Invented Math Strategies

How do children develop their own strategies?
I have come to believe that children develop their own strategies. Sometimes, it happens as a result of confusion. A student could be taught one strategy from one teacher and another strategy from another teacher. This could cause one child to incorporate two ways into one, resulting in confusion and the wrong answer. This is with a student at my intern school.We were working on regrouping with subtraction. I showed them a way of subtracting that includes showing the number of 10's in the tens place. For example, with the the number 43, there are 4 ten's. These tens would be wrote on the bottom left side of the math problem. I asked him to cross out a ten when he is adding it to the one's place. When he got the chance to use this method, with no help, he would take the ten away but would add it to the number that is being subtracted. He would put that in the place of the regrouped tens place (I hope that makes sense). I taught him how to demonstrate the adding of 10 to his tens place and another teacher taught him to add that ten to the ones place ( I realized this when asking him to subtract and regroup before I tried to teach him any strategies). I acknowledge the fact that I taught him this strategy poorly. But I wrote about this to show that students create strategies from ones that they've been taught. He also incorporated using blocks for his problems to help him maintain the placement of the numbers.

An example: 21 - 7 = 14. In this picture, he got it wrong at first. It showed another mistake that he was making, but I will discuss this later in the blog. He first added 10 to the 7 down at the bottom of the paper. You can't really tell because he erased his first response. He eventually added the 10 to the 1 in the tens place and gotten the problem correct.

When subtracting, one of the students would use tally marks. This was a strategy used by another student. With 2 of the 4 problems given using tally marks, he had gotten wrong (picture below). Noticing this, I introduced him to a new strategy. I called it the 'Count Up' strategy. When subtracting smaller numbers, take the subtracted number and count up until you get the bigger number (I can't remember the proper name for the subracting the number and the 'original number'). In other words, if c - b = d, I told him he could take 'b' and count up (not including b) until he got to 'c'. The number it took to get to 'c' was the answer or 'd'. This was also a problem that the other student was making in the first picture. I showed him the 'Count Up' strategy as well. (Below) His tally marks caused him to get the problem wrong. I tried to help him perfect this way, but it was much easier to explain the 'Count Up' method to both of them.

Math Talk Moves

What was the specific mathematical content being taught? This week I taught them a lesson on how to use a specific method for word problems. After I modeled filling out this special I had them to read a word problem and underline any information that they thought would be important. After that, we began to fill out the Math Problem Solving Guide. With this guide, we answered three basic questions and followed one command; what do I need to know, what do I know, what do I need to do, and check your answer.

Any specific talk moves being used? Were they effective? I noticed that I used a lot of wait time. There was one student who caught on really fast. When I realized this, I tried to spread out my questions among the other students. By doing this, I had to use more wait time because they didn't catch on as fast. As the book stated, some of them needed more time to get their thoughts together. I believe they were effective. The other student contributed with correct responses. Their responses gave me an opportunity to use two of the other Talk Moves. After a student would suggest something, I would ask them if other had more to add to the other comments. I also restated their suggestions to make sure that I understood what they were saying before I wrote them down on the board. I didn't allow them to say if they agreed or not because I wasn't sure if they could respectfully disagree. I wasn't sure if any of them would get offended if someone openly disagreed with them.


If I could redo this lesson, what Talk Moves would I use? I would probably allow them to agree or disagree with one another. Before I did that, I would have made sure that they know its OK for others to disagree with them. I am curious to see if it would make them want to put out more comprehensive suggestions if they knew someone would be listening. This would help when we are discussing extra information that is word problems. I can ask the question 'is this something we need to know'. Based on their answers, other students can comment on whether or not they believe it should be written on our Problem Solving Guide. After that, I would probably ask them to restate what another person had just said. I believe putting it in their own words would help them in understanding their peers' comments as well as other students. This could be helpful when I am asking 'why' should a statement be on our guide. When they present a rationale, I could ask another student to restate what the other student just said to clarify.

Teaching Space

These past few teaching days actually took place outside. Because of that, I wasn't able to take direct pictures of the children in the classroom. However, I was able to find pictures of my students uploaded to a site that I could use. Here are some pictures from their field day.


