I thought this was very interesting, and will try to make more use of it. My school does the MYP, which is exclusively calculator, but embedded in the computer-based response programme, so has different functionality.
I cam to the comments to say that if teachers were concerned about students "cheating" by using a scientific calculator, there are many occasions where a basic calculator could be offered for the arithmetic, without this being able to skip steps in the larger process.
Completely sensible. Before being a Maths teacher I had worked in coal mines, building and installing machinery. It was thought macho to crawl down a coal face 3ft high without kneepads. In the US where productivity was far higher no such macho nonsense prevailed and people had decent kneepads. I always felt that letting students in study and learning use calculators allowed them to use their own initiative and feel ownership. Where in any work environment now would you trust someone who said he would ignore his calculat(or/ing machine) and do the sum in his head? I wouldn’t even trust myself if I had a calculator to hand. If we are to teach students to go into a joined up world and work in teams, non-calculator is simply non-sense!
In my experience as a primary and secondary teacher it is evident that those who have just been shown the method don’t really understand what is happening with the concept. It’s not understood or embedded. What all children really need is to be shown what is happening with the concept - the why. They can see this through the use of manipulative such as cuisenaire rods. Once they understand the why the rest is easy and they don’t tend to forget the steps because they make sense. Maths is abstract and we need to visualise it to understand more deeply.
Oh, I completely agree. But these kids had been shown the why at least 5 times since Year 5, including with manipulatives. The arithmetic can still get in the way
I'd be asking how to show they why differently instead of deciding that understanding was just getting in the way --> UNLESS (but I'd want evidence) they do really understand the stuff. (I come from experience at a middle/high school for folks w/ language learning disorders and the privilege of having the training and resources to do this; sadly, I recognize this isn't always possible.)
Oh, and https://resourceroom.net/devmath/ is where I stuck some of the 'intro to fractions' things I did online. We'll see how it goes this semester ;)
“There is a second reason to encourage regular calculator use. Students who most need a calculator are often the least proficient in using one. Anyone who has marked calculator papers at GCSE will see low-achieving students use written methods for multiplications that they could just put in their calculator. Repeated practice is the only way to build this much-needed awareness and proficiency.“
The thing with this as well is that it does the opposite of what your headteacher said - it teaches a life skill, not just teaches what comes up in a test. This, really, is a much better use of a child’s time in education.
As much as I don't want to agree with this, I do. In my southern California (Los Angeles) students I've experienced everything you described with my students. I have year 8's (13-14), 25-40% of them, who begin the year thinking ⅔ +⅕ = ⅜, or -6-2 =-4. I can re-teach the algorithms, but indeed the arithmetic of solving a multi-step linear equation in one unknown, for example, impedes their success.
I think most, if not all of us do. The challenge is 13-14 yr olds willing to invest the extra time to master it. Extra time means less TikToking, less Call of Duty-ing, etc.
Interesting assumption for the top set - that using a calculator to check answers means they won't have to wait until the end of the lesson. Why wouldn't the answers be available earlier (eg: back of textbook; displayed on board; on worksheet)?
Oh they definitely could be. But I think it is a nice way to give students a bit of extra calculator practice - especially for things like fractions, indices, etc where they need to know the special buttons to press
The issue with groups like this is that every student is struggling in a different place and every student needs different amount of processing time for different part of the process. So how do you ensure that the whole class is actually feeling that they are all learning at the right pace (not feeling bored or rushed) not just in one lesson but every lesson every day?
I think that is a separate - and definitely tricky - issue. Even the kids who weren’t struggling in this lesson would forget what they had “learned” a few days later. I’ll try to write about the issue you raise in a future newsletter
I definitely agree with this Craig. There are so many instances of pupils repeating the same going through the motions of topics like this year after year, but never being able to put the whole thing together because their attention is on their struggle with the arithmetic.
I'm sure that low motivation is linked to this. They rarely feel successful, so lack the motivation to try.
