Blog # 19: Curriculum Plan Reflection

When I first read over the requirements for the Curriculum Plan, I was extremely nervous. The task that lied ahead for my group and myself seemed daunting and overwhelming. However, once my group and I began working on the project we realized that it was not as scary as we originally thought. After the first work day in class, we were able to get a significant portion of the assignment completed and our progress continued from there. After the final work day in class, our group decided to split up the workload in order to finish the assignment and begin filming the video. Each group member completed their portion of the workload by the deadline decided upon by the group and we were able to film on the final meeting day.

I was very satisfied with the final result of our curriculum plan project. We were able to create a complete and thorough curriculum plan for three years. We determined the most logical method of separation was by dividing our curriculum plan based upon how the common core standards were arranged. The standards of the same categories were kept together and we combined each semester based upon the two larger standard categories that best aligned with one another. In the end our curriculum plan was logically organized and would be successful in a real classroom.

Blog #18: Technology Blog

The technology that was the most useful from this class was the SMARTboard presentations. The SMARTboard is one piece of technology that has been neglected in our Bradley education, in my opinion. We have received some instruction here and there when questions have come up or a classmate has used the SMARTboard in a presentation but nothing much beyond that. It was nice to be able to learn more about the special features of the SMARTboard and to practice doing them in class. Most of the teachers that I have met in the field do not use the SMARTboard because they have not been taught how to use it. In one experience, I asked to use the SMARTboard to teach a lesson and the teachers did not know how to turn it on! I think we all need several opportunities or seminars to be taught and to practice with the SMARTboard, so that all teachers can make use of them in the classroom.

The videos that we watched online of teachers presenting mathematics lessons were useful to generate ideas on how one might possibly teach particular mathematics lessons. However, the teachers who were in the videos did not use technology at all in their lessons! They used the traditional paper and pencil types of methods instead. Therefore, I was highly disappointed after watching them. If I were to teach those lessons as did the teachers in the videos, I would not want to be a student in my classroom. They not only lacked the use of technology but student engagement and participation as well. The videos on that website need to be updated with teachers teaching fun, engaging, and authentic lessons.

Finally, we also researched math apps and applets in this class. I researched a map applet, a graph applet, and a counting app for middle school and grade school students. Out of these applications, I think the map applet would be the most useful because it can provide students with a great deal of statistical information in which they can use to perform a variety of calculations. The graph applet can be useful for helping early elementary students learn how to create basic graphs and the counting game can be a great reinforcement tool for young primary students learning to count as well.

Blog #17: Assessment in Math Reflection

Assessment in Math Reflection

                Teachers should take great care when creating assessments for their students. They should expect to make numerous revisions to their assessments. After each time a teacher gives students an assessment, the teacher should use student work to determine if there are any problems or bias within the assessment and revise it if necessary. Assessments should be as objective as possible. Additionally, teachers should make sure the directions to the assessments are written very clearly and are as specific as possible. The key to an effective assessment is continuous reflection on the part of the teacher.

                There is also the issue of asking students to evaluate the work of their peers. Can students be as objective as possible when evaluating the work of their peers, if students are not experts in the concepts they are assessing? The general consensus of teachers is that students cannot be objective and therefore should not be asked to evaluate the work of their peers. If the teacher believes students will gain insight by assessing the work of others, teachers should only ask students to critique the work of their classmates and the teacher should be responsible for providing the numerical score based upon student feedback.

Blog #16: Error Analysis Reflection

Error Analysis

            Student errors can be useful tools for teachers to determine how to tailor their curriculum and instruction to meet the needs of their students. Student errors reveal where previous instruction failed and where further instruction is needed for students to grasp important mathematical concepts. It is often the case that students understand the general premise of how to calculate particular problems but continuously make the same slight errors that cause them to get the wrong answers. For example, a student can multiply numbers correctly but position the products in the wrong place values and therefore get the wrong answers. Another example occurred when a student divides correctly but is dividing the wrong numbers and calculates the answer incorrectly. Teachers need to practice locating errors from student example problems and determining the necessary skills that need to be retaught in order to effectively aid students in learning mathematical skills.

