Math Tutoring Project
Contents
- 1 Introduction
- 2 Human and Computer Tutors
- 3 Initial Design of This Website
- 4 Math Tutoring
- 5 Tutoring and Teaching to Math Testing
- 6 Toddlers-PreK
- 7 Grades K-2
- 8 Grades 3-5
- 9 Grades 6-8
- 10 Grades 9-12
- 11 Post High School
- 12 Multi-grade Level Resources
- 13 Interesting Areas for Exploration
- 14 General References and Resources
- 15 Authors
- “Education is a human right with immense power to transform. On its foundation rest the cornerstones of freedom, democracy and sustainable human development.” (Kofi Annan; Ghanaian diplomat, seventh secretary-general of the United Nations, winner of 2001 Nobel Peace Prize; 1938-.)
Introduction
This Web Page is one component of the Information Age Education Math Tutoring Project. The project is in start up mode. The IAE Math Tutoring Project has several major goals.
- To support the development of local, regional, and national Professional Learning Communities (PLCs) of people interested in the use of math tutoring as an aid to improving math education. Please note that we are not trying to compete with the National Council of Teachers of Mathematics and/or any of its Affiliates. Rather, we strongly urge these organizations to include math tutoring as a regular component in their conferences and to take advantage of the work being done by the IAE Math Tutoring Project.
- To provide vehicles for collecting and sharing information to help people interested in math tutoring.
- To foster and make use of the general theme of play together, learn together as a vehicle for helping students learn math and their tutors get better at math tutoring. Our definition of play together, learn together is quite broad. Obviously it includes people playing games together. People engaged with others in a social networking environment can be thought of as playing together. Students working together in a cooperative learning environment or in project-based learning can be considered as playing together, learning together.
The project was inspired by the rapid and widespread acceptance of the following free IAE book.
- Moursund, David and Albrecht, Robert (9/2/2011). Becoming a better math tutor. Eugene. OR: Information Age Education. The PDF file is available at http://i-a-e.org/downloads/doc_download/208-becoming-a-better-math-tutor.html The Microsoft Word file is available at http://i-a-e.org/downloads/doc_download/209-becoming-a-better-math-tutor.html. If you want to just view the TOC, Preface, the first two chapters, and the two Appendices, go to http://iae-pedia.org/Math_Tutoring.
The Moursund and Albrecht book views math tutoring in a very broad sense. For example:
- A parent helping his or her three year old child learn to count or a parent helping an older child with math homework is doing math tutoring.
- In precollege education, many Individual Education Plans (IEPs) include math tutoring as part of a plan. Still more generally, math tutoring both in and outside of school is a proven excellent aid to helping students make more rapid progress in their studies of math.
- Math Help Sessions in secondary school or higher education can be considered as a type of tutoring. Indeed, one student helping another student with homework is a type of tutoring.
Human and Computer Tutors
The Brilliant Report by Annie Murphy Paul is a free monthly newsletter, with each issue addressing an important topic in education. Here is an issue on tutoring:
- Paul, A.P. (September, 2014). The Feeling of Learning. Retrieved 9/18/2014 from http://us2.campaign-archive1.com/?u=bc04df008d4705e4e77c2eb35&id=85d33b2c54&e=9fa2a82024.
Quoting from the newsletter:
- Human tutors—teachers who work closely with students, one on one—are unrivaled in their ability to promote deep and lasting learning. Education researchers have known this for more than 30 years, but until recently they haven’t paid much attention to one important reason why tutoring is so effective: the management of emotion. Studies show that tutors spend about half their time dealing with pupils’ feelings about what and how they’re learning.
- Now the designers of computerized tutoring systems are beginning to make sensing and responding to emotions a key part of the process, and they’re finding that users learn more as a result. At the same time, researchers are using the data generated by these programs to make new discoveries about emotion and its central role in learning.
- One such discovery is that the feelings that dominate psychology’s conventional theories of emotion—such as psychologist Paul Ekman’s six “basic emotions” of anger, disgust, fear, joy, sadness and surprise—are not, by and large, the feelings that are involved in learning. In educational settings, it’s the “academic emotions” that occur most frequently: curiosity, delight, flow, engagement, confusion, frustration and boredom.
Initial Design of This Website
This initial Web Page is a placeholder for what we hope will grow into a collection of related Web Pages. The topic of math tutoring can be divided up in many different ways. Initially we have divided the field by grade level of tutees. Eventually there will likely be separate Web pages for these various grade-level groupings.
Grade level tends to be an indicator related to the level of a tutee’s cognitive development. A typical human’s level of cognitive development and intelligence increases until about age 25. (Remember that a child's intelligence grows year after year during this same time period, but that Intelligence Quotient is calculated in a manner that tends to keep the IQ number nearly constant as a child grows in level of cognitive development and intelligence.)
Howard Gardner’s Theory of Multiple Intelligences posits that logical/mathematical is a type of intelligence different from the other eight areas of intelligence he has identified. Others argue that General Intelligence (g) correlates quite highly with each of the nine individual intelligence areas Howard Gardner has identified. In any event, we know that math is an important component of the content and thinking in many different disciplines. A math tutor needs to be aware of the math cognitive development and math intelligence of his or her tutees. It is not enough just to know what grade a tutee is in as one works to individualize tutoring to a tutees individual needs.
Math Tutoring
Tutoring is a type of individualized or very small group teaching, and math tutoring can have a number of different goals. A good starting point in goal development is to consider the individual needs of a tutee with respect to goals in math education.
Goals in Math Education
Nowadays in the United States, the math Common Core State Standards (CCSS) dominate in terms of specifying math curriculum content at the various grade levels. See http://www.corestandards.org/. In addition, state and national assessments dominate in specifying the levels and types of content knowledge, skills, and understanding that students are expected to achieve. The school math curriculum has moved strongly toward teaching to the tests and preparing students to take the tests.
