Review: “A Local Linearity Approach”, by Teague

This is a short review of “A Local Linearity Approach to Calculus” by Dan Teague.  The article is available for download at:

I enjoyed reading this 8 page paper.  The author discusses an approach to teaching calculus by using linear approximations of functions.  Since this is what the derivative is, this approach is logical.  The reason he likes the approach (and I agree with him) is that it is intuitive and therefore will help students get over the feeling of having to take things on faith.  He gives 3 detailed examples of uses of the approach.

The first is to argue for the weak form of L’hospital’s rule.  Here I found the use of local linearity to be very persuasive in giving intuitive support to L’hospital’s rule.  The approach serves its purpose very well, changing the proof from a technical argument to a visual and intuitive one.

The next example was the product rule in differentiation.  His method is to use linear approximations for the functions f and g at the point a, and calculate the derivative of their product h=(f)(g) based on these approximations.  This works pretty well, but his statement at the bottom of page 5: “at x = a, the approximations are exact” might do with some further clarification: It is clear that the approximation to h(x) is exact at a, but for h'(x) it is less clear.  It would also be helpful if the calculations to go along with figure 3 were included.

The last example was using a sequence of linear approximations to prove the Fundamental Theorem of Calculus.  This technique is known as Euler’s method.  This yields a nice proof of the fundamental theorem.  However, it is a challenge to complete the assertion that the limit of the sum is the values equals f(b) – f(a).  This may leave the intuition of some students unsatisfied.

This paper will benefit teachers of calculus as well as students.  And also has other significance.  Discoveries are often made through intuition, and this is probably the method that the inventors of calculus used.  The analysis that we learn today wasn’t developed till hundreds of years after the discovery of calculus.

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Wolfram | Alpha query engine

Wolfram Alpha is a math query engine, designed by the maker of Mathematica, the high powered computer algebra that does some nifty things such as solving advanced math equations and rendering 3D plots and doing derivatives and integrals.

You can type in anything on the command line, and it will try to figure out what you want.
E.g., if you type in: solve linear equations and press equals, you get the following back:

A menu of things you can do with some preset examples.  If go to the line that says:
solve a system of linear equations, and press equals you get the aglebraic solution, plus
a graph of the solution set.  You may change the equations on the command line to anything you want.

This tool can make math more of an exploration.  Students could make various queries, read the answers, then make hypothesis about mathematical rules that explain their discoveries.

It also has the potential to inhibit deeper learning, since students can do their homework without understanding the mechanics behind the solutions they receive from entering their homework problems.

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Linearity Research Project

Applied Math Linearity Research Project

Subject:  Calculus, Statistics, or Applied Math

Level:  Advanced

Objectives:  To develop research and independent problem solving skills.
Students will establish ground facts, then form an opinion which is supported by argument and research.

Materials:  Computers with access to the internet.

The Project:

In statistics, a linear approximation is used, called the least squares method, which finds a line, L, that minimizes the distance to L from a set of data.

In calculus, a linear approximation called the derivative is used.

In this research project, you will compare and contrast the benefits of these two methods of linear approximation.

This research project has several stages:

Part 1:  Get ground facts.
Learn about the basics of the two methods.  Research how they are defined, for why, when and by who they were invented, and what they were originally used for.  List reputable references to back up your findings.

Part 2:  Write an opinion.
Write an opinion about what the advantages, and disadvantages of each method are.  Give examples or cite scholarly references (or both) to back up your opinion.  The opinion should be clearly written.  It should have substance.  It should be followed with specific details and examples so it is clear what your meaning is.

Part 3:  Research Project: The U. S. Social Security system.
Nowadays, there is frequent conjecture about when our social security system will run out of money.  Choose one of the linear models you have researched, and use it to analyze the rate of Social Security collections and expenditures (two separate items).  Get your data on expenditures and collections from reputable sources (e.g. the U. S. government).  From your analysis, predict
(a) The year expenditures become greater than collections.
(b) The year Social Security runs out of money.


Part 1    50 pts

40 pts      Factual information      (20 pts for each method)
Factual information will be graded for accuracy, clarity and completeness.
10 pts      Quality of references
References should be from standard text books, peer reviewed journals, or expert sources (e.g., a scholar in the field).

Part 2    50 pts

10 pts    Opinion is substantial.
10 pts    Opinion is clearly written.
30 pts    Opinion is supported by argument and examples.