Motivation is 100% an issue. They struggle in the lesson, then essentially forget everything the next time they try a question. Building from a foundation of success is key to this
for adding and subtraction of fractions, I like to get students to write the second line of working as one fraction - in your example 2/5+3/4 = something/20 ie, LCM under a long line, then fill in the top line once that is written in.
then ask them 'how did we get from 5 to 20 ––> 'times by 4' , then whatever we do to the bottom, we do to the top and repeat for 'from 4 to 20'
2/5+3/4 = (4x2+5x3)/20 ––> for those that struggle with the multiplication or the process or jump straight to
2/5+3/4 = (8+15)/20
=23/20
obviously ––> working with vertical fractions. One of the points I stress is to ignore the plus or minus initially. So go through the process and put the +/- in last
2/5+3/4 = something/20
=(8 5)/20
=(8+15)/20
=23/20
possibly easier to show with a picture of the whiteboard - I will use a different colour when putting in the plus or minus
This is students can start cementing the process with steps linked to a repeated phrase
Whats the LCM
how do I get from 4 to 20
do the same to the top as I did to the bottom
put the +/- in
check the answer with the calculator
Having a calculator help with checking answers - ie "write what you think THEN check it with the calculator" is my approach.
Seems like you've now chosen to treat the symptom and not the problem. A year 10 student who can't multiply 4X2 of 3X4 is a disgrace to the educational system that caused this and no number of calculators can help this!
I think Craig's point is that you can choose to deal with the multiplication issue OR you can choose to deal with the adding fractions issue (within the one period of practice). Choosing to have the student focus on multiplication takes their attention away from how to add fractions. It doesn't preclude you from dealing with the times tables issues on another occasion, even within the same lesson.
I think that both perspectives have a point. Students that are not fluent with rudimentary arithmetic have been failed by the system; yet the statement of this does not resolve the problem for those students.
At the point in time where a student in their 7th (or higher) year of study of mathematics who has been promoted without core pre-requisite skills, who is expected to perform these higher skills, a work-around is necessary.
In some (short) number of years, they will leave school, and will need (benefit from) fundamental numeracy skills - you may as well teach them to use whatever tools they have access to. To this end, it is probably wise to ensure they use the calculator they have access to: their smartphones.
Nonetheless it is an enormously critical concern - that students could pass through a decade or more of weekly mathematics class and not attain fundamental fluency. I do not think it is correct to apply the blame to mathematics teachers - but the system of promotion/graduation has to be questioned!
I am amazed to see no visuals with this post. With a visual, they see why. They can count to get the common denominator. Don't worry about the lowest common denominator. We are focusing on the process. We use the same process for addition, subtraction and division. No reciprocals in division, this introduces another idea and therefore strays from the process.
See my earlier comment. These kids have been shown visuals and manipulatives for fractions several times over the years. The arithmetic still gets in the way.
Really interesting article. I had a similar experience recently with my bottom set Year 10s with ratio problems (especially simplifying). They could often spot a common factor, but were coming unstuck on doing the division. After giving them calculators we were able to focus more on interpreting the questions, problem solving, and the procedures over the arithmetic. We let them use calculators in a test this week on it and they did really well - it was much easier to determine where the issues were once their arithmetic skills weren’t a barrier
This is mindset in our secondary schools as well. Don’t bother to teach multiplication facts to automaticity in the
primary grades and then just give them calculators. If the answer is calculators, why bother teaching them how to add fractions with unlike denominators? Just teach them to convert the fraction to a decimal (using a calculator) and then add.
This post goes to the core of issues I experience as a Secondary Maths Teacher. Prior to this I was an upper Primary Teacher.
I agree that it is worth giving them a calculator or number square or something to take that part out of the way, if they do not know their number facts. Fixing the latter is desirable but not easy.
I also agree with Paul Kirschner that this is treating the symptom and not the problem although this is still useful advice to help tackle current and future difficulties. Where I would disagree with Paul is that it isn't necessarily the Year 10 Student who is at fault but rather the educational system that produces this predicament. The first fault is the poor standard of arithmetic fluency and number facts (number bonds, times table etc.) which undermines subsequent Maths expected to be learned. The other part that is wrong is that education systems shouldn't promote Students to courses who are not ready for them.
With respect to conceptual learning and understanding, the key part is that it is not embedded. When I first started teaching there was a swing to conceptual learning away from so called rote learning. Having used some useful structured approaches that went from manipulatives, to written methods and then finally to problem solving, experience has taught me to teach them the simple procedural part and then demonstrate the concept, often using manipulatives or diagrams when they have the basic written process. These then need to be properly embedded where Primary Pupils should not need to study too much for assessments - they should practice thoroughly over years in a structured manner such that they know it fluently. Another point here is that just because someone was taught rote (it's never that binary) doesn't mean that they do not figure out the concept. I use both progressive and traditional approaches.