For more information about CCSS , see:
- Moursund, D. and Sylwester, R. (March 2013). Common Core State Standards for K-12 Education in America. Eugene, OR: Information Age Education. Access the Microsoft Word file at http://i-a-e.org/downloads/doc_download/249-common-core-state-standards-for-k-12-education-in-america.html and the PDF file at http://i-a-e.org/downloads/doc_download/249-common-core-state-standards-for-k-12-education-in-america.html.
- Substantial progress is occurring in developing support materials for the CCSS Math. See http://ime.math.arizona.edu/progressions/.
However, there is much more to the discipline of math than what is specified by the CCSS and measured by the various state and national tests. Some of this "much more" comes under the heading of Math Maturity and is a crucial component of math tutoring. See, for example:
- Moursund, David (2011). Increasing the math maturity of K-8 students and their teachers. A detailed self-study course and materials for a two semester hour graduate-level College of Education course. Free download at http://i-a-e.org/downloads/doc_download/211-using-math-games-and-word-problems-to-increase-the-math-maturity-of-k-8-students.html.
- Moursund, David and Albrecht, Robert (2011). Using math games and word problems to increase the math maturity of K-8 students. Eugene, OR: Information Age Education. Download the PDF file from http://i-a-e.org/downloads/doc_download/211-using-math-games-and-word-problems-to-increase-the-math-maturity-of-k-8-students.html. Download the Microsoft Word file from http://i-a-e.org/downloads/doc_download/210-using-math-games-and-word-problems-to-increase-the-math-maturity-of-k-8-students.html.
- IAE (n.d.) Math maturity. Retrieved 9/28/2011 from http://iae-pedia.org/Math_Maturity.
- Moursund, David (August, 2010). Syllabus: Increasing the Math Maturity of K-8 Students and Their Teachers. A course developed by David Moursund. Retrieved 12/16/2011 from http://i-a-e.org/downloads/doc_download/201-extended-syllabusfor-prism-course.html.
A Tutoring Team
A tutoring team may have a number of potential members.
- The tutee. The overall goal is to help improve the education of the tutee, so the tutee should be the center of attention in a tutoring team.
- The “lead” tutor. This may be a paid professional, a volunteer, a same-age peer tutor, or a cross-age peer tutor.
- Parent, grandparents, guardians, and/or other responsible adults. They may help provide both informal and formal tutoring.
- The overall environment and in which the tutee lives, and people within the environment that have routine contact with the tutee. This include siblings, close friends, school counselors, personnel in religious institutions, and so on.
- Computers, audio and video materials, edutainment, printed materials, and other aids to learning as well as distractions from learning. This component is steadily growing in capabilities.
Desirable Qualifications of a (Human) Math Tutor
Human tutors have a wide variety of qualifications. Here is a list of qualification areas that should be considered in selecting a tutor.
- 1. Problem solving. Have good math problem solving knowledge and skills over the range of his or her math content knowledge.
- 2. Math education experience. Have considerably experience in helping students learn math. If your child has particular math-learning challenges, you want a tutor who is experienced in dealing with such challenges. In any case, you want a tutor who understands both the theory and practice of teaching and learning math.
- 3. Knowledge of Standards. Know the school, district, and state math standards below, at, and somewhat above the level at which one is tutoring.
- 4. Communication skills. This includes areas such as:
- a. Being able to “reach out and make appropriate contact with” a tutee.
- b. Being able to develop a personal, mutually trusting, human-to-human relationship with a tutee.
- c. Being able to make good "on the fly" decisions relating to personalizing the tutoring to better fit the needs of the tutee.
- 5. Empathy and patience. Knowledge of “the human condition” of being a student with a challenging life in and outside of school. A good tutor can help a tutee build self-confidence as a learner.
- 6. Learning skills. A math tutor needs to be an ongoing learner in a variety of areas relevant to math education. Computer technology, brain science, and learning theory are rapidly changing areas that are relevant to math tutoring.
- 7. Diversity. A math tutor needs to be comfortable in working with students of different backgrounds, cultures, race, creed, and so on.
It is interesting to look at this list in terms of the capabilities and limitations of computer-assisted learning systems (computer tutoring systems). While there are many things that a computer can do better than a human being, within the field of math tutoring there are many things that a human tutor can do better than a computer. The challenge in each math tutoring situation is to put together and orchestrate a tutoring team that is both effective and cost effective.
It is also interesting to compare the list to the following quoted material:
- "How do you go about being an effective tutor? [You follow the] five Cs. You foster a sense of control in the student, making the student feel that she has command of the material. You challenge the student—but at a level of difficulty that is within the student's capability. You instill confidence in the student, by maximizing success (expressing confidence in the student, assuring the student that the problem she just solved was a difficult one) and by minimizing failure (providing excuses for mistakes and emphasizing the part of the problem the student got right). You foster curiosity by using Socratic methods (asking leading questions) and by linking the problem to other problems the student has seen that appear on the surface to be different. You contextualize by placing the problem in a real-world context or in a context from a movie or TV show . . . Tutors range from virtually ineffectual to extremely helpful. Expert tutors have a number of strategies that set them apart. They do not bother to correct minor errors like forgetting to put down a 'plus' sign. They try to head the student off at the pass when she is about to make a mistake and attempt to prevent it from happening. Or sometimes they let the student make the mistake when they think it can provide a valuable learning experience. They never dumb down the material for the sake of self-esteem, but instead change the way they present it. Most of what expert tutors do is ask questions. They ask leading questions. They ask students to explain their reasoning. They are actually less likely to give positive feedback than are less effective tutors because this makes the tutoring session feel too evaluative. And finally, expert tutors are always nurturing and empathetic."—Richard E. Nisbett, Intelligence And How To Get It: Why Schools And Cultures Count.
Tutoring and Teaching to Math Testing
One way to prepare for a test is to learn the content that one expects to find on the test. Teaching to the test is the process of helping students learn this content. A second way to prepare for a test is to learn and practice skills of test taking. This section discusses teaching to the test and teaching test-taking strategies.
Testing and Teaching to the Test
One way to think about math education is that the major goal is to prepare students for math tests. A teacher is successful if his or her students do well on various high stakes tests. These may be teacher-developed tests, such as major midterm tests and a final exam. Often a teacher draws heavily on tests she or he has previously developed and used. They may be school district, state, national, or international tests. They may be college entrance or graduate school entrance tests. The expected content of the test drives the curriculum.
