Thanks for another though-provoking article, Craig!
What is the purpose of school?
I love Maths, and I love imparting knowledge to other people. Being a teacher is a no-brainer for me. But I have a love/hate relationship with schools. On the one hand, I think schools should stick to education and not be so focused on other things. For example, I want to teach mathematics. I don't want to be explaining to a teenager why throwing scissors across the classroom is bad (and then having to write a report about it too). I'm not their parent. I get that school have rules - uniform, mobile phones, etc. But if my students are engaged with my teaching, I don't want to have to send a student out of my class because they are wearing the wrong footwear.
So, on the one hand, I'm not too bothered about the things like "social equity" or things that can be learned from a mixed-attainment class (most of which can be learned in sets).
On the other hand, I want to get to know my students. Obviously, I want to know how they learn, so that my teaching will have the greatest impact. But I also want to know what their aims are in life. What level of Maths do they actually need to achieve those aims?
I am still in contact with many students who have now left school. I see some working at the local supermarket. I see some students mountainbiking trails I use. I see some students at local sporting events I'm involved in. One student (who would listen to music on her phone through headphones when completing her work, not disturbing anyone else, and I would check on her to make sure she understood) now runs the best pizza shop in my area. They are happy to talk to me because I took an interest in their lives.
But the purpose of school is not for me to be able to chat with students when they leave.
I believe that if schools (i.e. the leadership) encouraged teachers to ask the question "What level of [my subject] does each student need to achieve their aims in life?" then the quality of teaching would be better and student perception of school would improve. A lot of what schools do doesn't particularly benefit students later in life. I once wrote a list of nine things schools do where the reality is different. I showed it to a class and said, "I understand you. I understand why you get frustrated."
I honestly don't know what it would take to "change my mind", but my thoughts do definitely evolve over time.
I feel the same way - some of my students will never need to understand (and sadly probably never will understand) how to get the Y intercept of a straight line graph. How much time should they and the teacher spend trying to help them understand it? And for what purpose? No doubt there are students who are on the boderline and with the right gudiance and encouragement can achieve more than they or we imagine. Is that the main reason we spend time on maths that students will never need for their futures?
I am interested on your "nine things". Please share if you want.
In reality... Great ideas are often the work of a group
.
Schools teach... Copying and helping other people is not the way forward
In reality... Working together and reproducing what someone else has done is the foundation for progression
.
Schools teach... You must know how to reach the answer
In reality... The answer may be elusive, but a step along the way is a great achievement
.
Schools teach... Your exams are extremely important
In reality... The older you get, the less your exams mean
.
Schools teach... Only the qualified will get jobs
In reality... Realising your talents and knowing how to use them will provide many opportunities
.
Schools teach... English and Maths are the most important subjects you will ever study in school
In reality... Whilst many jobs have a requirement of English and Maths, realising and honing your talents will surpass that
.
Schools teach... Having everyone knowing the same stuff and thinking the same things is what society needs in order to progress
In reality... Progression is attained by many people thinking differently and creatively in order to help the different facets of society
.
Schools teach... Age is the most important thing we have in common, and people will learn best within their own age group
In reality... People are grouped according to their level of understanding of a particular subject, and the different ages provide additional creativity
.
Schools teach... If you are unable to do something, you are at a disadvantage
In reality... Stephen Hawking, Nick Vujicic
(just because you can't do something, that does not make you useless; working with other people where strengths complement each other is a huge advantage)
I believe math is worth learning even if students won't use it beyond their schooling. For instance, I think learning the rules of solving multistep equations are worth learning. It's reasonable to me that a significant number of students won't need to solve multistep equations after they leave school. To me, it's worth learning because it is challenging, and learning challenging, abstract concepts is a chance to show students what they are capable of, to show them that they can learn hard things, which helps them develop valuable beliefs about their ability as learners in the future.
This would be a hard hypothesis to gather evidence on, because the outcome is far in the future. I wonder if there is an impact of effective intervention on students' future self-efficacy or growth mindset beliefs? I'd be curious what evidence could address this, it's just a hard thing to test directly.
When done well, Mathematics teaches logical deduction. (The word "mathematics" is related to the Greek word for "discipline" and "disciple", i.e. people who follow a particular concept.)
And thinking logically is of huge importance to life. So I agree with you!
I'm a bit skeptical that math actually teaches a broad, transferable form of logical deduction. What evidence might convince you that this isn't actually a transferable skill?
Perhaps not so "broad". Transferable? Well, it depends on how far you take the Maths. I often talk to students about the "working" being like an essay. You start with your introduction, which is turning the question into a mathematical equation. Then each line of working is like a paragraph in your essay, logical statements flowing from the introduction. It ends with your conclusion, which is the answer to the question.
I know it's a bit of a reach, and I use that example not so much to tell students that Maths is important, but more to explain the importance of showing working!