I have tried to teach addition of fractions using Primary manipulatives in the Secondary but it is not very successful so I agree with the 5 years since Year 5 point. That repeated failure has a big impact where I suspect those Pupils may have been better off not bothering with fractions down the school. They appear to believe that they can't do Maths and it becomes: 'Whether you think you can, whether you think you can't, you are probably right'.
What could be done to improve matters?
- Highlight this issue with leadership in School, regionally and nationally if necessary as it is vital for success in Maths and probably has significant economic repercussions.
- Work with other parts of the Educational System to sort the arithmetic shortcomings and properly prepare Students so that subsequent Maths learning is not stymied.
- Reconsider System Pathways that attempt to remedy arithmetic fluency, ideally only until the point above is tackled. This would require some decent resources, suitable for the age of Pupils struggling that help to change repeated or embedded failure to success and motivation.
Welp, saying "use the calculator for the arithmetic" when .... the arithmetic is what they're going to be tested on... seems to be an issue to me, especially since most calculators will just let them plug fractions in and get a fraction answer.
Last semester I spent about half an hour w/ a group of students needing to boost their placement test scores ... building the concept of fractions from the bottom up, as "parts and wholes." It's different... but it worked b/c they totally knew that 1 - 1/8 was 7/8 and what denominators were.
YES, they could use times tables charts though, so they were freed up from that. Wetalked about what to do on the assessment to generate the facts they'd need (b/c once in our classes, they'd either be allowed to use the charts or allowed to use calculators. )
They *all* (sample size 5, I know :P ) shifted mindsets to wanting to figure out and understand the stuff and yes, they all boosted those scores to the placement they needed.
do not use a calculator. Have a sheet with the multiplication tables. And some decimal fractions like 0.375 Then let them check their answer by adding the decimal fractions.
If they can use a calculator. Why not Wolfram Alpha?
I thought this was very interesting, and will try to make more use of it. My school does the MYP, which is exclusively calculator, but embedded in the computer-based response programme, so has different functionality.
I cam to the comments to say that if teachers were concerned about students "cheating" by using a scientific calculator, there are many occasions where a basic calculator could be offered for the arithmetic, without this being able to skip steps in the larger process.
Completely sensible. Before being a Maths teacher I had worked in coal mines, building and installing machinery. It was thought macho to crawl down a coal face 3ft high without kneepads. In the US where productivity was far higher no such macho nonsense prevailed and people had decent kneepads. I always felt that letting students in study and learning use calculators allowed them to use their own initiative and feel ownership. Where in any work environment now would you trust someone who said he would ignore his calculat(or/ing machine) and do the sum in his head? I wouldn’t even trust myself if I had a calculator to hand. If we are to teach students to go into a joined up world and work in teams, non-calculator is simply non-sense!
In my experience as a primary and secondary teacher it is evident that those who have just been shown the method don’t really understand what is happening with the concept. It’s not understood or embedded. What all children really need is to be shown what is happening with the concept - the why. They can see this through the use of manipulative such as cuisenaire rods. Once they understand the why the rest is easy and they don’t tend to forget the steps because they make sense. Maths is abstract and we need to visualise it to understand more deeply.
Oh, I completely agree. But these kids had been shown the why at least 5 times since Year 5, including with manipulatives. The arithmetic can still get in the way
Agreed.
I'd be asking how to show they why differently instead of deciding that understanding was just getting in the way --> UNLESS (but I'd want evidence) they do really understand the stuff. (I come from experience at a middle/high school for folks w/ language learning disorders and the privilege of having the training and resources to do this; sadly, I recognize this isn't always possible.)
Oh, and https://resourceroom.net/devmath/ is where I stuck some of the 'intro to fractions' things I did online. We'll see how it goes this semester ;)
“There is a second reason to encourage regular calculator use. Students who most need a calculator are often the least proficient in using one. Anyone who has marked calculator papers at GCSE will see low-achieving students use written methods for multiplications that they could just put in their calculator. Repeated practice is the only way to build this much-needed awareness and proficiency.“
The thing with this as well is that it does the opposite of what your headteacher said - it teaches a life skill, not just teaches what comes up in a test. This, really, is a much better use of a child’s time in education.
