I agree with the following statement by Duval (2006):

From an epistemological point of view there is a basic difference between mathematics and the other domains of scientific knowledge. Mathematical objects,2 in contrast to phenomena of astronomy, physics, chemistry, biology, etc., are never accessible by perception or by instruments (microscopes, telescopes, measurement apparatus). The only way to have access to them and deal with them is using signs and semiotic representations. (pp. 107)

If one accepts that statement, it seems reasonable to conclude that representations are particularly important in the teaching and learning of mathematics. However, it does not imply that **multiple** representations should be at the core of teaching, as several scientific papers and recent official documents place them.

One of the justifications often presented to support the use multiple representations is the following:

Because no single visual representation perfectly depicts the complexity of mathematical concepts, instructors often use multiple visual representations, where the different representations emphasize complementary conceptual aspects. (Rau and Matthews, 2017, pp. 531)

I fundamentally disagree with this view because I understand that some representations are very powerful and may be able to communicate a wide enough (for educational purposes, for instance) range of conceptual aspects of a given concept. Two examples: Hindu–Arabic numeral system to represent quantities and flat drawings made with pen, paper, ruler and compass for euclidean plane geometry.

A second common argument is the idea that multiple representations promote conceptual understanding. The problem with this argument is that since there is no instrumental definition of conceptual understanding, been able to use multiple representations to present a given concept became the definition of conceptual understanding. So, it is not a matter of multiple representations promoting conceptual understanding, but conceptual understanding being multiple representations.

From my perspective, multiple representation is a matter of curriculum: we, teachers, teach multiple representations because they are included in the curriculum directly, as a topic on its own, or indirectly, as a pre-requisite for another topic. That is my stand point in the paper Implications of Giaquinto’s epistemology of visual thinking for teaching and learning of fractions, where I defend the adoption of a carefully chosen visual representation (instead of multiple representation) especially when it comes to low achieving students.

Curiously, my position finds support in the paper published by Rau and Matthews (2017), where the authors draw some recommendations to promote learning through multiple representations. When discussing the limitations of their recommendations, they state that "some visual representations may be intuitively more accessible than others because they align with the structure of human cognitive architecture" (pp. 540). The authors call these representations privileged and point out that "deploying them as anchor representations might help optimize the web of meaning that emerges from use multiple representations" (pp. 540).

That is my point! For some representations there are reasons to use them that go beyond curriculum. Therefore, these representations should stand out of the pool as tools that can actually support learning and, then, the other representations may come to complement specific aspects (if some) or to cover curricular goals.

Duval, Raymond. ‘A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics’. Educational Studies in Mathematics 61, no. 1–2 (2006): 103–31. doi:10.1007/s10649-006-0400-z.

Rau, Martina A., and Percival G. Matthews. ‘How to Make `more’ Better? Principles for Effective Use of Multiple Representations to Enhance Students’ Learning about Fractions’. ZDM 49, no. 4 (August 2017): 531–544. doi:10.1007/s11858-017-0846-8.

The paper **Promoting Broad and Stable Improvements in Low-Income Children’s Numerical Knowledge Through Playing Number Board Games** explores the effect of playing linear number board games in young children’s numerical knowledge paying particular attention to the socio-economical status of the children.

The paper reports two experiments. The design of the **first** study is pretty straight forward: experimental design (randomized controlled trial) with a control group (playing a similar game with colours instead of numbers) taking into account age, achievement in a pre-test and socio-economic status as dependent variables and achievement in a post-test and delayed post-test as independent variables. The conclusion is that playing the linear number board games for as little as 1 hour (divided into 5 sessions) affected positively the results in the post-tests and these results remained statistically relevant after 9 weeks.

The **second** study had a different aim: investigate the correlation between playing linear number board games at home and results in the pre-test. The motivation behind this study is related to the perception that low-income students have less experience with some sort of activities at home that could affect positively their learning (such as linear number board games, as shown by the first study). The data came from the pre-test of the first study and from a self-report from the students considering some popular games for children. The authors confirmed the positive effect: students that reported playing games that could be considered a linear number board game presented higher scores in the pre-test.

