kind of signal for "I found it!" (smoke signal? A whistle? Make a sound like a dying giraffe?)
You have to do all this in the space of 15 minutes which can be tricky, and sometimes involves running and looking like a loon. But me and my husband went out to Coombe Abbey country park and Coventry's War Memorial Park last week and managed to use this method successfully several times. The handy thing is, pokemon can't spawn just anywhere, there are a set number of spawn points in a given area. So once we identified some of the spawn points, we didn't even have to use the method, we could run to the nearby spawn point we found earlier. There are still enough different spawn points around to keep the game challenging though.
In case you were wondering, my Pokedex is now up to 94. I promised my tutor group that I would have caught 'em all before term starts again in September, which is looking very unlikely. Then again, my tutor group promised me they wouldn't fail their AS levels, so I might have a bit of leverage there...
I hope this method helps you hunt down some rare pokemon and maybe understand the relevance of circle geometry a bit more. (Oh my gosh, did I just come up with a "real life" application of geometry??? Noooooo! Keep Maths pure, people!)
Emma x x x
Wednesday, 1 June 2016
Maths Shouldn't Be about "Getting it Right"
If I had a penny for every time I heard a highly-educated, professional adult say "I can't do Maths", I'd be rich (and if I had £1.74 for every time I heard it, I'd have the makings of a GCSE Maths question).
I've talked before in previous posts that I don't think it should be socially acceptable to say "I'm rubbish at Maths!" and I've received a little bit of criticism from them for being snobbish and condescending because, after all, not everyone can be good at Maths and some people just can't do maths, it's not their fault. What utter rubbish. However, I do understand why so many of you think this, and I'm going to make it my life's mission to undo this damage and encourage teachers to join me in the fight towards making Maths a likeable subject.
When people are asked to recall Maths in secondary school, most people say things like "I was bad at it" or "I was good at it", whereas when asked to recall another subject, most people say things like "it was boring" or "it was fun". For some reason, Maths is seen to be so much more about performance and ability than any other subject. And for Maths, perceived ability and enjoyment are very closely correlated. I think that Art is kind of similar in that ability is more of a factor than in other subjects, but for some reason, Art is still fun to do even if you're "bad" at it. Painting stuff is fun, even if the end product is kind of ugly. In Maths, people seem to believe there is only the end product. The only thing that matters in maths is getting the right answer, and hence getting the right grade. So there is not really anything to enjoy, unless you get all the answers right.
Solution: take the emphasis off "getting it right".
I was looking through some Primary school work belonging to one of my year 13 students, and I found this completely hideous drawing of what was apparently a monkey eating a banana. It looked like nothing. It was just crazy scribbles all over the page. And there in the corner, was a large, very prominent, tick. A tick as in, yes, this is correct. A tick as in an acknowledgement that this piece of work is perfectly acceptable. And that made me think: in art, you can tick something without saying that it is "correct" or "perfect", whereas in Maths, you can only tick something that is correct. If a student writes something like 2.3 + 4.5 = 6.7, I can't tick it. But it's only slightly wrong, and the student has done some good mathematical thought to get to that answer. They have understood place value (a tricky concept), they just counted wrong. So let's just eliminate ticks altogether. Let's use something else that basically means the same thing but doesn't have the same connotations of "correctness", like a smiley face perhaps. I can put a smiley face next to 6.7, even if I can't put a tick. And I can write them a note explaining that they miscounted but that they have understood place value and I'm proud of them for that.
During a whole-class question and answer session or discussion or review, let's not ask questions that have correct answers. Let's ask questions about methods and thinking and pattern spotting. Instead of asking, "What's 67.8 divided by 4?" let's ask, "How could you go about dividing 67.8 by 4?" and let's actually listen to the student's explanation, not just wait for the numerical answer. So often teachers will ask for the numerical answer first, then follow up with a "and how did you get that answer?" when it should be the other way round.
Let's not give year seven students a Maths test on their first week of secondary school and use this to put them into "ability" classes. Let's give them a "thinking styles" test instead - one that assesses the way they think about maths and their approach to solving problems. Then we can use this data to put them into classes instead (or at least let them think that's what the classes are based on, if you can't actually stomach doing this).
Let's teach our students that Maths is about more than just getting a correct answer. After all, we all have calculators and Wikipedia. Learning Maths needs to be about learning problem solving techniques, pattern spotting, making connections, and communicating all of these things with multiple representations. Let's forget about the notion of being "good at maths" or "bad at maths" and focus instead on everyone "doing maths".
And whether you agree with me or not doesn't matter because this debate has no correct answer :)
Emma x x x
I've talked before in previous posts that I don't think it should be socially acceptable to say "I'm rubbish at Maths!" and I've received a little bit of criticism from them for being snobbish and condescending because, after all, not everyone can be good at Maths and some people just can't do maths, it's not their fault. What utter rubbish. However, I do understand why so many of you think this, and I'm going to make it my life's mission to undo this damage and encourage teachers to join me in the fight towards making Maths a likeable subject.
When people are asked to recall Maths in secondary school, most people say things like "I was bad at it" or "I was good at it", whereas when asked to recall another subject, most people say things like "it was boring" or "it was fun". For some reason, Maths is seen to be so much more about performance and ability than any other subject. And for Maths, perceived ability and enjoyment are very closely correlated. I think that Art is kind of similar in that ability is more of a factor than in other subjects, but for some reason, Art is still fun to do even if you're "bad" at it. Painting stuff is fun, even if the end product is kind of ugly. In Maths, people seem to believe there is only the end product. The only thing that matters in maths is getting the right answer, and hence getting the right grade. So there is not really anything to enjoy, unless you get all the answers right.
Solution: take the emphasis off "getting it right".
I was looking through some Primary school work belonging to one of my year 13 students, and I found this completely hideous drawing of what was apparently a monkey eating a banana. It looked like nothing. It was just crazy scribbles all over the page. And there in the corner, was a large, very prominent, tick. A tick as in, yes, this is correct. A tick as in an acknowledgement that this piece of work is perfectly acceptable. And that made me think: in art, you can tick something without saying that it is "correct" or "perfect", whereas in Maths, you can only tick something that is correct. If a student writes something like 2.3 + 4.5 = 6.7, I can't tick it. But it's only slightly wrong, and the student has done some good mathematical thought to get to that answer. They have understood place value (a tricky concept), they just counted wrong. So let's just eliminate ticks altogether. Let's use something else that basically means the same thing but doesn't have the same connotations of "correctness", like a smiley face perhaps. I can put a smiley face next to 6.7, even if I can't put a tick. And I can write them a note explaining that they miscounted but that they have understood place value and I'm proud of them for that.
During a whole-class question and answer session or discussion or review, let's not ask questions that have correct answers. Let's ask questions about methods and thinking and pattern spotting. Instead of asking, "What's 67.8 divided by 4?" let's ask, "How could you go about dividing 67.8 by 4?" and let's actually listen to the student's explanation, not just wait for the numerical answer. So often teachers will ask for the numerical answer first, then follow up with a "and how did you get that answer?" when it should be the other way round.
