Measurement Article

1. The Main points of the article:

• Measurement instruction in the classroom has been ineffective.
• 72% of 2nd graders in a study were able to relate the whole length of the line to the corresponding part of the ruler, demonstrating transitive reasoning. This means that 3rd graders should have been able to "relate the whole length of the line to the corresponding part of the ruler;" the low-level response given by so many 3rd graders can thus be "attributed to ineffective teaching."
  
• Students in early childhood are measuring with the purpose of coming up with a number; they are not being taught the concept and purposes for measurement.

2. A study showed that almost a third of seventh graders did not know what a unit of length was. Units of length is a concept that should be taught in early childhood, and clearly instruction of this concept has been ineffective. Children should be cognitively able to compare two or more objects in early childhood, and they should be taught to make measurement comparisons with a purpose. They should not feel as if the purpose of measurement is simply to come up with a number for the teacher.

3. When teaching measurement, I will need to make sure I am not limiting my students' understanding. I will make sure my students are able to indirectly compare two or more objects to ensure that they understand the underlying principles of measurement.

Geometry

Here is your reflection question for the week:

After learning about the Van Hiele Geometric Levels of Thinking- where do you think you generally fit into this framework? How will you use this information in your instructional practice?

After learning about the Van Hiele Geometric Levels of Thinking, I think I generally fit into Level 2: Informal Deduction. I can see relationships between shapes and recognize that some shapes are "subclasses of other shapes." I remember a significant portion of the vocabulary that I have been taught about geometry and shapes. I am able to determine relationships between shapes using the information I know about their properties.

I will use this in instructional practice by pointing out similarities and differences between shapes. I will try to guide students to think about relationships between shapes. For example, I might ask what a square and a rectangle have in common (equal angles) and what they have different (length of sides). I will assess what level of thinking students are in and will determine my instruction to meet their understanding. When students are in Level 0, I will begin to gradually introduce vocabulary and provide 3-D manipulatives. When students are in Level 1, I will also introduce ideas about relationships to help students begin to think about concepts that are understood in the 2nd level. I will use the Van Hiele Geometric Levels of Thinking to guide my instructional practice in mathematics. By using this system, I can determine what students understand about shapes and I can guide them to more complex understandings.

Chapters 11 & 12

Chapters 11 and 12 deal with content related to data analysis and probability.

1. How does the task presented in class (examining fair tests) compare to the content covered in chapter 11?

The task presented in class where we measured our wrists is suggested in Ch. 11, because each student got to contribute "one piece of data." The task presented in class was also a comparison, because we compared our wrist circumferences. We discussed that we could make a bar graph from the measurements we obtained. Chapter 11 also discusses Descriptive Statistics, which we related to our task in class. We determined the range of our wrist measurements, the average wrist circumference, the mode, and the median. We discussed what these terms meant and how we determined them.

2. What are you seeing related to data analysis and probability in your own classroom settings?

The Kindergarten class I am in incorporates data analysis and probability into daily activities. The class uses a bar graph to analyze weather conditions. There are about 5 different weather categories that students can pick from to add on to the graph every day. Then the class counts how many are in the selected conditions and determines which weather occurs the most frequently for that month. Bar graphs were also recently used to compare which type of apple students preferred. They could choose between 3 varieties and stacked an apple picture above their selection. Cluster graphs are also used in my Kindergarten classroom. For example, during a science lesson, students sorted items into two categories on the SmartBoard: "living" and "not living." I have not seen too much related to probability yet. The teacher does ask some questions to introduce students to the idea of probability, such as: "Juice and Popsicles are both offered at lunch today but you can only take one. So if you take a juice, can you take a Popsicle too?" The class also sometimes determines whether or not something is likely to happen, similar to Activity 12.1. For example, during weather, the teacher may ask students in August if it's likely to snow tomorrow.

3. Examine the SC early childhood content standards (K-3) for data analysis and probability. How do the state standards compare to chapters 11 and 12?