Blog Post #15: Problem Project Reflection

Problem Project Reflection

 

The most difficult aspect of the problem project was integrating three separate content domains within a single project. As my group was brainstorming ideas, we each could come up with multiple projects to assess our particular content domains but found it difficult to create a project that incorporated all three domains. Once we decided upon a project idea, it was significantly easier to complete the remainder of the requirements for the assignment. My group members and I finished our assignment relatively quickly and it would definitely be a project that I would use in my future classroom. As a student, I loved assignments that enabled me to be creative and to create or build final products at their conclusion. This project will gain student attention and keep them engaged throughout its entirety. 

Blog Post #14: Content Standards

Supplemental Material Needed:

                The selected textbooks I reviewed for grades 6-8 are fairly standard textbooks. The information within the textbooks were generally more concrete, knowledge based questions with a so-called “application” section within each chapter. What these textbooks and those similar to them really need are more high order, open-ended types of questions that apply to students real lives. Many of the application sections discussed ideas that were not relevant to the lives of middle school students and typically included a short paragraph describing how a professional uses math in his or her everyday life. Additionally, the majority of the standards that addressed making connections between concepts were not covered in these textbooks. The textbooks included the concepts separately but did not attempt to build connections for students.

                The textbooks for the other grade levels assessed by my classmates, were similar to those I assessed for the middle grades. In general, the textbooks only included the skills of the content addressed in the standards but lacked application and review. In many circumstances, the standards were not addressed at all in the textbooks. This is further evidence that the textbooks that are currently in schools are outdated and do not correspond to the new Common Core standards.

 

Concept Development:

                There was very little differentiation within the textbooks themselves that showed the differences within the grade levels. All three textbooks had similar topics covered throughout the entire book. It can be assumed that each of the concepts covered gradually increase in difficulty as students’ progress from grade level to grade level. The standards, however, do not include a great deal of overlap as do the textbooks. Students will need to understand the 6^th grade content listed in the standards before moving onto the 7^th and 8^th grade standards of course, but there is not a whole lot of repetition of common geometry topics from one grade level to the next. The 6^th grade standards focus on area, surface area, and volume; the 7^th grade standards focus on three dimensional objects; and the 8^th grade standards focus on rotations, reflections, and translations. These concepts build upon one another rather than repeat one another.

Blog Post #13: Assessment Articles

Assessing Problem Solving Thought

As the emphasis on problem solving increases, teachers need to prepare themselves to assess student problem solving understanding. The easiest way to assess student understanding of story problems is through a well-developed rubric. To create the rubric, teacher must first locate or write a high order thinking problem and work through the problem to reach a solution. Next, Teachers need to decide what are the various thinking skills that students need to display in their own solutions. The authors of the article decided to assess students on the following criteria: 1) understand or formulate the question in a problem, 2) select or find the data to solve the problem, 3) formulate sub-problems and select appropriate solution strategies to pursue and 4) correctly implement the solution strategy or strategies and solve sub-problems. These criteria may change based upon the problem(s) teachers are giving to their students.

After determining the grading criteria, the teacher should determine how many levels each criteria should be divided into. The authors suggest three levels for each criterion. Next, point values are assigned to each level within the criterion. The lowest level should be zero for students who do not attempt or do not show evidence of the criteria. Additionally, teachers should consider placing greater weight on harder level skills.

When assessing student work using the created rubric, the teacher should take great care not to assume they know what students were thinking as they were solving the problem. However, there will be instances when score must be given based upon inferences. It should be avoided whenever possible. The teacher should work through the problem as the student and assess the student on each criteria. The teacher may notice that his/her rubric needs adjusting for the next time he/she uses it while assessing the whole group of students. It is important for teachers to always give honest and complete feedback to students in order to help them improve.

This article provided valuable insight into how mathematics teachers (and teachers of other subjects too) should think about assessment. If an assessment tool is used only to have written documentation of whether a student did or did not do something correctly, it is not really assessing much. The well thought out rubrics that the article describes enables teachers to truly get at the root of student understanding and knowledge.  Once this types of rubrics are created, the teacher can use them over and over again and tweak it when necessary. Therefore, it may take some time initially to create these kinds of rubrics but it will be well worth it in the end.

