Invented Math Strategies

My student at Houston Elementary tended to use a direct model approach to almost all of the problems. This really worked for her in solving problems, however I do think that she was able to use other math skills to solve some of the problems.

Problem - Multiplication Equal Groups
Jesses has 3 pockets. He puts 2 pennies in each pocket. How many pennies does Jesse have?
The student used a direct model approach with this problem. She first counted out three "pockets" by using red manipulatives. She then counted out 2 black manipulatives and placed them in the "pockets". She then counted 1 at a time to get to 6.
While her strategy was effective, I think my student could have solved the problem differently in the following ways:
-Instead of counted from 1 to 6, she could have added 2+2+2 to get 6.
-Instead of direct modeling, she could have used math fact knowledge to know that 3x2=6.

I could tell that my student really wanted to show how she was able to solve a question instead of finding the answer. This was great because I got to see how her mind was working, however at times she was confused throughout her showing process and was unable to solve the problem without guidance.

Math Talk Moves

This week my students were tackling the topic of decomposing numbers. I assisted my CT with instruction in order to get a better feel of the students I would be working with. We began by coloring lima beans on one side only. Once the students had finished this, we began to play a "game". The students were asked to pull out a set number of beans, shake them in their hands like a dice and drop them on their paper. They were then instructed to place the number of colored beans on one side of a column and the number of white beans on the other. Once we had done this several times, it was pointed out to the students that the number of colored beans plus the number of white beans equaled the total amount of beans they were shaking. They had been decomposing numbers all along! Throughout this lesson, I noticed my CT revoicing students' explanations and reasonings and I noticed wait time that was appropriate for the group. When I was leading the class, I found the wait time to be difficult to measure and depended on my CT for help. Because of the difficulties this class sometimes has, I don't think talk moves 2-4 were used because they would not engage the whole class and could lead to disruptions or off-task behavior. Once the students get used to what is expected from them in math class, I think more discussions would be appropriate and less dangerous.



Looking back, I feel like I should have been more efficient with steps 2,3 and 4: asking students to restate someone else's reasoning, asking students to apply their own reasoning to someone else's reasoning and prompting students with further participation. Because this class only contains students with behavior issues, I found it really hard to monitor students' behavior while also engaging in more productive talk moves. The content that we were going over was not complex, but keeping tabs on the students made it difficult to extend their knowledge further than understanding or applying. I think if I had asked them to go further with their reasoning behind their answers and explanations I would have helped them become more comfortable with their abilities; they may have been able to understand the concept of decomposing numbers, not just finding numbers that add up to 20. I think that if the students were able to explain their reasoning to others and understand others reasoning, it would not only help them with this concept, but hopefully they would be able to apply those reasoning skills to other concepts and situations. As far as participation goes, the students really only participate as much as they are prompted to. I think requiring more out of them would keep them engaged and also further their knowledge. In the future, I would like to engage the students in more discussions about their ideas and reasoning behind the math solutions they come up with. I think that when they are able to explain their answers they will become more efficient in tackling problems they haven't seen before. Participating in discussions would also force the students to become more active participants and this would hopefully get those wheels in their heads' turning. It might also get them to move to more pen and paper tasks, and leaving the manipulatives for practice only.

Math Pictures

My Math Teacher


Student working out a problem.


Problems made for the students.


Small Group Instruction


The Tree of Knowledge

Stepping into Teaching

This is not my first time at my school. I was here a year ago, during my general education teaching semester. The school is very much the same. Throughout the building there are areas where different content is displayed throughout the year. At one end of the school there is a science area, where displays show the planets, different ecosystems and weather displays. There is another area that displays various cultures and histories of Texas. There are displays for science, social studies, art, music, language arts and PE. Unfortunately there is not a math display. In addition to the displays of content material, there are also reminders to stop at crossing, hallway behaviors, and student conduct codes throughout the school. The school itself is very warm and exciting for students. There are hallways that feel like tunnels and passageways. The teachers are very friendly and seem to know every student’s name. Because there are so many displays, students can always find some of their work on the wall. The ethnic make-up is 80% Hispanic, 8% African American, 10% Caucasian and 2% “Other”.

During Math class the students are situated around a small table with the teacher. Group sizes vary from 2-6. The teacher is very involved with students problem solving and invites the use of manipulatives, white boards and colors. She reminds students of strategies learned to help them move through their work. Word problems are done everyday. Right now, most students are working on some sort of addition and subtraction. Students are engaged with manipulatives and word problems that include their names and likes/dislikes. At times I jump in to show how I might work a problem or to be an example of a mistake. We really try to work on using mistakes as a learning experience.