However, math education is much more than passing tests. We want students to learn to solve a wide range of problems. We want students to understand what they are doing as they solve math problems. We want them to be able to check the accuracy of their math thinking and their implementations of the thinking. We want students to recognize situations in which math is a useful aid to problem solving—and then to make effective use of math in addressing such problems. We want students to develop math knowledge and skills that transfer to other settings—both in and outside of the classroom. We want students to learn to learn math—and learn to effectively deal with relearning what they have previously learned and forgotten.
In teaching and learning math, it is helpful to distinguish between training and education. Consider two general types of course methodologies and goals:
- Training. Students must learn to be able to accomplish the types of tasks being taught and tasks very similar to those being taught. There is close alignment between the training, the test, and what the students will be expected to do after they complete the course of instruction. It may be that a student is told quite specifically what task to accomplish, or it may be that part of the learning challenge is to recognize when the task needs to be accomplished. Training may involve memorizing procedures such as 1) a pilot's pre-flight check list before taking off, 2) an astronaut's check list before launch or before going EVA, 3) a mechanic's list of lube points for a specific car – transfer of knowledge required to lube an unfamiliar car, 4) procedure for dissembling, fixing, and reassembling a gadget, and so on.
- Education. Students must learn higher-order thinking and problem-solving skills to deal with problems and tasks that are non-routine—that are different than those emphasized in training. There is considerable emphasis on dealing with unfamiliar tasks and problems, and on using higher-order thinking skills.
Typical math courses and math tutoring seek an appropriate balance between these two general approaches. In both situations we can talk about authentic assessment and the alignment between what is taught and what is tested.
It is much easier to do authentic assessment in training than in education. We have driver training courses, and the goal is to learn to drive a car. The training and the assessment can be closely aligned—indeed, may be nearly identical. For example, suppose we want to prepare a car repair person to use a quite sophisticated computerized diagnostic instrument, decide what needs to be repaired, and then do the necessary repair or replacement, and check the results using the diagnostic instrument. The needed training and education can successfully be strongly oriented toward training in a hands-on environment. The (authentic) assessment can be done in the same hands-on environment. Note, however, that a good mechanic can transfer his or her knowledge and skills to relatively unfamiliar settings. This represents educaiton (perhaps self education) and use of higher-order thinking and problem solving skills.
Let’s look at an example from math. We want students to learn to add a pair of positive proper fractions each having a denominator larger that its numerator, and simplify the results. Thus, we want students to handle situations such as 1/3 + 2/3 = 3/3 , which simplifies to 1, and 1/2 + 1/3 = 5/6.
A computer—or a calculator that includes provisions for doing exact arithmetic with fractions—can solve such computational problems. The computer or calculator has no “understanding” of what it is being asked to do or is doing.
A student can be trained to accomplish such fraction addition tasks and be able to do so with little or no understanding. For a human, this is by no means a trivial learning task. To solve a particular computational problem may involve determining a common denominator, converting each of the addends so that it has this common denominator, adding the two fractions, and simplifying the result if it can be simplified. The smallest error likely leads to an incorrect result.
There are many common math tasks of the same ilk. Some require more steps than others, and some of the steps may be built on steps covered earlier in the curriculum. Suppose that a math curriculum is mainly built on this training approach. At each stage, students are expected to perform well on tests of the individual components of their training. However, success rates vary considerably.
However, this rote memory and training approach to learning math proves totally inadequate as students are faced by math problems requiring understanding, higher-order thinking, and transfer of learning.
Now, add sophisticated calculators and computers into the math environment. These tools are far better than students at "learning and doing" the lower-order, rote memory components of math. Our math education system finds itself needing to reconsider the balance between training and education. If the majority of math teaching and assessment is at a training level, not enough teaching and assessment activity goes on at the problem solving, understanding, higher-order thinking, and transfer of learning level. However, our math education system seems loath to make a major commitment to use of these tools as an aid to problem solving.
Teaching to Math Testing
Since this Web Page is focusing mainly on math tutoring, let's examine this situation from a tutoring point of view. There are many possible tutoring goals. Sometimes a goal will be a combination of education and training to supplement the student's classroom instruction.
However, sometimes the tutee's goal is to pass a test or to make a higher score on a test. It may be that the tutee has this single goal in mine. This situation encourages tutors to teach tactics and strategies of test taking.
How can one score well on a test without good problem solving skills, knowledge, and understanding? There is considerable literature on how to do this. The literature focuses on techniques for "outsmarting" the test developers. These techniques can be taught to tutees, and the tutees can be given lots of experience—just like the hands-on experience discussed above that are so useful in training. Classroom teachers are aware of this and many spend class time teaching such techniques.
The net result is that part or even quite a bit of math education and training focus on test taking. This wouldn't be a serious problem if math tests were sufficiently authentic, and the authenticity was strongly oriented toward the problem solving, understanding, and higher-order thinking.
The tests being developed to accompany the Common Core State Standards have an increased emphasis on problem solving, understanding, and higher-order thinking. I view this situation as somewhat of a game. A change in the curriculum and a change in the assessment will make some of the teaching —and teaching test-taking strategies—out of date. The people who write about how to outsmart the test developers may fall behind for a while, and will try to develop new strategies and techniques.
It will be interesting to see how this works out during the next decade.
Here are some Web references for test taking strategies. Their inclusion here is not intended to be a recommendation for the value of their content or a recommendation for their use. You may find them helpful if you want to look more deeply into teaching or tutoring methods for improving one's test scores.
- Text Document. http://www.scholastic.com/teachers/article/no-pain-high-gain
- Text document. How to Take a Math Test or Quiz. http://www.tc3.edu/instruct/sbrown/math/test.htm
- Text document. How to Work a Math Problem. http://www.tc3.edu/instruct/sbrown/math/prob.htm
- Text Document. http://www.tc3.edu/instruct/sbrown/math/test.htm
- Text Document. http://www.tc3.edu/instruct/sbrown/math/prob.htm
Cheating
Some students and some educators resort to cheating in order to increase test scores. This has long been a problem.