Assessment Design: Helping Pre-service Teachers Focus on Student Thinking

This article describes an activity for pre-service teachers that was designed for such teachers to think more deeply and critically about assessments. This critical thinking included not only the content of the tests but also the type of assessments and the time the assessments are given to students. The project began by introducing pre-service teachers to literature about effective assessments and assessing student understanding. Next, the pre-service teachers selected an NCTM standard and created an assessment with the intention that it would align to such standards. Pre-service teachers discussed their rough draft assessments with their peers and made any necessary changes from the feedback they received. Finally, the pre-service teachers implemented the assessments to a group of students and reflected on the outcomes. Many of the pre-service teachers were amazed about the small details they did not consider when originally creating their assessments. The main point received was that just because students compute the right answer, it does not necessarily mean that the student understood the material or could use it in a real-life situation.

I find assessment to be one of the hardest parts of lesson planning. It is so easy to throw several multiple choice problems on a sheet of paper and give them to your students rather than taking the time and the effort to think about the best method and time to assess them. It would be fantastic if all teachers could share their assessments with their colleagues to get feedback but unfortunately that is not really possible in a real-life situation. When I create my next assessment, I hope to use the ideas of this article especially the section that discussed when to deliver assessment. Assessment should not just be at the end of a lesson or a unit but throughout the lesson in order to tailor instruction to meet the needs and skill levels of all the students in the classroom. Additionally, I hope to remember to consider the best method of assessing my students and if the assessment I created actually matches the original goal and standard.

Assessing Students' Mathematical Problem Posing

As the emphasis on problem solving is increasing in today's  schools, problem posing is receiving more and more attention. Problem posing can be integrated into assessments or it can be assessed as its own entity. When problem posing is integrated into assessment, teachers could potentially provide students with mathematical statements and ask them to create questions based upon those statements. The articles example of this type of problem included: Pose problems that all can be solved using the same division statement 540 ÷ 40 = 13.5 ?. When problem posing is assessed, teachers can potentially provide their students with a set of information and ask them to create specific questions using that information. Their example included this problem: Ann has 34 marbles,  Billy has 27 marbles, and Chris has 23 marbles. Write and solve as many problems as you can using this information. Assessing these types of problems can be difficult for teachers and it largely depends on the instructional goals of the lesson. The authors of the article, however, suggest the following three criteria for assessment: quantity, originality, and complexity. Quantity refers to the number of questions students can create. Originality refers to the unique quality of the questions students posed. Complexity refers to the mathematical concepts and skills within the questions .

I think it is a great idea for teachers to ask students to participate in the question generating process. This allows the teacher to know what types of ideas or skills students are comprehending and those that students are struggling with. Additionally, it allows students to be creative. However, I do not think that students should be assessed based upon the quantity of questions they can come up with. This type of assessment seems a bit superficial and does not take into consideration any kind of mathematical ability. The criteria of originality and complexity seemed more appropriate for these types of activities. 

Blog Post #12: Journal Summaries 2

Hede, J.T., & Bostic, J.D. (2014). Connecting the threads of area and perimeter. Teaching Children Mathematics, 20 (7), 418 - 425.

Summary

In order to get 6th grade students to see the value and importance of learning the concepts of area and perimeter, the authors of this article decided to give students a more meaningful experience by asking them to design and create a quilt square. The lesson began with the teachers displaying various quilt patterns to the students and discussing the mathematical qualities that each quilt possessed. Students noticed that the majority of the quilts had geometric patterns that repeated one another. In phase 2 of the lesson, students  were provided with grid paper and instructed to design their own quilt square that contained only regular polygons and had a total area of 64 inches squared. Then students calculated the area and perimeter of each of their shapes to ensure that the sum was as close as possible to 64 inches. Finally, students cut out their shapes from wallpaper and created their final products. The quilt squares of all the students were displayed in the hallway for the rest of the school to see.

