(Pictured: Sally leading a small group Science investigation)
I hope you had a restful week! It went by so quickly and the rain made for some very cozy and quiet days for our family. Hope you enjoyed your break too. A special thanks to everyone for all the valentine cards and gifts this past week. They made me feel appreciated and loved. Thank you!
Last Tuesday, we celebrated Valentine's Day. We delivered valentines to our big buddies, cooked with Jennie and Meital to make valentine's cookies and studied the heart in science with Sally and Michelle. At the end of the day all the students finished delivering their cards and opened their valentines treats and gifts. Check out the photo gallery...so much fun!
In the next few weeks we will be winding up our thematic unit on the human body and moving on. For the rest of the year we will be focusing on the garden, developing observational skills (e.g. examining items closely, documenting) and studying the solar system. If you have any expertise or resources in this area, let me know! We can always use experts to help us.
In math, we have been slowly differentiating our curriculum to meet the needs of each student. Up to now, this has taken the form of small group work to develop strong number sense and flexibility in addition and subtraction. But recently we have begun to add problem solving and documenting to the mix. In TK, this has taken the form of asking students to write down their understanding of a single number. Students are given a focus number (e.g. Show me 4) and are asked to document their thinking on paper. This would seem to be a simple process of drawing 4 objects, but there are many ways to demonstrate this understanding. It could be demonstrated by simply drawing 4 objects. Or, it could be 10 objects with 6 crossed out (subtraction) or it could take the form of a number sentence (e.g. 2+2 - the part/part/whole concept). It could even be 2 groups of 2 (e.g. 2x2 - multiplication). The beauty of this type of "open-ended" work is that each student works at their own level of understanding, essentially self-differentiating based on where they are in their mathematical development. This type of work also allows students the opportunity to share their thinking with their peers and has the added benefit of providing the opportunity to practice their academic language, hone their communication skills and mentor each other. For the adults, a student's documentation is an informal assessment of their mathematical thinking. Some students require dictation to articulate their thoughts, but it is very interesting to see how each approaches and copes with challenge. This process not only gives each child a chance to think critically, but it also gives them the opportunity to "prove" their theories and work.
'Number Sense Series: Developing Early Number Sense' printed from http://nrich.maths.org/
What is number sense?
The term "number sense" is a relatively new one in mathematics education. It is difficult to define precisely, but broadly speaking, it refers to "a well organised conceptual framework of number information that enables a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms" (Bobis, 1996). The National Council of Teachers (USA, 1989) identified five components that characterize number sense: number meaning, number relationships, number magnitude, operations involving numbers and referents for numbers and quantities. These skills are considered important because they contribute to general intuitions about numbers and lay the foundation for more advanced skills.
Researchers have linked good number sense with skills observed in students proficient in the following mathematical activities:
- mental calculation (Hope & Sherrill, 1987; Trafton, 1992);
- computational estimation (for example; Bobis, 1991; Case & Sowder, 1990);
- judging the relative magnitude of numbers (Sowder, 1988);
- recognizing part-whole relationships and place value concepts (Fischer, 1990; Ross, 1989) and;
- problem solving (Cobb et.al., 1991).
How does number sense begin?
An intuitive sense of number begins at a very early age. Children as young as two years of age can confidently identify one, two or three objects before they can actually count with understanding (Gelman & Gellistel, 1978). Piaget called this ability to instantaneously recognize the number of objects in a small group 'subitizing'. As mental powers develop, usually by about the age of four, groups of four can be recognized without counting. It is thought that the maximum number for subitizing, even for most adults, is five. This skill appears to be based on the mind's ability to form stable mental images of patterns and associate them with a number. Therefore, it may be possible to recognize more than five objects if they are arranged in a particular way or practice and memorization takes place. A simple example of this is six dots arranged in two rows of three, as on dice or playing cards. Because this image is familiar, six can be instantly recognized when presented this way.
Usually, when presented with more than five objects, other mental strategies must be utilized. For example, we might see a group of six objects as two groups of three. Each group of three is instantly recognized, then very quickly (virtually unconsciously) combined to make six. In this strategy no actual counting of objects is involved, but rather a part-part-whole relationship and rapid mental addition is used. That is, there is an understanding that a number (in this case six) can be composed of smaller parts, together with the knowledge that 'three plus three makes six'. This type of mathematical thinking has already begun by the time children begin school and should be nurtured because it lays the foundation for understanding operations and in developing valuable mental calculation strategies.
What teaching strategies promote early number sense?
Learning to count with understanding is a crucial number skill, but other skills, such as perceiving subgroups, need to develop alongside counting to provide a firm foundation for number sense. By simply presenting objects (such as stamps on a flashcard) in various arrangements, different mental strategies can be prompted. For example, showing six stamps in a cluster of four and a pair prompts the combination of 'four and two makes six'. If the four is not subitized, it may be seen as 'two and two and two makes six'. This arrangement is obviously a little more complex than two groups of three. So different arrangements will prompt different strategies, and these strategies will vary from person to person.
If mental strategies such as these are to be encouraged (and just counting discouraged) then an element of speed is necessary. Seeing the objects for only a few seconds challenges the mind to find strategies other than counting. It is also important to have children reflect on and share their strategies (Presmeg, 1986; Mason, 1992). This is helpful in three ways:
- verbalizing a strategy brings the strategy to a conscious level and allows the person to learn about their own thinking;
- it provides other children with the opportunity to pick up new strategies;
- the teacher can assess the type of thinking being used and adjust the type of arrangement, level of difficulty or speed of presentation accordingly.
