Building Assessment into Instruction
OLIV: Welcome. My name is Oliv Klingenberg, and in this lecture, I'm pleased to share with you an activity I have used when trying to understand how children with a visual impairment think about numbers. I have used this activity to assess pupil's understanding of mathematical language and number concept within a learning context. The lecture is divided into four parts. Firstly, my colleague, Ole Erik Jevne, and I will show how the activity can be carried out. Ole Erik specialises in secondary level mathematics, but today, he will be playing the role of a pupil and will respond as if he were around six years old. After we demonstrate the activity, we will, in part two, show how the material, the manipulatives used in this activity, can be introduced to pupils. The third part concerns the misconceptions or immature conceptions about numbers. Finally, I've raised five educational approaches that can enhance numeracy and number understanding. Ole Erik, in your first character, you represent a young pupil with what I characterise as the expected mathematical knowledge level of a six year old.
OLE ERIK: In front of me, I have 55 small plastic cubes. All except one are interlocked as blocks. We have on block with two cubes, one with three cubes, one with four, et cetera, up to a block with 10 interlocked cubes. We refer to the blocks as towers and these towers are arranged from one to 10 in a stair frame.
OLIV: Now, Ole Erik, please hand me the tower made of six cubes.
OLE ERIK: Here they are.
OLIV: Because of infection control, in this presentation, Ole Erik is now handling me, is not handling me the tower, I have my own tower. So thank you, Ole Erik. I'm now going to divide the six-block tower into two smaller blocks. Listen. Can you hear? I have hid one section of the tower in my left hand and the other in the right hand.
OLE ERIK: To know what she is talking about, I have my hands on the top of her hands.
OLIV: How many cubes can you guess that I have in the hand that I am shaking?
OLE ERIK: Three.
OLE ERIK: Four.
OLE ERIK: Two.
OLE ERIK: She opens a hand and I have now the block made of two cubes in my hand.
OLIV: And how many cubes do I have in the other hand, this hand? I'm now shaking the other hand.
OLE ERIK: Four.
OLIV: Yes. How did you know or figure that out?
OLE ERIK: Because four and two is six.
OLIV: This answer indicates that Ole Erik has grasped the idea of a basic arithmetic structure. Written operations like four plus two equals six can make sense if he has learned to symbolise numbers using numerals of course. However, if when asked how he figured it out, Ole Erik response...
OLE ERIK: Because there were six.
OLIV: This answer that's not reflect linear notation. Perhaps, this Ole Erik pupil has too little experience talking about adding sets or taking away objects from a collection. My advice is to let the pupil talk and explain because when performing operations on objects, language including the use of mathematical word symbols, raises the pupil's mathematical thinking to successively higher levels. I have estimated that I have played this game or test with around a 100 pupils and have learned that many six-year-old children with VI can think in terms of groups of six elements, and mentally divide these numbers into two parts. So, Ole Erik, you have now responded as many six-year-old children with a visual impairment would have. We'd now like the move on to how the material can be introduced to the pupil. The Ole Erik pupil and I can communicate about the towers because he has spent time getting to know the material. Research literature tells us that pupils with VI need time to familiarise themselves with the material before the material can be used as a learning tool. Ole Erik got 55 cubes in the tray, and he has played with the cubes and interlock them. I have copied his way of playing with the cubes and verbalised that I have, what I have been doing, and have referred to the interlock cubes as towers. Ole Erik, I also made a tower. How many cubes do you have in yours?
OLE ERIK: one, two, three, four, five, six, seven.
OLIV: So it was seven. What I wanted to observe is how the pupil counts the cubes. Ole Erik passes his finger down the seven blocks tower and shows me that he can count and point in a coordinated manner. We now have, and you've seen in this activity that Ole Erik pupil and I have interlocked the cubes into two cubes, three cubes, et cetera, up to 10 cubes towers. To organise these towers in order, I use the frame, a one to 10 blocks stair frame. You can take them out. Yes?
OLE ERIK: I use a one to 10 blocks stair frame, and stair frames can be fabricated in 3D printing. Ole Erik is now manipulating the frame in which the towers can be placed.
OLIV: After a little while, I take some control and instruct him how to organise the towers. And I verbalise using brief comments and provide words to describe the steps of the process. It helps to sit on my hand so that I'm not tempted to help the pupil physically. If a pupil is able to organise the material by himself, the teacher can be more confident that the pupil understand what the dialogue is about. Ole Erik, I'm now handling you a one-cube tower. And please place this one-cube tower in the first column.
OLE ERIK: We have talked about the columns as if they are beds that hold different lengths or towers.
OLIV: And here, you have a two-cube tower, and the three-cube tower, and the five-cube tower. By deviating from the ordinary sequence of counting, I can get an indication of whether the pupil is confident that each number has a specific position along a number line. Here, you have a tower made of eight cubes. Okay. We now continue the part whole activity, and I will talk about immature conceptions of numbers. At pupil's development or early numerical competencies, is not always linear, and pupils differ in their timeline of for acquiring these skills. I have learned that if the biggest part of the tower is hidden in the second hand, some pupils struggle to determine how many cubes there are in the second part. This is the difference whether the second part is small or not. This is a difference in functioning that we must be aware of because conceptualising a number as being made up of two or more parts is the most important relationships that can be developed when it comes to numbers. Unfortunately, many textbooks proceed directly from beginning ideas of numbers to addition and subtraction, leading pupils with very limited knowledge about numbers that they can apply with new symbol mathematics. In the introduction of the activity when the pupil is building towers and we are talking about numbers, I get an impression of the child's number understanding. I try to permit every pupil regardless of his or her mathematical process to demonstrate some knowledge, skill or understanding. Sometimes, I do not start to divide six. A pupil can do this activity successfully if I divide a small tower, a tower made only three or four cubes. But, I have also met pupils that do not think of numbers as cardinal numbers, and number that specific how many cubes there are in the collection. And then, of course, they do not understand how to divide numbers into two parts, and here is an example. Ole Erik, can you give me the four-cube tower? Thank you. I now divide this four-cube tower. Can you hear? Did you hear?