Let's not give year seven students a Maths test on their first week of secondary school and use this to put them into "ability" classes. Let's give them a "thinking styles" test instead - one that assesses the way they think about maths and their approach to solving problems. Then we can use this data to put them into classes instead (or at least let them think that's what the classes are based on, if you can't actually stomach doing this).
Let's teach our students that Maths is about more than just getting a correct answer. After all, we all have calculators and Wikipedia. Learning Maths needs to be about learning problem solving techniques, pattern spotting, making connections, and communicating all of these things with multiple representations. Let's forget about the notion of being "good at maths" or "bad at maths" and focus instead on everyone "doing maths".
And whether you agree with me or not doesn't matter because this debate has no correct answer :)
Emma x x x
Saturday, 28 May 2016
It's about Time!
Secondary teachers: have you been astounded in recent years by the sheer number of students who cannot tell the time, and, in many cases, seem to have absolutely no concept of time itself?
I certainly have.
Now if you're a non-maths teacher, you might be thinking: "why haven't those lazy maths teachers taught them how to tell the time?" If you're a maths teacher, you might be thinking, "why didn't those lazy Primary school teachers teach them how to tell the time?" And if you're a Primary school teacher, you might well be thinking, "why don't parents teach their children to tell the time these days?"
Of course it is the joint responsibility of parents, Primary teachers, and (I would argue, to a lesser extent), Secondary Maths teachers, to teach students how to tell the time. However, some responsibility also lies with non-maths Secondary teachers too. The sad thing is, some teachers (Maths teachers included) are actually un-teaching students about time. That is to say, they're actually worsening students' understanding about time. Let me explain.
I have noticed that the only classrooms in my school that have wall clocks are the Maths classrooms. When I was in charge of numeracy across the curriculum at my school, I asked some representatives from other departments about why that was, and whether they would like me to order clocks for their classrooms (from the Maths budget!) and I was surprised to hear a resounding "no thanks". Their reasoning was that if there is a big, visible clock in the room, students will spend the lesson clock-watching, and will hence be less engaged.
Whilst I do sympathise with this, having experienced my fair share of disengaged students, I do think that we need to rethink this. I have sixteen year old students who have absolutely no concept of how long five minutes feels like, who have no idea what quarter of an hour feels like compared to three-quarters of an hour. And I believe this is because they do not do enough clock watching. Think about it: the youth of today are more likely to watch on-demand TV, Netflix, or youtube instead of scheduled programming, so they don't really have the experience of waiting for 7:30pm for Top of the Pops to start. It is experiencing things like this that teach us about time.
Have you ever told your class, "you have three minutes to finish this activity, and then we'll discuss it as a class"? I'm sure you have, because time limited activities are recommended by teaching and learning experts. But do you actually wait three minutes exactly? Or do you wait an arbitrary period of time, until the noise level has risen just enough to tell you most students have finished, or the time it takes you to give out the glue sticks? Sometimes we jokingly refer to these periods of time as "teacher minutes", but what's really not funny is that these "teacher minutes" may be the only experiences our students have with time periods, and when they build up an understanding of a minute based on this, they are going to be left with a completely warped impression. You could actually be damaging their conceptual understanding of the passage of time by doing this.
Have you ever sat with a student in a ten minute detention and had them squirm in their seat and ask after two minutes, "can I go yet?" They aren't being rude exactly, they just honestly have no idea what ten minutes feels like. And with no clock in the room, they may feel completely lost, the way a directionally-challenged person like myself might feel in the middle of a large homogeneous field with a map but no compass. What makes this all the worse, is that often we get bored of the detention, or we remember we have a meeting to get to, and we cut the detention short, knowing that the student won't realise. Thus reinforcing dodgy concepts of time.
Teachers are not the only ones to blame. When I was a kid I referred to a bus timetable to know when my bus was due, and my watch to know what the time currently was, and I worked out from there how long I had left to wait. Today, I simply look at the electronic display inside the bus shelter that says "6A - Pool Meadow - 8 minutes" and watch the time count down to zero, and then the magic word "Due". National Express minutes are even worse than teacher minutes. Sometimes the time remaining increases. Sometimes it stays on the same number for several minutes. I am aware of this because I have built up a good understanding of time over the years. Many young people today, however, have not, and hence this is just more misleading information about time to warp their understanding.
Now I'm going to stop complaining and start offering solutions.
1. If you're a teacher, get a clock for your classroom, and don't discourage clock-watching. If a student asks how long it is until the end of the lesson, point at the clock and get them to work it out. Help them with this if they can't read it. Don't be surprised if some of your year elevens cannot read an analogue clock. What would be even better is putting some words around the edge of your clock saying "o'clock", "quarter past", etc. This way students get used to this vocabulary.
2. When you use time-limited activities, time them properly. Don't use "teacher minutes". If possible, display a countdown clock or a large analogue clock on your interactive whiteboard during the activity, so that students can monitor the time themselves as they do the activity.
3. When doing exam practice, get students used to timed conditions by having a large analogue clock on the interactive whiteboard and refer to it frequently. For example, saying something like, "It's quarter past at the moment which means you've had twenty minutes and you've got another twenty left, because we finish at twenty-five to" and pointing to the position of the minute hand as you do so.
4. If you have a student who is perpetually late and you think may not have a good understanding of time, give him or her a time-based job to do so that they have to be aware of the time. For example, you could say, "Daniel, at two o'clock can you do me a favour and go and give this note to Mr Edwards? It's really important so can you remind me just before two o'clock so that he gets it on time". Now maybe you're thinking that if half of Daniel's attention is on the clock, he's not going to be able to learn as much or be fully engaged in the lesson. But I do think it's important that students learn to be time-conscious in order that they can be effective working adults.
Am I over-reacting here or do you also think this is a big issue that needs addressing? Do you strongly agree or disagree with any of the points I've made? Let me know in the comments.
Emma x x x
I certainly have.
Now if you're a non-maths teacher, you might be thinking: "why haven't those lazy maths teachers taught them how to tell the time?" If you're a maths teacher, you might be thinking, "why didn't those lazy Primary school teachers teach them how to tell the time?" And if you're a Primary school teacher, you might well be thinking, "why don't parents teach their children to tell the time these days?"
Of course it is the joint responsibility of parents, Primary teachers, and (I would argue, to a lesser extent), Secondary Maths teachers, to teach students how to tell the time. However, some responsibility also lies with non-maths Secondary teachers too. The sad thing is, some teachers (Maths teachers included) are actually un-teaching students about time. That is to say, they're actually worsening students' understanding about time. Let me explain.