As much as I don't want to agree with this, I do. In my southern California (Los Angeles) students I've experienced everything you described with my students. I have year 8's (13-14), 25-40% of them, who begin the year thinking ⅔ +⅕ = ⅜, or -6-2 =-4. I can re-teach the algorithms, but indeed the arithmetic of solving a multi-step linear equation in one unknown, for example, impedes their success.
We could also teach the concept.
I think most, if not all of us do. The challenge is 13-14 yr olds willing to invest the extra time to master it. Extra time means less TikToking, less Call of Duty-ing, etc.
Interesting assumption for the top set - that using a calculator to check answers means they won't have to wait until the end of the lesson. Why wouldn't the answers be available earlier (eg: back of textbook; displayed on board; on worksheet)?
Oh they definitely could be. But I think it is a nice way to give students a bit of extra calculator practice - especially for things like fractions, indices, etc where they need to know the special buttons to press
The issue with groups like this is that every student is struggling in a different place and every student needs different amount of processing time for different part of the process. So how do you ensure that the whole class is actually feeling that they are all learning at the right pace (not feeling bored or rushed) not just in one lesson but every lesson every day?
I think that is a separate - and definitely tricky - issue. Even the kids who weren’t struggling in this lesson would forget what they had “learned” a few days later. I’ll try to write about the issue you raise in a future newsletter
I definitely agree with this Craig. There are so many instances of pupils repeating the same going through the motions of topics like this year after year, but never being able to put the whole thing together because their attention is on their struggle with the arithmetic.
I'm sure that low motivation is linked to this. They rarely feel successful, so lack the motivation to try.
Motivation is 100% an issue. They struggle in the lesson, then essentially forget everything the next time they try a question. Building from a foundation of success is key to this
for adding and subtraction of fractions, I like to get students to write the second line of working as one fraction - in your example 2/5+3/4 = something/20 ie, LCM under a long line, then fill in the top line once that is written in.
then ask them 'how did we get from 5 to 20 ––> 'times by 4' , then whatever we do to the bottom, we do to the top and repeat for 'from 4 to 20'
2/5+3/4 = (4x2+5x3)/20 ––> for those that struggle with the multiplication or the process or jump straight to
2/5+3/4 = (8+15)/20
=23/20
obviously ––> working with vertical fractions. One of the points I stress is to ignore the plus or minus initially. So go through the process and put the +/- in last
2/5+3/4 = something/20
=(8 5)/20
=(8+15)/20
=23/20
possibly easier to show with a picture of the whiteboard - I will use a different colour when putting in the plus or minus
This is students can start cementing the process with steps linked to a repeated phrase
Whats the LCM
how do I get from 4 to 20
do the same to the top as I did to the bottom
put the +/- in
check the answer with the calculator
Having a calculator help with checking answers - ie "write what you think THEN check it with the calculator" is my approach.
Seems like you've now chosen to treat the symptom and not the problem. A year 10 student who can't multiply 4X2 of 3X4 is a disgrace to the educational system that caused this and no number of calculators can help this!
I think Craig's point is that you can choose to deal with the multiplication issue OR you can choose to deal with the adding fractions issue (within the one period of practice). Choosing to have the student focus on multiplication takes their attention away from how to add fractions. It doesn't preclude you from dealing with the times tables issues on another occasion, even within the same lesson.
Yes, that is exactly it.
I think that both perspectives have a point. Students that are not fluent with rudimentary arithmetic have been failed by the system; yet the statement of this does not resolve the problem for those students.
At the point in time where a student in their 7th (or higher) year of study of mathematics who has been promoted without core pre-requisite skills, who is expected to perform these higher skills, a work-around is necessary.
In some (short) number of years, they will leave school, and will need (benefit from) fundamental numeracy skills - you may as well teach them to use whatever tools they have access to. To this end, it is probably wise to ensure they use the calculator they have access to: their smartphones.
Nonetheless it is an enormously critical concern - that students could pass through a decade or more of weekly mathematics class and not attain fundamental fluency. I do not think it is correct to apply the blame to mathematics teachers - but the system of promotion/graduation has to be questioned!