Reflection

I thought this lesson was a creative way to teach area and perimeter. Generally, real-world application of these concepts involve arranging furniture in a room, but never have I heard of students creating their own quilt squares.  I am not convinced, however, that the students found this to be a meaningful, real-world application project. How many students actually plan on sewing their own quilts with the knowledge gained from this lesson? Students might be exposed to the difficulty of quilting if a grandparent had it as a hobby but otherwise this lesson ended in a fun art project. This is obviously a step up from practice worksheets but not necessarily a project I want to try out in my own classroom one day.

Patterson, L. G. & Patterson, K.L. (2014). Problem solve with presidential data. Mathematics Teaching in the Middle School, 19 (7), 406-413.

Summary

This lesson provided students with a integrated lesson of Mathematics and Social Studies. The teachers began the lesson by getting students excited about the current and past presidents by introducing them to presidential facts and trivia. They also showed students a presidential rap on YouTube. Then, the teachers integrated the mathematical concepts through a bell ringer at the beginning of the next class, asking students to define mean, median, and mode. Students were placed into small cooperative groups and each member of the group was given a group role (i.e. scribe, spokesperson, runner, etc.). In these cooperative groups, the students calculated the age of each of the presidents when they were inaugurated. These ages were graphed using a stem and leaf plot and then students calculated the mean, median, and mode of this data. The following class period, students created different representations for the data and discussed patterns and/or trends they noticed in the data.  

Reflection

Being a History buff myself, I always appreciate incorporating social studies into the curriculum of other subjects. However, it is unclear to me as to the purpose of this history lesson. If the objective of the lesson was to get students to discuss the idea of age in the effectiveness of the presidency, this lesson simply asked them to determine what is the average age of the American presidents. There was little discussion over how age is important to the presidency or why the majority of our presidents have been in their 50s.  The lesson seemed more mathematically based than social studies based even though it was taught in a social studies classroom. I think it  could be useful if the teachers continued on to discuss the factors that contribute to an effective president that includes a discussion on age. Then both the mathematics and the social studies objectives will be met.

Blog #11: Video Analysis Part II

Video #2: Number Operations (4^th Grade)

1.      Planning

            A video that corresponded to this lesson in which the involved teachers explained their planning process was not included for this lesson. Therefore, the written reflection of the planning process was the only insight the viewer could receive for the planning of this lesson. The goal of the lesson was to teach how to work through and solve story problems involving division through the Singapore Bar Model. The Singapore Bar Model provides a visual representation to aide students in working through the problem, especially those who are struggling. The teachers planning this lesson decided to teach similar lessons in two separate classrooms simultaneously. The other lesson, not shown on the video, was taught with the traditional “direct style” lesson format and the two were going to be compared after each was completed.  The teachers wanted to see if presenting students with an alternative method of solving a story problem (i.e. the Singapore Bar Method) would increase their understanding.

2.      Lesson

            The lesson was introduced through the idea that “a picture is worth a thousand words”. The teacher wanted her students to represent story problems using pictures so that someone who did not know the problem could view the picture and understand what the story problem was about and how the student arrived at his or her answer. The teacher also introduced students to the goals of the lesson and the protocols for appropriate behavior in the classroom.

 

            The teacher asked her students to form a bit of background knowledge before solving story problems by describing to her what they knew about multiplication and division. Students were able to tell her that multiplication was like addition as well as the relationship between multiplication and division but were highly confused when the teacher began discussing the idea of equal groups. The teacher attempted to force the idea of equal groups by providing more examples but students continued to solve the problems using their own strategies instead. This posed a problem for the teacher once students were introduced to the Singapore Bar Model. Students were asked to solve a problem using this method. Unfortunately, only a few of the students in the class ended up providing the correct answer to the problem with this strategy.

 

            Students were given the following information: Maria saved $24. She saved three times as much as Wayne. They were asked to think about how they might determine how much money Wayne and Maria have using this information. The majority of the students inaccurately calculated that Wayne had $72 after multiplying 24 times 3. Only a handful of students divided 24 by 3 and determined Wayne had $8. The teacher wanted the students to use the equal group strategy (Singapore Bar Model) to aide them in solving the answer to the problem. They were to draw a math picture that told the complete story of the information that they were given. The students who used the grouping strategy or who calculated the correct answer often drew pictures that were incomplete or that did not tell the entire story. The majority of the students used their own strategies to solve the problem and still arrived at the wrong answer.

