The Classification of Word Problems: Compare Problems

In past blog posts I have concentrated on the linguistic complexities of problem solving and, indeed, there is much research to attribute to this (Dellarosa, Kintsch, Reusser and Weimer, 1988), equally, however, ineffective instruction has also been cited (Carpenter & Moser, 1984, Verschaffel, 1987 and Essen & Hamaker, 1990).

A common discussion point for teachers originates from pupil outcomes in KS1-KS3 text books and exam papers; the separation of a student’s mathematical knowledge and their application of it to a word problem structure differs. The many research reports I have read in this area, often concludes in the importance of an awareness of problem categories, the teaching of them and highlighting, the difficulty of compare problems, particularly where addition is involved (Kintsch & Greeno, 1985; Valentin, 2004; Okomoto, 1996).

I started looking into categories for word problems in more detail a few years ago. Taking the 2018 and 2019 KS1 and KS2 papers and attempting to categorise the word problems within the reasoning papers, particularly compare problems, this proved more problematic and not so revelatory as I first thought. These were my findings:

  • In 2018 and 2019 in KS1 there were two-three compare problems in paper 2 and just three over both paper 2 and 3 in KS2.
  • Many problems at KS2 were a combination of problem types within the steps.
  • Most of the compare problems for KS1 were towards the back of Paper 2.
  • The most common problems at KS2 in 2019 were a combination of problem types, however, ‘Change 2’ and ‘Combine 1’ were very common (see TABLE 1). On the surface, this seems rather simplistic for KS2 but often, these problems were based upon the presumption of knowledge of mathematical concepts such as BODMAS, mean, no. of sides of shapes, rounding, conversions, inverses, heaviest/lightest, scale, days in months and fractions of amounts (the role of division).
KS2 – 2019 Paper 3
  • At KS1, the 2019 Paper 2 Problems could be categorised as X4 Combine, X2 Compare and X3 Change. The distribution within these types were even, for example, within the combine problems the combination was unknown and subsets unknown. Similarly, as with some KS2 problems, this was combined with a previous step (see example below):
KS1 – 2019 Paper 2

My focus on some comparison problem types in this article is not simply one of test analysis. From an early age, children naturally engage in the process of comparison, it is often connected with a sense of fairness. Are we getting the same? Comparison is seen in symbols used for inequalities, the way we categorise shapes, angles, data, calculations, ratios, percentages, measurements, and fractions too. This blog looks at 11 tried and tested ideas to explore with a range of Key Stages. Hopefully , you’ll find one or more approaches for September you’d like try.

A few years ago, I started some intervention work to include the teaching the problem types in isolation. This resulted in greater word problem confidence and identification for pupils but as mentioned in my findings above, pre-requisite knowledge is key to problem solving, regardless of identification of problem type. It is something I want to continue to explore further next academic year.

Word problem categories are by no means a panacea for children’s difficulties. In all of the examples I have shared, some thought must by given to the importance of the application of arithmetic operations, children’s mental representations of the problem and the relationships explored between the quantities and the text.

Idea 1 – Same/Different & Would you Rather?: Activities such as same/different or would you rather would be another good place to start. In these activities children are immediately asked to compare and justify their reasons for the comparison. The two hyperlinked websites provide a bank of ideas to get you started.

Most problems have an owner (of objects), quantity (the value of objects), type of object and the intended actions (goal). Idea 2 – Parts of a Problem (see diagrams below): I found it was helpful using the diagram below with children when exploring all of word problem. This helps identify whether we know some/all of the quantities. Is there an owner? Do we know the goal? Of course, we know, the goal is often in the highlighted blue strip of test questions but, there may be a series of calculations (in this case the cost of the potatoes and carrots) we need to carry out before achieving the goal.

This image has an empty alt attribute; its file name is qunatity-object.png
Parts of a Problem
This image has an empty alt attribute; its file name is carrots-1.png
Owner highlighted in red, quantities in blue, object in black and goal highlighted in yellow.

Rather than solve a problem in its entirety, I have used a spinner (a paperclip will suffice as a spinner tool) to get children to identify these parts of the problem. Later, these will be helpful when identifying problem categories; where the different arrangements of these parts distinguishes them from each other.

Idea 3 – Changing the Quantities of Objects using More-Same-Less: For many children, the test format is cognitively overloaded and returning to my blog on goal free problems, I think there is more potential in focusing on the quantity and object in the first instance, and just looking at the picture. Later, we could increase the difficulty and play with the quantities of the objects and see if this is still possible given the goal (to have change from £5.00) using the more/same/less structures I have been working on with Pete Mattock and Ashton Coward in https://www.more-same-less.co.uk/. This would also give rise to natural opportunities for comparison. For example, in which scenario did Jack get the most change?

KS2 – Original Problem
Goal Free
More-Same-Less

Do we teach all the problem categories?

One conclusion I would draw from my focus on categories was an increased awareness on the variety. It is easy to assume we cover all types in our teaching but this is simply not the case. As teachers, we all need to review our coverage of problem types as tests and school Curriculum resources will not necessarily provide the range we require. Tables 1 and 2 also provide a sort of hierarchy of difficulty and within this we can pitch our problems appropriately (as long as we also assume difficulty can also be increased through ideas such as assumed knowledge, the number of steps and step size too).

Research tells us change problems are meant to be easier than combine and compare, particularly for subtraction (Riley et al, 1983). The reasons given is that change problems often have dynamic relations which make the modelling easier and the combine and compare problems have static relations.