To begin with, early number activities are best done with moveable objects such as counters, blocks and small toys. Most children will need the concrete experience of physically manipulating groups of objects into sub-groups and combining small groups to make a larger group.
What games can assist development of early number sense?
Games can be very useful for reinforcing and developing ideas and procedures previously introduced to children. Although a suggested age group is given for each of the following games, it is the children's level of experience that should determine the suitability of the game. Several demonstration games should be played, until the children become comfortable with the rules and procedures of the games.
Deal and Copy (4-5 years) 3-4 players
Materials: 15 dot cards with a variety of dot patterns representing the numbers from one to five and a plentiful supply of counters or buttons.
Rules: One child deals out one card face up to each other player. Each child then uses the counters to replicate the arrangement of dots on his/her card and says the number aloud. The dealer checks each result, then deals out a new card to each player, placing it on top of the previous card. The children then rearrange their counters to match the new card. This continues until all the cards have been used.
- Each child can predict aloud whether the new card has more, less or the same number of dots as the previous card. The prediction is checked by the dealer, by observing whether counters need to be taken away or added.
- Increase the number of dots on the cards.
Memory Match (5-7 years) 2 players
Materials: 12 dot cards, consisting of six pairs of cards showing two different arrangements of a particular number of dots, from 1 to 6 dots. (For example, a pair for 5 might be Card A and Card B from the set above).
Rules: Spread all the cards out face down. The first player turns over any two cards. If they are a pair (i.e. have the same number of dots), the player removes the cards and scores a point. If they are not a pair, both cards are turned back down in their places. The second player then turns over two cards and so on. When all the cards have been matched, the player with more pairs wins.
- Increase the number of pairs of cards used.
- Use a greater number of dots on the cards.
- Pair a dot card with a numeral card.
What's the Difference? (7-8 years) 2-4 players
Materials: A pack of 20 to 30 dot cards (1 to 10 dots in dice and regular patterns), counters.
Rules: Spread out 10 cards face down and place the rest of the cards in a pile face down. The first player turns over the top pile card and places beside the pile. He/she then turns over one of the spread cards. The player works out the difference between the number of dots on each card, and takes that number of counters. (E.g. If one card showed 3 dots and the other 8, the player would take 5 counters.) The spread card is turned face down again in its place and the next player turns the top pile card and so on. Play continues until all the pile cards have been used. The winner is the player with the most counters; therefore the strategy is to remember the value of the spread cards so the one that gives the maximum difference can be chosen.
- Try to turn the spread cards that give the minimum difference, so the winner is the player with the fewest counters.
- Roll a die instead of using pile cards. Start with a set number of counters (say 20), so that when all the counters have been claimed the game ends.
- Use dot cards with random arrangements of dots.
Bobis, J. (1991). The effect of instruction on the development of computation estimation strategies. Mathematics Education Research Journal , 3, 7-29.
Bobis, J. (1996). Visualisation and the development of number sense with kindergarten children. In Mulligan, J. & Mitchelmore, M. (Eds.) Children's Number Learning : A Research Monograph of the Mathematics Education Group of Australasia and the Australian Association of Mathematics Teachers. Adelaide: AAMT
Case, R. & Sowder, J. (1990). The development of computational estimation: A neo-Piagetian analysis. Cognition and Instruction , 7, 79-104.
Cobb, P., Wood, T., Yackel, E., Nicholls, J., Wheatley, G., Trigatti, B., & Perlwitz, M., (1991). Assessment of a problem-centred second-grade mathematics project. Journal for Research in Mathematics Education , 22, 3-29.
Fischer, F. (1990). A part-part-whole curriculum for teaching number to kindergarten. Journal for Research in Mathematics Education , 21, 207-215.
Gelman, R. & Gallistel, C. (1978). The Child's Understanding of Number. Cambridge, MA: Harvard University Press.
Hope, J. & Sherril, J. (1987). Characteristics of unskilled and skilled mental calculators. Journal for Research in Mathematics Education , 18, 98-111.
Mason, J. (1992). Doing and construing mathematics in screen space, In Perry, B., Southwell, B., & Owens, K. (Eds.). Proceedings of the Thirteenth Annual Conference of the Mathematics Education Research Group of Australasia . Nepean, Sydney: MERGA.
Ross, S. (1989). Parts, wholes, and place value: A developmental view. Arithmetic Teacher , 36, 47-51.
Sowder, J. (1988). Mental computation and number comparison: Their roles in the development of number sense and computational estimation. In Heibert & Behr (Eds.). Research Agenda for Mathematics Education: Number Concepts and Operations in the Middle Grades (pp. 192-197). Hillsdale, NJ: Lawrence, Erlbaum & Reston.
Presmeg, N. (1986). Visualisation in high school mathematics. For the Learning of Mathematics , 6 (3), 42-46.
Trafton, P. (1992). Using number sense to develop mental computation and computational estimation. In C. Irons (Ed.) Challenging Children to Think when they Compute . (pp. 78-92). Brisbane: Centre for Mathematics and Science Education, Queensland University of Technology.