Chapter 2 Reflection

One thing I noticed in the reading was when it talked about the spatial relationships and how children should be able to recognize objects in patterned arrangements and however many objects there are without counting them. This stood out to me because in one of the videos, the children were deciphering which set of dots were easier to identify the number of, depending on the pattern they were in.

I really liked Activity 2.8 called, "Learning Patterns." This activity allows the children to get practice in recognizing patterns to count 'dots' or objects instead of actually counting them out. In the activity, the teacher holds a dot plate out for about 3 seconds and then the children make the pattern that they saw using their own dots on the counter or mat. This seems like a neat activity as well because it also helps children to remember what they see better (recognition and memory skills).

Challenge 5

The Role of Mathematical Tools

"Concrete Materials and Teaching for Mathematical Understanding" by Patrick W. Thompson

This article stressed the importance of the question as a teacher, what do I want my students to understand, instead of what do I want my students to do? This article argues for the use of concrete materials in a more judicious and reflective manner. Because although the use of concrete materials has always been appealing, just using them is not always enough to guarantee success. One thing the article mentioned that stood out to me was the part about the 3/5 circles chart. It talked about all the different ways that at student could look at that chart, or that concrete material, and interpret it. We as teachers, tend to want to encourage them to interpret it in a particular way, or in a few different ways, but there are so many ways that he/she can interpret it and we need to be open to that. The article says, "Mathematics, like beauty, is in the eye of the beholder--and the eye sees what the mind conceives" (pg 52).

"And the Answer is... Symbolic" by Mary Lou Witherspoon

This article talks a lot about the misconceptions that happen in math. One misconception that happens a lot is with the equal sign. Many children think that it can only mean, "and the answer is...." The article goes on to talk about then how to foster the communicative role of mathematical symbols in an elementary school. The way to do this is by communication! When students are communicating, they are talking about how they solved that math problem, using different manipulatives, such as their fingers, the chalkboard, etc.

"Using Math Manipulatives to Aid Learning" by Dottie Oberholzer

The main goal of this article was to create students who were flexible problem solvers who could apply math ideas to all kinds of situations! It talks about when teaching a new math concept, you should always start with a concrete stage (math manipulative), then progress to a semi-concrete stage, then to an abstract stage. This teaching helps the students be active participants in learning.

Challenge 4

A few weeks after the teaching of the unit, the first graders were interviewed again about their understandings of number concepts and of addition and subtraction involving sums less than 20. What shifts or changes do you see in the students’ thinking from the initial interviews?

Derek: After a few weeks of instruction on addition and subtraction, you can definitely see a change in Derek's attitude towards the subject. He keeps saying to the teacher stuff like, "that's too easy" and talking very confident-like. His ability to solve the higher number concepts has definitely gotten a lot better, because without the instruction before, he was not able to do that.

Elizabeth: Elizabeth definitely learned and grew so much after her time of instruction with addition and subtraction. She was adding and subtracting bigger numbers in her head and a little bit on her fingers as well. Before, she could only add and subtract very small numbers and now she can add/subtract much bigger numbers and you can definitely tell that her understanding of the number concepts has definitely grown.

Jim: Jim also got a lot better at adding and subtracting after a few weeks of instruction on the unit. Before, he would add/subtract numbers and give an answer that sometimes would be right and sometimes would be wrong and sometimes he could explain how he came up with his answers and sometimes he couldn't explain it at all and it or it wouldn't make any sense at all. I definitely saw that Jim is starting to understand the number concepts a lot better now. He was just a little bit of a slower learner than maybe some of the others.

Lauren: Lauren was very good at adding/subtracting before the instruction unit on it. However, she did learn a new technique to use and did memorize her doubles well. She was able to explain how she got her answers really well using the new technique she learned. She seems to have gained a little bit better understanding of the larger number concepts.

All of the students, during the instructional unit, seemed to have been taught their doubles in addition/subtraction. They knew them very well. They also knew how to add/subtract by multiples of 5 and 10 really well and whenever they got into the larger number concepts, that is what pretty much all four of them would do. They would round to the closest multiple of either 5 or 10 and then go from there how they knew best. It was interesting to see and a really neat way to watch how they had gone from their own individual strategies before the instructional unit to the new technique they had learned afterwards.

Challenge 3

At the beginning of first grade, several students in the class were interviewed about their understandings of number concepts and addition and subtraction of numbers up to 20 prior to the teaching of these concepts. What are your impressions of each student? What do they understand about number concepts and addition and subtraction of numbers up to 20?

Derek: Derek seemed to understand pretty well the concept of addition and subtraction. I was surprised at how well he could add and subtract the harder numbers. He counted on his fingers and used an interesting strategy that it seemed he came up with himself and I was very impressed with that, such as when he subtracted down from 11 to 7 and got 4.

Elizabeth: Elizabeth could add and subtract the really easy numbers such as 3 plus 3 and 5 minus 2. But the numbers that got a little more difficult, she would mix up a bit. The strategy she used was all counting in her head. She didn't use her hands at all, she just tried doing all the math in her head.

Jim: Jim could solve the really easy addition and subtraction. He didn't necessarily have a strategy it seemed. He just kept saying that he thought about it in his head. Some of his thought processes when he would give his answers wouldn't make sense at all, if the answer wasn't close, so I could tell that he definitely didn't grasp addition/subtraction in the higher numbers. But he did understand the number concepts between about numbers 1-6.

Lauren: Lauren definitely understood number concepts. She was very quick to answer addition and subtraction problems, a lot quicker than the others, even the bigger problems. The easier problems she did in her head and the harder ones, she counted out loud or on her hands. I was very impressed with how good she was at math.

Challenge 2

What instructional activities can you use to support the students’ learning of basic number concepts and beginning ideas in addition/subtraction of sums up to 20?

-Come up with fun ways to practice it
-Matching the numbers and words
-Learning time and money (place values)
-Calling out different categories and having kids count how many kids are in each one

Challenge 1

What can first graders do in terms of basic number concepts and of addition/subtraction of sums less than 20?

-Recognize numbers (1-100)
-Simple addition (single digits)
-Count to 100
-Count by 10s, 5s, evens and odds
-Greater than/less than
-Add/Subtract using manipulatives
-Split a number into equal groups (total in two sets or more)
-Sequencing (5, ? , 7)

Week 1 Reflection Questions!

1. What does the term early childhood mathematics mean to you?
When I was growing up, I always loved math! I was just naturally pretty good at it. I guess I got that from my dad, luckily. He's a math whiz. But I also think it was because my teachers made it fun. It was never just a "chore" or "busy work" to me. But I enjoyed doing it. Most of my math teachers were peppy and made math enjoyable and relatable. Therefore, I feel that it is extremely important for teachers to make early childhood mathematics fun! If it isn't fun while they are young, then it won't be when they get older. It's like a sport. If they don't like playing the first few times, then they aren't going to continue trying to! I want my students to love learning. And math is a great place to start because it is a subject than can be taught in so many different ways, with so many different fun games too!

2. What key points did you take from chapter one that inform your understanding of how to teach mathematics for young children?
A few key points that stood out to me in the reading from chapter one was that everything in math makes sense. Because math deals with numbers, there is always a right or wrong answer and so children are very capable of making complete sense of mathematics. Another thing that stood out to me in the reading was when it talked about the fundamental core of effective teaching of math: a combination of us teachers understanding how our children learn and grasp ideas, using problem solving to nourish that learning, and how to daily act on and assess that learning. These were a couple of the things that caught my attention in the reading of chapter one on how to better teach mathematics for young children.