I have noticed that the only classrooms in my school that have wall clocks are the Maths classrooms. When I was in charge of numeracy across the curriculum at my school, I asked some representatives from other departments about why that was, and whether they would like me to order clocks for their classrooms (from the Maths budget!) and I was surprised to hear a resounding "no thanks". Their reasoning was that if there is a big, visible clock in the room, students will spend the lesson clock-watching, and will hence be less engaged.
Whilst I do sympathise with this, having experienced my fair share of disengaged students, I do think that we need to rethink this. I have sixteen year old students who have absolutely no concept of how long five minutes feels like, who have no idea what quarter of an hour feels like compared to three-quarters of an hour. And I believe this is because they do not do enough clock watching. Think about it: the youth of today are more likely to watch on-demand TV, Netflix, or youtube instead of scheduled programming, so they don't really have the experience of waiting for 7:30pm for Top of the Pops to start. It is experiencing things like this that teach us about time.
Have you ever told your class, "you have three minutes to finish this activity, and then we'll discuss it as a class"? I'm sure you have, because time limited activities are recommended by teaching and learning experts. But do you actually wait three minutes exactly? Or do you wait an arbitrary period of time, until the noise level has risen just enough to tell you most students have finished, or the time it takes you to give out the glue sticks? Sometimes we jokingly refer to these periods of time as "teacher minutes", but what's really not funny is that these "teacher minutes" may be the only experiences our students have with time periods, and when they build up an understanding of a minute based on this, they are going to be left with a completely warped impression. You could actually be damaging their conceptual understanding of the passage of time by doing this.
Have you ever sat with a student in a ten minute detention and had them squirm in their seat and ask after two minutes, "can I go yet?" They aren't being rude exactly, they just honestly have no idea what ten minutes feels like. And with no clock in the room, they may feel completely lost, the way a directionally-challenged person like myself might feel in the middle of a large homogeneous field with a map but no compass. What makes this all the worse, is that often we get bored of the detention, or we remember we have a meeting to get to, and we cut the detention short, knowing that the student won't realise. Thus reinforcing dodgy concepts of time.
Teachers are not the only ones to blame. When I was a kid I referred to a bus timetable to know when my bus was due, and my watch to know what the time currently was, and I worked out from there how long I had left to wait. Today, I simply look at the electronic display inside the bus shelter that says "6A - Pool Meadow - 8 minutes" and watch the time count down to zero, and then the magic word "Due". National Express minutes are even worse than teacher minutes. Sometimes the time remaining increases. Sometimes it stays on the same number for several minutes. I am aware of this because I have built up a good understanding of time over the years. Many young people today, however, have not, and hence this is just more misleading information about time to warp their understanding.
Now I'm going to stop complaining and start offering solutions.
1. If you're a teacher, get a clock for your classroom, and don't discourage clock-watching. If a student asks how long it is until the end of the lesson, point at the clock and get them to work it out. Help them with this if they can't read it. Don't be surprised if some of your year elevens cannot read an analogue clock. What would be even better is putting some words around the edge of your clock saying "o'clock", "quarter past", etc. This way students get used to this vocabulary.
2. When you use time-limited activities, time them properly. Don't use "teacher minutes". If possible, display a countdown clock or a large analogue clock on your interactive whiteboard during the activity, so that students can monitor the time themselves as they do the activity.
3. When doing exam practice, get students used to timed conditions by having a large analogue clock on the interactive whiteboard and refer to it frequently. For example, saying something like, "It's quarter past at the moment which means you've had twenty minutes and you've got another twenty left, because we finish at twenty-five to" and pointing to the position of the minute hand as you do so.
4. If you have a student who is perpetually late and you think may not have a good understanding of time, give him or her a time-based job to do so that they have to be aware of the time. For example, you could say, "Daniel, at two o'clock can you do me a favour and go and give this note to Mr Edwards? It's really important so can you remind me just before two o'clock so that he gets it on time". Now maybe you're thinking that if half of Daniel's attention is on the clock, he's not going to be able to learn as much or be fully engaged in the lesson. But I do think it's important that students learn to be time-conscious in order that they can be effective working adults.
Am I over-reacting here or do you also think this is a big issue that needs addressing? Do you strongly agree or disagree with any of the points I've made? Let me know in the comments.
Emma x x x
Wednesday, 27 April 2016
Should I Encourage My Students Not to Fast During Exam Season?
You may or may not have noticed that Ramadan largely coincides with A2 exam season this year. All of the A2 maths exams this year are during Ramadan. As I have 18 Muslim students in my A Level maths class (and one Christian) this could have a big impact on my results. As a teacher (and form tutor), should I try to gently persuade my students that fasting might not be a good idea?
During Ramadan, Muslims do not eat or drink (even water) during daylight hours. This means staying up pretty late (in summer) to eat, and going hungry and thirsty most of the day. All exam advice I've ever read definitely says make sure you get plenty of sleep, eat a good nutritious breakfast, and keep your brain well-hydrated. Therefore surely fasting is going to lead to worse results?
Kids do not have to fast, technically, only adults, but most of the young people I know do choose to fast (or possibly their parents force them?) and are often very proud of the fact that they do. In case you are worried for their safety, I should point out that Muslims break their fast when they are ill (and make up the time afterwards) and girls do not fast when they are menstruating. Fasting is not about self-punishment or about inflicting suffering on yourself.
So should I try to convince my students not to fast? I do believe that fasting makes a difference to exam performance, and obviously I do want my students to do their best (because I care about them, not because I may get a good performance-management rating, for once). However, I don't want to encourage them not to fast. I can see that fasting is important to them, and helps them connect to God. Surely that is more important than exam results at the end of the day (or the end of life, in fact)?
As an Atheist, I don't believe that pleasing God is going to have any kind of positive effect in this life or the next, but I do believe that spirituality has an overall positive effect on a person's happiness, far more so than getting a good exam result. As a non-religious person, I'm almost jealous that these guys have something they have dedicated their life to, that they can practise in such a simple yet meaningful way. Who am I to take that away from them? As a white, non-Muslim person, who am I to advise Muslims on what parts of their religion (or even their culture) they should follow and which they shouldn't? And so what if non-Muslims have an unfair advantage in this year's exams? I'm sure Islam teaches that life doesn't always seem fair, but God will make sure you're all right in the end (Insha Allah, as my students would say).
So I'm not going to encourage my students not to fast, even if their results suffer.
I'm just going to hope that all of my female students happen to be on their periods for their C4 exam.
Emma x x x
During Ramadan, Muslims do not eat or drink (even water) during daylight hours. This means staying up pretty late (in summer) to eat, and going hungry and thirsty most of the day. All exam advice I've ever read definitely says make sure you get plenty of sleep, eat a good nutritious breakfast, and keep your brain well-hydrated. Therefore surely fasting is going to lead to worse results?
