Promoting Equity Through Reasoning, Teaching Children Mathematics

This article, written by 2 university professors, focused on increasing mathematical reasoning, and justification in minority students. The authors implemented an informal, after-school mathematics program. During the program they focused on a variety of fraction based concepts and had the choice to work individually, in pairs, or in groups. The goal for students was to problem solve then convince their peers of their answer. Two specific instances were explained in which this was successfully accomplished. The article concludes with 5 suggestions for teachers to out these ideas into action: Give students options for grouping, differentiate instruction through giving adequate time and extension options, let students share their ideas with the class, choose tools and tasks carefully, and hold high standards for all students.

I think that this article makes very excellent points. I am a huge proponent of the belief of the belief that every student can learn and these teachers should not only hold high expectations for all students, but also make these expectations known to students. No wonder many minority students are not achieving at the same level of their minority counterparts, since they often are being held to lower standards in the classroom and are not being given adequate opportunities to share in "thoughtful mathematics opportunities". I particularly liked the 5 suggestions to teachers at the end of the article. So often, I read educational journal articles and am left wondering, "How exactly can I implement this in a way that will work for me?" This article spells it out for teachers.

Examining Math Manipulatives

1. How do you hold every student accountable for learning while using manipultives?

Teachers must put purposeful effort into holding each student accountable for their learning with manipulatives. There are a number of effective ways to do so. First, you could have students record some or all of the work that they do with the manipulatives onto a sheet of looseleaf paper or onto a teacher made "record sheet". This is different from a work sheet in that students are recording what they do with the manipulatives, not simply using the manipulatives to solve teacher made problems. Another way to hold students accountable is to have students journal after using manipulatives. Journal topics may vary and may include things such as "what did you learn at this center?", "What was your favorite and least favorite thing you did at this center?", or "did this manipulative help you with a math concept? Explain how it did or did not, and which concept". Finally, teachers could have students share what they have learned through using the manipulatives with the class. This is a great way to get students talking about math.

2. Why is the new emphasis on "Hands on, Minds on", instead of simply "Hands on".

All manipulatives are hands-on by nature. However, this does not mean that because students are using the manipulative, they will learn math. Manipulatives should be used as a tool for instruction, the key word being instruction. Students should be instructed or guided in some fashion in order that they are using the manipultive to help them meet the math objectives of the lesson. Only then is the activity both hands on and minds on. Handing first graders a set of base-ten blocks does not guarantee that they will use the manipulative in any way involving math. In fact, I would predict that given this situation (with no prior instruction using base 10 blocks) many first graders would build intricate buildings and towers. Building towers has educational value, though it is not mathematics and likely does not meet the learning standards of the lesson. In order to achieve a true hands on, minds on experience for students, teachers must couple good instructional strategies with the use of manipulatives. Similarly, as stated above teachers must hold all students accountable for their learning while using manipulatives.

3. How are the process standards used with the use of manipulatives?

Representation immediately sticks out to me when considering the use of the process standards in learning with math manipulatives. The nature of a manipulative is such that the manipulative itself is a way to represent math concepts. Showing students multiple representations, including 3 dimensional objects, is a key to a solid understanding for many students. Communication can very easily be incorporated into the use of manipulatives by having students work together in small groups to explore math concepts with manipulatives. Also, sharing ideas and findings with the class after an investigation activity using manipulatives is a way to improve students' communication of math concepts. Manipulatives can also incorporate the process standard of connections. This can be done especially with real world manipulatives, such as money. Problem solving is included in the use of manipultives when the teacher allows students to investigate how to use the manipulative to solve a problem, rather than providing direct step-by-step instructions on what to do and how to do it. Lastly, students should use manipulatives as tools to explain their reasoning of and to show proof that they have the correct answer.

Errors Reflection

I think that analyzing common math errors that students make is an important component of a high quality teacher education program. Too many math teachers either were never taught how to carefully analyze student work to find the origin of their student's errors, or they do not take the time to do so. Too many teachers use the very ineffective teaching methods of "Louder and Slower" and "More of the Same", without spending the time and energy to understand where the student's misconception is. Though collecting adequate samples of a each student's work and carefully searching for common errors is very time consuming, I believe it will save in time spent teaching the student to understand the concept and achieve the correct answer. Once a teacher has identified a mistake that a student continually makes, he or she can hone in on correcting that particular piece of the puzzle.

Studying common student errors in math methods class helped make me a better future math teacher in two ways. First, it helped make me aware of particular mistakes that students commonly make. Second and more importantly, it taught me the skill of analyzing students work in order to identify a common error. I now know the types of errors students often make. Similarly, I know ways to prevent and correct these errors, such as teaching estimation and using manipulatives.