The authors of this Web Page do not condone cheating. We realize, however, that sometimes there is a very fine line between cheating and complete honesty.
Some students display a high level of ingenuity in methods they develop to cheat on a test. Electronic technology has added to their repertoire. This creates an ongoing battle between these students and their teachers or exam proctors.
One of your authors—Dave Moursund—recalls when he was an undergraduate at the University of Oregon and a department used the same exam for multiple sections of a course that met at different times during the day. Some of the students in the earlier-in-the-day sections passed on questions to students taking the test later in the day. Needless to say, class averages on the test got higher as the day progressed. The department should have been ashamed of using this approach to testing in a multiple section course.
But, at the same University, some of the fraternities and sororities kept files of old exams. Students having access to these old exams tended to have an advantage over students not having such access. Was that cheating, or merely an unfair practice. Hmm. Why didn't the library collect and make available all of the exams? Why did faculty members draw so many exam questions from their previous tests (perhaps with small modifications to the questions)?
As you can see, it is easy to find fault with both students and faculty.
Here is another type of example. Suppose that a teacher is a proctor for an annual high stakes test that students take. The teacher sees some of the exam questions, and later spends some class time using these and other examples to "teach to the test."
Now, how about a less blatant variation on this example. A teacher spends time browsing the Web to locate sample questions from previous tests. The teacher analyzes the test questions, looking for topics and patterns that the teach then uses in teaching to the test. While an individual teacher may do this, how about when a whole school or whole school district does it? How about when this is done by a company—perhaps a tutoring company? This is commonplace. What message does this send to the teachers and students?
Here is a reference that you might find useful.
- Strauss, Valerie (2/17/2012). How to stop cheating on standardized tests. The Washington Post. Retrieved 2/18/2012 from http://www.washingtonpost.com/blogs/answer-sheet/post/how-to-stop-cheating-on-standardized-tests/2012/02/16/gIQAPF0nIR_blog.html. Quoting from this article:
- Over the past three academic years, the National Center for Fair & Open Testing has confirmed cases of standardized test cheating in 32 states and the District of Columbia (see attached list). The root cause of this epidemic is clear from in-depth investigations into some of the most egregious scandals. Misuse of standardized tests mandated by public officials has created a climate in which increasing numbers of educators feel they have no choice but to cross ethical lines.
Here is a 10 minute British comedy video. It has one scene that is slightly raunchy, but overall you are likely to get a laugh out of this test-taking skit.
Toddlers-PreK
This is a place holder and initial rough outline for the components in this section. Eventually this section will become a self-contained Web Page.
Math is a broad, deep discipline. The discipline includes both a language and ways of thought. As a young child is learning one or more "natural" languages, the child is also learning bits and pieces of the language of mathematics and some mathematical ways of though.
The nature and extent of this initial math learning is highly dependent on a child's home and childcare environments. The analogy with learning a natural language is useful. If math-oriented language and thinking are a routine part of a young child's environment, the child will make good progress in laying a foundation for future math learning.
Math Content Goals
Math Maturity Goals
Play Together, Learn Together Resources
This includes both off-computer and on-computer resources.
Computer-Assisted Learning Resources
References for This Section
Grades K-2
This is a place holder and initial rough outline for the components in this section.
Many students enter kindergarten or the first grade substantially behind their peers in terms of math knowledge, skills, and understanding. They do not meet the prerequisites that are assumed in the math textbook and/or math curriculum that is designed for the math instruction being provided to the class. Their progress is slower than the class average and they steadily fall further behind their peers.
For many of these students, extra emphasis/time on math instruction along with individualization to meet their specific needs can bring them up to the level of their peers.
Math Content Goals
Common Core Mathematics Standards [ http://www.corestandards.org/the-standards/mathematics ]
Common Core Mathematics Standards, Grade K
- Introduction [ http://www.corestandards.org/the-standards/mathematics/kindergarten/introduction/ ]
- Counting & Cardinality
- Operations & Algebraic Thinking
- Number & Operations in Base Ten
- Measurement & Data
- Geometry
Common Core Mathematics Standards, Grade 1
- Introduction [ http://www.corestandards.org/the-standards/mathematics/grade-1/introduction/ ]
- Operations & Algebraic Thinking
- Number & Operations in Base Ten
- Measurement & Data
- Geometry
Common Core Mathematics Standards, Grade 2
- Introduction [ http://www.corestandards.org/the-standards/mathematics/grade-2/introduction/ ]
- Opearations & Algebraic Thinking
- Number & Operations in Base Ten
- Measurement & Data
- Geometry
Math Maturity Goals
Play Together, Learn Together Resources
This includes both off-computer and on-computer resources.
Computer-Assisted Learning Resources
AAAMath is one of our favorite places to recommend to tutees, grades K-8 and beyond.
- AAAMath Home http://www.aaamath.com/index.html
- AAAMath, Grade K Table of Contents http://www.aaaknow.com/kinder.htm
- AAAMath, Grade 1 Table of Contents http://www.aaaknow.com/grade1.htm
- AAAMath, Grade 2 Table of Contents http://www.aaastudy.com/grade2.htm
References for This Section
Grades 3-5
This is a place holder and initial rough outline for the component in this section. Eventually this section will become a self-contained Web Page.