Kids do not have to fast, technically, only adults, but most of the young people I know do choose to fast (or possibly their parents force them?) and are often very proud of the fact that they do. In case you are worried for their safety, I should point out that Muslims break their fast when they are ill (and make up the time afterwards) and girls do not fast when they are menstruating. Fasting is not about self-punishment or about inflicting suffering on yourself.
So should I try to convince my students not to fast? I do believe that fasting makes a difference to exam performance, and obviously I do want my students to do their best (because I care about them, not because I may get a good performance-management rating, for once). However, I don't want to encourage them not to fast. I can see that fasting is important to them, and helps them connect to God. Surely that is more important than exam results at the end of the day (or the end of life, in fact)?
As an Atheist, I don't believe that pleasing God is going to have any kind of positive effect in this life or the next, but I do believe that spirituality has an overall positive effect on a person's happiness, far more so than getting a good exam result. As a non-religious person, I'm almost jealous that these guys have something they have dedicated their life to, that they can practise in such a simple yet meaningful way. Who am I to take that away from them? As a white, non-Muslim person, who am I to advise Muslims on what parts of their religion (or even their culture) they should follow and which they shouldn't? And so what if non-Muslims have an unfair advantage in this year's exams? I'm sure Islam teaches that life doesn't always seem fair, but God will make sure you're all right in the end (Insha Allah, as my students would say).
So I'm not going to encourage my students not to fast, even if their results suffer.
I'm just going to hope that all of my female students happen to be on their periods for their C4 exam.
Emma x x x
Friday, 5 February 2016
Forget about Becoming a "Better Teacher"
Earlier this week I had a small epiphany. From reading The Teaching Gap, which I talked about a lot in my previous post, I discovered that in Japan, teachers are not judged in the way they are in the UK, as "outstanding", "good", "requires improvement" or "inadequate". In Japan, individual teachers are not judged at all. Instead, the lessons themselves are scrutinised routinely to make sure they are as effective as possible. Ofsted inspectors say they inspect lessons, not teachers, and your senior management team at school probably claim to do the same, but if your lesson is bad, the implication is that you are a bad teacher and you need to improve as a teacher. In Japan, as the lessons are planned by a group of teachers, and based on a previous group of teachers' lessons, the teacher who delivers the lesson is only held partly responsible for the effectiveness of the lesson, and if the lesson is bad, this is not seen as a reflection on the teacher.
Over the course of the five years or so I've been writing this blog, I've contemplated my development as a teacher and how I've become a better teacher over the years. Perhaps for my first five years as a teacher that was useful, but I've decided to move away from that kind of thinking. I will no longer attribute external circumstances to my internal failings (a very female way of thinking, according to Sheryl Sandberg), or even attribute successful learning to my skill in the classroom, and instead approach teaching from a more objective point of view. If a lesson does not work, I will look at how to change the lesson. I will not tell myself I'm a bad teacher. I will ask other teachers for lessons that worked well for them, and try them out. I will not tell myself, "well it worked for them because they're a better teacher than me", and will instead use a scientific approach: "can the results of their experiment be replicated with a different class?"
I am also going to teach myself to respond to all setbacks (by which I mean ineffective lessons) in the manner of Boston Philharmonic's Ben Zander and simply say "how fascinating!"
Here's a before and after as an example:
Before: "the students learned nothing this lesson, I'm a completely useless teacher."
After: "how fascinating! The students learned nothing this lesson. I wonder which aspects of the lesson could be changed to improve this?"
I believe that this way of thinking would really help departments to improve together as a team. Some departments have a few "outstanding teachers" who are often asked to share good practice or be observed, and often one or two "inadequate teachers" who are given a support plan and are under a lot of scrutiny. Wouldn't it be better if we removed these labels and stopped thinking individually, and instead focused on improving all lessons by planning together, testing out lessons, and continually refining these lessons until learning improves throughout the school? How often do these "outstanding teachers" actually share with their department detailed lesson plans for others to use? It is more likely that they share general techniques (like assessment strategies, or three ways of differentiating a lesson) rather than specific learning episodes (like: this is how I explain carrying remainders in bus stop division).
My department has started to move towards this way of working. We all split into pairs and worked on planning a series of three or so lessons on specific topics. These lessons have been saved in a central location so everyone can access and use them. What we really need to do now is make sure that every time a teacher uses one of these lessons, they document how effective the lesson was, what bits didn't quite work, and give suggestions for adaptations so that the next teacher can try a slightly refined version. If we repeat this process every time, by the end of the year we will have a good set of lessons, and after five years we'll have an amazing set of lessons.
Does your department do something similar to this and how effective is it? Do you have any other ideas for implementing a more Japanese approach to professional development?
Emma x x x
Wednesday, 3 February 2016
Changing the UK's Teaching Culture
This week I have been reading another excellent book about mathematics teaching. This one is called The Teaching Gap by James W Stigler and James Hiebert and it's about a large-scale study that was carried out on year nine maths lessons in the United States, Germany, and Japan. The authors sought to find out why America was lagging behind in the international comparisons of school mathematics achievement, and why Japan was doing so well. All quotations in this post are from this book.
Let me quickly summarise what they found from studying dozens of filmed lessons from each country:
I would say that in the UK we are probably somewhere between Germany and the US. Clearly the Japanese method is working best, as the PISA results show that mathematics achievement is best in Japan. What's more interesting to me, however, isn't the Japanese approach to teaching maths, it's the Japanese approach to improving teaching and learning.
Stigler and Hiebert talk about the fact that teaching is a cultural activity, much like a family dinner. Cultural scripts are learned by observing and participating in the activity. We know how to behave in a Sunday lunch situation because it is cultural. We don't study this, we just absorb the knowledge from being there. For some people of the older generation, using a computer is not a cultural activity, and they had to sit down with a manual and learn how to use it. For children in this day and age, using a computer is a cultural activity, because it is something they have just picked up from observation and by being around computers often. Teaching is a cultural activity and it is learned from spending thirteen years in a classroom as a student. We only spend one year training to be a teacher formally, and this pales in comparison to the thirteen years we spend informally learning how to teach.
Cultural activities never suddenly change, they evolve slowly over time. The UK cannot suddenly say, "Hey, Japan teaches better than us, let's now teach like Japan!" This would be too inconsistent with our current mental cultural "script" that we have about teaching. Teaching systems are made up of elements that interact with each other. If we change one significant element, we can throw the system off-balance. The system then rebalances itself by shifting things around. The result is that the change that was made is adjusted so that the system can function as it did before, so no real change has taken place. An example of this would be for your school to tell all of the maths teachers they must use a problem-solving approach in every lesson. This huge change throws the teacher's system off-balance. The teacher then subconsciously redresses the balance to their system by giving too much direct instruction to the students whilst they are problem-solving. Thus the huge change looks like it has been implemented when in fact it has been completely swallowed up by the system and nothing has changed.
The students are part of the system too, and they also have a script in their minds about what their role in the classroom is. When teachers change their teaching system, often they fail to account for the fact that their students have not changed their learning system. The students' internal systems will over-compensate to deal with the teacher's new system, leading to normality.