Part 3    50 pts

10 pts    Reliability of data
30 pts    Choice and execution of model
10 pts    Conclusion

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Mathchat 7/26/10 — Making Maths Real

I followed along the mathchat on Monday 7/26.  The topic was: How can we make classroom mathematics more real?  There were approximately 5 active participants.  Here are some notes:

One participant raised the question: Is real concrete or authentic?The following site was suggested: (actually they typed; I’d like to know how does one derive this compressed address?)
This seems like a site with worthwhile suggestions for teaching.  From the site:

The purpose of the MDA is to evaluate what your students understand about targeted mathematical concepts you teach…

Mathematics Dynamic Assessment (MDA) is a reliable process that provides teachers with in-depth information about their students’ mathematical understandings, that can be practically completed in a classroom context, and that is flexible enough to be used with any mathematics curriculum. The MDA integrates four research supported effective assessment approaches in mathematics:  1) Determining student interests for the purpose of imbedding instruction in meaningful and authentic contexts: 2) C-R-A assessment, 3) error pattern analysis, and 4) flexible mathematics interviews.

There was another interesting site mentioned.  This one dealt with surprising research which showed that not teaching math in primary school had a beneficial impact on student’s math skills:

Two of the British teachers participating (both were primary school teachers) noted that the range of numbers being introduced at the early grades was being greatly reduced and emphasis was being placed on conceptual topics.

There was talk of changing the label of maths to “functional numeracy” in order to help make maths real.

An interesting high (middle?) school curriculum was mentioned at:

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Lesson Plan Critique

I critiqued a lesson plan found at the following address:

My review starts with a synopsis, then a critique. Finally, I’ve copied the full plan at the end of the review.

Title – Solving Linear Equations – Review Game
By – Rebecca Hooper
Grade Level – 9-10

Synopsis of the plan:
The class breaks up into small groups.  Each group is responsible for developing a board game, where points are won based on correctly answering linear algebra questions.  The students are responsible for finding the questions (on the web), creating an answer key, and designing the board game.  The questions are to cover varying levels of difficulty:
1. Equations using addition and subtraction.
2. Equations using multiplication and division.
3. Two or more step linear equations.
4. Equations that have variables on both sides of the equation.
5. Equations involving decimals and fractions.

Once the games have been developed, the groups exchange and play each others games.  The plan is envisioned as an end of the unit review tool.

I like this plan because of the active learning (or reviewing in this case) that it promotes.  Students are on their own to find linear algebra problems which fit different categories and create an answer key for those problems.  They also are able to use creativity in designing the game.  In order to carry out this plan, students will have to differentiate between different types of linear algebra problems.  This will cause them to be mentally engaged in the activity.  They also have a chance for physical activity as well as design activity in creating the games.  So the lesson plan engages different types of learning styles and mental reinforcement pathways.

On the other hand, they will play the games after they have created them.  This will exercise their knowledge of the topic.  With its social and competitive aspects, this seems like a fun way to review for a test.

One negative I see in this plan is the expectation that it will take 4 hours of class time.  This seems like much too much.  Do teachers have that kind of time for a unit review?  My perception is that reviews typically are planned for a single period.  I would suggest some adjustments to the plan to reduce the amount of class time needed for it.  Some aspects of it could be assigned as homework before the unit is completed, such as finding the problems on-line, and thinking about game designs.  Then for class, one period could be used for the groups to finalize their game designs and create the games, and another where they play the games.  This is still 2 class periods rather than 1, but its less than 4.

From the point of problem solving and posing, this task is good that it gives students a chance to create problems, which will engage them in higher level mental activity of synthesis and creation.  This, in turn, will lead to a more permanent understanding of the material.  From the point of Bloom’s Taxonomy, the plan is good in that it engages at least 3 of the 4 higher level categories in the taxonomy:
Analyzing – which the students need to do to categorize the problems
Evaluating – since creating the game requires checking, experimenting and testing
Creating – since they will be designing, planning, and constructing.


Title – Solving Linear Equations – Review Game
By – Rebecca Hooper
Primary Subject – Math
Grade Level – 9-10


Solving linear equations is a cornerstone of Algebra and other higher level math classes. The skills involved are critically important to the students’ confidence and success within high school mathematics. In this project, students develop a board game that help their peers review solving linear equations. This project requires internet access as the students will use various websites to collect sample problems of varying degrees of difficulty. The entire project should use about 230 minutes of instruction time. Adjust as needed.


Students work in groups of 2 or 3 to create a board game that reviews solving linear equations. The students use the internet to research math problems as well as develop a key of correct answers for each question in the game. After games have been constructed, the students play each other’s games.


Students will work together to research, plan, and construct a board game.
* Students will follow the teacher’s guidelines to complete the project.
* Students will solve linear equations of varying degrees of difficulty.


Students have the opportunity to:
* work in groups
* review skills needed for all higher level math classes Algebra and above
* review for chapter/unit test
* review for exam
* review for end of course test

SC state standards:

Algebra 1/Math Tech 1&2
Standard EA-1…all indicators
Standard EA-4…indicator 4.7


1. The student will solve one step linear equations using addition and subtraction.
2. TSW solve one step linear equations using multiplication and division.
3. TSW use two or more steps to solve linear equations.
4. TSW solve linear equations that have variables on both sides of the equation.
5. TSW solve linear equations involving decimals and fractions.