 

            Finally, the teacher attempted to get students to use and understand the bar model strategy by asking students to copy and solve the problem using a fictional “Charlie’s way”. The students were provided with the basic structure of the problem and were asked to fill in the empty boxes with the appropriate answers. Again, the majority of the students were confused and calculated Wayne’s total money as $72. Only a handful of students provided the appropriate answer and filled in the boxes accurately.

 

3.      Faculty Debrief

            After the completion of the lesson, the teachers discussed several issues that caused the lesson to not be entirely effective for the students. First, the idea of equal groups did not come readily to the students. The teacher had to press the issue and chose to continue to roll with the idea throughout the lesson even without the students fully comprehending what equal groups actually meant in relation to the story problem. Next, students were able to describe to the teacher that multiplication was like addition but could not verbally describe why division was related to subtraction. Thirdly, students’ pictures to represent the provided story problem often did not match the story problem or were incomplete. Students generally forgot to label the individual parts of their pictures so that the reader could fully understand the process the student went through to solve the problem. Finally, there was a problem with students changing their answers due to peer pressure. Many of the students originally wrote down the correct answers but after discussing the problem with their peers, changed their answers.  

 

4.      Overall Reactions

            Many of the ideas and strategies discussed in ETE 339 at Bradley were included in this lesson. The teacher did not simply present the Singapore Bar Method to the students and then ask them to work through several story problems with the method. Instead, the teacher sought to incorporate all student ideas and methods into the lesson. She emphasized that the Singapore Bar Method was only one way to solve the problem but other methods were just as valid. Additionally, there were no worksheets for this lesson. The teacher provided much of the practice problems and examples on the white board rather than a worksheet. Students were given a sheet of paper with the story problem written out on the top just for easier visibility. Overall, the lesson was presented well by the teacher even if the students did not fully reach the objectives of the lesson by its conclusion. Additional lessons on this subject will be necessary for this group of students to fully grasp the Singapore Bar Method and to solve story problems involving division.  

Blog #10: NAEP Student Analysis

Blog # 10: NAEP Student Work Reflection

                One thing that was evident after every group had completed their presentations is that there are multiple ways to assess student work using the provided rubrics. In many circumstances, group members disagreed on how to assess a student and there was almost always disagreement among the members of the entire class. The NAEP problem that seemed to be the easiest out of the four to assess was the Graphs of Pockets problem. In this problem, the reviewer did not have to analyze student work but only their written explanation of how the student arrived at the answer. Often the work shown and the written explanation of the problem belonged in separate categories on the rubric for the other problems and it was difficult to determine where the student fell. The rubric for the Graphs of Pockets problem is clearly written and much more simplistic than the other three rubrics and therefore the easiest to assess.

                The problem with the second lowest difficulty to assess was the Marcy’s Dots problem. In this problem, the reviewer is looking for whether or not the student found the pattern and accurately calculated each step using that pattern to arrive at the correct answer. The student either wrote out the details of the pattern or showed the pattern through their work. The rubric, however, was slightly vague in its explanation of each category. For example, the explanation under the partial category simply states, “a partial correct explanation”. This statement can be interpreted a million different ways which might be helpful to the reviewer if a particular problem does not meet the criteria of the other categories or it might be difficult for the reviewer to determine exactly what that means.

                The third problem in terms of difficulty in assessment was my group’s problem called Number Tiles. We struggled when trying to assess this problem because the rubric was very complex, having several different components for each category. This included separate identifiers about the rubric listed with letters. This often caused confusion when trying to assess a problem due to multiple answers being included in a single category.

                Finally, I felt the Radio Stations category was the hardest of the four problems to assess because a diagram was involved in the answer. A visual that makes sense to one person might not necessarily make sense to another person. Additionally, this problem seemed to be the hardest problem out of the four to solve for it required multiple steps and higher order thinking. The rubric for this problem includes an example diagram and is very detailed for what aspects of the diagram needed to be labeled for each category.