The SC early childhood content standards for data analysis and probability are directly related the concepts found in chapters 11 and 12. The Kindergarten standard states that "The student will demonstrate through the mathematical processes an emerging sense of organizing and interpret data." In chapters 11 and 12, different ways to organize data, such as bar graphs and cluster graphs, are described. In the text, there are also assessment suggestions that suggest how the teacher can guide students towards interpreting data. For example, the teacher may ask the students to explain what their graph tells them. The first grade standard, which states is the same as the Kindergarten standard with the addition that the student will "make predictions on the basis of data." Probability is related to making predictions. Determining whether the likelihood of something is "certain, impossible, or possible" is a probability concept. When students determine the likelihood of an event, they are also making a prediction about whether or not it will happen. The standard for 2nd grade has the same core as the K and 1st, with the addition of that the student will describe trends of a data set. After students begin to make their own graphs, as described in Ch. 11, they can analyze the information and eventually observe trends and patterns.

Chapter 3 Reflections

What are the key ideas presented in Chapter 3?

• The key ideas presented in Chapter 3 had to do with how math problems are approached.
• Children should not learn the term addition as "joining" or "putting together" because this will limit their understanding of the concept.
• In "story problems" children often focus on getting the answer in the way that the teacher wants; in context problems, students can see how math applies to the real world.
• Contextual problems and models are the two main tools that teachers use to help students understand addition and subtraction.
• Models can be used for problem solving, even when there is no context involved.
• Multiplication and division should be combined as soon as possible so students see how they are related.
• Solving problems should be a means to helping students learn.

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

I thought it was interesting that the one of the problems presented in the chapter sounded like addition but was solved through subtraction, and another problem that sounded like subtraction was solved through addition. Students that are taught concrete ideas of what addition and subtraction are may be limited in their ability to solve math problems in the future. Therefore, I have learned that children should not be told how to solve a math problem; instead, they should use methods that they can understand. If math problems are worked out using manipulatives and real life examples, students may find more purpose in mathematics. Students should be able to use the method they understand best when solving problems.

I have also learned that teaching numbers and operations should not be taught with the sole intention of finding the answer. Children should see that math is useful and applies to everyday situations. If students see how they can apply math to everyday life, they will likely be more interested in the subject.

Week 3 Reflection Questions

How does the information and the tasks presented in chapter two connect to the videos of lessons you viewed as part of challenge 5?

The information and the tasks present in chapter two are about understanding number concepts and number relationships. The video that I watched about classroom norms discusses children's different approaches to solving the same math problem. The teacher emphasizes to the students that they need to understand their classmates approaches to solving the problem, as there are several different ways to combine numbers to reach the same answer. In this video, the teacher's objective is to have students understand that numbers can be combined in different ways to make the same result. In Chapter 2 of the text, there are four methods described to help students develop relationships among numbers 1 through 10. This relates specifically to the video I watched; in the video, one of the students points out that "5+0", "1+4", "4+1" etc all add up to 5. The text states that "the principal tool that children will use as they construct these relationships is counting" but counting will become "less and less necessary as children construct new relationships," which they were doing in the classroom norms video. The teacher in the video wanted students to understand each other's solutions to the problem, because if they understood that the different number combinations (ie 4+1, 2+3) all added up to 5, they also understood something about number relationships.

What task (activity) in chapter two was most interesting to you? Why?

One of the activities in the chapter that captured my attention was Activity 2.10: One-Less-Than Dominoes. I thought this activity would be great to use when teaching number concepts because it is a game. Students will likely be engrossed in playing the game and will not be focusing on being in math class. In this activity, students play dominoes in the regular way (which may need to be explained before starting) but instead of matching ends, a new domino can only be added if its end is one less than the end of the one on the board. There are also variations of this activity that can be played, which are mentioned in the book. For example, the game could be played for one more, two more, or two less. I think students will enjoy this activity and it can also help them develop their understanding of number concepts. It will help students think of numbers in different ways. For example, they may start thinking of 3 as "one less than 4" and also as "two more than 1."

