Learning mathematics often involves small problems, linked to concrete everyday situations. For example, pupils have to add up quantities of flour to make a recipe or subtract sums of money to find out what’s left in their wallets after shopping. They are then led to translate words into solving strategies using a mental representation of numbers, the arithmetic operation to be performed, and non-mathematical information, such as the context of the problem. A team of scientists at University of Geneva (UNIGE), in collaboration with CY Cergy Paris University (CYU) and University of Burgundy (uB) wanted to have a clearer idea of these mental representations to enable a better understanding of the choice of calculation strategies. They analyzed drawings made by children and adults when solving simple problems.** **Their results, published in the journal *Memory & Cognition*, open up new perspectives for the teaching of mathematics.

In the study 10-year-old children and adults were asked to solve simple problems with an instruction to use as few calculation steps as possible. The participants were then asked to produce a drawing or diagram explaining their problem-solving strategy for each statement. The contexts of some problems called on the cardinal properties of numbers - the quantity of elements in a set - others on their ordinal properties - their position in an ordered list.

The former involved marbles, fishes, or books, for example: ‘‘Paul has 8 red marbles. He also has blue marbles. In total, Paul has 11 marbles. Jolene has as many blue marbles as Paul, and some green marbles. She has 2 green marbles less than Paul has red marbles. In total, how many marbles does Jolene have?’’. The latter involved lengths or durations, for example: ‘‘Sofia traveled for 8 hours. Her trip started during the day. Sofia arrived at 11. Fred leaves at the same time as Sofia. Fred’s trip lasted 2 hours less than Sofia’s. What time was it when Fred arrived?”

Both of the above problems share the same mathematical structure, and both can be solved by a long strategy in 3 steps: 11 – 8 = 3; 8 – 2 = 6; 6 + 3 = 9, but also in a single calculation: 11 – 2 = 9, using a simple subtraction. However, the mental representations of these problems are very different, and the researchers wanted to determine whether the type of representations could predict the calculation strategy, in 1 or 3 steps, of those who solve them.

### Identifying Mental Representations

‘‘Our hypothesis was that cardinal problems - such as the one involving marbles - would inspire cardinal drawings, i.e. diagrams with identical individual elements, such as crosses or circles, or with overlaps of elements in sets or subsets. Similarly, we assumed that ordinal problems - such as the one mentioning travel times - would lead to ordinal representations, i.e. diagrams with axes, graduations or intervals - and that these ordinal drawings would reflect participants’ representations and indicate that they would be more successful in identifying the one-step solution strategy,’’ explains Hippolyte Gros, former post-doctoral fellow at UNIGE’s Faculty of Psychology and Educational Sciences, associate professor at CYU, and first author of the study.

These hypotheses were validated by analyzing the drawings of 52 adults and 59 children. ‘‘We have shown that, irrespective of their experience - since the same results were obtained in both children and adults - the use of strategies by the participants depends on their representation of the problem, and that this is influenced by the non-mathematical information contained in the problem statement, as revealed by their drawings,’’ says Emmanuel Sander, full professor at the UNIGE’s Faculty of Psychology and Educational Sciences. ‘‘Our study also shows that, even after years of experience in solving addition and subtraction, the difference between cardinal and ordinal problems remains very marked. The majority of participants were only able to solve problems of the second type in a single step."

### Improving Mathematical Learning

The team also noted that drawings showing ordinal representations were more frequently associated with a one-step solution, even if the problem was cardinal. In other words, drawing with a scale or an axis is linked to the choice of the fastest calculation. “From a pedagogical point of view, this suggests that the presence of specific features in a student’s drawing may or may not indicate that his or her representation of the problem is the most efficient one for meeting the instructions - in this case, solving with the fewest calculations possible,” observes Jean-Pierre Thibaut, full professor at the uB Laboratory for Research on Learning and Development.

‘‘Thus, when it comes to subtracting individual elements, a representation via an axis - rather than via subsets - is more effective in finding the fastest method. Analysis of students’ drawings in arithmetic can therefore enable targeted intervention to help them translate problems into more optimal representations. One way of doing this is to work on the graphical representation of statements in class, to help students understand the most direct strategies,’’ concludes Hippolyte Gros.

### StepUp Note

By analyzing the illustrations produced by both children and adults when tasked with solving simple mathematical problems, researchers discovered that, whatever the age of the participant, the most effective calculation methods were correlated with certain types of drawings. This research suggests drawing analysis as a novel method to enhance mathematical learning and instruction. StepUp programs include Vision Blocks exercises for seeing number sets and number dictation exercises for knowledge of spoken and written number names. The Quick Check Math program adds exercises which integrate math computation and math operation. Altogether, these programs provide useful tools for evaluating and practicing mathematical learning for early learners, and create a foundation for learning and understanding higher math operations.

Note by Nancy W. Rowe, M.S., CCC/A

Reposted from University of Geneva