It was during Stuart Welsh's session during MathsConf23 that I came across the notion of arbitrary and necessary knowledge. Intrigued, I decided to look up and read the article Arbitrary and Necessary Part 1: a Way of Viewing the Mathematics Curriculum by Dave Hewitt.
Dave Hewitt creates a distinction between two types of knowledge in the mathematics curriculum:
Dave Hewitt creates a distinction between two types of knowledge in the mathematics curriculum:
- knowledge that you cannot arrive at by yourself, you need an external source of information (words, symbols, notation, and conventions). That is, arbitrary knowledge.
- knowledge that, given the right prior knowledge, you can arrive at (or work out) by yourself because it is necessarily true (properties and relationships). That is, necessary knowledge.
I quite like having the vocabulary to be able to distinguish between these two types of knowledge. It means that I can consider the similarities and differences between the two types of knowledge and how they interact with each other.
Mathematics has such a large number of conventions, vocabulary, symbols, and notations that students need to become familiar with. That is, "the student who, perhaps unfairly, needs to accept and adopt in order to communicate with the mathematics community" since "students are unlikely to convince the mathematics community to change the names and conventions already established, even if they had the platform to attempt to do this." Note, I said they become familiar with this knowledge. Over time and through exposure, these arbitrary choices become less foreign to students and they are able to use them to communicate with other mathematicians because they have the same "language" to speak in. This is no different to learning another language. You learn any new symbols needed for reading or writing words and how to say them, then you build up from there to words, phrases, sentences, paragraphs, and beyond.
These words, symbols, notation, and conventions are purely the product of what has survived the course of history. If events had transpired differently, maybe we would be using different arbitrary knowledge. What if the library of Alexandria didn't burn down? What if there was greater communication between mathematicians of the East and West? What if languages developed differently? What if Greek and Latin weren't the big languages in use by many mathematicians? What if we used base-60 or base-12 instead of base-10? What if we used tau instead of pi? What if Donna turned left instead of right? What impact would those changes have on the arbitrary and necessary knowledge we know today?
What about the future? Will all notations that have been developed stay as they are? Is it not possible that someone comes along with a notation or convention that works better and supersedes our current arbitrary knowledge? Look at grid method for example, and how much that has taken off. That is an arbitrary structure for performing multiplication and division but has in built checks to make sure all partial products are determined and connects the algorithm to the understanding of multiplication as the number of items in an array or the area of a rectangle. Arbitrary knowledge is not set in stone insofar as it is only what everyone currently agrees upon a shared meaning of. There are a number of mathematical notations that are not consistent such as multiplication using × or ⋅, or derivatives using f'(x), dy/dx, D_x(f). And what about the reverse, where different ideas use the same symbols such as an apostrophe for image, derivative, and complement, or an index of negative one indicating the reciprocal and the inverse function.
Later in the article, Hewitt puts forward the idea of generates: thing that have been generated from names and conventions, using awareness. Generates have two forms and can be described as: "if I adopt this convention, then this is a property I can state" and "if I make this change to the convention, then this could be so". That is, building concepts up using existing conventions to work out more mathematics, and seeing what mathematics we can uncover if we make changes to fundamental conventions we rely on. Many concepts in mathematics have been uncovered in these ways usually in a pure maths setting before later potentially finding an application, such as modular arithmetic and cryptography.
Mathematics also has a large number of properties and relationships that have been established over time. The great thing about mathematics is that once something is proven to be true, then it will remain true forever (provided the basis for the proof is all true). These facts that are necessarily true are the big ideas we generally want to teach students, as Hewitt also argues.
Some arbitrary and necessary knowledge act as prerequisite knowledge to understand even more mathematics. Arbitrary knowledge often acts as a lens we can use to observe a concept in action (such as using algebra or a Cartesian plane) and necessary knowledge also leads to more necessary knowledge (such as addition and repetition leading to multiplication).
Hewitt quotes Henri Poincaré and summarises his point as this:
A teacher's explanation is often based upon the teacher's awareness, and so may use things which students do not find evident - things which are not in the students' awareness and so the explanation will not be one which will help those students to educate their own awareness.