Over the last 40 years there have been several models that have tried to categorise word problem types. The conceptual differences have been attributed to the difficulties children have with them. Word problems can be placed in categories based upon relationships between objects, persons and events described in the problem. In this blog post I mainly look at the categories offered by Riley et al (1983) and include ‘equalize’ problems from Carpenter and Moser (1982). There is a particular focus on what is commonly identified in research as the most challenging on the three main categories, compare problems. However, in summary, there are three main problem types:

  • Combine two or more quantities.
  • Change in a quantity.
  • Compare between two or more quantities.

Both example lists below are not exhaustive and further variations are available.

TABLE 1
TABLE 2

Depending on the position of unknown quantities, problems differ in their difficulty (Carpenter et al., 1981; Riley, 1981; Riley et al., 1983). An example of this is with change problems; where we know the start and change, it is much easier than if the start or the change is unknown.

Scaffolds and Models to Support Comparison

Idea 5 – The Blank/Empty Bar Model – Noticing & Constructing: It is often helpful to begin our comparison in reverse. That is to say, build one from a blank model to support children’s understanding of the process of comparison and components of the problem involved rather than one that is complete. Using Cuisenaire, Multilink or Foam Squares on a large sheet of A3 paper is perfect for this.

In this example the squares are marked out. What do they immediately notice? With the class we can ask them to label the owner of each of the bars, which objects do they represent? Does one part equal one object? Then we can begin the ask questions of comparison, before making further changes (add another owner; a third bar, changing the value of the parts) and asking further questions of comparison. We could write out or discuss all of the possible comparison problems the model could represent.

Idea 6 – The Blank/Empty Bar Model & Story: We could then move onto a blank bar model. The situation can be broken up into smaller parts, however, in this example, a problem is read out to the children by the teacher and the pupils must label the bars correctly and consider the significance to the problem:

There are 34 cakes left at the bakery.

There are 14 more cookies than pastries at the bakery.

How many cookies and how many pastries are in the bakery?

In both examples, teacher modelling of the process and explaining exactly why the bars have been labelled this way requires input.

The models below may also be supportive to your teaching of comparison.

The simple compare scaffold bar model
Multiplicative Comparison and the next step, ‘Equalize’ – to make the quantities the same. Often these are problems where one character in the problem gives the other some of their objects so that they have equal amounts.

In Asha Ditendra’s Article ‘Teaching Students Math Problem-Solving Through Graphic Representations’ (2002), an adapted model by Marshall, S. P (1995) is used:

Idea 7 – Number Line Comparison: I also find number lines helpful, either one or two (one directly above the other) when comparing two quantities. Bead strings work just as well or counters in two rows. In the example below, I’ve used the Number line ITP from Mathsframe to compare two quantities. A problem such as Rohan has 6 buttons, Kelly has 14 buttons. How many more buttons does Kelly have than Rohan?

Mathsframe ITP

Idea 8 – Making Changes for Further Comparisons: In this idea I have taken the original problem and looked at how we might explore changing it to explore further opportunities for comparison:

The scaffolds above are available to use/adapt through this PowerPoint:

Idea 9 – Focus on the Vocabulary of Comparison: There are implications for the emphasis we place on mathematical vocabulary when focusing on compare problems; more/less, fewer, increase/decrease, larger/smaller and confusing cross-problem constructions; ‘more than’ as opposed to ‘then ….. more’. In the multiplicative ‘times as much’ problems, additive structures of ‘more’ are also helpful, ‘how many times more ….’. It is also important for children to realise they may not be making the comparison using a consistent unit, this may vary (often seen in measurement problems at KS2, for example comparing the cost of 250ml of orange juice to 3/4 of a litre of lemonade).

Idea 10 (similar to Idea 4): Using a Goal Free Approach to Increase Compare Opportunities.

In the example below we have a KS2 SATs question that originally asked for the total number of hours English were taught on the timetable, however, there are clearly many opportunities for comparison here, for example, hours different subjects are taught and time taken for break v lessons

KS2 2017 – Paper 3

Idea 11 – Using Graphs, Charts and Tables: Often compare problems sit within the questions involving graphs, charts and tables, however, even when this is not the case, opportunities are abundant. Comparison is one of the many tools applied to data sets. In this KS1 Bar Graph, the opportunity for comparison has been used but there are many more opportunities we could ask of the children to open up the problem further:

KS1 Paper 2 – 2019

Key References

  • Semantic structure of arithmetic WPs (Riley et al., 1983)
  • Standard/non-standard WP
    (Jimenez and Verschaffel, 2014)
  • Jitendra, A. (2002). Teaching students math problem-solving through graphic representations. TEACHING Exceptional Students, 34(4), 34–48.
  • Valentin, J et al (2004) Roles of Semantic Structure of Arithmetic Word Problems on Pupils’ Ability to Identify the Correct Operation
  • Okamoto, Y (1996) Modeling Children’s Understanding of Quantitative Relations in Texts: A Developmental Perspective

Plans & Autumn 2020 onwards..

After 14 years of research and one NCETM innovation, I’ve finally decided to write a book of research underpinned activities for KS1-3 to support mathematical word problems and associated vocabulary.

This will be my second book, the first, a long while ago, at 29, a book on Calculators in KS2 at a time calculator usage had greater emphasis due to its inclusion in the Primary Mathematics Curriculum and end of Key Stage tests, published by BEAM.

I’ve been back in the classroom for almost three years and now, it would be hard to extract me. Quite simply, I love it. It has also given me time to innovate, experiment and assure staff in the potential of these ideas. I am absolutely passionate and committed to benefits from them.

So what will the book look like? The ideas shared over conferences in the last three years including the recent #MathsConference23 amounts to a few of a larger bank written/adapted over the years. I feel strongly it has to be a resource teachers can pick up and use instantly in the classroom.

So one more blog post by the end of next week looking at problem categories and then, at least for a while, posts will look a little different.