Teachers can sometimes be sceptical of new ideas and reforms. This is because the cultural beliefs about teaching are so fully integrated into a teacher's mindset that they are not questioned or even noticed. There are some things we would not even consider changing about maths teaching in this country, and there are some things we may like to change but we view them as unchangeable. For example, in the UK we put our students into ability sets, often from year 7. Many teachers I have spoken to about the subject have said it would be impossible (or at least very very difficult) to teach mixed ability classes in maths. And yet, we're pretty much the only country in the world who doesn't have mixed ability classes!
So is it impossible for us to change our teaching in any significant way? No, we just have to approach change very differently.
Stigler and Hiebert suggest we start by becoming more aware of the cultural scripts we are using. We need to find out: what are our fundamental beliefs about mathematics teaching? We can change these beliefs and we can change our scripts, but only if we are aware of what they are like in the first place.
Despite years of reform, new textbooks, excellent materials provided by the government and mathematics teaching groups (like the Standards Unit), maths teaching in the UK hasn't really changed that much, and neither has our standing in the international league tables. However, in Japan, teaching has changed greatly over the past fifty years, and so has their students' achievement levels. This is because Japan uses a very different model for improving teaching. The Japanese system leads to slow, steady, maintained improvement over a long period of time.
Japanese maths teachers take responsibility for improving lessons. And the key word there is lessons: they do not strive to improve themselves as teachers, they strive to improve individual lessons. Lessons which can be used by the other teachers in their school and even in their country. This continuous professional development is called Kounaikenshuu, and consists of teachers working together in small groups within their school to plan a specific lesson together, in great depth, try the lesson out with a class and observe how it goes evaluate the lesson afterwards, then revise the lesson and teach it again, to a different class. Notice that the focus is on improving a lesson, not on improving an individual teacher. When you develop a good teacher, their impact will only last the forty years they teach for and only affect their students, whereas developing great lessons can have impact for hundreds of years and benefit thousands or even millions. The lesson observations in Japan are focused completely on the structure of the lesson and how the learning takes place. No judgement would be given to the teacher themself.
Because Japan has a national curriculum, and because Japanese students are taught in mixed ability classes, these lessons that are created can be used by many different teachers year in, year out. So it makes sense to invest heavily in getting the lessons right. These lessons will be improved every year through these research groups, and the teachers involved write reports that are published in compilations of lesson studies for other teachers to read. These are kept within the school unless they are particularly ground-breaking, in which case they are sent to the authorities to publish nationally.
What I like so much about Kounaikenshuu is that the teachers feel as if they are contributing to the teaching profession itself, not just themselves. Yeah, it's great when you receive an "outstanding" rating, but isn't it better to feel as though you have improved mathematics learning in your entire country, even just a little bit?
This idea of lesson study does exist in the UK and is happening in many schools. But let's make sure that this is not just another fad that will be absorbed into our systems and thus make no difference to our practice. My school introduced lesson study last year as the main part of our CPD. However, this year it has vanished, which is a shame. I think it was a great idea and I'm so glad we did it, but I didn't engage in it fully, and I didn't get much (if anything) out of it, and I believe that is because of the following reasons:
Of course, you don't need a group in order to conduct lesson studies, although it will not be as effective alone. I have decided to start my own lesson studies, on a much smaller and less impressive scale. I have decided to plan one lesson each week in great depth (not my usual back-of-an-envelope planning) and then evaluate the lesson afterwards. I'm keeping the lesson plans and post-lesson notes in a special notebook (#stationerynerd) and I'm going to keep these notes to refer to the next time I come to teach the same topics. The problem is, of course, that as a teacher of setted children, my lesson plans aren't as widely applicable as they would be in Japan. This year I am teaching year seven set eight. Next year I will most likely be teaching set four. The year after that will be set two. The classes will probably respond completely differently to the same lessons. But I'm going to try. My plan is to change my practice one lesson at a time, using the Japanese art of Kounaikenshuu.
Let me quickly summarise what they found from studying dozens of filmed lessons from each country:
- American lessons usually involved a teacher showing the class an example, followed by the students copying the method used to solve very similar problems. There was a big emphasis on learning definitions and procedures. Lessons always started by going through the previous homework.
- German lessons involved very challenging mathematics, and usually involved the teacher going through a very difficult maths problem on the board with the class contributing. There was little time for practice in the lesson, and this was set as homework instead. Lessons always started by going through the previous homework.
- Japanese lessons started by recapping the main point of the previous lesson (they did not see any homework in Japanese lessons) and then the teacher would give a problem for the students to solve without any kind of indication of a method. The students would then work to solve it, sometimes working in groups. The teacher would then ask students to explain their methods and the teacher would discuss the merits of each method.
I would say that in the UK we are probably somewhere between Germany and the US. Clearly the Japanese method is working best, as the PISA results show that mathematics achievement is best in Japan. What's more interesting to me, however, isn't the Japanese approach to teaching maths, it's the Japanese approach to improving teaching and learning.
Stigler and Hiebert talk about the fact that teaching is a cultural activity, much like a family dinner. Cultural scripts are learned by observing and participating in the activity. We know how to behave in a Sunday lunch situation because it is cultural. We don't study this, we just absorb the knowledge from being there. For some people of the older generation, using a computer is not a cultural activity, and they had to sit down with a manual and learn how to use it. For children in this day and age, using a computer is a cultural activity, because it is something they have just picked up from observation and by being around computers often. Teaching is a cultural activity and it is learned from spending thirteen years in a classroom as a student. We only spend one year training to be a teacher formally, and this pales in comparison to the thirteen years we spend informally learning how to teach.
Cultural activities never suddenly change, they evolve slowly over time. The UK cannot suddenly say, "Hey, Japan teaches better than us, let's now teach like Japan!" This would be too inconsistent with our current mental cultural "script" that we have about teaching. Teaching systems are made up of elements that interact with each other. If we change one significant element, we can throw the system off-balance. The system then rebalances itself by shifting things around. The result is that the change that was made is adjusted so that the system can function as it did before, so no real change has taken place. An example of this would be for your school to tell all of the maths teachers they must use a problem-solving approach in every lesson. This huge change throws the teacher's system off-balance. The teacher then subconsciously redresses the balance to their system by giving too much direct instruction to the students whilst they are problem-solving. Thus the huge change looks like it has been implemented when in fact it has been completely swallowed up by the system and nothing has changed.
"It has now been documented in several studies that teachers asked to change features of their teaching often modify the features to fit within their pre-existing system instead of changing the system itself".
The students are part of the system too, and they also have a script in their minds about what their role in the classroom is. When teachers change their teaching system, often they fail to account for the fact that their students have not changed their learning system. The students' internal systems will over-compensate to deal with the teacher's new system, leading to normality.