I wish the topic of proof came up earlier in the curriculum... most students hate the proof that comes with congruent triangles. But mathematical proof provides a good introduction to logical fallacies (x implies y does not therefore mean that y implies x). And I, personally, think that an understanding of that is highly useful for life (but I can accept that I'm probably biased in that!). And it's probably why many of the great philosophers are also mathematicians.
I would say that logic only transfers sometimes, and math isn't necessary to be logical. Sure, math can help. But knowing math doesn't make one fully logical, and people who don't know math can be logical in contexts they are familiar with. I know lots of people who know lots of math and believe illogical things.
I've read research by people like Jo Boaler that advocates mixed ability teaching. I've taught in that way in primary classes and can see it worked in her examples. However, in the secondary school where I teach now, the range of attainment is so wide, I think mixed ability would be inefficient, demoralising and make it difficult to prepare for qualifications. I would be interested to see the alternative in action though.
Have you done the drill down into P8 data for mixed attainment teaching or is that a thought experiment? My understanding from EEF was that setting made no positive impact. I believe that mixed attainment teaching has the benefits described by your American colleague here *and* doesn’t negatively impact attainment of most students. Would change my mind if you presented the data you describe
I believe feedback is a critical part of the teaching dynamic - as critical as instruction, higher-order questioning, checking for understanding - all the aspects of the process that most educators agree on. I have left the classroom and give private lessons in students' homes. My students attend various international schools in my area, three of which I have taught at. I am dismayed to find that without exception, students are not receiving actionable feedback on their work. A grade (in any format) conveys very limited information. Since I teach maths and have managed this process myself, I find I have little patience with this change in practise. I used to design tests and then immediately write comprehensive solution guides showing how to solve each problem (sometimes using different approaches). I would post the solutions whilst students were taking the tests since there is no moment when they will be more interested than that immediately following the completion of the test. Students would hand in the tests and go directly to whichever portal the school was using and see my feedback. Each time, rigorous conversations would emerge as they left the testing room. If I had waited until the next day - or week!!!! - they would not have engaged so enthusiastically in the process of diagnosing their performances. SO - I really feel strongly about this and I would really like to hear why I should not be getting wrinkles over the fact that my current students are not benefitting from better feedback. I provide it - but that is not at all as beneficial as having the teacher provide it. IMHO
I agree this is great. But there's pretty much always someone away. Or the other class isn't doing the test til tomorrow. Or whatever else. I always find that you have to hold it back. How did you never have anyone absent?
I had at least three versions of each test and used them interchangeably year after year. So there was always a different version for the absent students, and because that version might have been used as the main test in a previous year, there was already a solution guide. If I'd had to start from scratch each year, it would have been too much for sure.
I believe that copying something down into a book is a waste of time and has no learning benefit. I believe this because when I stopped asking pupils to copy down things into their book, this did not impact pupil understanding or recall of key facts on whiteboard responses, and it also meant we had more time for practice of application. I also believe this because back when I have asked pupils to copy things down in the past, checks for listening/understanding showed that this didn't appear to have much impact on whether they could recall what they had written down or whether they even really understood what it meant.
To change my mind, I would have to see several solid instances of a lesson in which time spent copying something down into the book had a tangible impact on pupil learning that wouldn't have been achieved by doing something more interactive such as engaging with a whiteboard response to a check for understanding.
Hi Lucy - I can only speak for my own experience as a student. I am a visual learner to the extreme. Nothing a teacher said to me during class planted in my mind. Even group discussions did not accomplish lasting learning. I realised this as a child and copied down everything the teacher said during the lesson without trying to learn it in the moment. Then I went home and recopied everything I'd written into a duplicate set of notebooks and whilst I did that I learned the material - because I was looking at it, not listening. Mine is an extreme situation but it may be that some of your visual learners benefit from writing everything down.
Hi Callie, thanks for sharing your thoughts and experiences! I can see that this may be the case for some students, although I do think that is likely to be a minority in my school based on my experiences.
Do you not think the same outcome could be achieved then by them being given a handout, for example, of a worked example and key definitions? It would certainly take less time than hand writing out everything from the board, and they would then still be able to take these home and copy them out if that is how they felt best to revise independently.
I was going to respond to your first post, saying how having students copy out a worked example can be helpful for when they are answering questions themselves and get stuck... but you answered that in your reply - give them a handout! It requires a little extra preparation and is less spontaneous (I often make up examples on the spot, intended to directly address problems or questions students have, but I guess this can still be done with a handout).
So then, my question becomes... why have handouts rather than just go back to working out of textbooks? Textbooks have worked examples and questions, plus an explanation of the topic.
Thanks for another though-provoking article, Craig!
What is the purpose of school?