In response to supporting an engaging and exciting curriculum, short ideas that combine other curriculum areas and the real world (purposeful connections). Beyond word problems, my next area of interest is Statistics. Many of the ideas will contain statistical elements. We often forget, the data handling cycle is an inquiry cycle, rich in opportunities for reasoning and problem solving. It also has a place right across the Curriculum.

The first idea will look at recycling. Here’s a clue. One of the activities we will discuss:

Using More Same Less in the Primary Classroom & Experimenting with ‘Always, Sometimes & Never’

See the source image

In this blog post I report back about my own use of the ‘More-Same-Less’ grids I put together on the website https://www.more-same-less.co.uk/ (for the Primary Grids go to ‘Grid Collection’ on the menu bar at the top. There are now over 50 for EYFS-KS2!) and Rose Maini, Mathematics Coordinator at one of my schools, Holy Family Catholic Primary in Langley trials both the ratio and money grids. Finally, Alison Hogben (https://twitter.com/AlisonHogben) trials the variation on this theme I put together for ‘Always-Sometimes-Never’ with Odd and Even Numbers. I have reposted a full set of these at the end of this post should you want to experiment with them!

I would first of like to say (I know Alison found this too) teaching with the careful, school safety considerations for COVID-19 does make group work and checking responses tricky. Thankfully, whiteboards and visualisers go a long way in making a temporary fix for the sharing of responses/interaction. We hope our findings inspire you to give both of these approaches a try.

More-Same-Less

An adventure in ratio & money – what is the most notepads you can buy with a given amount?

The M-S-L Grid for our Lesson

I’m a great believer that if you are going to try something new with a class tell them. Tell them it might work brilliantly, need a few tweaks or really not work at all, not the first time. Tell them we are all learning together and you will be guided by their feedback and outcomes. Experimentation is so exciting!

As it had been a while since I’d covered Ratio and Proportion with this Year 6 Class, I wanted to make sure they remembered the terms and start to identify with the context in the lesson with a pre-learning picture task. Particularly now, the need to consolidate what children do know rather than rush to fill gaps/move on, supports a much more supportive and positive outcome for the children.

For retrieval, we looked at the term ‘Ratio’ and ‘Proportion’ using the vocabulary grid explored in my #MathsConf23 session. I wanted to make sure they didn’t just give a definition or an example but a much deeper understanding of the terms. Again, ‘acting out’ the definition in the current climate proved tricky!

Starter Activity from my #Maths Conference 23 Session.

Then we explored the definitions through some posters I’d put together many years ago and explored any gaps in their understanding.

Ratio and Proportion Posters

Next, we discussed the picture and models below; the process of making squash, which bar would represent the strongest drink? What are the ratios/proportions of squash to water? I found this activity particularly supportive to the grid they would face. We could then draw similarities and differences in the context and the children could more easily apply their knowledge to the new task. If I was to do this again, I would certainly use a pre-model/picture task until their familiarity and confidence with the grids grew.

Picture/Model Task

Next, we started to look at the grid. We started by answering the initial, middle question.

Original Problem

Using purple multi-link to represent the blackcurrant cordial and blue multi-link to represent the water proved a supportive model and link to their work on bar models.

Multi-link Bar Model

Given each of the parts were worth 40ml, it was then much easier to see the total drink = 160ml. The children knew this drink tasted ‘just right’ so any change may make the drink weaker or stronger and they had to comment on this with each change and justify their answers.

The original multi-link model was left out on display, the children were always referring to it as each change in the remaining boxes were made. By doing this, the children could see what was the same and what was changing in square of the grid.

Focus on one column at a time

We started by looking at just one column first. To reduce the cognitive load, I didn’t want the children distracted by the other parts. By looking at one column, we established there would always be more water than our original problem but it would be the blackcurrant cordial that would either change or remain the same.

I modelled an example problem from the top left (More/More) showing this:

More/More

In this example we have more blackcurrant cordial and more water. There would be 240ml of blackcurrant squash in total. It works on one level but as the children discussed ‘…it would be a stronger drink than the original’ and one child added ‘…because blackcurrant here is 2/6 or 1/3 and in the original it was 1/4. 1/3 is more than 1/4.’ The wow moment came when one child said, ‘…I know a way it could taste just the same as the original and still have more parts water and blackcurrant’. They demonstrated confidently the use of equivalent ratios with, ‘..let’s just double the amounts.’

By doubling the original ratio, you still satisfy the demands of more water and more cordial but the taste remains the same as the ratios are equivalent.

The bottom right part of this column was equally interesting. On first inspection, many children thought it was not possible to have ‘less blackcurrant cordial’ than the original, that would leave just water they said! One child however, had a very creative idea. Add one water so there is more and rather than remove a whole blackcurrant part they said,

‘We could halve it but then we would need to halve all the water parts so that they are equal = 9 parts. So now blackcurrant would be 1/9 instead of 1/4. It would be a very weak tasting drink because 1/9 is less than 1/4.

The idea to halve the blackcurrant and by doing so, halving all the other parts (water)

Within the lesson, there were many opportunities for reasoning and the range of responses and possibilities each square could bring, together with the taste of the blackcurrant drink brought into question a comparison of fractions; the original to the adjusted problem.

The children then chose any number they liked to work on the remainder of the boxes with the children on their table or alone.

In the feedback and when presenting their problems, it was clear that most of the children understood how the grid worked and the freedom they had to change quantities (as long as they were within the more/same/less parameters each square requested). A few children could only demonstrate this in a ratio (but could discuss the taste of the drink and the total millilitres of drink correctly in class).

Independent task

I really think this activity has lots of potential and I’m excited to try more next September. I do think the use of manipulatives alongside the activity was a huge benefit in supporting procedural variation and something I’d recommend to anyone wanting to use these for the first time.