Blog Post #9: Math Applet Review

1.       State Data Map (6-8 grade)

·         http://illuminations.nctm.org/Activity.aspx?id=3580

·         Data Analysis and Probability

·         In this applet, students are provided with a visual representation of the 50 states of our country. They can manipulate the map so that it shows various comparisons of the states including land area, population, representatives in Congress, etc. Students can also change the colors of the states to better see the contrasts between them or they can enter their own data into the map. The map will calculate the mean and the median of the data shown as well.

·         This applet would be a useful tool in the classroom for both the mathematics classroom and the social studies classroom. Math teachers can use this tool when teaching about data analysis and probability. Students can be asked to compare states of their choice given the provided data or answer teacher generated questions using the data. Additionally, this tool might be helpful in reviewing the concepts of median and mode for students.

2.       Bar Grapher (3-5 grade)

·         http://illuminations.nctm.org/Activity.aspx?id=4091

·         Data analysis and probability

·         This applet enables students to create their own bar graphs. Students can select from pre-made data sets or input their own data into the applet. The axes of the graph can be manipulated using a scrollbar on the applet as can the scale of the graph. Additionally, students have the option of changing the colors of the graph to make it more visually appealing.

·         This tool might be useful in an elementary classroom with students who are struggling with or just learning how to create bar graphs. They can input their data into the applet to check their own work or to gain a better understanding of why the graph is set up as it is. Students can include their graphs in classroom presentations to share with the rest of the class. These computer generated graphs are much more simpler to create than they would be to create in a program such as Microsoft Excel and it would provide the rest of the class with a graph that is much more easier to read than a graph created by hand. Additionally, the computer generated graphs are much easier to manipulate or change than would a graph created by hand in case of any mistakes made.

3.       Okta’s Rescue (K-2 grade)

·         http://illuminations.nctm.org/Activity.aspx?id=3528

·         Number and Operations

·         This app is for early elementary students to practice their counting skills. The object of the game is for students to count the correct number of octopi when they are provided with a specific whole number. It consists of three levels. The first level asks students to count up to 6 octopi; the second level goes through 12, and the third up to 18. There is also the option for teachers to customize the numbers as well. Once the time period has expired, students are asked to count how many octopi they saved using a number line.

·         This app is an engaging and entertaining way to help young students practice their counting skills. It can provide differentiation for an entire class of students with the three levels and the customized level options. Therefore, students at higher levels will not be bored easily and students at lower levels can work their way through the levels at their own pace. Students will enjoy working against the clock as well as the colorful graphics and sounds that the app provides to them.

Blog Post #8: Teach Rich Task Reflection

Personal Learning for Teach Rich Task

1.      Difficulty in locating a Rich Task Activity

      As I was researching online, attempting to locate an appropriate activity that challenged students to use logic and reasoning to solve a real-world problem, I found it extremely difficult. The majority of the lesson plans that I was readily able to find were the traditional textbook, cookie cutter type of lesson plans that did little for students other than asking them to memorize a procedure and replicate that procedure multiple times. These lessons did not seek for students to reach high level thinking or even to engage students in an interesting lesson concept. I finally found a specific story problem that met the above criteria and decided to build a lesson plan around that particular problem rather than spending hours and hours looking for a pre-made lesson plan that did not measure up.

2.      Different ideas of what is a Rich Task Activity

      My group members and I were not always on the same page as to what constituted a rich task activity and what did not. A rich task activity in my perspective is a lesson that is hands-on, engaging, and requires students to think outside the box in order to come to solution. Students should be challenged by exploration and discovery on their own rather than the teacher simply providing the students with a formula or a step-by-step procedure. The rich task activity that my group chose had the potential to meet these criteria but my group members were fairly satisfied with the lesson plan as-is. I believed that this lesson plan could have been at least slightly improved by making it more interactive and student generated.

3.      Challenges for Gifted Students

      Even after reading the helpful hints on Sakai about ways to challenge gifted students in the same lesson, my group members and I found it difficult to decide on how to challenge gifted students for our particular lesson. The original ideas for this section of the lesson plan included asking the gifted students to help out the other students in the classroom who were struggling or to provide the gifted students more challenging questions at the end of the lesson. Both of these ideas were directed stated in the helpful hint worksheet as two things not to do for gifted students so it was back to the drawing board.