"Trying to improve teaching by changing individual features usually makes little difference, positive or negative. But it can backfire and leave things worse than before. When one or two features are changed, and the system tries to run as before, it can operate in a disabled state".
Teachers can sometimes be sceptical of new ideas and reforms. This is because the cultural beliefs about teaching are so fully integrated into a teacher's mindset that they are not questioned or even noticed. There are some things we would not even consider changing about maths teaching in this country, and there are some things we may like to change but we view them as unchangeable. For example, in the UK we put our students into ability sets, often from year 7. Many teachers I have spoken to about the subject have said it would be impossible (or at least very very difficult) to teach mixed ability classes in maths. And yet, we're pretty much the only country in the world who doesn't have mixed ability classes!
So is it impossible for us to change our teaching in any significant way? No, we just have to approach change very differently.
Stigler and Hiebert suggest we start by becoming more aware of the cultural scripts we are using. We need to find out: what are our fundamental beliefs about mathematics teaching? We can change these beliefs and we can change our scripts, but only if we are aware of what they are like in the first place.
Despite years of reform, new textbooks, excellent materials provided by the government and mathematics teaching groups (like the Standards Unit), maths teaching in the UK hasn't really changed that much, and neither has our standing in the international league tables. However, in Japan, teaching has changed greatly over the past fifty years, and so has their students' achievement levels. This is because Japan uses a very different model for improving teaching. The Japanese system leads to slow, steady, maintained improvement over a long period of time.
Japanese maths teachers take responsibility for improving lessons. And the key word there is lessons: they do not strive to improve themselves as teachers, they strive to improve individual lessons. Lessons which can be used by the other teachers in their school and even in their country. This continuous professional development is called Kounaikenshuu, and consists of teachers working together in small groups within their school to plan a specific lesson together, in great depth, try the lesson out with a class and observe how it goes evaluate the lesson afterwards, then revise the lesson and teach it again, to a different class. Notice that the focus is on improving a lesson, not on improving an individual teacher. When you develop a good teacher, their impact will only last the forty years they teach for and only affect their students, whereas developing great lessons can have impact for hundreds of years and benefit thousands or even millions. The lesson observations in Japan are focused completely on the structure of the lesson and how the learning takes place. No judgement would be given to the teacher themself.
Because Japan has a national curriculum, and because Japanese students are taught in mixed ability classes, these lessons that are created can be used by many different teachers year in, year out. So it makes sense to invest heavily in getting the lessons right. These lessons will be improved every year through these research groups, and the teachers involved write reports that are published in compilations of lesson studies for other teachers to read. These are kept within the school unless they are particularly ground-breaking, in which case they are sent to the authorities to publish nationally.
"Lesson study is a process of improvement that is expected to produce small, incremental improvements in teaching over a long period of time. It is emphatically not a reformlike process".
What I like so much about Kounaikenshuu is that the teachers feel as if they are contributing to the teaching profession itself, not just themselves. Yeah, it's great when you receive an "outstanding" rating, but isn't it better to feel as though you have improved mathematics learning in your entire country, even just a little bit?
This idea of lesson study does exist in the UK and is happening in many schools. But let's make sure that this is not just another fad that will be absorbed into our systems and thus make no difference to our practice. My school introduced lesson study last year as the main part of our CPD. However, this year it has vanished, which is a shame. I think it was a great idea and I'm so glad we did it, but I didn't engage in it fully, and I didn't get much (if anything) out of it, and I believe that is because of the following reasons:
- The groups we met in were too large (eight to fifteen teachers I think). If we had met in threes or fours I think we could have had more focused discussions.
- We worked on our own independent projects, not in groups. We observed each other in order to help each other with our projects, but it would have been more useful if we had worked on exactly the same project, as they do in Japan.
- We were not in subject groups. In fact, we were encouraged to split the faculty up into different groups, the idea being we could then bring back to the department a variety of findings. But I think it would have been more effective if we were in small groups of teachers from the same department, and teaching very similar classes.
- We were not focusing on lessons, but on general strategies. Some of our areas of research were: increasing independence, encouraging a growth mindset, using flipped learning, and closing the gap between Pupil Premium and non-Pupil Premium students. I feel that it would have been much more effective if instead we had been focusing on: how to teach adding negative numbers, or how to introduce differentiation to year 12s. By focusing on specific lessons that get taught again and again and again by many different teachers, our findings would be immediately applicable and have a wide impact.
Of course, you don't need a group in order to conduct lesson studies, although it will not be as effective alone. I have decided to start my own lesson studies, on a much smaller and less impressive scale. I have decided to plan one lesson each week in great depth (not my usual back-of-an-envelope planning) and then evaluate the lesson afterwards. I'm keeping the lesson plans and post-lesson notes in a special notebook (#stationerynerd) and I'm going to keep these notes to refer to the next time I come to teach the same topics. The problem is, of course, that as a teacher of setted children, my lesson plans aren't as widely applicable as they would be in Japan. This year I am teaching year seven set eight. Next year I will most likely be teaching set four. The year after that will be set two. The classes will probably respond completely differently to the same lessons. But I'm going to try. My plan is to change my practice one lesson at a time, using the Japanese art of Kounaikenshuu.
Thursday, 28 January 2016
The Elitist Nature of Mathematics
I have always received good grades in maths. This made me feel special. I remember that day in year six when I, along with just two other students from my large-ish Primary school, sat the level 6 Maths SATs paper in a special room. Why was it just us three? Because we were special. There was only a level 6 paper for maths, not for English or science. Why? Because maths is special. Because maths can reach levels of difficulty that English and science cannot.
I remember the day in year seven when in the very first week we had a maths test. Because maths is about showing how much you know. I remember the day, a week later, when we were put into sets for maths. Not any other subject, just maths. Why? Because in maths, ability is important. In maths, the clever students need to be separated from the bad students so that they can learn hard maths unhindered.
I remember the day in year nine when I sat my SATs, and again, English and Science only went up to level 7, but maths went up to level 8. Only the top class got to do that paper. It was a paper for special people, for the chosen ones who were good at maths.
I remember the day in year eleven when I chose my A-Level subjects. I remember being told, "you're good at maths, so you should do maths". Other people were told, "you enjoy English, you should pick English," or "you enjoy design, you should pick design". Because maths is something you do because you're good at it, not because you enjoy it.
I remember the day in year thirteen when I chose the courses to apply for at university. I considered doing Psychology. I loved Psychology. But lots of people are good at Psychology. I was the best in my school at Maths. So I chose Maths. Not many people are good at Maths. I chose my university because it was the best non-Oxbridge, non-London university for Mathematics research. The cleverest mathematicians work there, so it must be the best place.
I found university maths very hard. I was not that good at it. So I stopped liking maths. Maths is not fun when you're falling behind. There was no help given at my university. The professors lectured but they did not teach. Because if you need help learning university maths, you're not good enough to be here.
