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?