Reflection Question for Week 2

I read two articles that discussed Classroom Norms. The first article, Equity and Accessibility, by J. Hiebert, discusses the importance of participation and discussion in every mathematics lesson. The article stresses how important it is for the success of every child in the classroom to be able to discuss and participate in the explanation of math problems. I thought the author made an interesting point when he stated that "each child's thinking should be discussed and valued because all ideas and methods are potential learning sites." If one child is having trouble understanding how to reach the correct answer in a problem, it's possible that one of his classmates used a method he would understand. If all children in the classroom are going to be successful in mathematics, then all methods, whether correct or not, should be discussed and analyzed. I also thought the author's suggestions for modifying the discussion process would be useful in any classroom. Some students will be shy and uncomfortable with the idea of speaking in front of their entire class, so having students work in small groups, having them check their answers with the teacher first, or having them write their methods on paper for others to read are all suggestions that may help students gain confidence in their math abilities. The author also stresses the idea that all students, regardless of their intellect, can understand math. Therefore, each child should be considered when coming up with math problems. Problems that can be solved in different ways are good to use, because children of all different learning levels should be able to find a way to solve them. I have learned that discussion of math problems is important to have in the classroom, because you never know when the discussion of a method will help a student gain a better understanding of a concept.

The second article I read was "Creating a Problem-solving Atmosphere" by Yackel, Cobb, Wood and Wheatley. This article also emphasizes the importance of classroom discussion. The authors describe a "problem-solving atmosphere," which is created through discussion of methods. In this atmosphere, children view math problems as personal challenges and they feel accomplished upon completion of the problems. In this type of environment, children enjoy figuring out math problems for themselves. In the problem-solving atmosphere, the teacher explains that is is okay for students to make mistakes. Students can learn from mistakes, and there is nothing wrong with making an error and correcting it. I also like how the article discusses students working through problems, regardless of how long it takes them. Finally figuring out a math problem after working hard out it will be very satisfying for anyone, particularly a young child. I also think it's important to point out that the article explains how important it is for the teacher to have an open attitude towards students methods. If students are expected to volunteer their methods in front of the class and are expected to experiment with different methods, then the teacher needs to try to understand their train of thought and be open about their approach to the problem. There should never be one "right way" to get an answer.

Both articles show how to create a classroom environment that promotes discussion. Discussion about mathematical methods in the classroom can help students understand math approaches and open their eyes to methods they wouldn't otherwise think of themselves.

Challenge 5

Initial Thoughts-

The topic my group selected to look into was Classroom Norms. At first, I wasn't exactly sure what the topic was referring to. In the video, the teacher asked the students if their classmate was correct in his approach to a math problem. She asked them if they understood his approach and what he was talking about. Then another student confirms that she believes he solved the problem correctly. From the video, I think that the classroom norms refer to the students being actively involved in the learning process and making sure they understand different methods to getting answers in mathematics. I think that the classroom norms in the video show that the students are expected to explain how they reach answers in math, and are expected to question each others work.

Week 1 Reflection Questions

Reflection questions for week one:

What does the term early childhood mathematics mean to you?

I think of early childhood mathematics as the foundation for all math. During early childhood mathematics, children develop their first understandings about numbers, sequences, patterns, shapes, addition, and subtraction. They begin to understand the concepts of adding and subtracting, and should start to become comfortable with simple math. I also think that if a child has positive math experiences in early childhood, he or she will be more open to doing math in the future. I think early childhood mathematics can shape a child's future opinion of math.

What key points did you take from chapter one that inform your understanding of how to teach mathematics for young children?

I took several key points from chapter one that will affect how I will teach mathematics for young children. First, I will take the idea that social interaction and group work can help children develop understanding in math. I would normally think of math as independent work, but after reading the chapter, I can see how discussion and group work would greatly benefit children in learning math. I also like the idea of doing mini-lessons, which the chapter suggests. I think if students engage in mini-lessons over the course of several days, they will develop a greater grasp of a topic and will remember it better than if it is crammed into a single lesson. I also think the point that the chapter makes about focusing on big ideas, rather than small skills, is important. It will be easier to assess a student's understanding of bigger concepts, and will be more important for moving forward that a student understands main ideas.

Challenge 4

Shifts in the students-

Jim has a much better understanding of adding, subtracting, and sequence. Before, he could not count, but now he was able to correctly add 7+6. He also used addition concepts to make more difficult questions easier. He understands that taking away is counting down. He can count down fairly successfully, although a few times he would skip a number.

Lauren already understood the concepts of addition and subtraction, as well as number relationships. This time, she was even quicker in answering the questions. When asked 7+6, she knew the answer was 13, because 7+7 is 14 and 6 is one less than 7. This shows that she understands sequencing, and understands that 7 is one more than 6. She also broke down some of the larger problems into smaller groupings. She understands the concepts of sequences, addition, subtraction.