Initial Impression of School: When I first walked into the school, it reminded me of an old school I visited when I first started doing observations. It reminded me of Dawson Elementary, a school that's in central Austin. I could tell that it was one of the older schools in the city because of the way it was built. I saw pipes along the ceiling in the hallways. I saw a lot of student art work on the walls with posters about bullies. I thought it was sort of diverse. There seemed to be a lot of parental involvement. I saw a lot parents and this made me believe that they have strong parental support.

Math Class: She pulls out students for math instruction. Because it was so early in the semester when I observed the teacher, she hadn't established a system for what/when she would be working on math. She explained to me that one thing that they liked to do last year was something called 'fast math' or 'minute math'. This is something that she had began to do with the kids earlier during that previous week. 'Fast Math' is when she gives them a sheet with addition math problems and they have one minute to work out as many problems as possible. When they are done, she grades them on the spot and gives them an opportunity to correct their mistakes. This is something that is helping them become more comfortable with math at the beginning of the year. Also, she told me that the children see it as more of fun challenge than anything else.

Teacher's beliefs: After talking to my teacher, I have learned that she really enjoys working with her kids. She believes that with the child's full, undivided attention, they are a lot more productive. This is the reason why she has arranged her groups and my groups the way that they are grouped. For example, we have two sets of twins. Last year, when the brothers were in the same class, my cooperative teacher and their general education teacher report having a lot of problems with the student's behavior. They realized that they reacted to each others actions a lot of times. This wasn't helpful when trying to keep them engaged. As a result, they have put the brothers in two different classrooms and they are also pulled into two different groups at two different times of the day. My cooperative teacher has told me that they have made a lot behavioral improvements. As a result, they are getting more work done.

Me as a teacher: Being in this classroom is really important to me. This is the first time I am working in a resource room. I plan on teaching a resource room, but I have always leaned towards high school aged children. Being in this classroom, in this elementary school, is really starting to change my mind. They are not too mature to appreciate the special handshakes and not too young to have productive conversations. I like how they like a lot things, but are slowly transitioning to more 'teenage' things. This is important to me because I plan to use their interests to setup incentives for the students, especially for those with a BIP. This is also the first time I am working with so many Black students who share a similar household background with myself. This is the most important because by sharing a similar background, I hope this will help me build repor with the students in such a short time. Being that I plan to work with children who have similar background, I look forward to being the 'teacher' for the first time and learning everything I can.

Questions: problem solving vs teaching the skills

Q1: I think it takes less time to do problem solving. Also, using problem solving allows them to pull from their own understanding/background is more effective.

Q2: They will force me to seek out help from other teachers. From my experience, I have learned that some students get it better when someone else explains the concept. Often, other people can explain things in way that's easier than myself. Different strokes for different folks

Q3:  I would argue that they don't have to discover 'everything' and we as teachers have to be creative enough to pull children close enough to the answer and solutions without giving it to them. Usually, "reinventing wheel" for themselves proves to be better for them in the long run.

Q4: You can ask them to explain how they got to where they are in the problem. You can ask them questions that force them to think more critically about the problem.

Q5: They were asked to build on their prior knowledge and pull from previous mathematical experiences.

Special Education Teacher

Why do you want to be a special education teacher? What is your vision/belief about teaching mathematics to your students? 

When it comes to math, money is an important concept. I want my students to understand that eventually they will have to make their own decisions and this will include paying bills and working for a paycheck. Being that people with disabilities can be taken advantage of (at times more than most other people), it's important that they understand the value of computing their wages, bills, and other things that involve monetary exchanges. I sincerely believe that people, in general, are good people. At the same time, I am very aware that there are people who have no spines and will take the opportunity to take from people, even if they have disabilities. I want my students to understand this when I am done teaching them.

Greatest Challenge

What is the single greatest challenge that you have faced in math? How have you faced, handled or dealt with this challenge? Have other people assisted you in dealing with this challenge? How has the challenge had an impact on your experiences with math?