* access to the internet
* Algebra 1 textbook
* posterboard
* cardstock paper, notebook paper
* pencils, pens, markers, colored pencils
* rulers/straightedges
* dice
* game pieces
* calculators


1. Spend a few minutes introducing the project providing a handout if desired.
2. Explain guidelines and student responsibilities. Student responsibilities should be developed by the individual teacher. Sample guidelines are as follows:
1. Players will advance around the board.
2. Each player should have his/her own game piece.
3. How will the winner be decided?
4. Game questions will be various linear equations to be solved by the player.
5. Calculators and scratch paper will be allowed.
6. Directions and answer key will be provided.
3. Organize students into groups of 2 or 3. Allow the groups time to plan, organize, and use the internet to research game questions. Be sure the students use a variety of questions with various levels of difficulty.
4. Allow the groups time to construct their games.
5. After groups have finished making their games, allow the students to play other groups’ games.

* This game could really be used in a variety of settings and subject areas. Have fun!

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Research on Bloom’s Taxonomy

After some reviewing of articles on Bloom’s Taxonomy, I came to the opinion that the taxonomy is a tool, not a rule.  In one article, “Bloom’s Taxonomy in Developing Assessment Items”, the researcher found the taxonomy was a useful tool for developing assessment for an on-line precalculus class.

“By using tasks at higher levels of Bloom’s taxonomy, we are forcing students to move beyond the uninformed use of such rules. We hope that this will help students retain knowledge as well as improved understanding and attitude.

We found Bloom’s taxonomy to be a useful framework for developing multiple-choice, short-answer, matching, and essay questions that can involve students in complex cognitive tasks.”

The authors liked the taxonomy because it was helpful tool in choosing assessment items that would promote higher order thinking.

In another article,  ‘Application of Bloom’s Taxonomy Debunks the “MCAT Myth”‘, the taxonomy was used as a tool to show that, contrary to common perception, the MCAT has more emphasis on higher level reasoning than four other assessments which were evaluated: AP Biology, GRE Biology, Undergraduate Biology,  and First Year Medical School (which hands down had the least emphasis).  The author’s rated the individual exam questions for difficulty based on Bloom’s Taxonomy.  I would have liked if they had divulged more specific information about how they arrived at their ratings for the individual questions.  However, the general idea was clear enough: Only questions engaging the upper four levels of the taxonomy were considered to require higher level reasoning.

I also read an article “Was Bloom’s Taxonomy pointed in the wrong direction?” , which pinned responsibility for the shortcomings of high school history students on Bloom’s taxonomy.  In the article, the authors interviewed and tested top AP History students in the Puget sound area in Washington State.  They used material from old AP exams to question the students.  They discovered even these top students did not approach the questions as historians are supposed to.  The students mimicked Bloom’s hierarchy, going upwards as they progressed in their evaluation of historical material, but they did not ask the right questions along the way.  As a result, the knowledge they applied to the problem was the wrong knowledge.  In order to determine the correct knowledge, one has to be able to ask the right questions.  So, a good student needs to be able to progress both ways through the pyramid to really be doing history well.

My own opinion is that the views of all these articles have merit.  To me, the taxonomy is a tool which may be used along with other tools to help in evaluating assessments, lesson plans, and assignments.  I do not see it as a law of nature, and I believe that if I use it, I should be open minded and flexible in applying it.  I like the last article because it points out an important weakness in the taxonomy:  the taxonomy assumes there is a linear hierarchy to reasoning:  One needs knowledge first to progress to Understanding and so on.  In reality, there are many feedback loops and much jumping back and forth among the categories in the taxonomy.


D. Vidakovic, J. Bevis, and M. Alexander (2003). Bloom’s Taxonomy in Developing Assessment Items. Journal of Online Mathematics and its Applications.  The Mathematical Association of America, 2010.

S. Wineburg, J. Schneider (2009). Was Bloom’s Taxonomy pointed in the wrong direction? Phi Delta Kappan, 91(4), December 1, 2009, 56–61.

A. Y. Zheng, J. K. Lawhorn, T. Lumley, S. Freeman. Application of Bloom’s Taxonomy Debunks the “MCAT Myth”. Science, 319(5862), January 25, 2008, 414–415.

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Editing Wikipedia

Today I registered with wikipedia so I could make edits on their site.  I made a couple of minor edits to the class entry on multiple representations, and added a reference link to the GeoGebra info page.  It was easy to do.  I’d suggest to anyone writing their first entry to press the edit button on some pages to see how they are put together.

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