The entire grid can be quite overwhelming, therefore, in terms of reducing cognitive load in the first instance, I would cover up squares to focus attention on a specific part. Modelling and demonstrating the activity is essential and this will take time in the first instance.

The children reported back how they had enjoyed the task and certainly, it was a helpful activity in recalling and embedding their understanding of ratio and proportion.

Here are some of the wonderful outcomes from the pupil’s work:

Children’s work from Holy Family Catholic Primary School, Langley, Slough

I would also consider the following as relaxation of restrictions allow:

With huge thanks to the children and staff at Holy Family Catholic School, Langley, Slough

  • Opportunities for children to swap/discuss their grids.
  • Showing a problem with the more-same-less labels removed to see if they can identify what has happened to it.
  • Come up with more than one problem to fit a given square of the grid – For those able to do do this, the activity demonstrated a deeper understanding.
  • Give each pair a different square in the grid to share as jigsaw activity (this would make the activity quicker and may be something to consider if you did not want to use these as a whole class activity).

Rose Maini, Mathematics Coordinator at one of my schools, also tried the ratio grid but also looked at the notepad grid too. The children were asked how many notepads of a given cost could they by with a given note. Rose reported that after some introduction, how quickly the children picked up the idea. The results speak for themselves.

Original Grid: How many notebooks can you buy for ….?

Always-Sometimes-Never


Exploring Odd and Even Numbers in Alison Hogben’s Classroom

The children were introduced to the task by being given some simple calculations to do and asked about what they noticed. Although initially they were not told that the investigation was about odds and evens, they quickly established a pattern, refined by the aid of stem sentences. The activity sheet was introduced with the children being asked to place the statements they had already found into the correct boxes. They were then left to explore some of their own, being encouraged to use all four rules of calculation, and show proof by using examples.

Many of the children recognised that they could use something they had found out to fill several boxes. E.g. If ‘even subtract even is always even then it is never going to be odd.’ What the children had failed to consider however was division when there was a remainder, which would have meant that the ‘always’ and ‘never’ boxes could not be filled. 

The beauty of this task is the accessibility for all children and how easily it can be differentiated. The level of calculation is low, which reduces cognitive load.

Those who found it more difficult to know where to start were given a statement to investigate and then work out which box it fitted. Those more confident worked on their own choice of investigation, then were challenged to explain the reasoning behind what they found. E.g. why does an even number multiplied by an even number result in an even product?

They were further extended by being asked which boxes could not be filled and the reasons why.


Scribed pupil verbal responses – these two children used their knowledge from the addition and applied it to the multiplication.

If you are keen to try these ideas, resources for both formats are below for you to download.

Resources

MORE-SAME-LESS LESSON (INCLUDING GRIDS)

ALWAYS-SOMETIMES-NEVER GRIDS (FOR KS1 & KS2)

Developing Mathematical Explanations

One idea using a scale & a mark scheme

In 1998 ‘Mathematics Inside the Black Box’ was published (Wiliam, D and Hodgen, J) as part of a subject series originating from ‘Inside the Black Box’. At the time, the messages taken from this publication, developing assessment for learning strategies within the Mathematics classroom had a profound effect on my teaching. Many of the strategies suggested in that little, blue pamphlet have sustained the test of time; identifying easier and harder questions from a list reminds me of the brilliant ‘Increasingly Difficult Questions’ website by Dave Taylor, multiple choice with more than one correct answer has similarities to the publication ‘Concept Cartoons in Mathematics’ (Dabell, J et al, 2008), the focus on incorrect, as well as correct answers using multiple choice is also developed through diagnostic questions.

‘Mathematics Inside the Black Box’ highlights mistakes, often better for learning than correct answers and the use of mark schemes to support model answers. The Explanation Scale activity uses both of these, along with children talking about their ideas; using and constructing the language of Mathematics with their peers. Conversations and discussions supporting the later, written form.

The journey to a written explanation is a complex one. The experience of physical objects begins the process. The language, pictures and written symbols that follow do not guarantee clarity in written explanations. The process requires an understanding of the Mathematics and the vocabulary to express ideas succinctly and clearly. Mathematics is a social activity and here is one of many examples of that fact. Collaboration. Time for children to think alone about a statement and time to work with one or more groups of their peers.

Blog posts from David Didau and Tom Sherrington have highlighted the benefits of sharing a mark scheme in the classroom with children. Expectations can be clarified this way, however, it also supports worked examples and where errors are highlighted, discussions about ‘what went wrong?’

Back in 2005, our Science Adviser at the time was using a book by some of the creators of ‘Concept Cartoons’, called ‘Active Assessment in Science’ (Naylor, S et al – 2005), five years later the Mathematics version was published but it was the Science version that particularly interested me.

Mathematics and Science both share the need to need to discuss statements; true or false and justifying correct or incorrect statements. The book contained a scale I recognised an immediate application to Mathematics.

With a few adaptations, it looked a little like this:

Explanation Scale

What to put on the scale? My idea was to use the mark scheme to its advantage, extract the correct and incorrect responses, mix them up and label them. The children would first think about the statement presented to them alone and then, in pairs/groups, look at the mark scheme statements to decide where A-I should go on the scale. For the KS2 question at the start of this blog, it would look something like this:

Mark Scheme (top) & Extracted Statements (bottom)

There is a comfort for children in discussing work that is not their own and identifying effective/ineffective explanations together with those that simply do not have sufficient information. It was an inclusive activity and supported their confidence in making judgements.

The most interesting part of this activity came from the discussions children were having about where answers should be placed and why. There were many revisions before we came back as a class to discuss. Often, children would feel strongly about their choices and required lots of time before presenting them to the whole class!