My teacher feels that the basics of math are most important and will teach the students what they need to know in order to be successful in their classrooms. Even if their general education classroom is learning place value, she will not move to place value if the students have no number sense. I like that she is teaching them functional skills that they need to know in order to progress. I agree that it is important for these students to have a strong foundation before they move to more complex mathematical principles. I find that I rarely disagree with this teacher. We have similar teaching styles and very similar personalities. At times I do wish that she would push the students a little more than she does, but I don’t know them well enough to be sure of their potential. Because we are similar I find it to be easy to watch and learn from her teaching practices.

As a learning math teacher, I find my teacher incredible resourceful and I really want to pick her brain. Like me, she is a math person so I really look to her when explaining concepts the students don’t understand. I find that so difficult and watching her is really reassuring me that it can be done. Hopefully in the future I’ll know more about my math identity as a teacher, instead of just wandering around in my own ideas of math.

Reader's Response

1. How does taking a problem-solving approach to teaching math differ from first teaching children the skills they need to solve problems and then showing children how to use those skills to solve problems?

I think that by starting with the problem solving skills, the children can make their own inferences as they are learning the math skills.

2. How do you think your experiences, feelings, and beliefs about math will impact the kind of teacher of math that you will be or the kind of teacher of math that you want to be?

I think that because math has always been simple for me, I might find it difficult to translate my knowledge to my students. Its like Amy was saying earlier about how when she explains something to someone else it always makes sense in her head because she knows what she's talking about. Me too.

3. Not everyone believes in the constructivist-oriented approach to teaching mathematics. Some of their reasons include the following: There is not enough time to let kids discover everything. Basic facts and ideas are better taught through quality explanations. Students should not have to "reinvent the wheel." How would you respond to these arguments?

I think its important for students to be able to explore math and come up with their own ideas and then be given quality explanations.

4. We sometimes want to jump in and help struggling students by saying things like, "It's easy! Let me help you!" Is this good idea? What is a better way of helping a student who is having difficulty solving a problem?

Its not a terrible idea, but not a good one either. Instead, we should ask leading questions to help the student come up with answers on their own. This way they feel successful.

5. Reflecting on how tasks were defined in the Van de Walle chapters, how did the tasks presented in the Behrand article to Learning-Disabled students help in their mathematical development? Please give specific examples.

Giving the students manipulatives (Cal and the "bags") allowed them to see the problem and solution right in front of them. It was not just a mathematical concept, the numbers were represented by real life objects that they were able to group together to see how multiplication works. This tactile aid can help a student understand more abstract concepts.

Math Life Story

In 6th grade I got a perfect score on my math TAAS test, which was a high point for me. In the years previous I had gone to a private school and had no experience with the TAAS test and was very nervous to take it due to all the hype; but really, I had nothing to worry about.

My only low point in math was my spring semester of my freshman year, while I was taking geometry. I was out of school for two weeks due to a scheduled heart procedure. When I came back I immediately had to take a test over material I had never even seen in order to keep my eligibility to play softball and be in the UIL One-Act Play. I didn't do terrible, but was EXTREMELY embarrassed of the C I made. Luckily, I still made an A for the semester.

My M316L class at UT really made me look at math a different way. I rarely ask "why?" in math because the concepts seem so simple to me. This class forced me to look at problems from all angles and really take into consideration the struggles other students may face. So while I have always been a great math student, I'm worried about my ability to be a math teacher.

Two of my favorite math memories include nothing about school, but using math to solve a problem at home:

My dad is very handy and spends lots of time building or fixing things around his home. One summer while visiting, we were working on installing some cabinets in the garage when I noticed that my dad did not take into account the slope of the garage when measuring the height for the cabinets. Because of this the cabinets would slope up, eventually too high for my step-mother to reach. I remeasured and helped my dad fix his mistake.

Three years ago Hurricane Ike passed through New Caney, TX and toppled a tree onto my boyfriend's parent's garage. My boyfriend's father decided to do all of the repairs himself, however received a lot of help from me. I was able to use my geometry skills to draw out the plans and arithmetic to calculate the materials we would need. Everyone was very impressed with me, being a girl and all, and I really began to feel like a part of the family.

My biggest challenge in math has been teaching rounding to 5th grade students at Williams Elementary. It was my first lesson to teach and the students were still getting to know me. The students did not follow my instruction well and seemed to have no idea of what I was talking about. It was very frustrating for me and for them. My cooperating teacher helped us get through, but the material had to be covered again and again. I definitely felt discouraged with teaching.

I'm pretty sure I've always wanted to be a special education teacher. I've just been very interested in the diversity and the capabilities of this population. My students may not create a new math theorem, but being able to add two-digit numbers could be a huge milestone for them and should be regarded as so. I want my students to be able to be functional in math, like being able to balance a check book, calculate their spending, and understand their paychecks. I think this math skills can make them successful in living independently.