## Tactile Mathematics

MARIA: Hi. What kinds of learning materials will pupils who use braille need in the future? That is one of the questions that inspired the project you will be hearing about in this presentation. My name is Maria Sandhaug. I am the Project Manager for this project and I work as a Senior Advisor at the Department of Development and Research at Statped. Together with three of my colleagues, I will be talking to you about this ongoing project. To start, Ida will tell you a bit more about the project, after which Hilde will present some results from our survey among pupils, and finally Oddvar will be presenting a learning tool produced as part of the project.

IDA: My name is Ida Sødahl Utne and work at Statped at the development and production of learning resources for pupils who use braille. It is important for us to be sure that resources and learning materials we develop meet pupils' need and preferences. In response to feedback from pupils, parents and interest groups, we wanted to find out more about how our learning resources in mathematics can and should be produced so that pupils benefits the most from them. We also wanted to find out about how the resources are used, and which qualities and materials are required in production. This project is a collaboration between advisors who provide guidance to schools and the producers of learning resources. A new curriculum was introduced in Norway in the autumn of 2020, and consequently many new learning resources have been and are being created. Many of these are digital. It has been emphasised that these resources must be designed for universal accessibility. Even though pupils use computer and braille display effectively, the need for paper productions will be considerable both today and for a long time to come. Many people may think that digital tools will reduce the need for paper books both for sighted pupils and those who use braille. But it is important to point out here that even if many sighted people benefit from presentation on a screen, instead of in a book, the screen is a two-dimensional representation for them. Blind pupils on the other hand only gets one line on the braille display before having to scroll to the next line. Statped is responsible for the production of learning materials for pupils who use braille. The production takes place in two different locations. Trondheim produces materials for pupils in primary school while Oslo produces for pupils in secondary and upper secondary school. Learning materials consist of texts, examples and assignments of illustrated and explained using figures and tables. Not all learning materials can be presented on a braille display. So, booklets with figures and tactile illustrations are also produced, which are provided to the pupil together with the electronic book. Nowadays math books produced for pupils who use braille are permanently electronic books as Oddvar documents. These books are adapted for reading on a braille display, and the pupils can work on the material on their own computers. For pupils in the lower grades, the books are also produced as paper books. Producing textile mathematical materials requires considerable time and resources. All mathematical expressions must be written manually and set braille standards or notations have been developed for writing mathematics in braille. Some of the feedback we have received from pupils and parents is that they want more of the mathematics learning materials to be available on paper in braille, because their experience has been that in many contexts, it is easier to learn mathematical knowledge when it is presented on two dimensions. it is often easier to gain an overview of the layout of an information in first simple tables when there is a more comprehensive presentation of the material on paper. Some elements such as shapes, graphs, and diagrams are impossible to present on a braille display. If we do not choose to present this in two dimensions, the material must in most cases be converted into text chilled representations. This leads to a much larger amount of text for the pupil to read. One of our goal with this project is to find out which part of the mathematics learning materials for pupil in grades 5-10 must be presented on paper and which can be presented in a braille display. Another important goal of this project is to determine which form of presentation on paper is best suited in different contexts, swell paper versus paper braille, grid versus dots or dotted lines. In order to find the right answers to this question, it has been very important for us to have the opportunity to test the material and pupils in relevant age levels. Hilde will soon tell us more about this. An internal goal for us is to better co-ordinate our productions so that the pupils always know what to expect from Statped productions. In the digital school, the traditional textbook is becoming increasingly subordinate. This give teachers greater freedom in terms of method and each individuals can choose learning resources and methodology related to curriculum goals in keeping with each class or pupils' areas of interest and level of knowledge. We therefore believe that textbook independent resources will gradually become more relevant. In terms of this project, a textbook independent electronic rubric is being produced, containing a collection of tactile illustrations corresponding to their collection. Oddvar will tell us more about this later in the presentation.