The authors synthesizes their conclusion as follows:

All these findings converged on two conclusions. First, differing experience with board games is one source of the gap between the numerical knowledge of children from more and less affluent backgrounds when they enter school. Second, this gap can be reduced by providing children from low-income backgrounds experience playing number board games.

My questions would be:

- Why linear number board games are not being widely adopted in pre-schools?
- Why nobody is trying to replicate the results in different contexts?

In my opinion, this seems to be a study very replicable and with a great potential to impact classroom practices with relatively low costs. Therefore, it is hard to understand why it does not receive more attention from the academic community.

Ramani, Geetha B., and Siegler, Robert S. . "Promoting broad and stable improvements in low‐income children’s numerical knowledge through playing number board games". Child development 79.2 (2008).

Sometime ago, I fount the paper **Learning to “See” Less Than Nothing: Putting Perceptual Skills to Work for Learning Numerical Structure** by Jessica M. Tsang, Kristen P. Blair, Laura Bofferding & Daniel L. Schwartz. The paper is not directly related to my research, but I think it is one of the best papers in Mathematics Education I have ever read. Below, I will comment a bit about it.

This is the starting point of the paper:

“Our proposal [...] is that people recruit the distinct perceptuo-motor system of symmetry to make meaning of and to work with integer structure. If true, how can we use this knowledge to help children learn?” (pp. 157).

To answer that question, the authors utilized an experimental design: three groups, each learning integers with a different approach (two common in American textbooks and one emphasizing symmetry). They took a series of measures to ensure the basic premisses of experimental designs (something that is not common in educational research), but what I think makes this paper particularly good is how they defined the null hypothesis. Instead of only comparing the results in a pre and post-test, they used two measures: regular pre and post-tests + a post-test composed of generative questions.

If the results in the regular post-test showed differences between the groups, it would not mean that one of the approaches was better or worst than the others, but it would mean that the quality of the instruction received by the groups varied and this would be a problem in terms of their research question. Therefore, they were expecting similar results in the regular post-test and better performance in the generative questions by the group taught using symmetry.

This was the first time I saw a research using this approach. I think this is very distinctive and improves greatly the quality of the results because it neutralizes the interference of unexpected changes in engagement, excitement, expectations and instruction quality due to simply "being involved in a research project" due to the requirement of similar results in the regular post-test.

Every time I read a paper of a researcher or teacher trying two different approaches in a classroom and simply comparing pre and post-test results, I think: how can I know if the teacher was equally engaged in the lessons? I wouldn't! It is natural to expect that the involvement of the teacher in the research would affect his expectations and performance in the classroom. That is why I think the requirement of similar results in the regular post-test and a second measure to indicate the success of the intervention (generative questions, for instance) sounds very appropriate.

In fact, there are some issues related to the validity of the generative questions, but it is already a step towards more convincing experimental approaches in educational researches directly connected to classrooms.

PS: the paper has other merits beyond what was discussed here. It really worth reading.

Tsang, J. M., Blair, K. P., Bofferding, L., & Schwartz, D. L. (2015). Learning to “See” Less Than Nothing: Putting Perceptual Skills to Work for Learning Numerical Structure. Cognition and Instruction, 33(2), 154–197. http://doi.org/10.1080/07370008.2015.1038539

I am not a big fan of traditional uses of Tangram (related to Arts or Geometry), but I find it an interesting tool to introduce fractions through visual resources. This approach seems adequate to be used after rectangular models in order to introduce the idea that the pieces not necessarily have to have the same shape. But, it has the limitation of offering only fractions with powers of 2 as denominators. This characteristics can be positive at first, but can become a limitation after some time and that is the point when this "tangram of thirds" can be useful.

The pieces in the image above are fractions (of the whole square) with multiples of 3 as denominators. I decided to share this idea because I couldn't find anything like it online and because I believe it may be useful for enthusiasts of visual resources in mathematics education. You can download files with the Tangram of thirds in the links below:

Image with fractions | Image without fractions | Tangram made with Geogebra

Some tasks that I think could be interesting using this resource:

**Introduction:** given the whole square with the pieces drawn on it and 2 of the smallest triangles, find the fraction that each piece represents of the whole.

**Halves:** are you able to build half using one set of pieces in three different ways?