#10 Teaching Beginning Number

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NUMBER CARDS TO TEACH BEGINNING NUMBER

To teach students a beginning sense of numbers, teachers often have student make pictures of numbers. Students count and arrange 5 counters on their paper. Then they find another arrangement of 5 counters to draw. This activity is designed to teach students what the number 5 is. Then students are assigned the task of making number cards for 6. Then 7, and so on. Some questions arise: How does the teacher know when to move on to the next number? How high up in the numbers should students go? When should making number cards be stopped and learning number facts started? The answer lies in seeing and working with the cognitive system of number that is developing in students’ thinking. It is not learning each individual number that is the goal. It is the organization of mental operations into a system that finally understands number as a concept. The goal for teaching is to shift from teaching numbers to mental system that produce numbers. Let’s look at if from the student point of view.


STUDENTS’ ACTIVITY

When we ask students to draw, write, speak, or represent anything, what students must do to respond is to reconstruct and represent their thinking and the things or content of their thought in some kind of over activity like drawing a picture of 5 objects to represent what they see as five. But behind what they see is the machinery of thought, the cognitive system of mental operations or structure that is doing the work of organizing and producing a conception and representing it as the content of thought. Students don’t see and are not aware of their mental machinery, only the objects of thought that their mental system organizes. What students see as number is not their method of understanding number, it is only an outward, static element of thought. What that piece of content represents greatly differs among students according to how developed their system of understanding number is. Students all produce a drawing of 5 counters but the meaning of their drawing varies greatly.

Primary students initially have a perceptual/activity idea of number that is only a temporary image of the result of counting or arranging. They have not yet reconstructed these activities at the level of mental activity so they do not yet mentally construct their representations of number out of its relationships, only an image of counters. They are unaware of how its parts make wholes or the order of the different sizes of the wholes except through rote activities of counting. Number for them is in reality mostly tied to the activity of counting and its images; it is not a mental idea. So the number 7, or any other number, starts as something that results from counting or as a temporary image of the arrangement of counters. However, to understand 7 means to see how it is made up of parts and how it can be ordered in a series among 6, 5, 8, 9 etc. It means to understand a number as one element in a system of numbers. Numbers can only be understood and defined in relationships to other numbers. Seven comes before 8 and after 6. It has 4 and 3 as parts as well as other possible parts.

The number cards are activities that could encourage students to put together the various conceptual relationships that make up each number out of their arranging activities but since drawing an arrangement of 5 can be done as a simple behavioral activity that copies an image, it does not necessarily pose a mental challenge, it doesn’t necessarily promote the development of a mental concept, an understanding of number. The teaching challenge is not to have students produce accurate number cards; it is to help students actually see numbers in their thought as a whole made up of relationships and not just as the appearance of a an image of the collection they made from a learned counting procedure. In fact, the image is irrelevant to the concept of number since perceptually the image changes while conceptually number remains constant. These facts mean that instead of simply drawing a picture of an arrangement of 7 things, arranging and seeing 7 things in different subgroups could help the student see that 4 and 3 remain 7, particularly if students are asked to figure out a way to show that two different arrangements on the number cards are the same number, that is, that 7 is simultaneously the various arrangements of parts as well as a whole.

If the card activity challenges the students make and name these relationships in the physical display, then they push on students to relate the parts to the whole. By posing the problem of how many different arrangements of these 7 counters they could make, the teacher is essentially saying "find the different relationships inside 7." Rather than a static display of 7, the teacher is asking for the student to use overt activities guided by mental operations to undo 7 into its parts. When students are asked to transform something like 7 into another arrangement that is still 7 and explain their results, they are being asked to use mental operations that make up the number to explain how it is composed. If they haven’t yet reconstructed their physical activity to abstract and organize these actions as mental operations of thought, they can still do the activity physically by pushing 7 counters into two piles. Then the instructional task is to help them think back over their actions so as to "remember" how they started with 7 counters and pulled them apart, to undo what they did to get back to 7, and to see if the number has changed. "How many did you start with? How many do you have now? How would you draw what you did? Do you have the same number altogether? How could you show this card of 4 and 3 is the same as that card that has all 7?"