After much justification of where a given statement must be, the correct and completed version held equal learning value. What did they notice about all the effective explanations? What similarities did they have? What about the incorrect ones? How could we improve those in the insufficient part of the line to improve them? Can you write a better response to this statement now, given what you know? Can anyone give me another statement with insufficient information?

A rubric like this may help and is available through the GoogleDrive Resource Link below:

There is also important discussions to be had about the economy of words used in an explanation and what do we mean by ‘explanation’? Can you use a picture? Calculation? Particularly under a time limit. There were also opportunities for the children to redraft any initial explanations created at the start of the process (after going through the scale work).

This brings me to another point. We often redraft pieces of writing in English but very rarely do we do this in Mathematics. Our ideas often tumble onto the page as initial thoughts and therefore, the process of redrafting seems to make sense, particularly after some discussion and input.

If you’d like to try this in your own classrooms, don’t feel your explanation scale needs to be confined to a photocopied sheet; whole class whiteboard work and display border work perfectly (my initial trials were on a much larger scale and interactive; display border became an interactive scale across the length of the classroom with the labels displayed, allowing children to come up with post-it notes of their responses for later discussion).

Certainly, beautiful and ideal mathematical explanations do not just reside in the mark schemes of past papers, but given their formative assessment value, there’s an opportunity to use both the accepted (correct) and incorrect responses in a learning rich activity. I had taken this from a Science publication but with adaptations, its value could extend beyond Mathematics, into other areas of the curriculum where explanation is an equally valuable part.

Resources contain a scaffold for classroom use, the example used in the blog (PPT) & 2005/2008 KS2 statements to place on a scale.

GoogleDrive – https://drive.google.com/drive/folders/1FrhXTQT6CHoNctibKy9GcRZk6K-eHL6k?usp=sharing

Harry Beck’s Vision – A Mathematical Exploration

(Teaching ideas from my 2016 ATM session – Mathematics as Human Endeavour)

Harry Beck

“Looking at the old map of the railways, it occurred to me that it might be possible to tidy it up by straightening the lines, experimenting with diagonals and evening out the distances between stations.”

Harry Beck

In this short blog post I will be sharing a session I put together back in 2016 For an ATM Conference session on using the London Underground in Primary Mathematics.

The London Underground map has become one of Britain’s top 10 design icons. It was not always this way. When Harry Beck, an employee of what was colloquially known as, ‘the Tube’ designed the map in 1931, there was great scepticism of his design. Harry was paid just 10 guineas for his work. A mere 500 pamphlets produced for the general public in the first instance, followed by huge demand. The rest, they say, is history.

The purpose of the map was to show connections and fare zones; lines ran only vertically, horizontally or at 45 ͦ diagonals. It had been compared to a circuit diagram. The creation used for other underground maps across the world (Harry Beck was involved in the design of the Paris Metro).

Harry’s 1960 map is most similar to the one we have today; the interchange symbol, now a circle, some lines changed their Pantone colours, additional ones were added but always maintaining the status of the river Thames (the 2009 version removed the river but public support secured a fast return).

Ideas for Further Exploration in the Classroom

So, what to do with the map? Obviously, there is the ‘I want to go from station x to station y, how many ways can I do it? What is the shortest/longest route? How much will it cost?

The TfL website now has a marvellous ‘Top Gear-esque’ feature of allowing you to plot your journey see all the available transport options (including walking) for you to make a decision on which mode would be best.

However, here’s a few others I’ve used before that you might like to try:

  • Based on the average train speed (33km per hour), without stopping, how long would it take to travel from end-to-end?
  • Exploring the stations with steps; most/least and comparing stations.
  • I love the children’s book ‘London ABC’ Illustrated by Ben Hawkes.

I’ve used this to imagine we go through part/whole of the alphabet to find out how much this London discovery would cost and the underground routes you would take. For anyone that is familiar with the book, you will be aware of what each letter represents:

Those highlighted in yellow have set dates or simply may not be achievable on your visit!

There have been many interesting, alternative London Underground maps that have been produced over the years with great mathematical value. Here are a few of my favourites:

  • Fitness – What do the numbers above represent? Walking steps to stations. Created in response to the pedometer craze, London Underground offered an alternative for those wanting to exchange all/part of their journey for a stroll. Further conversion work could be considered; how many steps = mile? Which routes are equivalent to a mile? Added to this, not an exact science, but the calorie-burn from all this walking: https://media.timeout.com/images/102785052/image.jpg
  • Temperature on the London Underground – Data taken from one day in summer 2013:
  • The average cost per month of renting in London (ideal for money investigations and the harsh reality of London living!). The map was created by ‘Thrillist’ in 2015 (so expect some further increases to monthly costs):

https://www.thrillist.com/lifestyle/london/london-underground-rent-map

  • Finally, here’s an interesting map, proving that despite the term ‘underground’, it’s worth exploring how much of it is actually below ground.

Ideal for fractions and percentages estimation and investigation.

Additional sources of information

Here’s a GoogleDrive of useful resources too: https://drive.google.com/drive/folders/1teLQPCp37EKKYIT9tlIH79RhKEV4_XOy?usp=sharing

Goal Free-dom

Background – Why?

I’ve been using goal-free problems for the last 16 months in the classroom with some extremely encouraging results in pupil success and confidence when interpreting tables, charts and sentences. The approach encourages flexible thinking (Hasanah, R et al 2017)1, exploring different contexts for problems, improved recognition of similar problems and as a consequence, the language associated with it.

In the last 9 months, I’ve led a NCETM innovation with 8 fantastic London schools and their teachers. The project looked at mathematical vocabulary and deciphering word problems. Goal free being one of the many methods explored but all with some encouraging results in a short space of time in both KS1 and KS2.