That is, explaining a concept that relies on knowledge that either (1) students have learnt before but is not consolidated or readily accessible or (2) that students have not yet learnt is an ineffective way to teach the understanding of the concept. It treats the concept as arbitrary since the explanation does not link to what the students actually know and can call upon. We know that mathematics is an interconnected web of ideas, so why wouldn't we build on what they already know?
Hewitt puts forward the following quote from Merttens (1995):
To instruct children is to give them a series of "now do this, then do that" procedures. It is also to credit them with their own intelligence. It is to assume that, with our help, they will utilize these procedures as and when appropriate, that they will have the intelligence not only to adopt but to adapt them, phrasing them in their own terms and for their own reasons, articulating them (in all senses of the word) in their own contexts; [...] The much-talked-of understanding will either come as they use the procedure, or later on, or even not at all because they never have the need to relate that particular algorithm to any other aspects of the subject. To understand, in this sense, is to translate, to incorporate what one has been given into one's own story (p 7).
When I read this quote, I agreed with what Merttens was saying. Building the understanding through use and familiarising first and building on it with later explanations can help to marry prior knowledge with the new idea. As opposed to starting from the prior knowledge, building on it conceptually, and then using and applying it. This is not to say that we can just teach any old topic and work backwards continuously until we reach the prior knowledge they do have (i.e. teaching how to find the area under a curve using integration then working all the way back to area of a rectangle and basic algebra). But instead making a small leap forward then building the bridge back. Remember Poincaré and the ineffective nature of explaining using concepts that are not readily available by the students. However, it was Hewitt's next paragraph that left me uneasy.
By this point, Hewitt had used the fact that necessary knowledge can be worked out to say we should teach necessary knowledge by using well structured tasks that bring out those ideas. While I agree that this can be done, it generally requires an excellently and expertly planned task, which is not something that is necessarily readily available or easy for most teachers to create when they are not available.
Hewitt then comments on the Merttens quote as follows:
Students may well go on to use "their own intelligence" by becoming aware of why certain procedures must give correct answers, in which case, the received wisdom [necessary knowledge taught as arbitrary] will become something which is necessary and so "incorporate what one has been given into one's own story". However, I have seen too many classrooms where students are not given the time or indeed the encouragement to work on trying to understand the received wisdom they have been offered. Too often, mathematics lessons appear to be about receiving the teacher's wisdom and practising how to replicate it, before moving on to the next item of received wisdom the teacher passes on. So, I feel there are other reasons for why students do not understand procedures about necessary aspects of the mathematics curriculum other than never needing "to relate that particular algorithm to any other aspects of the subject".
Maybe it was the way I read this paragraph, but, to me, in the context of reading the paper, it came across as saying that because, in many classrooms, teacher's do not return to build the understanding, we should instead teach using an inquiry model. Whereas I would argue doubling down and ensure that the explanation or justification of the concept happens. The leap from making a small change in the teaching method (ensuring the explanation occurs) compared to completely overhauling it (changing from direct instruction to inquiry based) seems a little extreme.
I think my main bugbear with using tasks to teach a necessary concept comes down to this. A well designed task can elicit a new (related) idea as being true and works as revision and extension of previously learned concepts. But what is the focus of the task? Revision and extension of the prior knowledge or learning and practising the new (related) concept and/or skill. Is the cognitive load of recalling and applying prior knowledge in a new context combined with understanding the new concept going to overload the student's working memory? While I do not have a definitive answer (as it would vary wildly on how solid the prior knowledge is and how complicated the new concept is), I would argue that it is likely to be on the higher side and that the new concept could get lost in the prior knowledge if we are not directly using or applying the new concept (compared to indirectly using it through the prior knowledge or only arriving at the idea by the end of the task and not using it at all).
Instead, if by using necessary knowledge in an arbitrary way students familiarise themselves with the concept and/or skill, they will reduce the working memory cost of the new concept as well. So, when marrying the prior knowledge and the new concept together, both are part of prior knowledge and the learner can focus purely on how they connect.