THE INSTRUCTIONAL OBJECTIVE

The instructional objective is to help the student to arrive at the conclusion that 7 can be arranged as 4 and 3 such that 7, the total, hasn’t changed even though the number looks like it was changed by being subdivided into parts. When the student can mentally represent this taking apart and putting together of the whole, the whole becomes the concept of number, and we say the student now "conserves" number. We mean number has become a non-contradictory and stable thing of thought such that reasoning and problem solving with it is possible. So 4 and 3 come to be seen as the same as 7. Eventually we want students to be able to represent this activity first in little drawings and rebus like sentences and gradually to reconstruct these so they become equations of 7 = 4 + 3, 7 = 5 + 2, etc. So for instance, if the student starts by writing 7 on one side of the card and the two parts 4 and 3 are on the other, eventually the student will know 7 and 4 & 3 are the same so that from seeing the 4 and 3 side the student will know it is a 7 card. Once students arrive at these understanding, they can turn the facts into problems and, for example, play games with each other. One student could hide one part such as the three and show the other student the front and back. What is the hidden part? But the essential objective is to help the students see that all the seven cards are equal. The game could be extended. What other cards showing parts could they choose that would have 7 on the other side? In other words, you want them to see that they can substitute 5 and 2 or 4 and 3 and eventually see that 5 + 2 is the same as 4 + 3. Understanding this vicariance means students now understand that there are different ways to subdivide into parts and all of these are equivalent and do not change the overall whole.

When we use number cards, we want student to use math operations to make the parts and whole not make a drawing of what they see as an arrangment. We want students to see the numbers, not images, and to do this they must use the operations that are inherent in the numbers. We do not ask for these operations to be represented with + and – signs, only to be used implicitly. This implicit use is because students can see the numbers before they can abstract and see the methods (the operations) they use to compose numbers. The ends are apparent long before the means. After the number cards, the instructional task is to move the students toward making the operations of combining and breaking apart explicit as math operations. We challenge them to represent their combining actions by using a rebus, eventually to replace their drawings with the + sign and – sign. Once they understand the conservation of the whole while being subdivided into parts as they use number cards, they can represent this relationship replacing their drawings of moving objects together by using the = sign. We are pushing on students to gradually substitute representational symbols for their actions on the objects they attempt to represent in drawings and rebuses.


ACTIVITIES AS CONTINUOUS CONSTRUCTION

The number cards are not a single activity; they are not very useful if they just teach one thing. They are much more constructive if they are constantly challenge students to make a new construction. Learning activities for beginning number are a series of problems challenging students to act on a set of counters to transform them in various ways. As they get better at producing part/whole organizations, we want to move to the next step and push on them abstract what they did by asking for explanations. We also push on their representations by asking them to refine their drawings so as to gradually transform number cards in equations of the math facts. The number cards are the start of a continuous process of construction repeated mentally at successively higher levels of organization and representation. First the counters "show" the students the results of their actions on the objects, then the student draws or shows these actions on number cards, then the child uses numbers instead of the counters, then the child uses the + to represent the combining action and usually an arrow, —>, to represent the result. At one stage students might use 4 + 3 —> 7 to replace of the actions with objects. Obviously, the next instructional objective would be to help students see if they can turn a one way procedure around to show how to take 7 apart. Once students see that they can also start with 7 and subdivide it into 4 + 3 as well as or to put it together from its parts, and that during all of this the number remains constant, that is starting with 4 + 3 to make 7 is no different from starting with 7 and making 4 + 3 because it can go either way, then the two arrows now make an equivalence. They can replace the arrows with something to represent "you can go both ways." You want them to show these simultaneous two way actions can be represented as an = sign.


