The Future of Education

ED526b- Learning and Assessment in Secondary Math

Unit 5- Assignment 5.7

Robert Stock

     In this assignment I chose to take part in the web discussion “It Takes a Village” which was presented by Greg Whitby.  The forum was held on the Elluminate web facility at 7:30, 8/12/2010.  The topic of discussion was contemporary learning which is based on a variety of unconventional teaching strategies.  It is the vision of Mr. Whitby to radically alter the way in which instruction is provided for students in the Australian school system.  His belief is that students should be free to choose where, when, and how they acquire knowledge within the framework of the educational system.  By shifting the onus for learning from the teacher to the student the hope is that student engagement will improve and learning outcomes will be enhanced.  Although I am unfamiliar with the public school system of Australia I believe that there are several caveats to this utopian vision.  In this country students do not have a choice whether or not to participate in the educational process.  They are required by law to attend school.  This fact alone creates a completely different dynamic within the classroom.  Certainly for students who appreciate their educational opportunities and who are eager to increase their understanding this pedagogy could prove highly successful. 

      Changes come slowly to any area of human endeavor and that is not necessarily a bad thing.  Ever since the inception of public education there has been ongoing debate as to how this system should be operated.  What may work for one group may not meet the needs of another.  Cultures vary from school to school and from classroom to classroom.   As an employee of a public school system I am obligated to work within the framework of my school district and to uphold the policies and regulations that have been set forth by the powers that be.  Change will come, of that I am certain.   My hope is that we will be receptive to change when it is in the best interest of our students and that we will resist change when it is motivated by less altruistic motives.

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1.  Formative and Summative Assessment:

The following quote describes the differences between formative and summative assessment.  Although the quote refers to law students it is applicable to students in any educational setting.

“The difference between formative and summative assessment is often an area of concern for law teachers. The essence of formative assessment is that undertaking the assessment constitutes a learning experience in its own right. Writing an essay or undertaking a class presentation, for example, can be valuable formative activities as a means of enhancing substantive knowledge as well as for developing research, communication, intellectual and organizational skills. Formative assessment is not often included in the formal grading of work, and indeed many believe that it should not be.

In contrast, summative assessment is not traditionally regarded as having any intrinsic learning value. It is usually undertaken at the end of a period of learning in order to generate a grade that reflects the student’s performance. The traditional unseen end of module examination is often presented as a typical form of summative assessment.”

The use of both of these assessment methods would be included in my instructional protocol.  I believe that we must check for understanding as we teach to avoid losing touch with our students.  Certainly the summative assessment is necessary when determining student grades.


2. Neutrality- Objective and Subjective:

Subjective assessment is described by this reference; “In subjective assessments the teacher’s judgment determines the grade. These include essay tests. Essay tests take longer to answer and they take longer to grade than objective questions and therefore only include a small number of questions, focusing on complex concepts.”   Objective assessment can be defined by this quote; “Objective assessments (usually multiple choice, true false, short answer) have correct answers. These are good for testing recall of facts and can be automated. Objective tests assume that there are true answers and assume that all students should learn the same things.”    In everyday use the objective assessment is probably the assessment of choice for most educators.  The ease of scoring is a big factor in this case and since time is always in short supply this form of assessment will constitute the core of my assessment strategy.  The subjective assessment can be used in an open-ended format.  Due to the use of this type of problem on standardized tests I would include this type of testing as an alternative assessment technique.


3.  Self Assessment:

The concept of self-directed assessment is explained by this reference:  ” Students need the opportunity to evaluate and reflect on their own scientific understanding and ability. Before students can do this, they need to understand the goals for learning science. The ability to self-assess understanding is an essential tool for self-directed learning.  ”    The ability to self assess is a function of student maturity.  Any self assessment or peer assessment assumes a level of understanding on the part of the students that requires honesty and heart-felt evaluation.  In a 9th grade algebra class I am not sure I can utilize this format effectively.  Perhaps there will be opportunities to include this assessment type somewhere along the line but I will have to exercise extreme caution when implementing such a strategy.


4.  Constructed Response- Selected Response:

From :

“With selected response assessment items, the answer is visible, and the student needs only to recognize it. Although selective response items can address the higher levels of Bloom’s taxonomy, many of them demand only lower levels of cognition. With constructed response assessments (also referred to as subjective assessments), the answer is not visible — the student must recall or construct it. Constructed response assessments are conducive to higher level thinking skills.”  Once again I take the easy way out.  With ESL students I am happy when they are capable of selected responses.  There are those occasions when a constructed response would be nice but that is a rare occurrence.

5.  Ability and Performance:

Timothy Slater of Montana State University describes the rationale for performance assessment in the following reference;   WHY USE PERFORMANCE ASSESSMENT?