On the whole, I am very much a fan of the distinction between arbitrary and necessary knowledge and what that allows us to think about. But I do not fully agree that since necessary knowledge can be worked out from scratch that it should be every time. I cannot fathom what mental strain that would cause young people to be constantly recreating nearly the entirety of mathematical progress when historical mathematical figures in their prime could not necessarily determine some of these concepts. They took advantage of the existing work and stood on the shoulders of giants to build even higher. Sure, I will think twice about asking a student to "think about" an arbitrary concept they have forgotten and instead prompt them with a mnemonic or memory aid, and not just tell a student who has forgotten a necessary concept that they could rederive given the right prompt(s). But I do not think that it is productive to have students always (some inquiry is fine) recreating necessary knowledge when the concept is already, historically, well known.
Mathematics has such a large number of conventions, vocabulary, symbols, and notations that students need to become familiar with. That is, "the student who, perhaps unfairly, needs to accept and adopt in order to communicate with the mathematics community" since "students are unlikely to convince the mathematics community to change the names and conventions already established, even if they had the platform to attempt to do this." Note, I said they become familiar with this knowledge. Over time and through exposure, these arbitrary choices become less foreign to students and they are able to use them to communicate with other mathematicians because they have the same "language" to speak in. This is no different to learning another language. You learn any new symbols needed for reading or writing words and how to say them, then you build up from there to words, phrases, sentences, paragraphs, and beyond.
These words, symbols, notation, and conventions are purely the product of what has survived the course of history. If events had transpired differently, maybe we would be using different arbitrary knowledge. What if the library of Alexandria didn't burn down? What if there was greater communication between mathematicians of the East and West? What if languages developed differently? What if Greek and Latin weren't the big languages in use by many mathematicians? What if we used base-60 or base-12 instead of base-10? What if we used tau instead of pi? What if Donna turned left instead of right? What impact would those changes have on the arbitrary and necessary knowledge we know today?
What about the future? Will all notations that have been developed stay as they are? Is it not possible that someone comes along with a notation or convention that works better and supersedes our current arbitrary knowledge? Look at grid method for example, and how much that has taken off. That is an arbitrary structure for performing multiplication and division but has in built checks to make sure all partial products are determined and connects the algorithm to the understanding of multiplication as the number of items in an array or the area of a rectangle. Arbitrary knowledge is not set in stone insofar as it is only what everyone currently agrees upon a shared meaning of. There are a number of mathematical notations that are not consistent such as multiplication using × or ⋅, or derivatives using f'(x), dy/dx, D_x(f). And what about the reverse, where different ideas use the same symbols such as an apostrophe for image, derivative, and complement, or an index of negative one indicating the reciprocal and the inverse function.
Later in the article, Hewitt puts forward the idea of generates: thing that have been generated from names and conventions, using awareness. Generates have two forms and can be described as: "if I adopt this convention, then this is a property I can state" and "if I make this change to the convention, then this could be so". That is, building concepts up using existing conventions to work out more mathematics, and seeing what mathematics we can uncover if we make changes to fundamental conventions we rely on. Many concepts in mathematics have been uncovered in these ways usually in a pure maths setting before later potentially finding an application, such as modular arithmetic and cryptography.
Mathematics also has a large number of properties and relationships that have been established over time. The great thing about mathematics is that once something is proven to be true, then it will remain true forever (provided the basis for the proof is all true). These facts that are necessarily true are the big ideas we generally want to teach students, as Hewitt also argues.
Some arbitrary and necessary knowledge act as prerequisite knowledge to understand even more mathematics. Arbitrary knowledge often acts as a lens we can use to observe a concept in action (such as using algebra or a Cartesian plane) and necessary knowledge also leads to more necessary knowledge (such as addition and repetition leading to multiplication).
Hewitt quotes Henri Poincaré and summarises his point as this:
A teacher's explanation is often based upon the teacher's awareness, and so may use things which students do not find evident - things which are not in the students' awareness and so the explanation will not be one which will help those students to educate their own awareness.
That is, explaining a concept that relies on knowledge that either (1) students have learnt before but is not consolidated or readily accessible or (2) that students have not yet learnt is an ineffective way to teach the understanding of the concept. It treats the concept as arbitrary since the explanation does not link to what the students actually know and can call upon. We know that mathematics is an interconnected web of ideas, so why wouldn't we build on what they already know?