HILDE: My name is Hilde Havsjømoen and I work as a Senior Advisor at Statped specialising in braille and mathematics. I would like to tell you about the work we have done with pupils during this project. As Ida explained, one of our goals has been to learn more about how tactile images should be designed to give pupils the greatest possible learning outcome. To do this, we planned a survey among several pupils who use braille ages 10-17, during the first six months of 2020. But due to the Corona virus outbreak, the tests were only conducted among five pupils, all at ages 11-13. We wanted to test different ways to design tactile images. We chose three different illustrations on the topic of graphs. Figure number one shows two linear graphs in the first quadrant of a co-ordinate system. There is a marked point where the two graphs cross each other. Figure number two is a non-linear graph, also in the first quadrant. The graph shows how a feature develops over time. Figure number three shows all of the four quadrants of a co-ordinate system. There are five marked points in the co-ordinate system. The learning resource department at Statped produced tactile adaptations of the three images. Several illustrations of each figure were made and tested according to the different designs. This resulted in 19 different tactile versions of the three images. Since we were only able to conduct a test among five pupils, the results I will now be presenting are very preliminary. I will be focusing on the feedback from pupils concerning the production method and how to design the co-ordinate system with regard to grid lines and labelling. In Norway, we mainly use two different production methods. The one is embossed braille, and the other is swell paper images. We wanted to test whether one of these methods offers pupils an advantage. Embossed braille images are cheaper and faster to produce. So, it is worth knowing when images produced this way can be used. Our preliminary results showed no evident differences between the swell and embossed versions. Which one of the pupils reported to be best, depended on their past experiences to some degree. In the film, you can see a pupil comparing a swell and embossed version of figure number one, the two graphs. With this illustration, all of the pupils reported that the swell versions were best. In this next film, we see a pupil comparing swell and embossed versions of figure two, the non-linear graph. With this image, the tests implied that the embossed versions were as good as those on swell. Based on the results, we can assume that swell is preferable when it is important to discriminate between different tactile structures, as was indicates in figure one. In figures two and three other features were relevant. One pupil commented that the quality of the braille was best in the embossed version. We also wanted feedback on how to design the grid lines in the co-ordinate system. In this film, the pupil has three different versions of figure one, one with fully drawn grid lines, one with dotted, and one with a combination. There is no clear difference between whether a grid is dotted or made with fully drawn lines. In the film you are now watching, the pupil received a version of figure one, in which the grid lines were replaced by single dots in each grid line crossing. The feedback from all of the pupils indicated that the distance between the dots is critical. If too long, it was difficult to follow them to the axis. The results can therefore imply that grid lines, either fully drawn or dotted, are easier to follow. On the other hand, one pupil reported that it is easier to read the graph when the grid lines were replaced by dots. It seems that this gives the graph a better appearance. We also wanted feedback on where to place the information along the axis. In the picture, you can see two different versions of the swell version of figure three with the four quadrants. In the first one, the values are placed along the axis, while on the second the values are placed outside the co-ordinate system. Only three of the pupils gave us feedback on this question. Two of them reported that they prefer the values to be placed outside the co-ordinate system. The third pupil was indifferent. Similar studies have shown the importance of labelling not interfering with the diagram and with illustrations in which the values along the axis interfere with the graph, it should be examined further whether the values should be placed outside the diagram. Finally, we made a number of interesting observations during the tests on how the pupils use their hands when exploring the tactile graphics, and which words or concepts they used when talking about the different tactile qualities. We think there is a potential for greater awareness of these things when teaching pupils about tactile illustrations. When it comes to illustrations that combine the foreground and background, such as the non-linear graph in the co-ordinate system, not all of the pupils had a clear understanding of grid lines as a background. More systematic educational materials may be needed to learn about the co-ordinate system and grid lines separately.

ODDVAR: My name is Oddvar Øyan, and I work as a Senior Advisor in the field of computers for the visually impaired, as well as with mathematics and braille. I would now like to present another outcome of an analysis of the parts of the curriculum that are better when presented two dimensionally. To make sure we do not omit any of the mathematical themes, we decided to make a book of rules for pupils in the grades 8-10. This book of rules comprises the mathematical areas of knowledge in the curriculum, such as numbers and an understanding of numbers, algebra, functions, geometry, statistics and calculations or probability. This book of rules consists of two parts, a word document that presents the mathematical topics, and a braille book in seven volumes, consist of tactile materials. I will now show you the two parts of the book of rules. On my computer now, I have the word document with the text from the printed book. And the texts is of course, in Norwegian, and my programmes are also in Norwegian but I'll show you this anyway, The topic today is linear functions, (speaks Norwegian), and I used to search books in word to find the topic in the document. (computerised voice speaks) I find my topic, linear functions, close the search button. I close the search dialogue and starting to read in the book. I now I scroll on the down, four lines, and find the code corresponding to the braille materials. And the code today is FU/FT/LF 01. Here I have one volume of the tactile material to this book of rules. And now you find my code and find the drawing to this topic. At the bottom right is the keys from the word document, and they are sorted alphabetically. So, I can find the code FU/FT/LF/ 01. And here is the graph corresponding to the topic I find in the word document. This I have shown you now is still a prototype. So, I hope soon we have the whole book of rules ready. The book of rules is provided to the pupil together with the textbook used in the pupil's class. It can be used together with different books. This means that it is not necessary to produce all graphics and diagrams in the print book, making production faster. The tests we conducted in the project provided input to the design of the tactile material we have produced in the book of rules. There has also been an ongoing discussion and testing in the project's group. We still believe that we are in the starting phase of this work, and there are many themes we still want to test, for example, which themes in the mathematics subject are easier to read in swell paper format or embossed on a braille printer. We will be carrying out more work in this area in the months to come.

MARIA: My name is Maria. And I hope you find our preliminary results to be of interest in terms of the kinds of mathematical learning materials pupils using braille will need in the future, and whether they need paper play or tactile images in a digital school. Thanks to this project, a fruitful partnership has evolved between the Department of Visual Impairment and the Department of Development and Production on Learning Resources. As result, Statped has rapidly increased its development and production rate of braille books. We are talking weeks here, instead of months, which truly benefits the pupils. We will continue to gain a better understanding of pupils needs for adapted learning materials in mathematics. So, please feel free to contact us if you have any comments or ideas you'd like to share. Our email addresses are listed in the conference programme and can also be found on our website, www.statped.no. We look forward to hearing from you and thank you for your attention.