Look at my bolded statements. These are the beliefs that I developed as a result of my education. These beliefs are shared by many whether they consider themselves good at maths or not. If you ask someone what maths was like for them at school, they almost always answer by commenting on their performance, e.g. "I was always good/bad at maths". If you ask someone what, say, Geography, was like for them at school, they almost always comment on their enjoyment of it, e.g. "I found it boring/interesting". This is because maths is seen to be a subject that is all about ability and performance.
I remember the day, not too long ago, when I was looking at year eleven students who had chosen to do maths A-Level next year.
I said, "She's in set four [out of nine], is she deluded? She can't do maths A-Level!"
I said, "Students shouldn't be encouraged to pick maths unless they're in set one"
I said, "Maths isn't like Psychology, not just anyone can do it!"
I said, "They need to realise maths is really hard, it's not like other A-Levels"
I said, "I'm sure she'll try really hard but she just doesn't have what it takes"
I cannot believe that I said these things just a week ago. Since reading Mathematical Mindsets by Jo Boaler, I have completely changed my beliefs about maths as a subject. No one is born good at maths. No one is born bad at maths. Maths is not about answering questions correctly. Maths is not about passing tests. Maths is about connections and communication. I almost want to petition to have maths re-named to that in my school. While I'm at it, maybe I should re-name my blog to "not just tests".
Here's what I think we should do to overcome this problem:
-Stop testing students in maths in the first week of year seven.
-Don't put students into sets in maths until you absolutely have to (do we ever have to? Finland doesn't).
-Stop praising students for getting answers right in maths lessons.
-Don't ever tell a student they are good at maths (I hope it goes without saying to never tell a student they're bad at maths).
-If your top set year eleven study Additional Maths or Further Maths GCSE, don't make it just for top set, make it an opt-in subject for any student who loves maths.
Let's eradicate this ridiculous notion that maths is different from every other subject. As maths teachers, it is up to us to end this.
Emma x x x
I remember the day in year seven when in the very first week we had a maths test. Because maths is about showing how much you know. I remember the day, a week later, when we were put into sets for maths. Not any other subject, just maths. Why? Because in maths, ability is important. In maths, the clever students need to be separated from the bad students so that they can learn hard maths unhindered.
I remember the day in year nine when I sat my SATs, and again, English and Science only went up to level 7, but maths went up to level 8. Only the top class got to do that paper. It was a paper for special people, for the chosen ones who were good at maths.
I remember the day in year eleven when I chose my A-Level subjects. I remember being told, "you're good at maths, so you should do maths". Other people were told, "you enjoy English, you should pick English," or "you enjoy design, you should pick design". Because maths is something you do because you're good at it, not because you enjoy it.
I remember the day in year thirteen when I chose the courses to apply for at university. I considered doing Psychology. I loved Psychology. But lots of people are good at Psychology. I was the best in my school at Maths. So I chose Maths. Not many people are good at Maths. I chose my university because it was the best non-Oxbridge, non-London university for Mathematics research. The cleverest mathematicians work there, so it must be the best place.
I found university maths very hard. I was not that good at it. So I stopped liking maths. Maths is not fun when you're falling behind. There was no help given at my university. The professors lectured but they did not teach. Because if you need help learning university maths, you're not good enough to be here.
Look at my bolded statements. These are the beliefs that I developed as a result of my education. These beliefs are shared by many whether they consider themselves good at maths or not. If you ask someone what maths was like for them at school, they almost always answer by commenting on their performance, e.g. "I was always good/bad at maths". If you ask someone what, say, Geography, was like for them at school, they almost always comment on their enjoyment of it, e.g. "I found it boring/interesting". This is because maths is seen to be a subject that is all about ability and performance.
I remember the day, not too long ago, when I was looking at year eleven students who had chosen to do maths A-Level next year.
I said, "She's in set four [out of nine], is she deluded? She can't do maths A-Level!"
I said, "Students shouldn't be encouraged to pick maths unless they're in set one"
I said, "Maths isn't like Psychology, not just anyone can do it!"
I said, "They need to realise maths is really hard, it's not like other A-Levels"
I said, "I'm sure she'll try really hard but she just doesn't have what it takes"
I cannot believe that I said these things just a week ago. Since reading Mathematical Mindsets by Jo Boaler, I have completely changed my beliefs about maths as a subject. No one is born good at maths. No one is born bad at maths. Maths is not about answering questions correctly. Maths is not about passing tests. Maths is about connections and communication. I almost want to petition to have maths re-named to that in my school. While I'm at it, maybe I should re-name my blog to "not just tests".
Here's what I think we should do to overcome this problem:
-Stop testing students in maths in the first week of year seven.
-Don't put students into sets in maths until you absolutely have to (do we ever have to? Finland doesn't).
-Stop praising students for getting answers right in maths lessons.
-Don't ever tell a student they are good at maths (I hope it goes without saying to never tell a student they're bad at maths).
-If your top set year eleven study Additional Maths or Further Maths GCSE, don't make it just for top set, make it an opt-in subject for any student who loves maths.
Let's eradicate this ridiculous notion that maths is different from every other subject. As maths teachers, it is up to us to end this.
Emma x x x
Saturday, 23 January 2016
Is Homework Racist?
The other day I started reading Jo Boaler's new book Mathematical Mindsets and I think it is so far the book that that has had the biggest impact on my beliefs about teaching and learning in mathematics. When I have finished reading it I will give a full review and summary on this blog but for now I just want to talk about one of the things that really stood out to me as I was reading.
Homework is racist!
At the school where I teach, homework is given very high importance. We set a lot of it, especially in maths, where the policy is that homework is set after every lesson. We have a whole-school detention in place for students who do not complete homework. One quarter of every student's end of term report to parents is about homework quality and quantity. I tend to spend at least 25% of every lesson going through the previous day's homework. I used to say all of these things proudly. But now, knowing what I know now, I feel uneasy, and even slightly ashamed. And, well, racist.
So let me explain. Students from disadvantaged backgrounds and belonging to some ethnic minority groups are less likely to do their homework in the first place, are likely to spend less time on their homework, are less likely to have help on their homework from family members, are less likely to have the tools they need to do their homework at home (like a desk, a quiet place, stationery supplies), and are less likely to have the time to spend on homework, due to looking after younger family members, doing house work, or working a part-time job.
If I set homework and a student does not do it, that 25% of the lesson I spend going through the homework has little benefit to that student. If that student never does their homework, they are effectively only receiving 75% as much teaching as the other students (and that's assuming the homework itself was of no educational value). You might say it's the student's fault they didn't do their homework, or it was the student's choice. However, when you consider on a national scale that some ethnic groups are less likely to do their homework, we are actually indirectly discriminating against those ethnic groups.