Math Content Goals
Common Core Mathematics Standards [ http://www.corestandards.org/the-standards/mathematics ]
Common Core Mathematics Standards, Grade 3
- Introduction [ http://www.corestandards.org/the-standards/mathematics/grade-3/introduction/ ]
- Operations & Algebraic Thinking
- Number & Operations in Base Ten
- Number & Operations - Fractions (denominators 2, 3, 4, 6, 8)
- Measurement & Data
- Geometry
Common Core Mathematics Standards, Grade 4
- Introduction [ http://www.corestandards.org/the-standards/mathematics/grade-4/introduction/ ]
- Opearations & Algebraic Thinking
- Number & Operations in Base Ten (whole numbers less than or equal to 1,000,000)
- Number & operations - Fractions (denominators 2, 3, 4, 5, 6, 8, 10, 12, 100)
- Measurement & Data
- Geometry
Common Core Mathematics Standards, Grade 5
- Introduction [ http://www.corestandards.org/the-standards/mathematics/grade-5/introduction/ ]
- Opearations & Algebraic Thinking
- Number & Operations in Base Ten (whole numbers less than 1,000,000)
- Number & operations - Fractions (denominators 2, 3, 4, 5, 6, 8, 10, 12, 100)
- Measurement & Data
- Geometry
Math Maturity Goals
Play Together, Learn Together Resources
This includes both off-computer and on-computer resources.
Computer-Assisted Learning Resources
AAAMath is one of our favorite places to recommend to tutees, grades K-8 and beyond.
- AAAMath Home http://www.aaamath.com/index.html
- AAAMath, Grade 3 Table of Contents http://www.aaastudy.com/grade3.htm
- AAAMath, Grade 4 Table of Contents http://www.321know.com/grade4.htm
- AAAMath, Grade 5 Table of Contents http://www.worldplenty.com/grade5.htm
References for This Section
Grades 6-8
This is a place holder and initial rough outline for the component in this section. Eventually this section will become a self-contained Web Page.
Math Content Goals
Common Core Mathematics Standards [ http://www.corestandards.org/the-standards/mathematics ]
Common Core Mathematics Standards, Grade 6
- Introduction [ http://www.corestandards.org/the-standards/mathematics/grade-6/introduction/ ]
- Ratios & proportional Relationships
- The Number System
- Expressions & Equations
- Geometry
- Statistics & Probabiliity
Common Core Mathematics Standards, Grade 7
- Introduction [ http://www.corestandards.org/the-standards/mathematics/grade-7/introduction/ ]
- Ratios & proportional Relationships
- The Number System
- Expressions & Equations
- Geometry
- Statistics & Probabiliity
Common Core Mathematics Standards, Grade 8
- Introduction [ http://www.corestandards.org/the-standards/mathematics/grade-8/introduction/ ]
- The Number System
- Expressions & Equations
- Functions
- Geometry
- Statistics & Probabiliity
Math Maturity Goals
Play Together, Learn Together Resources
This includes both off-computer and on-computer resources.
Computer-Assisted Learning Resources
AAAMath is one of our favorite places to recommend to tutees, grades K-8 and beyond.
- AAAMath Home http://www.aaamath.com/index.html
- AAAMath, Grade 6 Table of Contents http://www.321know.com/grade6.htm
- AAAMath, Grade 7 Table of Contents http://www.worldplenty.com/grade7.htm
- AAAMath, Grade 8 Table of Contents http://www.321know.com/grade8.htm
References for This Section
Grades 9-12
This is a place holder and initial rough outline for the component in this section. Eventually this section will become a self-contained Web Page.
Math Content Goals
Common Core Mathematics Standards [ http://www.corestandards.org/the-standards/mathematics ]
Common Core Mathematics Standards, High School: Number & Quantity
- Introduction [ http://www.corestandards.org/the-standards/mathematics/hs-number-and-quantity/introduction/ ]
- The Real Numnber System
- Quantities
- The Complex Number System
- Vector & Matrix Quantities
Common Core Mathematics Standards, High School: Algebra
- Introduction [ http://www.corestandards.org/the-standards/mathematics/high-school-algebra/introduction/ ]
- Seeing Structure in Expressions
- Arithmetic with Polynomials & Rational Expressions
- Creating Equations
- Reasoning with Equations & Inequalities
Common Core Mathematics Standards, High School: Functions
- Introduction [ http://www.corestandards.org/the-standards/mathematics/high-school-functions/introduction/ ]
- Interpreting Functions
- Building Functions
- Linear, Quadratic, & Exponential Models
- Trigonometric Functions
Common Core Mathematics Standards, High School: Modeling
- Introduction [ http://www.corestandards.org/the-standards/mathematics/high-school-modeling/introduction/ ]
Common Core Mathematics Standards, High School: Geometry
- Introduction [ http://www.corestandards.org/the-standards/mathematics/high-school-geometry/introduction/ ]
- Congruence
- Similarity, Right Triangles, & Trigonometry
- Circles
- Expressing Geometric Properties with Equations
- Geometric Measurement & Dimension
- Modeling with Geometry
Common Core Mathematics Standards, High School: Statistics & Probability
- Introduction [ http://www.corestandards.org/the-standards/mathematics/hs-statistics-and-probability/introduction/ ]
- Interpreting Categorical & Quantitative Data
- Making Inferences & Justifying Conclusions
- Conditional Probability & the Rules of Probability
- Using Probability to Make Decisions
Math Maturity Goals
Play Together, Learn Together Resources
Fraction Games at Math Playground
At Math Playground, you can learn math by playing games and solving puzzles.
- Decention – a game of fractions, decimals, and percents http://mathplayground.com/games.html
- Bridge Builders http://www.mathplayground.com/FractionGame/FractionGame.html
We suggest that you play these games before suggesting them to your tutee, then observe while your tutee plays the game, and be ready to answer a question, provide a gentle hint, or otherwise enhance your tutee’s game-playing experience.
Fraction Activities at NCTM Illuminations
- Fraction Game http://illuminations.nctm.org/ActivityDetail.aspx?ID=18
The object of the game is to get all of the “racers” to the right side of the game board, using as few fraction cards as possible. Click on Instructions, read, and then play. After playing a few games, click on Explorations, read, and then explore.
Vary the gear ratio of a bike. The distance traveled by a half-pedal is determined by the ratio of gears. Can you capture all five flags on a course? Click on Instructions, read, and then play. After playing a few games, click on Explorations, read, and then explore.
Community View: Explore new ways to teach math, not 'to the test'
See http://www.lohud.com/article/20120925/OPINION/309250039/Community-View-Explore-new-ways-teach-math-not-test-?gcheck=1&nclick_check=1. A distinguished author recounts experience in working with students at the high school level. He used real-world-based examples involving probability (and dice, etc.) to engage students in problems that required doing arithmetic with fractions.
