Week 3, Chapter 2

Some of these concepts could easily be practiced with the Quick Image activity using ten frames. Quick Images are a great way for students to learn the concept of a number, and, therefore, also a great way to learn the concept of "more than" or "less than." In fact, ten frames in general would be a great way to study more than versus less than, because if more boxes are filled in on one ten frame, the visual image helps students understand.

Early counting is made easier with activities such as the Double Decker Bus video. Starting with small amounts of something that is tangible is a great idea. As in the video, the teacher started with 3 students. When all 3 students were sitting on the "bus" she asked how many were on the second level, how many were on the first, and how many there were total.

Relationships among numbers relates to the video about finger patterns. Children can easily learn their doubles and the significance of five and ten with their hands, as well as concepts such as "two more than", etc.

Activity 2.9 really is my favorite activity. It is similar to Quick Images, but uses paper plates instead of ten frames. I like this version better because it is easier to show that five dots in a row is five, but so is five dots configured like the dots on a die. Before doing the Quick Images activity, students could even be responsible for making the plates. The teacher could put the students in 10 groups and assign each group a number; then each group has to think of all the possible ways/patterns to draw their number on the plates. After that, the teacher could flash the plates to the whole class or students can flash the plates to each other during center time or in small groups. I think it is a very versatile and efficient activity.

Week 2, Challenge 5

"Kindergarten is More Than Counting" is about helping students become fluent with numbers. As long as students have manipulatives, they are more likely to count by ones to reach an answer. The author suggests using a Quick Image format to help students learn patterns and number characteristics. By flashing images of dominoes for fewer than 3 seconds, students learn to rely on pattern recognition because they do not have enough time to count the individual dots. After doing this, the author suggests using story problems. She says that students need to learn patterns and the qualities of numbers before they can learn how to take numbers apart and put them together, the ultimate sign of fluency.

"Understanding Children's Reasoning" states that the most effective way to understand a child's thinking is to ask them to explain how they reached an answer. The author says that the teacher must always be analyzing comprehension by asking questions such as, "How did you get that?", "Why did you start there?", and "How did you know to ____?"

"Thinking Strategies: Teaching Arithmetic Through Problem Solving" says to begin by learning the students' different conceptual levels, such as using fingers, using manipulatives, working with number lines, using pencil and paper to draw, using pencil and paper to write, or creating their own way. To teach thinking, this author also suggests using Quick Images to help students recognize patterns and learn step-by-step thinking. The author recommends using a balance (tangible or hypothetical) to practice balancing problems as well as relating known problems to new problems. She says it is always beneficial to see what the students can come up with on their own first and work with that because children should be given more responsibility in the learning process in order to retain information.

Challenge 4

The students progressed with their addition and subtraction between interviews, especially learning their doubles, fives, and tens. All four students seemed to use these three skills often, especially when working with larger numbers.

Derek: When solving 5+4, he said he knew that 4 is one less than 5, so he just doubled 5 and took away 1. Derek always gets the right answer to the questions, but his explanations are long-winded and confusing. He works very well with his doubles.

Elizabeth: For the questions that Elizabeth doesn't know immediately, she solves them by using her fingers. For 11-7, she says she pretended there was one more finger, then took away 7. She got confused, though, and ended up with 3. When solving 9+6, she said immediately that the answer was 15 because 10+6=16, so 9+6=15. She is good with logic, and she uses her tens, fives, and doubles whenever possible.

Jim: Jim uses dominos when subtracting, flipping them over as he counts down. He also uses his tens, fives, and doubles. For 15-7, he knew that 7+7=14, and one more is 15, so 1+7=8. I thought his thinking process for solving 7+6 was more complex. He took the 6 and broke it into 3 and 3, then added one 3 to the 7 to make an even 10. Then he had 3 left over, and 10+3=13. His logic is present, but he also changed his answers multiple times, either because he didn't know or because he solved the problem quickly and made a careless mistake.

Lauren: She was very good at addition and subtraction before the elapsed time, but now she always gets the right answer. For 8+9, Lauren took 1 away from the 8 and added it to the 9 to make 10, then added the remaining 7 to get 17. For 12-4, she used her fingers. She pictured 2 imaginary fingers that she immediately took away, then she took away 2 real fingers to give her the answer of 8.

All four children gained logic and problem-solving skills throughout this elapsed time. They all learned to use their finger less and use easy numbers (5 and 10) whenever possible.

Challenge 3

Each child went through the same basic pattern of explaining their thoughts: they began by saying they "just knew how" or they guessed, then would say they counted or used their fingers.

Derek: In the beginning, when asked how he knew the answer, he said he "just did." When prompted to explain further, he said he decides to stop when he gets to big numbers, so he stopped at 14. Derek said he used his fingers to count up from 7 to 15, and he got the correct answer.

Elizabeth: When doing addition with fictitious apples, Elizabeth always reached the correct answer but her explanation was lacking; she would say she didn't know how she got her answer, she just guessed. Toward the end of the video, she would either say that she reached her answer by "counting in her mind" or "counting up."

Jim: Jim always said "I just thought about it for a minute." In the beginning, he said he didn't count, he just saw it in his mind. When doing subtraction with apples (6-2) he said he thought about it for a minute then took away Five and Six, as if they were proper nouns. When completing a different problem, he said that he knew if there were 4 cows and he took away 1, there were 3 left because 3 comes just before 4. Jim had trouble with larger numbers because he did all of his math in his mind.

Lauren: In the beginning, Lauren used the dots on the dominos to count up. She was not always accurate, but she used the patterns on the dominos to help her. Even if there wasn't a tangible domino there, she imagined it, suggesting she was good at abstract thinking. Throughout the video, she would explain her answers by using her fingers and counting backwards in her mind. When Lauren understood and came up with the correct answer she could always verbalize it.

Week 1, Chapter 1

1. When I think of early childhood math, I don't think so much about learning skills as I do about recognizing patterns and learning how to think logically. Especially in the younger grades, I think children should focus on learning their shapes, making patterns with colors, learning how the shapes fit together, and learning how to solve problems on their own. Problem solving is the basis to all math.

2. One major point this chapter made is that children learn through reflective thinking, social interactions, and hands-on learning. They need time to work together and learn from each other, but also to think quietly to understand what they have learned and make sense of it. It is important to establish an open, trusting classroom so that children feel comfortable making mistakes and sharing their ideas with me and the other children.

Chapter 1 also suggests helpful ways to teach by solving problems, but before even beginning, the problem must "begin where the students are" in both life experience and skills attained so that the problematic portion is the math, not the question.

Not only that, but you have to prepare the children for problem-solving. Before problem-solving, the teacher must establish expectations and explain the problem thoroughly. During the problem-solving, the teacher should never show a student how to solve a problem, only provide hints or "pieces" of the solution and talk the student through the solving process with encouragement. Praise can be negative feedback for the ones not receiving it, so use it cautiously. Instead of one-on-one time, perhaps engage the students in a full class discussion so they can agree on the solution together and not rely on the teacher.

Letting students explain things to each other allows for a more open and positive experience because teachers words are often taken concretely; when talking to other students, they can question and disagree without feeling disrespectful or "wrong."

Assessments such as rubrics and interviews are useful in mathematics. Rubrics are a good way to grade assignments based on understanding and accomplishment; i.e. how much of this project was completed, and how much was completed correctly? Diagnostic interviews are one-on-one discussions to gauge a child's thinking process about a subject.