“Although facts and concepts are fundamental in any undergraduate SMET course, knowledge of methods, procedures and analysis skills that provide context are equally important. Student growth in these latter facets prove somewhat difficult to evaluate, particularly with conventional multiple-choice examinations. Performance assessments, used in concert with more traditional forms of assessment, are designed to provide a more complete picture of student achievement.”  A performance assessment in a secondary mathematics class would probably be constructed in a project format.  Allowing the students to utilize their creativity would be an interesting and exciting way to measure student achievement.  The design of such an assessment could provide a differentiated approach to the curriculum goals.  Here the key is student motivation.  For the engaged student a performance assessment may well prove to be a rewarding experience.

6.  Authentic and Standardized:

     In Developing Authentic Assessment: Case Studies of Secondary School Mathematics Teachers’ Experiences Christine Suurtamm writes,”Since traditional tests often focus only on the answer or the use of a suitable algorithm to reach the answer, authentic assessment techniques need to be employed to provide a broader range of measures. In this article, the term authentic assessment is used to describe assessment of this type: assessment that involves students in tasks that are worthwhile, significant, and meaningful and that resemble learning activities. Such assessment activities also encourage risk taking, allow for mathematical communication, and provide the opportunity to demonstrate the application of knowledge in unfamiliar settings.”  This definition of authentic assessment provides the basis for the design of assessment strategies that can have a significant effect on student understanding.  By creating assessment tools that have relevance to the world in which our students live we can build bridges that can connect our students to the goals of our curriculum.  It is not always easy to accomplish this goal.  Mathematics like most subjects demands a level of commitment on the part of the student. However by providing assessment that is challenging and engaging the chance of a successful outcome is certainly increased. 

     Choosing the appropriate assessment for a particular group of students is a task that requires a thorough understanding of your students.  In an ESL classroom it is important to design assessment with realistic goals in mind.  Communication is paramount in working with students who are struggling with language as well as content.  Often assessment must be modified to meet these challenges.  

     As a teacher I am acutely aware of the need to assess my students.  The entire system of public education revolves around one form of testing or another.  Every student wants to know how they are performing and it is the job of the teacher to provide that information.  Whether it is a paper and pencil test, an alternative assessment, a performance based evaluation, or any other measure of student knowledge the fundamental purpose is the same.  Teachers must deliver instruction that is tied to a specific program.  They must then determine how well their students have learned this material and they must do it fairly and reliably.  Assessment lies at the heart of the educational process.  It is the very art and science of teaching.


Suurtamm,C,Developing Authentic Assessment: Case Studies of Secondary School Mathematics Teachers’ Experiences,Canadian Journal of Science, Mathematics, & Technology Education, Oct2004, Vol. 4 Issue 4, p497-513, 17p; (AN 15400415)

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Data Manipulation and Visualization

     In this task I have chosen to consider the lesson plans that were compiled by NSDL Middle School Portal for math on the topic of statistics and probability.  This resource contains seven separate lessons and a number of activities that complement the lessons.  Although the lessons are geared to a middle school level I believe they would be useful in a secondary ESL math classroom.  The first section covers posing and formulating questions.  Section two deals with representation of data.  The third section is on the interpretation and evaluation of data.  Section four introduces student designed data collection and analysis.  The fifth lesson looks at probability.  Lesson six works with computations for probability.  The last section reviews other data collection ideas. 

     Included in the site are a number of templates that can be used to create the materials needed for some of the activities.  Because the site is geared to an introductory level  it is ideal for differentiating the instruction in a mixed ability classroom.  For many of the ESL students that I have worked with in the past these types of activities would provide an effective way to gain understanding.  The web address is:

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Scatter Plots

  In this assignment I have created a lesson plan that could be used to teach algebra 1 or algebra 2 students about scatter plots.  the plan includes the use of flip charts which have been created for use with the interactive white board.  These flip charts are created using the active inspire program.

ED526b- Learning and Assessment in Secondary Math

Unit 5- Assignment 5.2 – Data Representations

Robert Stock

Lesson Plan

Lesson Topic:  Scatter Plots

Essential Question:  What is a scatter plot and how do I interpret the data?

Standards:  2.1, 2.6, 2.7

Resources:  Textbook, Internet search, interactive web resource, worksheets.

 Web Resources:


   1.  Please Do Now:   Construct a graph by plotting the points given in this table;

                                                X coordinate                    Y coordinate

                                                           1                                      10

                                                           2                                      22

                                                           3                                      35

                                                           4                                      49

                                                           5                                      62

     2.  Guided Practice:    Flip chart presentation of scatter plots and data correlation.

     3.   Internet search:   Find definitions and examples of scatter plots, positive correlation, negative correlation, and correlation coefficients.