Converting Fractional Expressions Video (34:59)
This is a 35-minute video. We watched it and jotted the approximate time at which each fraction alakazam occurred. You can use our notes below to skip ahead to any topic that interests you.
- 00:35 Improper Fraction to Mixed Number
- 03:19 On the Calculator (TI-84)
- 04:41 Mixed Number to Fraction
- 05:58 On the Calculator (TI-84)
- 07:01 Common Fraction to Decimal Form
- 09:35 On the Calculator (TI-84)
- 10:57 Terminating Decimal to Fraction
- 13:14 On the Calculator (TI-84)
- 13:45 Repeating Decimal to Fraction (not in this lesson)
- 14:09 On the Calculator (TI-84)
- 16:40 Decimal to Percentage
- 18:21 On the Calculator (TI-84)
- 19:08 Common Fraction to Decimal
- 20:16 Percentage to Decimal
- 21:14 On the Calculator (TI-84)
- 21:42 Percentage to Common Fraction
- 24:07 On the Calculator (TI-84)
- 24:50 Complex Fraction to Simple Fraction (not in this lesson)
- 25:10 Converting Ratios
- 25:50 Exercise #24: Convert 27/5 to a mixed number
- 26:38 Exercise #25: Convert 2 3/7 to an improper fraction
- 27:13 Exercise #26: Convert 3/8 and 5/11 to decimals
- 28:07 Exercise #27: Convert 0.2 and 3.12 to fractions
- 29:34 Exercise #28: Convert 3/4 to a percentage
- 30:14 Exercise #29; Convert 0.228 to a percentage
- 30:52 Exercise #30: Convert 100% to a decimal
- 31:34 Exercise #31: Convert 0.205% to a decimal
- 32:42 Exercise #32: Convert 14% to a decimal
- 33:26 Exercise #33: Convert 0.001% to a common fraction
Computer-Assisted Learning Resources
Arithmetic with Fractions
Alas, alack, and oh heck, many students have difficulty doing arithmetic with fractions. This is a big problem in pre-algebra, algebra, geometry, and beyond. You and your tutees can practice arithmetic with fractions at Internet sites suggested in this section.
Your tutees can practice arithmetic with fractions at AAAMath http://www.aaamath.com. They can become wizzes at adding, subtracting, multiplying, and dividing fractions. Serendipity! Homework becomes easier, they do better on tests, and they think that you are a terrific tutor.
AAAMath Fractions – Table of Contents http://www.aaastudy.com/fra.htm#topic1
We have selected items from the table of contents that we especially recommend to our tutees.
Equivalent fractions http://www.aaastudy.com/fra42ax2.htm. This is the right stuff for understanding addition and subtraction of fractions.
Adding Fractions
- Adding fractions with the same denominator http://www.aaastudy.com/fra57ax2.htm
- Adding fractions with different denominators http://www.aaastudy.com/fra66kx2.htm
- Adding mixed numbers http://www.aaastudy.com/fra66dx2.htm
Subtracting Fractions
- Subtraction with the same denominators http://www.aaastudy.com/fra57bx2.htm
- Subtraction with different denominators http://www.aaastudy.com/fra66lx2.htm
- Subtracting mixed numbers http://www.aaastudy.com/fra66ex2.htm
Multiplying Fractions
- Multiplying fractions http://www.aaastudy.com/fra66mx2.htm
- Multiplying fractions by whole numbers http://www.aaastudy.com/fra66nx2.htm
- Multiplying mixed numbers http://www.aaastudy.com/fra-mul-mixed.htm
Dividing Fractions
- Dividing fractions by fractions http://www.aaastudy.com/fra66px2.htm
- Dividing fractions by whole numbers http://www.aaastudy.com/fra66ox2.htm
- Divide mixed numbers http://www.aaastudy.com/fra-div-mixed.htm
Simplifying fractions http://www.aaastudy.com/fra66hx2.htm
Fractions and Mixed Numbers
- Converting improper fractions to mixed numbers http://www.aaastudy.com/fra66hx2.htm
- Converting mixed numbers to improper fractions http://www.aaastudy.com/fra66hx2.htm
References for This Section
Post High School
This is a place holder and initial rough outline for the component in this section. Eventually this section will become a self-contained Web Page.
Math Content Goals
Math Maturity Goals
Play Together, Learn Together Resources
This includes both off-computer and on-computer resources.
Multi-grade Level Resources
This is a place holder and initial rough outline for the component in this section. Eventually this section will become a self-contained Web Page.
Play Together, Learn Together Resources
This includes both off-computer and on-computer resources. Here is an example from Bob Albrecht.
Ahoy Mathemagicians:
You are a small winner on a TV game show.
Your prize is a bag of pennies, but there is a catch.
You may have as many pennies as you can carry 1 kilometer in 1000 seconds or less in one trip. [Average speed greater than or equal to 1 m /s]
The pennies are all dated 1983 or later, so the mass of each penny is 2.5 grams. [A penny dated earlier than 1982 has a mass of 3.1 grams. The mass of a 1982 penny might be 3.1 grams, 2.5 grams, or maybe another mass that we don't know about.]
How many pennies will you take as your prize?
What is the value in dollars of your prize?
How much would you pay a person to carry your bag of pennies 1 kilometer in 1000 seconds or less?
Answer the same questions, but for a bag of nickles, a bag of dimes, and so on.
Here is a handy table of masses of coins and the dollar bill.
Money Mass penny 2.500 g nickel 5.000 g dime 2.268 g quarter 5.670 g half dollar 11.340 g Presidential $1 coin 8.1 g Native American $1 coin 8.1 g dollar bill 1 g
Why I like this activity:
- It intertwingles math and science.
- It doesn't have a single "right" answer.
- The answer is personal. How much can I carry 1 km in 1000 s?
- Relates to grade level. Older, bigger, stronger kids can carry more.