Elizabeth has also greatly improved in her addition, subtraction, and sequence understanding. She understands that "taking away" is the same as subtraction, or counting down. She understands number relationships, which was demonstrated through the way she solves problems. For example, when solving 6+ _ =15, she knew that 10+6 was 16, and 9 is one less than 10, so 9+6 much be one less than 16, which is 15.

Challenge 3

Initial Thoughts-

Jim did not seem to understand the concepts of addition and subtraction. He was not able to add. One of the two problems he answered correctly was the first one, which asked, "How many do you have if you take away 4 from 6?" He said he "took away 5 and 6," leaving him with four. This shows that he does understand that numbers go in a sequence, and 5 and 6 come after 4. When asked other questions, such as 7+6 and 8+5, Jim guessed random numbers. He was also asked "4-1" and he was correct, saying that "3 comes before 4." Again, this shows that Jim knows the concept of number sequence, but probably only up to about 6 or 7.

Lauren understood the concepts of addition and subtraction. She knew that 7+6=13, by "counting." She next said that 9+6 was 16, and when explaining how she reached that answer, she realized is was actually 15. She understands number sequence, and understands the concept of adding. Lauren could also count backwards, from at least 12 and possibly higher. She understood that subtracting meant taking away, which required counting down.

Elizabeth understands some basic addition, although she had difficulty explaining procedure. For example, she knew that 3+3 was 6, but could not explain it on her own without a prompt. Later, she answers that 4 cows+2 cows is 5 cows, but then changes her answer to 6 when counting out loud. Elizabeth typically tried to answer the questions "in her mind" but when she checked her work using her fingers, she was often able to correct her mistakes. She seemed to understand the concept of addition, but not subtraction. She was not able to answer the subtraction problems correctly. She did not seem to understand that "taking away" was subtracting, which meant counting backwards. She did seem to understand number sequence.

Derek seems to understand the concepts of addition and subtraction. He understands counting sequence, and can count backwards and forwards. However, he said he did not like "high numbers" and probably does not feel comfortable with adding and subtracting past 15. He understood that subtraction meant taking away, and was able to count backwards successfully.

Challenge 2

Challenge 2: How will you get them from where you think they are to where they should be?

There are several instructional activities that I would use to help students move from where they are to where I would like them to be. I would use manipulatives whenever possible so the students can physically see how many of something each different number represents. I could give each student ten flat marbles, and ask them to place two marbles on the right side of their desk. Then I could say, "Add 2 more marbles." Then ask, "How many do you have on the right side now?" I would continue this activity with larger numbers and incorporate subtraction as well. This will be an engaging way for the students to practice simple addition, subtraction, and number relations.

Another activity that I think students would enjoy, which would also help with addition and subtraction, would be having them come up with about five to ten addition and subtraction problems on their own, using numbers 1-20, which they will trade with a classmate. Then, after they trade and complete the problems, they will trade back and check their classmates work. This will allow them to be interactive.

Other activities might include incorporating everyday objects into addition and subtraction. Using numbers from student's personal lives, such as how many siblings they have, and letting them compare will make them more interested than if they were just working with numbers without any personal significance.

As time progresses, I will observe how students understanding of numbers changes. I can do this by using the same activities but making them more challenging.

Challenge 1

Challenge 1: What do you think first grade students can do related to numbers, number relationships, and addition and subtraction with sums less than 20 coming into 1st Grade?

I think students coming into first grade should be able to count to twenty and should be able to understand that 2 is more than 1, 3 is more than 2, etc. They should be able to put cards with a different number of objects in order, meaning if they were given three cards, the first with one picture of a bear, the second with two pictures of bears, and the third with three, they should be able to determine which one has the most bears and which one has the least. They should be able to tell how many more bears are on the card with three than on the card with two by doing simple subtraction.

Students coming into first grade should understand addition in the way that if you ask them how many pets they have if they have one cat and one dog, they should be able to say "two." They should be able to do the same with subtraction. They should also be able to recognize what the numbers one through twenty represent and what they look like, and understand that the symbols represent a number.