I love Maths, and I love imparting knowledge to other people. Being a teacher is a no-brainer for me. But I have a love/hate relationship with schools. On the one hand, I think schools should stick to education and not be so focused on other things. For example, I want to teach mathematics. I don't want to be explaining to a teenager why throwing scissors across the classroom is bad (and then having to write a report about it too). I'm not their parent. I get that school have rules - uniform, mobile phones, etc. But if my students are engaged with my teaching, I don't want to have to send a student out of my class because they are wearing the wrong footwear.
So, on the one hand, I'm not too bothered about the things like "social equity" or things that can be learned from a mixed-attainment class (most of which can be learned in sets).
On the other hand, I want to get to know my students. Obviously, I want to know how they learn, so that my teaching will have the greatest impact. But I also want to know what their aims are in life. What level of Maths do they actually need to achieve those aims?
I am still in contact with many students who have now left school. I see some working at the local supermarket. I see some students mountainbiking trails I use. I see some students at local sporting events I'm involved in. One student (who would listen to music on her phone through headphones when completing her work, not disturbing anyone else, and I would check on her to make sure she understood) now runs the best pizza shop in my area. They are happy to talk to me because I took an interest in their lives.
But the purpose of school is not for me to be able to chat with students when they leave.
I believe that if schools (i.e. the leadership) encouraged teachers to ask the question "What level of [my subject] does each student need to achieve their aims in life?" then the quality of teaching would be better and student perception of school would improve. A lot of what schools do doesn't particularly benefit students later in life. I once wrote a list of nine things schools do where the reality is different. I showed it to a class and said, "I understand you. I understand why you get frustrated."
I honestly don't know what it would take to "change my mind", but my thoughts do definitely evolve over time.
I feel the same way - some of my students will never need to understand (and sadly probably never will understand) how to get the Y intercept of a straight line graph. How much time should they and the teacher spend trying to help them understand it? And for what purpose? No doubt there are students who are on the boderline and with the right gudiance and encouragement can achieve more than they or we imagine. Is that the main reason we spend time on maths that students will never need for their futures?
I am interested on your "nine things". Please share if you want.
I wrote this 10 years ago.
.
Schools teach... You must work independently
In reality... Great ideas are often the work of a group
.
Schools teach... Copying and helping other people is not the way forward
In reality... Working together and reproducing what someone else has done is the foundation for progression
.
Schools teach... You must know how to reach the answer
In reality... The answer may be elusive, but a step along the way is a great achievement
.
Schools teach... Your exams are extremely important
In reality... The older you get, the less your exams mean
.
Schools teach... Only the qualified will get jobs
In reality... Realising your talents and knowing how to use them will provide many opportunities
.
Schools teach... English and Maths are the most important subjects you will ever study in school
In reality... Whilst many jobs have a requirement of English and Maths, realising and honing your talents will surpass that
.
Schools teach... Having everyone knowing the same stuff and thinking the same things is what society needs in order to progress
In reality... Progression is attained by many people thinking differently and creatively in order to help the different facets of society
.
Schools teach... Age is the most important thing we have in common, and people will learn best within their own age group
In reality... People are grouped according to their level of understanding of a particular subject, and the different ages provide additional creativity
.
Schools teach... If you are unable to do something, you are at a disadvantage
In reality... Stephen Hawking, Nick Vujicic
(just because you can't do something, that does not make you useless; working with other people where strengths complement each other is a huge advantage)
I believe math is worth learning even if students won't use it beyond their schooling. For instance, I think learning the rules of solving multistep equations are worth learning. It's reasonable to me that a significant number of students won't need to solve multistep equations after they leave school. To me, it's worth learning because it is challenging, and learning challenging, abstract concepts is a chance to show students what they are capable of, to show them that they can learn hard things, which helps them develop valuable beliefs about their ability as learners in the future.
This would be a hard hypothesis to gather evidence on, because the outcome is far in the future. I wonder if there is an impact of effective intervention on students' future self-efficacy or growth mindset beliefs? I'd be curious what evidence could address this, it's just a hard thing to test directly.
When done well, Mathematics teaches logical deduction. (The word "mathematics" is related to the Greek word for "discipline" and "disciple", i.e. people who follow a particular concept.)
And thinking logically is of huge importance to life. So I agree with you!
I'm a bit skeptical that math actually teaches a broad, transferable form of logical deduction. What evidence might convince you that this isn't actually a transferable skill?
Perhaps not so "broad". Transferable? Well, it depends on how far you take the Maths. I often talk to students about the "working" being like an essay. You start with your introduction, which is turning the question into a mathematical equation. Then each line of working is like a paragraph in your essay, logical statements flowing from the introduction. It ends with your conclusion, which is the answer to the question.
I know it's a bit of a reach, and I use that example not so much to tell students that Maths is important, but more to explain the importance of showing working!