In America, some students take an Algebra course in eighth grade (year 9) then if they pass it they can move on to Geometry in the first year of high school (year 10). However, whether they are put into Geometry or back into Algebra in high school is at the teacher's discretion. In 2012, the Noyce Foundation studied student placement in nine school districts in San Francisco and found that 60% of students who should have moved onto Geometry were actually placed into Algebra. This is damaging for those students because it means they would not be able to do AP Calculus in their final year, and hence may not get into college to study a STEM course. When the Noyce Foundation looked more closely at the data, they found that of the 53% of African American students who passed Algebra in eighth grade, only 18% were put into Geometry, and of the 50% of Latino/a students who passed Algebra in eighth, only 16% took Geometry in ninth. (For Asian students, 52% became 52%, for white students, 59% became 33%). Given these facts alone, you would conclude that the teachers who were deciding which classes to put students in were racist. The Silicon Valley Community Foundation hired lawyers who found that the schools were actually acting illegally. But the teachers did not see themselves as racist, nor did they realise they were discriminating. They had simply been choosing which students to advance by looking at test scores and homework completion. What they did not take into account was that the homework completion criteria disproportionately impacts minority students, and is hence still illegal discrimination.
Something similar may be happening at your school. Have you ever seen a student move down a set for not completing their homework or for having poor behaviour? As some ethnic groups are less likely to do their homework and more likely to have poor behaviour, this may be a case of illegal racial discrimination.
PISA, the organisation which compares mathematics achievement of students around the world, did a study in 2015 into homework and achievement. They had data from 13 million students, and found that homework perpetuates inequities in education.
Interestingly, Finland and Korea, two of the highest achieving countries in terms of maths, set the least amount of maths homework.
You might be tempted to say that instead of reducing the achievement gap by eliminating homework, we should instead focus on getting ethnic minority or Pupil Premium students to do their homework. After school homework clubs, often made compulsory for certain students, could work to address this. However, after reading around the topic, I am starting to doubt whether there is actually any point in maths homework at all.
Here are a list of studies that have started to convince me to ditch homework:
Baker and LeTendre (2005) found no positive link between amount of maths homework and maths achievement.
Mikki (2006) found that the countries that set the most homework have the lowest achievement.
Kistantas, Cheema and Ware (2011) found that the more time students spent on homework, the lower their maths achievement across all ethnic groups.
Author Alfie Kohn has written extensively on the subject.
So setting maths homework might not have much of a positive effect on the students that do it, but it does have a negative effect on the students who don't do it. This is almost contradictory, but if it's true, it makes setting homework a completely ridiculous thing to do.
I am going to do some more research before throwing out homework altogether (not that my school would let me, anyway). In the meantime, I am going to change the type of homeworks I set so that they are more in line with my new beliefs about growth mindset and mathematics learning, which I will talk about more in future posts.
Emma x x x
Homework is racist!
At the school where I teach, homework is given very high importance. We set a lot of it, especially in maths, where the policy is that homework is set after every lesson. We have a whole-school detention in place for students who do not complete homework. One quarter of every student's end of term report to parents is about homework quality and quantity. I tend to spend at least 25% of every lesson going through the previous day's homework. I used to say all of these things proudly. But now, knowing what I know now, I feel uneasy, and even slightly ashamed. And, well, racist.
So let me explain. Students from disadvantaged backgrounds and belonging to some ethnic minority groups are less likely to do their homework in the first place, are likely to spend less time on their homework, are less likely to have help on their homework from family members, are less likely to have the tools they need to do their homework at home (like a desk, a quiet place, stationery supplies), and are less likely to have the time to spend on homework, due to looking after younger family members, doing house work, or working a part-time job.
If I set homework and a student does not do it, that 25% of the lesson I spend going through the homework has little benefit to that student. If that student never does their homework, they are effectively only receiving 75% as much teaching as the other students (and that's assuming the homework itself was of no educational value). You might say it's the student's fault they didn't do their homework, or it was the student's choice. However, when you consider on a national scale that some ethnic groups are less likely to do their homework, we are actually indirectly discriminating against those ethnic groups.
In America, some students take an Algebra course in eighth grade (year 9) then if they pass it they can move on to Geometry in the first year of high school (year 10). However, whether they are put into Geometry or back into Algebra in high school is at the teacher's discretion. In 2012, the Noyce Foundation studied student placement in nine school districts in San Francisco and found that 60% of students who should have moved onto Geometry were actually placed into Algebra. This is damaging for those students because it means they would not be able to do AP Calculus in their final year, and hence may not get into college to study a STEM course. When the Noyce Foundation looked more closely at the data, they found that of the 53% of African American students who passed Algebra in eighth grade, only 18% were put into Geometry, and of the 50% of Latino/a students who passed Algebra in eighth, only 16% took Geometry in ninth. (For Asian students, 52% became 52%, for white students, 59% became 33%). Given these facts alone, you would conclude that the teachers who were deciding which classes to put students in were racist. The Silicon Valley Community Foundation hired lawyers who found that the schools were actually acting illegally. But the teachers did not see themselves as racist, nor did they realise they were discriminating. They had simply been choosing which students to advance by looking at test scores and homework completion. What they did not take into account was that the homework completion criteria disproportionately impacts minority students, and is hence still illegal discrimination.
Something similar may be happening at your school. Have you ever seen a student move down a set for not completing their homework or for having poor behaviour? As some ethnic groups are less likely to do their homework and more likely to have poor behaviour, this may be a case of illegal racial discrimination.
PISA, the organisation which compares mathematics achievement of students around the world, did a study in 2015 into homework and achievement. They had data from 13 million students, and found that homework perpetuates inequities in education.
"Homework may then have the unintended consequence of widening the performance gap between students from different socio-economic backgrounds". (PISA, 2015)Read the full paper here: Does Homework Perpetuate Inequities in Education?
Interestingly, Finland and Korea, two of the highest achieving countries in terms of maths, set the least amount of maths homework.
You might be tempted to say that instead of reducing the achievement gap by eliminating homework, we should instead focus on getting ethnic minority or Pupil Premium students to do their homework. After school homework clubs, often made compulsory for certain students, could work to address this. However, after reading around the topic, I am starting to doubt whether there is actually any point in maths homework at all.
Here are a list of studies that have started to convince me to ditch homework:
Baker and LeTendre (2005) found no positive link between amount of maths homework and maths achievement.
Mikki (2006) found that the countries that set the most homework have the lowest achievement.
Kistantas, Cheema and Ware (2011) found that the more time students spent on homework, the lower their maths achievement across all ethnic groups.
Author Alfie Kohn has written extensively on the subject.
So setting maths homework might not have much of a positive effect on the students that do it, but it does have a negative effect on the students who don't do it. This is almost contradictory, but if it's true, it makes setting homework a completely ridiculous thing to do.
I am going to do some more research before throwing out homework altogether (not that my school would let me, anyway). In the meantime, I am going to change the type of homeworks I set so that they are more in line with my new beliefs about growth mindset and mathematics learning, which I will talk about more in future posts.
Emma x x x
Subscribe to:
Posts (Atom)