Chapters 4 and 8

In my first grade practicum experience, I have not seen many of the strategies discussed in the book for addition facts. The one thing my teacher does use is the 36-fact-charts. She has students fill these in for their addition facts (0+0, 0+1, 1+0, etc.) and to learn doubles and doubles-plus-one facts. She does not use ten-frames or dice, but maybe they are not ready for these manipulatives yet. It is a good sign that my teacher uses the "most widely promoted strategy": counting on. She has taught the students to choose the biggest number in an addition problem and count upwards to reach their answer. Unfortunately, like the chapter suggests, this is not the most effective strategy because it is not appropriate with bigger numbers.

I have never been in a classroom higher than first grade, so I have never seen subtraction, multiplication, or division taught or practiced in a classroom.

This past week I saw an excellent lesson about measurement. My teacher asked the students to bring in a "pet rock." She allowed the students to decorate their rocks as a special activity with me and as we decorated them I asked them to compare their rocks: which one was bigger, which one was heavier, which one was smoother, etc. Although this was impromptu and informal, the activity introduced the concept of weight and making comparisons between objects.

Mrs. Gajda has also done morning work with measurement concepts. On Halloween, the students used candy corn as an informal unit of measurement to measure line lengths and were then able to eat their candy. This was the first time I had seen a measurement lesson; they have not yet covered estimation or standard units of measurement. The candy corn lesson was a good introduction to units of measurement, as well as the concepts of longer/shorter.

Mrs. Gajda has not talked about area, but she does have tangrams in her classroom for early finishers so I suppose that can be seen as an informal introduction to area. Later this year she will use the tangrams to figure out which combination of shapes match up with others (i.e. have the same area.) Volume and capacity are also concepts above the first grade level that I have not seen taught, directly or indirectly.

The students in my classroom cannot tell time yet, but Mrs. Gajda is using calendar math to introduce time. For each school day, the students color in another minute on an analog clock. When they reach sixty days, she will talk about how the spaces on the clock represent minutes and that one fully colored clock is 60 notches/60 minutes, or one hour.

Chapters 11 and 12

1. The wrist measurement activity we did in class works with chapter 11 because the data is student-based. Chapter 11 is all about helping the students learn to use data; students learn everything better when it is based on themselves. Allowing students to come up with the different questions they have about their wrists gets students involved in the process and the data will be different and specific to each student, which personalizes the activity. Whereas we used a tally chart to document the different wrist circumferences, the activity could use whichever form of graph the class is learning or needs help with. If this activity was done in a first grade classroom, it would be a fantastic activity to teach how to tally and then how to transfer the data in a tally chart to a bar graph. In class we also focused on finding the most appropriate average. This corresponds better with the standards for higher elementary; in fact, the measures of central tendency are not even mentioned until fifth grade. However, the book says it is appropriate to discuss the "middle" of a set of data even if the MoCT are too advanced.

2. I am in a first grade classroom. We have learned a lot about different types of charts and graphs. One of the first days of school, we made a pictograph of months and birthdays on the back wall. The months start with January at the top of the wall and end with December at the bottom of the wall. We spent about an hour working on this graph one day. Each child received an index card on which they were instructed to write their name and birthday and to draw a cake. Then we started with January birthdays and taped each card onto the wall. This was fantastic according to chapter 11 because it was a student-centered form of data. Other than this activity, the students practice tally charts almost daily during math. Several weeks ago they learned how to transfer data from their tally chart to a bar graph. When I taught my lesson I focused on this concept. I gave each student a bag of M&M's; they sorted their candy by colors, made a tally chart of how many candies they had per color, then colored a bar graph to represent this data.

3. In Kindergarten, the standards focus on beginning to organize and interpret data. Sorting and tallying activities discussed in chapter 11 would relate to this. Even if the teacher did not discuss the proper way to tally (crossing number 5) it would be a start to learning to organize data, as the standard states.

Grade 1 says students should be able to collect, organize, and interpret data as well as make hypotheses about it. The book does not really discuss hypothesizing, but all of these pieces of grade 1 could be covered in a leaf activity. Before going outside the students could hypothesize about which color leaf they will find the most of as a class. Students can collect leaves, sort them by colors, make a tally chart, and then make a pictograph or a bar graph to help them determine if their hypotheses were correct.

Grade 2 includes everything from standard one and adds that students should be able to describe the trends of data. The book also does not focus specifically on data trends. A good activity to cover data trends is to have students graph their individual test scores for each subject and compare them from week to week and subject to subject.

Grade 3 adds that students should be able to understand the benefits of multiple representations of data, as well as basic concepts of probability. This is the first time the content in chapter 12 has even been mentioned. Chapter 12 is about basic probability and how to use tree charts, predictions, the range of impossible to certain, and various games to practice probability. In third grade, students should also know that different graphs and charts can make the same set of data appear different. Chapter 11 goes over the benefits and differences of multiple types of graphs.

Chapter 3 Reflection

Key Ideas:

• Addition and subtraction are connected by means of "part" and "whole." Addition is used to name the whole when the parts are known, and subtraction is used to name one part when the whole and the other part are known.
• Students need to learn these more broad definitions rather than "put together" or "take away."
• When teaching addition and subtraction, teach lesson using context/stories. Don't focus as much on the symbols involved.
• Multiplication and division are connected and relate back to addition and subtraction.
• For all math, as long as the students know the numbers in a problem (and their answer is a known number) they can answer the problems. Problems should not be "dumbed down."
• Models can be used to visualize math problems.
• There are properties in addition, subtraction, multiplication, and division that should be taught to help students remember how to do each operation.

How do these ideas inform your understanding of teaching numbers and operations?

Before reading, I had heard that we should teach addition and subtraction together and multiplication and division together, but I didn't fully understand why. This chapter explained how the operations are more similar than they are different and that teaching them together can actually help students learn more efficiently and effectively. The explanations and examples of different models you can use for teaching the operations were helpful and I will definitely use models in my class because I know children learn in different ways and some might need this visual example. Others might need an auditory or written example of the properties, so those will also be included in my classroom teaching.

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.