The accompanying resources to this blog have been created out of need! It will now be my go-to when planning (rather than an internet search with the snipping tool). It’s been a strange journey, revisiting past papers with a purpose, going back to 2001 to present day. It seems for the first nine years there were an abundance of questions on soup, bird watching and cheese!

Sweller (1998)2 championed goal free problems; problems lose their specificity and they become non-specific, opened-up and flexible. Supporting knowledge acquisition and cognitive load. Craig Barton talks about this in more detail in his book ‘How I wish I’d taught Maths’ (2018)3 a must read for Primary and Secondary teachers alike.

Without a goal (often highlighted in light blue on a test paper), a word problem becomes an opened-up ground of possibilities. With just a picture, sentence, table or graph the goal is left to the children. They are the creators of what this will mean. Without a goal there is no overwhelming amount of text to distract the reader and increase cognitive load. It’s mathematical freedom.

A large number of problems children will see in published materials are goal led, particularly those presented in end of Key Stage tests.

Goal Led
Goal Free

Some years ago, at the start of my word problems research, I presented goal led problems in tables and charts to a Year 5 test group. I asked them to simply highlight the parts of the table/graph they would require to answer the question and make notes of any part they were not sure of.

What fascinated me in particular, was the highlighting. A large number of children were either unfocused or unable to identify information required by a question. Tables and posters were more problematic. My assumption at this stage could be we are au fait with teaching word problems using graphs and related tally/frequency charts as it’s a taught strand in Mathematics, Statistics and given emphasis as a result. Tables and posters stand alone in Mathematics and these, yielded the highest errors. Many of these reside in real life; price lists, hire price/time and price savings. They have a future purpose.

I first came across ‘goal-free’ through Pete Mattocks ‘Goal Free Problems’ website4. Some of these were perfect for Year 6 teaching, however, an approach across the Primary range was something the innovation group really wanted to consider. Clare Sealy’s article for Third Space Learning, ‘How I wish I Taught Primary Maths – Focusing Thinking and Goal Free Questions’5, exemplifies Primary examples.

‘….with the first type of question, concerns about the final goal can intrude upon their thinking. The pupil’s attention is split between thinking about the first step and thinking about the other steps that they need to take next. Therefore, even if they successfully answer the question, they may not have learnt anything generally that could be transferred to new kinds of problems.’

This is something many teachers can relate to, I often found my classes looking at step two of a problem before trying to answer step one, as mentioned, unable to decipher the ‘set-up’ information a sentence/table/chart would give. Take that away and you have no alternative but for deep exploration.

In KS1 I’ve started using them as a whiteboard enquiry, ‘what can you tell me about this …?’ The teacher acting as scribe for ideas discussed. In KS2, they can be discussed one at a time or 2-3 enlarged to A3 on different tables; children rotate and add comments to further explore under the visualiser or in a presentation.

KS1 Example

Goal free provides an activity all children can access at their own ability. Their discussions can reveal so much about the learner as well as the metacognitive possibilities, ‘I noticed you decided to …. with this table, can you explain a little more about that?’

Care must be taken in teacher questioning; probing thinking without directing one line of thought over another. Ideas must come from the children. It’s all about paying attention, noticing.

A Goal Free Example

In this example (1), children may consider the total number of chocolates in both the large and small box, they might just look at the large or small box in isolation or consider how many more chocolates there were in the large box compared to the small box. Children may notice, the small contains half the amount of chocolates than the large. By clearing away the detail of the goal, there emerges some clarity and a chance for real exploration. The inclusivity of this sort of activity is clear.

We could end our journey there, return to the original problem (2), From experience, I know the children would always feel more fully equipped to solve. However, often, a third step (3) could be this; we could take away even more goals/information, open-up the opportunity for even further exploration, connection making and creativity.

Supposing there were no Ken? We start to think about the possibilities. What if we gave the boxes some dimensions? What if we priced them? What would be reasonable? What if there was a special offer on? All ideas must come from the children but you can see how easily one problem could take half your lesson (if explored properly), as in my last post, it’s quality over quantity. Slow learning.

If you’ve never tried goal free, like all new approaches, it must involve an introductory lesson. Children soon become familiar with the idea.

So, when to use it? If I were starting a unit on teaching perimeter and area, I’d start with the core knowledge and models, then I’d use goal free in the application to deepen understanding,

Clare Sealy’s article mentions its use,

‘to further cement knowledge, using them after you have taught pupils a principle. This will encourage focused thinking within the classroom.’

I have also used them as homework (at the same point in the learning) and got the class to discuss their findings as part of a lesson later that week.

Here’s some points to consider:

  • You only need 2-3 a lesson or one for a starter you could keep revisiting throughout the week. Don’t be tempted to over plan. It’s all about the quality discussion and developing layers of understanding.
  • Encourage annotation – What do you notice? In SATs we are told to write in the box; answer and working. A goal free activity is meant to be messy. I often paste a problem in the centre of a sheet of A3/A1 paper with a bunch of marker pens.
  • In KS2, you could start with 3 goal free problems, rotated from table to table before a whole class discussion. Each table has the responsibility to identify something else the last didn’t.
  • Younger children, may require shared/guided support with their observations whilst using goal free.

Going Deeper:

  • Comparing two goal-free problems – same/different (as the problems in the free resources are sorted into categories, you could easily do this). What is the same? What is different?
  • Goal free retains the existing context of the problem, however, what about this idea? Keep the question as similar as possible but change the context. Is it possible? Which context could you have? For the chocolate box example, it could be biscuits or seeds planted in big/small trays. A change of context can be just as challenging for some children.
  • Can they add another step? Introduce another area of mathematics?