     4.  Data Collection:  As a class activity we will measure each student’s height and record it in a table.   We will also determine each student’s shoe size and add this information to our table.  From the data we will construct a scatter plot.  From this plot we will try to determine if a correlation exists for our data.

     5.  Ticket Out the Door:  If the data points on a scatter plot rise from right to left, how can you describe the relationship between the x- coordinates and the y- coordinates?

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Data Visualizations

     Data is defined in Wikipedia as ,”The term data refers to groups of information that represent the qualitative or quantitative attributes of a variable or set of variables. Data (plural of “datum”, which is seldom used) are typically the results of measurements and can be the basis of graphs, images, or observations of a set of variables. Data are often viewed as the lowest level of abstraction from which information and knowledge are derived.”  The visualization of data can be represented in many ways.  In this assignment I will look at five methods to achieve that goal and because the task is to, “Find five less common data visualization tools you think should be more common.”, my choices will reflect my personal opinion on the matter. 

     The Periodic Table of Visualization Methods is a wonderful catalog of  visualization techniques.  Since my background was chemistry I find this table to be particularly attractive.  On the left hand side of the chart we see a number of methods that can be used to represent data visually.  Of the eleven chart types listed here it is my belief that the cartesian chart, the boxplot, the scatterplot, the histogram, and the bar chart  should receive greater exposure in a secondary math curriculum.  Although there is nothing extraordinary in these five data representation methods they all are commonly found on the PSSA exams that are a critical part of the today’s educational environment.  Since we all “teach to the test”  students need to understand how to read a boxplot even though after high school they may never encounter one again.  Those students who  move on to higher levels in their study of mathematics will certainly encounter graphic representations such as scatterplots, histograms, and bar charts.  These common types of representations of data are not so common among my ESL students.  When I gave my algebra 2 students an alternative assessment two years ago I asked them to produce a graph that could be used to explain a relationship between groups of data.  The only requirement for the assignment was for the students to create something visual that could serve as a way to describe the information being displayed.  My thinking was that my if students could understand the use of graphs in everyday life they would be able to make the connection between graphic representations of data and algebraic concepts.  Sadly only a few of my students were able to complete the assignment.  Maybe they were just lazy or maybe they just did not know how to read a graph.  At any rate the concept of graphic representations was elusive for many students. 

     It is a sobering fact of life as a high school math teacher in an urban setting that you cannot take anything for granted.  Making assumptions about student understanding often leads to problems.  Here I must beg the reader’s indulgence as I have strayed somewhat from the main theme of the assignment but I believe that the real value in any assignment must be found in it’s usefulness.  In other words if I can’t use this stuff in the classroom, why am I doing it? 

     Graphic visualizations that are less common are frequently more fun to look at.  The Map of Calculus created by Don Cohen is a visual gem.  Confusing at first glance but visually interesting.    Other types of graphs that are imaginative and fun can be found on the rest of the Periodic Table of Visualization Methods.  Some of my favorites are the radar chart cobweb, the argument slide, and the knowledge map ( complete with the ocean of oblivion). 

     I believe that we are all visual learners to some extent.  Graphic organization of ideas and information  is a major component of the current methodology that has been adopted  by our school district.  Certainly these types of representations play an enormous role in the teaching of mathematics.  All students must understand how to create graphs and how to interpret them. Having some fun with the process can liven up an otherwise dull activity.

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Calculus for Second Graders?

     It turns out that teaching calculus to seven-year olds may not be as far-fetched as it sounds, at least not according to Don Cohen who is known as “The Mathman”.  His presentation last night on Math 2.0 showed sceptics that there really are no areas of math that are beyond the understanding of motivated youngsters.  His “Map to Calculus” is an interesting but somewhat complicated collection of mathematical topics that resembles a flow chart for some complex industrial process.  Needless to say Mr. Cohen’s approach to teaching is unconventional.  His enthusiasm for the subject is apparent and his energy belies his age.  I am not sure just how much of what he does with his students will be applicable in my classroom but it is nice to look at teaching from a different perspective and to reaffirm the fact that math can be fun.

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The GeoGebra tool has been on my computer for the last two years and I have tried to experiment with this program and find uses for it in my lesson preparation.  So far I really have not had the time to get too involved with the use of GeoGebra so I have only used GeoGebra for simple graphs of linear or quadratic equations.  Like most of the software that came pre-loaded on my school computer I have not been able to get too deep into the use of this resource.  Maybe this year will be different and I’ll find a way to incorporate more of this great technology into my teaching.  At any rate I know that GeoGebra is a powerful program and once I become more familiar with the workings of this stuff I hope to do great things for my students.  For now I am able to provide a nifty little webcast on systems of linear equations.  Nothing earth shattering but useful nonetheless.

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