Computer-Assisted Learning Resources
Brain Science
Some research in brain science cuts across disciplines, while other focuses in specific discipline areas. For example, think about measuring intelligence. One can attempt to development a single measurement that captures a person's overall intelligence. This "overall intelligence" is often called "g". Or, one can do as Howard Gardner and others have done—study multiple intelligences.
The article Willis (10/5/2011) provides an example of recent research that cuts across all learning areas.
The areas discussed in the article all focus on developing and making use of the higher-order thinking skills of the pre-frontal cortex. In very brief summary, quoting from the article:
- For young brains to retain information, they need to apply it. Information learned by rote memorization will not enter the sturdy long-term neural networks in the pre-frontal cortex (PFC) unless students have the opportunity to actively recognize relationships to their prior knowledge and/or apply new learning to new situations.
References for This Section
Here are two free books written by David Moursund that reflect his insights into roles of computers in post secondary education.
Moursund, David (2007). College student's guide to computers in education. Eugene, OR: Information Age Education. Access at http://i-a-e.org/downloads/doc_download/1-college-students-guide-to-computers-in-education.htm.
Moursund, David (2007). Faculty member’s guide to computers in higher education. Eugene, OR: Information Age Education. Access at http://i-a-e.org/downloads/doc_download/15-faculty-members-guide-to-computers-in-higher-education.html.
Willis, Judy (10/5/2011). Three brain-based teaching strategies to build executive function in students. Edutopia. Retrieved 10/8/2011 from http://www.edutopia.org/blog/brain-based-teaching-strategies-judy-willis.
Interesting Areas for Exploration
This section is for general questions that may be worth exploring and where our combined readership may well have good answers.
Q1. Are tutees encouraged to take notes during tutoring sessions? What is gained and what is lost if tutees spend some of their time taking notes?
- The following is a response from Bob Albrecht based on his experiences.
- No, my students do not take notes other than the answers to exercises. Usually there is only enough time to do the assignment. Sometimes we don't get through the entire assignment. JB and JD will occasionally take notes about the fun stuff we do, alternative math.
- A Web search revels a lot of literature on the values of note taking. Note taking and use of the notes are an important component of study skills. The Website http://salsa.missioncollege.org/mss/stories/storyReader$22 summarizes some of the key ideas.
Q2. There are to major categories of intervention with students. One is the general category that includes behavior, impulsiveness, instant gratification, general creativity and independence of thought, and so on. Such areas cut across all academic disciplines. The other is discipline specific and/or interdisciplinary from a content point of view. In math for example, it includes learning math content, gaining in math maturity, and learning to use mathematical-type thinking across the disciplines. What's a math tutor supposed to do? What might help a math tutor in either or both of these general types of interventions?
- Comment from Dave Moursund. Both types of interventions are important, and it is reasonable to integrate the two in a manner to fit the needs of a particular tutee. Research suggests the the content content interventions are apt to have greater long term impact on a student's progress in math.
Q3. Should creativity (math-oriented creative thinking and exploration) be a part of a math tutoring intervention? If "yes," what are some good examples of these types of activities?
- Comment from Dave Moursund. I am thinking that creativity, independent exploration, problem posing, and so on could be lumped together to form a category of materials we need to find and/or develop for each of the grade-level bands listed earlier in this document.
Q4. Our nation's current approach to improving math education tends to be a top down approach. Seymour Sarason argued that the only way to improve our educational system is to give more power to students and their teachers—a bottom up approach. How can we use a bottom up approach to improve the effectiveness of math tutoring?
- Comment from Dave Moursund. I believe a tutor has more "freedom" than does a classroom teacher dealing with a classroom full of students. So, a tutor can experiment both in empowering tutees and in methods used while tutoring.
Q5. We know that the Internet (including the Web) is a valuable tool in each academic discipline. What should students in math classes be learning about the Internet as it pertains to math? What are they actually learning? What role should math tutors play in this overall endeavor?
- Comment by Dave Moursund. We are all familiar with the arguments about when and if students should learn to use calculators and are allowed to use calculators. Calculators can do some of the math that students are learning about as they study arithmetic. Now, we are faced by the same type of issue, but use of computers and the Internet (including the Web). Computer systems can solve or help greatly in solving a wide range of the types of problems students study in math classes.
- As an example, consider the task of organizing and graphing data, the task of graphing various functions, and the task of solving various types of equations. Computers are very good at this. Humans can learn to do these things by hand, but it takes considerable time and effort to become good at such tasks, as well as considerable time and effort to accomplish such tasks by hand.
- Now, consider a student who is particularly challenged by both arithmetic calculation and the various procedures taught in higher levels of math.How should this student's math learning time and math tutoring time be used? (Hmm. I see I attempted to answer a question by asking another question.)
General References and Resources
These will be general references and links to material that cut across multiple grade levels.
Garfunkel, Sol and Mumford, David (1024/2011). How to fix our math education system. The New York Times. The Opinion Pages. Retrieved 10/4/2011 from http://www.nytimes.com/2011/08/25/opinion/how-to-fix-our-math-education.html?_r=4&nl=todaysheadlines&emc=tha212.
There are many people who argue for our current math eduction, and many others who argue against it. In some sense, a tutor can be caught between a rock and a hard place. On the one hand, a tutor is hired (or volunteers) to support the current system and make it still more effective. On the other hand, a tutor may have person views that are quite opposed to the current system.
The Op-Ed piece referenced above provides arguments against our current math education system. Quoting the first two paragraphs of the article:
- THERE is widespread alarm in the United States about the state of our math education. The anxiety can be traced to the poor performance of American students on various international tests, and it is now embodied in George W. Bush’s No Child Left Behind law, which requires public school students to pass standardized math tests by the year 2014 and punishes their schools or their teachers if they do not.
- All this worry, however, is based on the assumption that there is a single established body of mathematical skills that everyone needs to know to be prepared for 21st-century careers. This assumption is wrong. The truth is that different sets of math skills are useful for different careers, and our math education should be changed to reflect this fact.