Overall Use of Technology in the Math Classroom

Using a wide variety of technologies throughout this course has helped open my eyes to the many ways that technology can assist in teaching and learning math. That said, it has also made me more critical of the use of technology in the classroom. It is my personal belief that everything should be done in moderation, especially when trying something new. New technologies should be used in conjunction with old methods that have proven to be effective teaching tools. Students should be learning using math applets, calculators, SMART boards, and other new technologies, as well as more conventional learning tools such as paper and pencil, base ten blocks (physical blocks to pick up in your hands, not on the computer), unifix cubes, and pattern blocks. Using many different learning tools that are both high and low tech is one way in which a teacher can teach to a variety of learning styles and preferences. For many students, the use of technology is motivating, while for others it is frustrating.

More specifically, I think that the SMART board can be a positive teaching tool, though I can think of very few applications of the SMART board that can not be done with other technologies that are more commonly found in classrooms (i.e. projector, overhead, dry erase board, computer). One application that I do see the SMART board as uniquely useful for is the ability to project a graph, make notes on it, and then save the page. One of my main frustrations in using the SMART board is the inaccuracy of writing with the pen. I think it is important to model good handwriting to students, and I can not do this with the SMART board pen.

Making Techonology Work

In this article, 3 authors from Croatia explain why technology is necessary to incorporate into math education and how it can enhance the math curriculum. In summary, technology is beneficial because it enables students to work on more complex, often real-life, tasks, motivates students, and helps students visualize math concepts. However, technology is only useful in teaching when it's use has been thoughtfully evaluated and planned out. The authors described several specific programs and tools and their suggested uses in the classroom. The Geometer's Sketchpad is particularly liked by students and teachers because it allows students to focus more on overall concepts, rather than spending too much time on measuring and computations. Also, the article described a game in which students learn to differentiate when they should and should not use a calculator. Finally, the authors describe various real-life problems that students are able to analyze using various computer programs. Without these programs assistance, these problems would often be too advanced or too time consuming to implement.

I really enjoyed reading this article, as it has brought me one step closer to embracing more forms of technology for use in the math classroom. The article was extremely well written and concise, yet included excellent points and provided useful examples. I particularly liked the game in which students race to see if a computation can be solved faster by using a calculator or by doing mental math. I think that this is an excellent way to show students, rather than tell them, when they should choose to use a calculator as opposed to mental math. However, the idea of racing bothers me, especially the idea of having two students race against each other. What if one student is simply faster than another? This would skew results not to mention likely be the cause of hurt feelings. To alleviate this I may consider having students time themselves with a stop watch doing the problem first mentally, then using the calculator. This would not eliminate being timed, but at least it would minimize competition.

Soucie T., Radovic N. and Svedrec R. (2010). Making Technology Work. Mathematics Teaching in the Middle School, 15 (8), 467-471.

Smorgasbord of Assessment Options

In this article a fifth grade teacher from Virginia explains the importance of student centered assessments which match the "target achievement" or goal. She uses a Virginia Standard of Geometry to focus her example assessments on. Assessment should produce useful information for both the students and the teachers and should influence future teaching. Student centered assessment means that the assessment centers around the concepts which have been learned, instead of what has been taught. Similarly, learning is a process which is scaffolded, and assessment should reflect this by assessing what has already been learned and pointing towards the "next step". Lastly, teachers must carefully choose assessment that provide the most valuable information in the least amount of time invested in assessing.

I most definitely agree with all of the main points of the article. I particularly identified with the point the author made on page 467 that thought the process and products may vary from classroom to classroom or student to student, the content should still remain the same in accordance with the standards. Likewise on page 468 the author outlines how student centered assessment is extremely beneficial in meeting the needs of all students, including those with special needs, those who are bilingual, and those who are gifted. As a special education major, I can see how creating student centered assessment is vital to gaining information which is useful to both the teacher and the student.

Bacon, K. A. (2010). Smorgasbord of Assessment Options. Teaching Children Mathematics.
458-469.

Portfolio Assessment

This article details how two middle school teachers began using portfolios as a way to enhance communication between parents, students, and teachers. The two teachers were looking for a way to emphasize student learning and growth, rather than letter grades. To do so, they designed two lessons on triangles using the "backward" design approach, and approach in which the teacher begins the lesson planning process by determining what they would like their students to gain from the lesson and which standards they will be meeting. The teacher then goes on to determine how they will assess the student and finally creates a procedure in which to teach students the material. After the two lessons on triangles were completed, students created portfolios, showcasing their work. The portfolios included a letter to their parents explaining their learning, examples of their work and assessments with commentary from students, a letter from the teacher, and a student/parent reflection sheet. Overall, both teachers felt that the portfolios and "backward" design was very beneficial. The portfolios increased parent understanding of what their children were learning as well as increased communication between parents and students. Another benefit is that students feel pride and ownership of their work when using portfolios. Also, the authors emphasized the fact that portfolios demonstrate whether or not students have met state and national standards more completely than some other methods assessment.

Britton, K. L. and Johannes, J. L. (2003). Portfolios and a Backward Approach to Assessment. Mathematics Teaching in the Middle School 9(2), 70-76.