Hewitt puts forward the following quote from Merttens (1995):
To instruct children is to give them a series of "now do this, then do that" procedures. It is also to credit them with their own intelligence. It is to assume that, with our help, they will utilize these procedures as and when appropriate, that they will have the intelligence not only to adopt but to adapt them, phrasing them in their own terms and for their own reasons, articulating them (in all senses of the word) in their own contexts; [...] The much-talked-of understanding will either come as they use the procedure, or later on, or even not at all because they never have the need to relate that particular algorithm to any other aspects of the subject. To understand, in this sense, is to translate, to incorporate what one has been given into one's own story (p 7).
When I read this quote, I agreed with what Merttens was saying. Building the understanding through use and familiarising first and building on it with later explanations can help to marry prior knowledge with the new idea. As opposed to starting from the prior knowledge, building on it conceptually, and then using and applying it. This is not to say that we can just teach any old topic and work backwards continuously until we reach the prior knowledge they do have (i.e. teaching how to find the area under a curve using integration then working all the way back to area of a rectangle and basic algebra). But instead making a small leap forward then building the bridge back. Remember Poincaré and the ineffective nature of explaining using concepts that are not readily available by the students. However, it was Hewitt's next paragraph that left me uneasy.
By this point, Hewitt had used the fact that necessary knowledge can be worked out to say we should teach necessary knowledge by using well structured tasks that bring out those ideas. While I agree that this can be done, it generally requires an excellently and expertly planned task, which is not something that is necessarily readily available or easy for most teachers to create when they are not available.
Hewitt then comments on the Merttens quote as follows:
Students may well go on to use "their own intelligence" by becoming aware of why certain procedures must give correct answers, in which case, the received wisdom [necessary knowledge taught as arbitrary] will become something which is necessary and so "incorporate what one has been given into one's own story". However, I have seen too many classrooms where students are not given the time or indeed the encouragement to work on trying to understand the received wisdom they have been offered. Too often, mathematics lessons appear to be about receiving the teacher's wisdom and practising how to replicate it, before moving on to the next item of received wisdom the teacher passes on. So, I feel there are other reasons for why students do not understand procedures about necessary aspects of the mathematics curriculum other than never needing "to relate that particular algorithm to any other aspects of the subject".
Maybe it was the way I read this paragraph, but, to me, in the context of reading the paper, it came across as saying that because, in many classrooms, teacher's do not return to build the understanding, we should instead teach using an inquiry model. Whereas I would argue doubling down and ensure that the explanation or justification of the concept happens. The leap from making a small change in the teaching method (ensuring the explanation occurs) compared to completely overhauling it (changing from direct instruction to inquiry based) seems a little extreme.
I think my main bugbear with using tasks to teach a necessary concept comes down to this. A well designed task can elicit a new (related) idea as being true and works as revision and extension of previously learned concepts. But what is the focus of the task? Revision and extension of the prior knowledge or learning and practising the new (related) concept and/or skill. Is the cognitive load of recalling and applying prior knowledge in a new context combined with understanding the new concept going to overload the student's working memory? While I do not have a definitive answer (as it would vary wildly on how solid the prior knowledge is and how complicated the new concept is), I would argue that it is likely to be on the higher side and that the new concept could get lost in the prior knowledge if we are not directly using or applying the new concept (compared to indirectly using it through the prior knowledge or only arriving at the idea by the end of the task and not using it at all).
Instead, if by using necessary knowledge in an arbitrary way students familiarise themselves with the concept and/or skill, they will reduce the working memory cost of the new concept as well. So, when marrying the prior knowledge and the new concept together, both are part of prior knowledge and the learner can focus purely on how they connect.
On the whole, I am very much a fan of the distinction between arbitrary and necessary knowledge and what that allows us to think about. But I do not fully agree that since necessary knowledge can be worked out from scratch that it should be every time. I cannot fathom what mental strain that would cause young people to be constantly recreating nearly the entirety of mathematical progress when historical mathematical figures in their prime could not necessarily determine some of these concepts. They took advantage of the existing work and stood on the shoulders of giants to build even higher. Sure, I will think twice about asking a student to "think about" an arbitrary concept they have forgotten and instead prompt them with a mnemonic or memory aid, and not just tell a student who has forgotten a necessary concept that they could rederive given the right prompt(s). But I do not think that it is productive to have students always (some inquiry is fine) recreating necessary knowledge when the concept is already, historically, well known.