The article then goes on to argue that we are forcing many students to study math that is irrelevant to their current and/or likely future needs. As an example, it raises the question: "… how often do most adults encounter a situation in which they need to solve a quadratic equation?"
Koebler, Jason (9/28/2011). Survey: STEM engagement begins early. US News. Retrieved 9/29/2011 from http://www.usnews.com/education/blogs/high-school-notes/2011/09/28/survey-stem-engagement-begins-early.
Quoting from the article:
- Many students decide to study science, technology, engineering, or math (STEM) early in their high school careers, according to a new survey released earlier this month by Microsoft.
- Almost four in five college students who are pursuing a STEM-related degree say they decided to go into their field in high school or earlier; about one in five say they decided in middle school or earlier.
- More than half of male students surveyed say that games and toys they played with as a child and the school clubs they joined initially sparked their interest in the field. For 35 percent of females, games and toys were also significant, but about half of female students surveyed say they are pursuing STEM because they want to "make a difference." A third of male students say that was the reason they're studying the field. [Bold added for emphasis.]
McNeil, Michele (10/10/2011). Flexibility on tutoring pleases districts, worries industry. Education Week. Retrieved 10/11/2011 from http://www.edweek.org/ew/articles/2011/10/12/07ses_ep.h31.html?tkn=SRCCzXIYYmzvGxvugT1lDRDKd5KbeS7WMrPD&cmp=clp-sb-ascd. Quoting from the article:
- The U.S. Department of Education’s plan to grant states broad flexibility under the No Child Left Behind Act will free up as much as $800 million in money school districts now must set aside for tutoring students, but may mark a significant financial blow to an education industry that has grown up around serving low-performing schools.
- Somewhere in the middle of this policy debate, an estimated 600,000 students nationwide, at least this school year, are taking advantage of free tutoring from providers of their choice because they go to schools that have failed to hit their academic goals under the law for at least two years in a row.
Moursund, David (12/4/2011). Teaching to the test. IAE Blog. Retrieved 2/13/2012 from http://i-a-e.org/myblog-admin/teaching-to-the-test.html.
- This short article focusses mainly on some teaching & learning situations that involve higher-order thinking and problem solving. Examples are given where it seems quite appropriate to teach to the test.
Moursund, David (2009). Becoming more responsible for your education. Access at http://i-a-e.org/downloads/doc_download/39-becoming-more-responsbile-for-your-education.html. Eugene, OR: Information Age Education.
- This 96-page book has an 8th grade reading level and is written specifically for young teenagers. Its goal is to help such students learn to take more responsibility for their own education. By age 13, many students are beginning to have the mental maturity to take a major role in their own education. Preservice teachers, inservice teachers, and parents will also find the book useful. For example, parents may want to read the book along with their young teen-age children, and use the reading to facilitate “serious” educational conversations with their children.
Sullivan, Missy (11/21/2011). Behind America's tutor boom. SmartMoney. Retrieved 12/25/2011 from http://www.smartmoney.com/spend/family-money/behind-americas-tutor-boom-1318016970246/?link=SM_mag_inside.
This article explores the booming business of tutoring. Quoting from the article:
- Whether they're seeking remedial help for their child or a leg up to the Ivy League, millions of parents are encountering a frustrating new homework project of their own: learning the intricacies of the tutoring-industrial complex. The "supplemental education" sector is now an estimated $5 billion business, 10 times as large as it was in 2001, according to Michael Sandler, founder of education-research and consulting firm Eduventures. Tutoring firms no longer offer just subject-specific help in, say, Latin or chemistry; increasingly, they're marketing a dizzying menu of test prep, study skills, enrichment tutorials, scholastic summer camps and prekindergarten readiness programs. And students looking for late-night homework assistance now have the formerly unthinkable option of typing in Mom or Dad's credit card number and connecting -- in real time -- to an anonymous tutor halfway around the world via text, Skype and online "whiteboards."
The design and quality of the tutoring varies considerably. For example, here is a quoted description of what the authors of this IAE-pedia page consider to be a terrible approach to education, whether it is tutoring or regular classroom instruction.
- Kumon students are encouraged to stick with the program until they're "completers." On the math side, that involves mastering 200 drill sheets in each of the curriculum's 21 levels, starting with two plus two and ending in calculus. It's a rigorous undertaking that Nativo says can require five years and relies almost exclusively on old-fashioned repetition and memorization -- all under the gun of a timer. Mike Zenanko, director of a respected tutor-training program at Jacksonville State University in Alabama, calls such a curriculum "one size fits all," while other critics use phrases like "mind-numbing" and "kill 'em and drill 'em." Suzuki says the system builds focus and stamina, and that the program achieves customization by allowing students to work through the material at their own speed. Still, the number of students who "complete" hovers around 1 percent, says Nativo
Sparks, Sarah D. (4/20/2011). Computer Tutors Prod Students to Ask for Help. Education week. Retrieved 10/3/2011 from http://www.edweek.org/ew/articles/2011/04/14/28help.h30.html?tkn=ZQTF5gvbohZSfU1KbKWGP4SAWHy%2FrFBe%2Bcqy&cmp=clp-sb-ascd.
Quoting from the article:
- In a series of studies presented at the American Educational Research Association’s annual meeting, held here April 8-12, researchers from the Vancouver-based university and Carnegie Mellon University in Pittsburgh found that students typically will go to extreme lengths to avoid asking for help when working on computer-based tutoring programs. Yet if they learn to think about when and how to ask for help, they are more likely to avoid simply cheating to get answers.
- In the classroom, it can be difficult to determine why a student does or doesn’t ask for help. Yet when students use an online program, the computer can record how fast and how often a student tries to solve a problem, uses a dictionary, or asks for help. In several studies since 2006, Mr. Roll and his research partners at Carnegie Mellon’s Human-Computer Interaction Institute—assistant professor Vincent Aleven, senior systems scientist Bruce M. McLaren, and professor Kenneth R. Koedinger—mined data from math-tutoring programs to gauge how 10th and 11th grade students used a help button, which offers progressively more in-depth hints and eventually gives the answer to the question.
Authors
David Moursund and Robert Albrecht.