In the resource I’ve created below, there are over 200 KS2 and 100 KS1 goal free problems sorted into categories and a few extra ideas and sources of information too. The resources are hyperlinked to a GoogleDrive as the files are large. They are available as a PPT with/without links to the last three years of original questions (found below the slide) or PDF for each Key Stage.

They are also hyperlinked in the contents to provide quick access. Some of the slides have part of the problem, some just an image (different levels of goal free as mentioned).

KS1 Menu
KS2 Menu

If you are searching for more sources of goal free inspiration here’s some of my favourites:

  • With younger children use images instead of written problems.
  • Use images from the ‘real-world’. I often use photographs I’ve taken whilst out of supermarket offers and car parking charges.
  • Infographics are a fantastic source of information; word problems can be sourced from them to be presented in a ‘goal free’ format.
Infographics from ‘Maths Shed’ – http://www.mathematicshed.com/infographics-shed.html
  • Bar Models – In this lesson example, I added children’s names to each row. In this ‘times as much’, multiplicative reasoning question. Discussing the possibilities for possible questions before similar and different contexts were applied.
  • An all-time favourite book of mine is by ATM’s publication by Jill Mansergh & Margaret Jones (2007) ‘Thinking for Ourselves’6. There is a section in it about children problem solving from initial presentations like the one below.

There are many problems for you to choose from. Using different posters, questions posed are labelled ‘straightforward, challenging & more challenging’ or simply discuss what you see. Every time I have used this, a limited experience of the theatre prompts a whole discussion about seating and the price of tickets.

Image from ATM’s publication by Jill Mansergh & Margaret Jones (2007) ‘Thinking for Ourselves’. pp30

  • Old, Science SATs papers with useful graphs and charts. For further cross-curricular sources, Geography and History have some good sources too.

Here’s where to find the sources I’ve mentioned and more:

Books/To Purchase

WEBSITES

Infographics

 Pictures

References

1 – Hasanah, R et al (2017) – ‘Can Goal Free Problems Facilitate Students Flexible Thinking?’- https://www.researchgate.net/profile/Endah_Retnowati/publication/318923144_Can_goal-free_problems_facilitating_students%27_flexible_thinking/links/5a820648a6fdcc6f3ead6fc3/Can-goal-free-problems-facilitating-students-flexible-thinking.pdf?origin=publication_detail – Accessed 3 June 2020

2 – Sweller, J., Merriënboer, J.J.G. and Pass, F.G.W.C. (1998) ‘Cognitive architecture and instructional design’, Educational Psychology Review 10 (3) pp.251-296

3 – Barton, C (2018) ‘How I’d wished I’d taught Maths’ pp161-165

4 – Goal Free Problems – http://goalfreeproblems.blogspot.com/

5 – Sealy, C (2020) Third Space Learning Article, ‘How I wish I Taught Primary Maths – Focusing Thinking and Goal Free Questions’ – https://thirdspacelearning.com/blog/how-teach-primary-maths-goal-free-question/

6 – ATM – Mansergh, J & Jones, M (2007) ‘Thinking for Ourselves’ – https://www.atm.org.uk/Shop/Thinking-for-Ourselves/dis019

Key Words? Everything Matters

Fuentes (1998) states that every word in a word problem is necessary for it to be solved.

I am fascinated by this medium we have come to apply our mathematics, the humble word problem. In my first blog, I’m going to begin to look at some considerations and ideas for the classroom. I say ‘begin’ because future posts will look at particular aspects in more detail, the aim of this post is to provide a brief overview of things to consider/ways in.

Three years ago, I returned to teaching part-time after some time out in advisory and lecturing roles. I put these ideas to work immediately in the classroom and later, worked with the NCETM on an innovation (the report is to be submitted this summer). I’m convinced they work; pupil confidence, esteem just as improved as any quantitative data. I’ve always disliked RUCSAC. The day it was introduced to me, a year into teaching, it lasted two months in my classroom and there started my journey to find something better. 14 years ago it was going to be the subject of a PhD I’m still promising myself to start, one day.

If only life could be that simple; to follow a set of instructions and successfully solve a problem. For my confident problem solvers, it had some success (when they chose not to rush the problem and check their work) but for remainder there was always a sense that the mathematical sieve that is RUCSAC left a jumble of words and numbers they did not know what to do with; the numbers were often interpreted incorrectly and there was still the tricky problem of what to actually DO with them and more worryingly, in doing this, the ‘story’ of the problem was summarised to a few words and numbers that had ignored the sentences they had been extracted.

Hegarty, Mayer, and Monk (1995) performed a study at the college level on the different types of reading errors made by two types of problem solvers—successful and unsuccessful. They found that unsuccessful problem solvers had more difficulties than successful problem solvers in translating the word problem to mathematical representations because they were more focused on the numbers than on informative words within the problem.

Hegarty, M. (1995) – Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers. Journal of Educational Psychology, 87, 18-32.

The simple truth is, to understand what is important in a word problem in the first place, you must understand all of the sentences presented to you. There is a shift to word problems losing many of the obvious ‘key words’ and replacing them with the implied; verbs for example, ‘planted’, ‘remove’, ‘collect’ or ‘ate’. The four operation posters that may cover many walls of our classrooms, their definition is not a simple case of an awareness of the vocabulary used to express a given operation but guided by the sentence arrangement. Here, the case for reading the whole problem is highlighted further, take these examples for instance:

Mike had 11 apples.  He had 9 more than Asif.  How many did Asif have?

or

Carla and Nathan had 18 stickers altogether.  Nathan had 7 stickers. 

How many stickers did Carla have?