I wish the topic of proof came up earlier in the curriculum... most students hate the proof that comes with congruent triangles. But mathematical proof provides a good introduction to logical fallacies (x implies y does not therefore mean that y implies x). And I, personally, think that an understanding of that is highly useful for life (but I can accept that I'm probably biased in that!). And it's probably why many of the great philosophers are also mathematicians.
I would say that logic only transfers sometimes, and math isn't necessary to be logical. Sure, math can help. But knowing math doesn't make one fully logical, and people who don't know math can be logical in contexts they are familiar with. I know lots of people who know lots of math and believe illogical things.
I've read research by people like Jo Boaler that advocates mixed ability teaching. I've taught in that way in primary classes and can see it worked in her examples. However, in the secondary school where I teach now, the range of attainment is so wide, I think mixed ability would be inefficient, demoralising and make it difficult to prepare for qualifications. I would be interested to see the alternative in action though.
Have you done the drill down into P8 data for mixed attainment teaching or is that a thought experiment? My understanding from EEF was that setting made no positive impact. I believe that mixed attainment teaching has the benefits described by your American colleague here *and* doesn’t negatively impact attainment of most students. Would change my mind if you presented the data you describe
I believe feedback is a critical part of the teaching dynamic - as critical as instruction, higher-order questioning, checking for understanding - all the aspects of the process that most educators agree on. I have left the classroom and give private lessons in students' homes. My students attend various international schools in my area, three of which I have taught at. I am dismayed to find that without exception, students are not receiving actionable feedback on their work. A grade (in any format) conveys very limited information. Since I teach maths and have managed this process myself, I find I have little patience with this change in practise. I used to design tests and then immediately write comprehensive solution guides showing how to solve each problem (sometimes using different approaches). I would post the solutions whilst students were taking the tests since there is no moment when they will be more interested than that immediately following the completion of the test. Students would hand in the tests and go directly to whichever portal the school was using and see my feedback. Each time, rigorous conversations would emerge as they left the testing room. If I had waited until the next day - or week!!!! - they would not have engaged so enthusiastically in the process of diagnosing their performances. SO - I really feel strongly about this and I would really like to hear why I should not be getting wrinkles over the fact that my current students are not benefitting from better feedback. I provide it - but that is not at all as beneficial as having the teacher provide it. IMHO
I agree this is great. But there's pretty much always someone away. Or the other class isn't doing the test til tomorrow. Or whatever else. I always find that you have to hold it back. How did you never have anyone absent?
I had at least three versions of each test and used them interchangeably year after year. So there was always a different version for the absent students, and because that version might have been used as the main test in a previous year, there was already a solution guide. If I'd had to start from scratch each year, it would have been too much for sure.
I believe that copying something down into a book is a waste of time and has no learning benefit. I believe this because when I stopped asking pupils to copy down things into their book, this did not impact pupil understanding or recall of key facts on whiteboard responses, and it also meant we had more time for practice of application. I also believe this because back when I have asked pupils to copy things down in the past, checks for listening/understanding showed that this didn't appear to have much impact on whether they could recall what they had written down or whether they even really understood what it meant.
To change my mind, I would have to see several solid instances of a lesson in which time spent copying something down into the book had a tangible impact on pupil learning that wouldn't have been achieved by doing something more interactive such as engaging with a whiteboard response to a check for understanding.
Hi Lucy - I can only speak for my own experience as a student. I am a visual learner to the extreme. Nothing a teacher said to me during class planted in my mind. Even group discussions did not accomplish lasting learning. I realised this as a child and copied down everything the teacher said during the lesson without trying to learn it in the moment. Then I went home and recopied everything I'd written into a duplicate set of notebooks and whilst I did that I learned the material - because I was looking at it, not listening. Mine is an extreme situation but it may be that some of your visual learners benefit from writing everything down.
Hi Callie, thanks for sharing your thoughts and experiences! I can see that this may be the case for some students, although I do think that is likely to be a minority in my school based on my experiences.
Do you not think the same outcome could be achieved then by them being given a handout, for example, of a worked example and key definitions? It would certainly take less time than hand writing out everything from the board, and they would then still be able to take these home and copy them out if that is how they felt best to revise independently.
I don't think that would have worked as well for me but perhaps for your students.
Hi Lucy,
I was going to respond to your first post, saying how having students copy out a worked example can be helpful for when they are answering questions themselves and get stuck... but you answered that in your reply - give them a handout! It requires a little extra preparation and is less spontaneous (I often make up examples on the spot, intended to directly address problems or questions students have, but I guess this can still be done with a handout).
So then, my question becomes... why have handouts rather than just go back to working out of textbooks? Textbooks have worked examples and questions, plus an explanation of the topic.
Good thoughts there. Thank you. With AI overtaking us, some of those points are even more relevant.