The authors of this article made a blunt statement that there is something missing in today’s math classes; reasoning and proof. In the past this has only been introduced and studied in high school geometry, but it has been noticed that it should be incorporated in math starting in kindergarten. Of course the rationale, support and reason will vary between kindergarten, fifth grade and twelth grade, asking students to understand why they get certain answers in math or set up a graph in that certain way is beneficial. By posing these open ended questions, students must think about the problem at hand, the answer they came up with and why it all matters. This will not engage every student who has a mental block towards math, but it will only help by relating math to real-life and putting it in different terms. This is done in a verbal or written reflection, discussion, question-and-answer, instead of this number plus this number equals this number. As teachers we try to tie subjects together, cross-curriculum. English teachers sometimes struggle with incorporating math, and visa versa. There is a fabulous way now—students can write about, create and distribute a survey, or interview each other to incorporate both language arts and math. Talking more about proving now, the article set forth examples of math problems that show contrasting answers and limitations that students can use to proof why an answer is the way it is. Two terms were defined that are important with this topic. Examples-based justification “means that students justify their generalization by stating that it worked for all the cases they tested” and structurally-based justification “which guarantees a proof”. After students worked through an activity dealing with making boxes out of toothpicks in different configurations, they contrasted the two justifications when debriefing the activity and talking about what worked and why.
Reasoning and proof is so important and needs to be re-implented for two reasons. More specifically for math, if students learn how to analyze and break down the math problems they are working with now, it will come more naturally to them later, and more generally, it is important for students to be able to research, question, test and support ideas, opinions and topics in school and the real world.

Knuth, E. J., Choppin, J. M. and Bieda, K. N. (2009). Proof: Examples and beyond. Mathematics
teaching in the middle school 15(4), 206-211.

Process Standards, Reasoning and Proof

Being able to understand how and why an answer was formed can help a child understand math in a less frustrating way, but reasoning and proof is also more specific to math problems themselves. This skill, of asking how and why and then finding the proof through patterns, structure and regularities in their real-life surroundings, is beneficial in the classroom and for life. It is often taught in the math classroom because patterns, structures and consistency is a normal occurrence within number lines, multiplication facts and elsewhere. Although in math class, these proofs are logical, the skill of reasoning and proving can lead to more creative answers in other paths of life. It is a skill taught over time through any lesson that it can be tied into. Children will not being able to successfully execute this skill after just one lesson, no matter how in depth. Questions such as “what do you think will happen next” and “Is this consistent no matter what” are questions that trigger thoughts in children. Their thought process or investigation is an informed one, or a mathematical conjecture. When open ended questions are posed to students, they may not even realize they are using “reasoning and proof” when answering, sometimes it will come more naturally then other times when a real “investigation” is taking place. When forming their answers, they can disagree with the information stated, agree with it, relate it to real-life, other math, or other content areas. They also may still question things that are not as measurable in the classroom.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000.

Video Blog

Purpose:The purpose of this set of video was to show the process of a lesson in a classroom with children that resemble those pre-service teachers will be working with. The video links were categorized into the different sections of a lesson including large group time, transitions, small group work and the products. Through these videos, viewers saw the interaction between students and teacher, the benefits of certain strategies and the weaknesses of others. It was not staged so human error on the child and teacher’s side occurred which only created a better learning moment for us as pre-service teachers.

Describe how the teacher’s questioning and the manner in which student responses are handled, contribute or do not contribute to a positive classroom learning environment.

I thought she did a great job of setting the students up for success rather than failure through her questions and discussion. She relayed a sense of support to the students before posing the question so that they felt confident volunteering and comfortable answering. She gave enough time between question and picking for students to formulate answers and rephrased questions if students seemed confused. There was a large amount of respect from the students, while the teacher was talking and they stayed quiet until she picked on someone to answer the questions. No one was ever made fun of for their answer, but rather redirected or encouraged by classmates or the teacher. The teacher, with the help of her students, created a very positive environment for group discussion which is hard, but so important in any classroom.

How do you determine if group work is appropriate and effective?