Context matters. Research tells us children perform better in reading comprehension with subject matter they have some personal experience1, the same could be true of word problems. Contexts exclude. I recall a Year 2 lesson based around a café and when asked, four of the children had never visited one. What followed was some discussion, drama and google searches before the lesson could resume. It’s not just context either. If we take our café, often, vocabulary particular to the setting is also presented; starter, tip, bill and so on. This is supported by Sethole (2004) and Nunes (1993) suggests this may render the mathematics within the problem as inaccessible and even, invisible.

Jo Boaler often talks of the mathematics world and the real world. I’ve had some interesting discussions in the classroom about how problems we explore in the classroom differ from real life? I compare them to the problems children see in their KS2 Tests, ‘NOT TO SCALE’, we are warned this is not the reality in order to solve the problem successfully, word problems require a suspension of some reality too in order for you to solve them and children need to be aware of those differences.

In order to understand a story, we would never dream of distilling it to a few key words to extract meaning. Of course stories differ greatly to those offered in word problems but there are often, some similarities; a beginning/middle/end, sentences and characters, again, this requires discussion, together with features they do not share with a story, for example, no plot or motives, sometimes grammatically incomplete.

When we are read a piece of fiction, our mental representations of the characters and situations support our understanding. In Mathematics, mental representations have been often highlighted as a successful strategy in comprehending and solving problems. In a classroom set-up we often present the text of the word problem and perhaps, any accompanying image. Imagining a problem is closely related to a child’s ability to understand the context (Caldwell and Goldin 1979, Nunes 1993, De Corte et al 1990), however, I have found by asking children to visualise and imagine a problem by breaking the problem into 2-3 parts and reading one part at a time, understanding of the problem is enhanced. This also supports the many insights we now have into cognitive load.

So what to do? I think the deciphering of word problems requires slow learning. There are only so many problem types (more of this in future posts). I’m a big fan of Jamie Thom’s ‘Slow Teaching’, quality over quantity, the exploration and annotation of just 1-2 problems with group discussion in contrast to a page or more. Adopting some of the strategies used in guided reading and because a word problem is relatively short, reading and re-reading. When I started this journey, I never thought I could make a lesson from two problems but explored at depth, it’s very easy. Benefits are more than just numerical outcomes but the ability to identify problem types from a mixture, confidence to start and visualise. For those of you concerned about challenge from this slower approach, for some able problem solvers the concern is not forming a number sentence to represent the mathematics involved but understanding some vocabulary, omitting steps and checking their result makes sense within the context of the question, errors, a slower approach may address

Many years ago, at the time of levels in education, I had to carry out a piece of research that would begin to answer why some children, predicted Level 5, lost out by a few marks. I followed a group of Year 6 children into Year 7, finally presenting them with their original paper and discussing the questions they found challenging/made errors. Three key findings were; barriers to some vocabulary (such as parallel and perpendicular), weaker aspects of number knowledge/application identified and further understanding of instructional vocabulary were required. There are many things we can presume from this but perhaps a slower approach may have supported gaps in understanding.

I carried out an analysis of the 2008 KS2 Mathematics Paper 2 and 3 into instructional vocabulary, verbs and contexts. The breakdown highlights the complexity of language structures required to solve them.

Mathematical reading is about precision (Fuentes, 1998) and very few children are taught explicit ways to read it (Kenney,  Hancewicz, Heuer, Metsisto & Tuttle, 2005).

So how do we begin to use a guided approach to the teach word problems? Here’s a few ideas and pointers (a few were mentioned earlier in the article):

  • Discuss how word problems are different from everyday problems and how their structure differs.
  • Replace some written word problems with activities where they will need to listen to them being read to support their visualisation.
  • Discuss context. Are all children familiar with it? If not, how can we support children to develop their understanding of it and the related contextual vocabulary? Drama? Video?
  • Read problems multiple times. I first came across the three reads strategy through Duane Habecker. How many teachers can grasp a problem from one read? Many of us know clarity comes with multiple reads of the problem but do children know this? Here’s an outline of what each read might look like:
sfusdmath.org
  • One of the most effective questions at KS2 I have found is to ask learners to summarise part or all of the problem in their own words. This clarifies the problem in their own minds, supports those learners who understand the pupil summary far better than the original and acts an assessment of pupil understanding of the problem.
  • Use scaffolds to explore structures mathematical word problems in your guided sessions. Here’s one of the many I have created over the years:
  • Use goal free problem approaches (more of this in future posts. I have plenty ideas for using these in a primary classroom).
  • If your pupils are preoccupied with the numbers within the problem, perhaps a numberless approach may support them. I first came across ‘Numberless Word Problems’ about three years ago through Brian Bushart’s wonderful links and articles in his blog. The blog has a growing number of numberless resources for you to use and adapt but essentially, the strategy can be adapted to any word problem, shifting the focus onto the language and supporting rich discussions and reasoning. If we take the KS1 problem below and apply the numberless strategy it might look something like this:
  • The numbers are replaced with the word ‘some’.
  • The problem is revealed in small steps.
  • Numbers are slowly reintroduced.

Therefore, the problem may start out as,

‘Some bananas are shared equally among some monkeys.’ (There are various numbers that could be used together with those that would not create an equal share).

Then, ’20 bananas are shared equally among some monkeys.’ (Again, there are still multiple possibilities, along with numbers this could not be used to create an equal share).

In this problem, the next stage reveals both numbers and more discussion takes place as the final part of the problem (what we must do with this information) is only revealed in the final stage.

Find out more about these and other strategies for supporting word problems in my session ‘Solving Mathematical Word Problems – 5 Activities to Build Understanding’ – The Complete Maths Virtual Mathematics Conference on Saturday 20th June 2020. #MathsConf23 

1 – Priebe. S, Keenan. J & Miller. A – ‘How Prior Knowledge Affects Word Identification and Comprehension’ (July, 2011)

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