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).

- 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):

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.

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?

## 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.

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.

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?

**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

**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:

**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