OLE ERIK: Yes. (laughs)
OLIV: How many cubes do your guess or think I have in this hand?
OLE ERIK: Five.
OLIV: I absolutely need more observation than this one answer, but it is possible that this Ole Erik pupil understands number words, not as word or quantities, but as names. In other words, the names of objects when talking about objects in a counting context. Ole Erik succeeded when giving me the four-cube tower. It could be that he succeeded because he has learned to use the counting sequence, the counting sequences while touching objects. I said, "Four, give me the four-cube tower." He has learned to count one, two, three, four, he has learned to count and touch objects, and he has learned that he shall stop the sequence and the moment when he says the same number word as what's in the question. One hypothesis is that pupils who think of numbers as names do not focus on groups or object, but instead, direct their attention towards the single object or bodily movement corresponding to the counting sequence. The question "how many" does not make sense, but the pupil has still learned to answer such questions with a number word. Ole Erik answered five, maybe because five is the next number word in the counting sequence. Mathematic cannot simply be taught the repetitive drilling or the basic number facts will never result in understanding, although some pupils can be quite successful in first grade, thanks to an amazing in the memory. As the tasks become more complex, they start to experience greater difficulties in performing at their potential. So, what can enhance numeracy and number understanding? There are a multitude of activities that form the basis for development of mathematical thinking. I'll end this lecture with five approaches to mathematical learning. Language is first. At the end, age of five, six, mathematics is a language. And as with any new language, the learning is in the reading, the writing, and most importantly, the talking. I wish we adults, talk less and the pupils more. Mathematic is a language because mathematical ideas are often grounded in everyday experiences. Experiences that humans have wanted to communicate. Numbers are grounded in experiences about collections of objects, physical objects, but also auditive. What have your pupil met before that make he or she think and talk about object collections? And when do small children with VI talk about numbers? Consider the wise words of Foulke & Hatlen who wrote, "The world seeks infants who can see, but infants who cannot see must learn to seek the world." The art or teaching a child with VI is to be aware of the child's experiences and to hear and know attention. At least the attention or what the fingers are touching. And you can ask about what the fingers are touching, and you can verbalise if the child do not own the words. The second approach is to distinguish between skills in doing mathematics and more cognitively complex tasks. I firmly believe that mathematics as plussing and getting answers are not the correct approach if a pupil has not yet grasped the idea of number being made up or two or more parts. But the pupil may be able to learn some of the skills while doing calculations. Learning to read and write in numerals is an example of a skill in doing mathematics. Even if a task type is too cognitively demanding for a pupil, there may be elements of an activity that the pupil can learn. So the dichotomy between skills and understanding can be essential for inclusion and the opportunity to participate in class activities. Approach number three, number sequences and counting. Counting exercises in class are useful for all pupils. Learning to produce a standard list of counting word in order is a skill, but this skill is an activity the pupil with VI can do together with the whole class. However, counting something is based on numerical understanding. My advice is to be alert situation in which counting comes naturally, but because it is difficult to sense what happens as the consequences of operation with object. Have a look at Ahlberg and Csocsán's publication "Grasping Numerosity Among Blind Children" or other research literature about counting among pupils VI. Four, numbers represented as auditory structures. In another presentation at the Tactile Reading conference, my colleague and I present the project aimed at developing learning resources in mathematics for braille reading pupils in first grade. One of the resources, but not mentioned in our presentation is (foreign language) which translates roughly as auditory numerical structures, auditory visuals or auditory numerical figures. I'm not sure for the English term, but let me give an example of of auditory structures of six. (claps) I made six out of three and three, and two and two and two. The different rhythms of six beats can serve as the starting point for talking about numbers, talking about part whole, et cetera. The fifth and final approach in this lecture involves attention to quantities in everyday life. In this, I use a metaphor of containers. What I want is to help the pupil perceive objects as a group or collection, to talk about these groups, collections using number words. The container metaphor is the teacher's tool for creating object collections in a context the pupil can mentalize or visualise. Because the context is familiar, it's clear, has a bounded region in space, and there are not too many objects in the collection. For example. The pupil's family is the context for talking about part part whole relationships because the family is familiar and consists of children, adults girls, the entire family. A pocket is a container. The pupil can have three pearls in the pocket, and you can create a story about the three pearls. Maybe the pupil is able to visualise them, maybe the pupil can talk about them using number words. A lunch table, and for well-known pupils around this table is a container context, as well as a car with passengers, and a hand with five fingers, and the story or a narrative about these five fingers. And do remember, there are not too many object in these containers, three, four is fine in the beginning. For a long time, I have tried to understand how children with vision loss learned to think mathematically, and how mathematics can be taught. I hope this lecture and the activity about numbers as part and holes have inspired you to become more knowledgeable about mathematics, education and pupils with VI. Thank you for your attention.