The single greatest challenge that I have ever experienced was in college. I had to take a college algebra class. This was the first class that I had ever taken while I was in college and I had given up on math in the 10th grade of high school. I not only had to go back and memorize formulas, but I had to remember how and when to apply them. Also, my teacher gave us (what seemed like) a ton of homework every night. It was summer class so we met everyday. I never had to work that hard memorizing and applying anything. I didn't know anyone and I didn't know how to ask for help. I am very happy because later realized office hours were made to help, so I started to go. It didn't make the work load lighter, but it did help me in my understanding. Thanks Mr. Garcia

Other Important Scenes

 Describe at least two more scenes involving math in your life that stand out as especially important or significant. Think in terms of scenes from your childhood, adolescence and adulthood. 

One scene that comes to my mind comes from the first book I learned to read. The title of the book was "How many ways can you cut a pie". My mother read it to me A LOT because it was the only book I wanted to read. I think I liked the book because it was about math. More particularly, the book was about fractions. I thought that was funny; the first book I learned to read was about math. 

Another scene that comes to my mind occurred when my brother lived with me. He used to have problems 8th grade algebra. At first, I couldn't understand why he was having problems. I felt like I was explaining everything so well. I then realized that he learns like me. He needs a model to constantly refer to to better understand mathematical concepts. We ended up making a tablet full of algebra concepts like adding/subtracting fractions and certain formulas. It made it easy once I realized that he needed something to refer to and that he needed a lot of foundational mathematical knowledge to better understand the work he had. He didn't have a good understanding of negative numbers in general and other things.

Turning Point

Sometimes when people look back over their lives they can identify critical points when some kind of important change occurred.  Was there a point in your life when you experienced a turning point, that is, you changed in some way regarding your understanding of or feelings about math?  Please describe a specific episode when you feel you experienced a turning point.  If you think you have not experienced a turning point, describe an event that comes closer than any other in qualifying as a turning point. 

My turning point happened when I first started to take geometry in the 10th grade. I had always felt like math came so easy to me. If I didn't understand a concept or anything, I wouldn't worry about it and ask a teacher to explain it to me. That would be the end. I would understand. Coming to this class changed my view. This was the first time when something was explained to me, one on one, and I had to pretend like I understood to keep the teacher from staying longer. She had other students who needed her just as bad as I did. It didn't stop that day. Later on in the school year, she would introduce more concepts and I would find myself not knowing what to do, with the explanation. She explained things well, I just couldn't get it. This changed me and showed me that I was no longer mathematically invincible. 

Nadir Experience

A ‘‘nadir’’ is a low point. A nadir experience, therefore, is the opposite of a peak experience. It is a low point in your experiences with math. Thinking back over your life, try to remember a specific experience in which you felt extremely negative emotions about math. You should consider this experience to represent one of the ‘‘low points’’ in your math story. What happened? When? Who was involved? What did you do? What were you thinking and feeling? What impact has the event had on you? What does the event say about who you are as a teacher?

A low point in my mathematics career happened during my 6th grade. I was sitting in class and my teacher was reintroducing division and my inner circle of friends all knew long division. I was really embarrassed because I managed to make it to the 6th grade without learning division and all of my friends could do it. I was even embarrassed to tell my teacher. Eventually I told my teacher and she made me feel so bad at first, but not on purpose. She couldn't believe it. She took me in and taught me division in that same hour. 

Peak Experience

 A peak experience would be a high point in your story about math in your life – perhaps the high point. It would be a moment or episode in which you experienced extremely positive emotions; like joy, excitement, great happiness, uplifting, or even deep inner peace after some math experience. Describe what happened, where it happened, who was involved, what you did, what you were thinking and feeling, what impact this experience may have had upon you, and what this experience says about who you are now as a teacher.

I had a peak math experience this past summer  with my little brother's best friend. He was hanging out at my place with my brother and I knew he was really into math, so I showed him some of my homework from class. I had no idea he was as smart as he was (or maybe I'm just not that bright #shrugs). He not only figured out how to do my homework, but he did it faster with more accuracy. I didn't look at him the same or math. For a second, he was teaching me. I was humbled by that experience so I invited him to class.