I am amazed to see no visuals with this post. With a visual, they see why. They can count to get the common denominator. Don't worry about the lowest common denominator. We are focusing on the process. We use the same process for addition, subtraction and division. No reciprocals in division, this introduces another idea and therefore strays from the process.
See my earlier comment. These kids have been shown visuals and manipulatives for fractions several times over the years. The arithmetic still gets in the way.
Really interesting article. I had a similar experience recently with my bottom set Year 10s with ratio problems (especially simplifying). They could often spot a common factor, but were coming unstuck on doing the division. After giving them calculators we were able to focus more on interpreting the questions, problem solving, and the procedures over the arithmetic. We let them use calculators in a test this week on it and they did really well - it was much easier to determine where the issues were once their arithmetic skills weren’t a barrier
This is mindset in our secondary schools as well. Don’t bother to teach multiplication facts to automaticity in the
primary grades and then just give them calculators. If the answer is calculators, why bother teaching them how to add fractions with unlike denominators? Just teach them to convert the fraction to a decimal (using a calculator) and then add.
This post goes to the core of issues I experience as a Secondary Maths Teacher. Prior to this I was an upper Primary Teacher.
I agree that it is worth giving them a calculator or number square or something to take that part out of the way, if they do not know their number facts. Fixing the latter is desirable but not easy.
I also agree with Paul Kirschner that this is treating the symptom and not the problem although this is still useful advice to help tackle current and future difficulties. Where I would disagree with Paul is that it isn't necessarily the Year 10 Student who is at fault but rather the educational system that produces this predicament. The first fault is the poor standard of arithmetic fluency and number facts (number bonds, times table etc.) which undermines subsequent Maths expected to be learned. The other part that is wrong is that education systems shouldn't promote Students to courses who are not ready for them.
With respect to conceptual learning and understanding, the key part is that it is not embedded. When I first started teaching there was a swing to conceptual learning away from so called rote learning. Having used some useful structured approaches that went from manipulatives, to written methods and then finally to problem solving, experience has taught me to teach them the simple procedural part and then demonstrate the concept, often using manipulatives or diagrams when they have the basic written process. These then need to be properly embedded where Primary Pupils should not need to study too much for assessments - they should practice thoroughly over years in a structured manner such that they know it fluently. Another point here is that just because someone was taught rote (it's never that binary) doesn't mean that they do not figure out the concept. I use both progressive and traditional approaches.
I have tried to teach addition of fractions using Primary manipulatives in the Secondary but it is not very successful so I agree with the 5 years since Year 5 point. That repeated failure has a big impact where I suspect those Pupils may have been better off not bothering with fractions down the school. They appear to believe that they can't do Maths and it becomes: 'Whether you think you can, whether you think you can't, you are probably right'.
What could be done to improve matters?
- Highlight this issue with leadership in School, regionally and nationally if necessary as it is vital for success in Maths and probably has significant economic repercussions.
- Work with other parts of the Educational System to sort the arithmetic shortcomings and properly prepare Students so that subsequent Maths learning is not stymied.
- Reconsider System Pathways that attempt to remedy arithmetic fluency, ideally only until the point above is tackled. This would require some decent resources, suitable for the age of Pupils struggling that help to change repeated or embedded failure to success and motivation.
Welp, saying "use the calculator for the arithmetic" when .... the arithmetic is what they're going to be tested on... seems to be an issue to me, especially since most calculators will just let them plug fractions in and get a fraction answer.
Last semester I spent about half an hour w/ a group of students needing to boost their placement test scores ... building the concept of fractions from the bottom up, as "parts and wholes." It's different... but it worked b/c they totally knew that 1 - 1/8 was 7/8 and what denominators were.
YES, they could use times tables charts though, so they were freed up from that. Wetalked about what to do on the assessment to generate the facts they'd need (b/c once in our classes, they'd either be allowed to use the charts or allowed to use calculators. )
They *all* (sample size 5, I know :P ) shifted mindsets to wanting to figure out and understand the stuff and yes, they all boosted those scores to the placement they needed.
do not use a calculator. Have a sheet with the multiplication tables. And some decimal fractions like 0.375 Then let them check their answer by adding the decimal fractions.
If they can use a calculator. Why not Wolfram Alpha?
A calculator demotivates weak students.