4.      Teaching Peers

      Though I have been asked to teach lessons in front of my peers every semester at Bradley, it is still a different dynamic than teaching children of the appropriate grade level as the lesson. I find teaching my peers to be an even more daunting task than teaching children the same lesson because mistakes are more easily noticeable and it is difficult to treat my peers as though they were years younger.

5.      Other Group Presentations

      The group presentations presented by my other classmates were much more informal than I was expecting when preparing for my own presentation. The majority of the lessons presented were described to the class in the teacher perspective rather than presented as if to a group of students. One group, due to weather conditions, was not even able to present part of their lesson so they ended up describing what they would have done with a group of students instead. The lesson that my group and I presented seemed to be the closest presentation of an actual lesson to a group of students.

Blog Post #7: Understanding Math Concepts & ThinkThru Lesson High Level Task

Understanding Math Concepts

            As discussed in class, it is difficult for teachers to look inside students’ minds to know if the students understand a mathematical concept. Even if students complete a traditional paper and pencil type of assessment, it does not guarantee that the student truly understood the concept because he or she could have just memorized a formula or made logical guesses on the assignment. In order to genuinely know students understand a concept, the teacher has to lead them through a series of progressive “moves” . The author of A Model for Understanding Understanding in Mathematics describes understanding as a continuum. It is not possible to create a one line definition of the word but it is possible to list certain characteristics or evidences of understanding for teachers to use as guidelines for instruction. Students who have a clear understanding of a concept can do tasks such as restating the concept in their own words, giving examples of the concept, recognizing the concept in multiple situations, identifying connections between the concept and other concepts or ideas, or stating what is opposite or contradictory to the concept. This definition is still a working definition of understanding for understanding can be shown in a myriad of ways.

            Teachers have to scaffold instruction in order for students to reach higher levels of understanding. Students may be able to give examples of the concept in the early learning stage but most likely would not be able to identify things that are true about the examples of the concept. Teachers must begin by asking students to display their understanding of concepts through less complex explanation and calculation and then slowly build toward the more complicated ideas. Students have to understand the ideas in the first level of understanding before moving onto the second level. However, there is not a specific order in which teachers have to get their students to reach before moving onto the next criteria of understanding. For example, students can display understanding by providing an example of the concept while simultaneously identifying a non-example of the concept in some instances. The structure of the curriculum is entirely dependent upon the concept being taught, the skill levels of the students, and the experience of the teacher.

Thinking through a Lesson: Successfully Implementing High Level Tasks

            The TTLP or Thinking Through a Lesson Protocol is designed for mathematics teachers to implement high level tasks for their students. The TTLP is a lesson planning process that consists of three steps: 1) selecting and setting up a mathematical task, 2) supporting students’ exploration of the task, and 3) sharing and discussing the task. The first step of the protocol asks teachers to decide on exactly what they want their students to learn at the conclusion of the lesson. Teachers need to be clear and concise when creating lesson objectives. Teacher need to consider student prior knowledge, expectations for when students are working on the task, challenges some students might face while working on the task, and how to introduce the task to students. The second step is concerned with how the teacher will monitor students while they are working on the assigned task. The teacher needs to consider how to get students started on the task, how to keep students engaged while working on the task, and how to advance students mathematical understanding while working on the task. Finally, the third step to the protocol asks teachers to determine how students will share the procedures they took to solve the problem, how to ensure every student in the classroom participates, and how to assess student understanding.

            Over time teachers who use the TTLP method, ask these questions automatically when they are creating their lesson plans and do not need to complete the entire protocol line by line. The purpose of the protocol is to change teacher thinking and planning of mathematics lessons so that they are focusing on advancement of student understanding rather than impromptu planning. Teachers who use this method have reported that lessons go smoother and students are able to take more away from lessons in which the teacher can accommodate all the diverse learning styles of the classroom. This is the result of teachers anticipating what procedures students are going to be using to solve the problem in advance.