SEE AND PROMOTING THE MENTAL SYSTEMS AT WORK

Of course this method of developmental instruction to promote student construction of number depends on the teacher being able to see and follow the mental system students currently are using. Obviously, seeing and working with mental systems is far too difficult to adopt instantly. So giving tasks is a more formal way to see students’ mental activity and determine to what extent students actually understand and have grouped the relationships that make up number into an organized system. Is the student still at the stage of see that combining two subsets makes a new number or is the student at the stage of seeing that a number is composed of all these relationships. If we show students a collection of objects and then break the collection down into subsets and ask if the number is still the same, students obviously may see that the whole remains constant no matter whether it is in parts or not or they may still be at the counting stage or even earlier. If we then make a different arrangement, say we make three or four subsets out of the counters so that for 7 we show them 1, 2, 3, and 1, and then ask the students if the number is still the same or not, if they understand number they will immediately insist that it is the same without counting or hesitation. If we ask them to explain it, they will somehow indicate they understand the idea that the number has these various relationships and these relationships that make up the whole as a matter of necessity, do not change the whole. Once they understand that number is made up of relationships, then the instructional problem is to get them to represent more and more of these activities in symbol form. We can formally assess this level of representation by asking them to generate a representation. "How would you show how we arranged the 7 counters this way so that they still make 7 by using some kind of symbols on a piece of paper?"

The instructional focus is always to help students understand that all numbers are composed of relationships, not just something they end up with after counting. We don’t teach number; we teach the relationships that make up number. Once they understand a number is composed of relationships, then they will be able to find the same kind of part/whole relationships in any number. They are not learning 7 they are putting together the idea that 7 and every other number has relationships that can come from how they organize their actions on counters to produce various organized displays. They put relationships into counters at first with overt actions of re-arranging counters so they are turned into numbers. Then they reconstruct what they did to the objects to produce the various displays of a number on the level of thought and representation. Of course the part/whole relationships that make up every number are only half of the number concept. The other half is the ordering of numbers. We want students to see also that there are orders relationships among numbers such that the position of one number is unique and different from all other numbers, that is, every number also occupies a single and constant position in a series when we order all the numbers. So we also have students do a sequence of activities so that they can put together the idea that 7 as number (and not as a memorized name in a series like letters are memorized in the alphabet) is set into an ordered series of numbers. Teachers often assign the number cards in order, but for students to put order into number, the students have to somehow organize the random number cards and order them so as to see numbers as a series. It is better to do number cards at random letting students choose the number of counters they will work with each session. Then these various number cards can be used in seriation activities. "How can you organize you number cards according to how different the numbers are? How can you transform your 7 number card so that it is an 8 card? What do you have to do to your 6 card to make it an eight cards? How would you change the parts of 7 in order to have the parts of 8? How could you show all the number cards in a series? How could you show on your number line how you showed me how to change your 6 into an 8?"


CONCLUSION

To return to the questions we posed at the start regarding when to move on to the next number or how long to do number cards, the answer is simple but very difficult to accomplish. It involves seeing and working with each constructive step the student must make in moving from physical actions and images of objects to abstracting, representing, and grouping the operational form of activities into an organized system of operations composing number at the level of thought. This objective means our instructional goal is to be able to see how students see and understand a situation and what his next constructive step should be. Then we pose a problem that challenges the student to take what he can already do and reconstruct and represent it at a little bit higher level of abstraction. Thus, for each activity the teacher is asking herself, "what mental system is the student using to produce his result, and how can I pose the problem for the next constructive step the student must make?" The basic outlines of the activities don’t change. What changes is the levels of meaning and representation of this meaning the student can accomplish. The teacher is essentially taking the student’s activity, having it do over what it just accomplished but on a little bit high level, and thereby constantly transforming it and moving it toward a more abstract one. Developmental methods for promoting student construction always take what students have previously successfully produced to add a bit more challenge so as to ask the student to go through the activity again but on a higher level of abstraction. Thus making the number seven starts as only a result of counting objects and stopping when arriving as the sound of "seven." But this putting objects into a collection one-by-one is the precursor to understanding much later that seven is always composed of the addition of parts, and that these parts of seven can themselves be enumerated so that seven is represented in terms of relationships. Thus, helping students construct number is a very complex and extended project that cannot be accomplished by simply transmitting number ideas by direct teaching or even by showing number to students by having them do assignments in which the follow a procedure. There must be a real intellectual challenge to students asking them for a reconstruction of their existing capability. Teachers there must work through students’ activity to get them to make the constructions. It is a very technical process and goes far beyond what parents and politicians think happens.