Before allowing the students to go off into small groups she asked volunteers to explain and repeat the instructions she had given. This was an opportunity for students to ask questions for clarification as well. By doing this, it created a structured transition into small group work and also give the students to start and do their work without having to stop for clarification. I think an obvious sign of group work being unsuccessful is if students are working alone, and not engaging in the group or if the entire group is off task. These things will happen but the work given at group time should be different enough from normal individual work that they stayed somewhat engaged. If repeating the instructions prior to engagement and students stay on task while engaged, this would show appropriateness of material and activity.

Propose a strategy for dealing with one or more of the student mistakes or misconceptions that arose during the lesson.

I think it is important to make sure a child feels comfortable being redirected and is comfortable with constructive criticism. Saying that I think some children would feel more comfortable being one-on-one with a teacher while getting assistance so that they could ask questions, where as some children might feel more comfortable in a group so that they do not feel singled out. There are most probably going to be more than one student who needs extra clarification or help but if a teacher knows their students they can assess the situation and see how to best group the students and help them. Also a student might just need to talk with another student to hear the information on a different level and that can be beneficial to both students.

Overall Reflection: During the videos on reflection I realized something new. When I think of reflecting, self-reflection, on the lesson I did with the class I thought more about how the students reacted, what went well, what could be improved, but I realized it is also important to reflect and make sure the key themes and curriculum ideas were hit. Debriefing with the students after a lesson is good also so that I know what they took away from it and how it can improve in the future. Overall I thought this set of videos was a great breakdown of the different aspects that goes into a lesson and the atmosphere. I liked the breakdown and reflection questions on misconceptions and responding because those are areas in which I could use help from good models.

Preschoolers' Number Sense; Article on Equity

Equity means to be equal, but with average class sizes being well over twenty, finding a group of students that come into the year with equal strengths, and capabilities would be hard. In this particular article the subjects were preschool aged. This means attention spans and abilities weren't comparable to a student in a higher grade. Traditional assessments would not show accurate data among four and five year olds because they wouldn't know how to take a test, or fill in a scantron. Modifying this traditional assessment to fit these needs resulted in what this article was written on. Students sat with an observer and handed the following materials; a bucket of bears and cards with dots or numbers. Goal of the activity; where there were many but for the child, it was their goal to match the number of bears with the number or gathering of dots represented on their cards. The observer guided students to communicated the number of bears they picked up, how many they had in total and how many they had compared to the second player. Students were observed on their abilities to count, compare, identify, use the strategy of quantification, and one-to-one correspondence, counting strategies and cardinality. As said earlier some students will be far ahead, some average and some below but by using this assessment consistently, children are can start out with the correct adaptions, accommodations and support to make it to the finish line. Not to mention this is an activity that disguises math in a fun game, re-constructing the connotation towards math.

Moomaw, S., Carr, V., Boat,M. and Barnett, D. (2010). Preschoolers' number sense. Teaching children mathematics 16 (6), 332-340

The Equity Principle in Mathematics

The term "equal" is a common term to students in a math classroom. Equity, "high expectations and strong support for all students" should and is becoming a common term among teachers. Although this concept should be applied to curriculum throughout, this speaks specifically towards mathematics. As I looked at the three points that stood out to me at the end of my readings, I saw a similarity among them. The three dealt with the treatment of children who are seen as "different". Having a soft spot for students/children who come in with uncontrollable setbacks, I constantly brainstorm how to support and help them. I didn't expect to find some answers in an article about Principle of Math. The article stated that students not only should, but must have the opportunity to study. This does not require all children to study the same way though, and that is important to note. The second point I highlighted related to the prior. Teachers need to provide an environment, materials and resources that can be used to accommodate students coming from different backgrounds, such as low-income families, non-English speakers, or students with disabilities. An interesting point was written about the stereotype towards those who have, for lack of better words, "math-brains"; those who people assume will succeed in math verses those whom it seems impossible. The authors compared this to our push for the importance of high literacy and verbal rates in the English Language. We set a high standard for that, why not for math? Are we setting our students, children up for failure because we don't give all students equal expectations and goals? Just as every child with a set-back is different, students even in a mainstream classroom are different. They learn, listen, think, discover, brainstorm and problem solve differently, and it is up to the teacher to learn and try to understand the strengths and weaknesses of their students to better adapt a subject that has such a negative pre-notion.