Inquiry-based learning occurs when students pose their own relevant questions and seek their own solutions. An interactive notebook becomes a portfolio of student work through the school year. It serves as both a formative and summative assessment. Keeping a notebook promotes writing, thinking and organization. Follow this link to see a summary slide show about interactive notebooks.

"Writing is how we think our way into a subject
and make it our own."William Zinsser, author of Writing To LearnLearning is a hands-on, minds-on experience. National Science Education Standards,1996

Math & science notebooks in school & the professional world

School

Workplace

The ability to thoroughly document and to keep clear, understandable records of one’s work is a key skill that the prospective scientist or mathematician must develop. In the classroom, the lab notebook is used to assess students’ progress toward this objective. The goal is to help students construct knowledge and cultivate critical student skills.

In the professional world, a good notebook is a priceless resource. The lab notebook documents the investigative journey of a scientist, telling the story of the progress, questions, barriers, solutions, failures and successes over the course of time…an evolution of learning. At the highest level it is a legally valid record and can be used as evidence during a patent application and to show what and when inventions or discoveries were made. On the other hand, it can be a career ending disaster. A well kept notebook provides a reliable resource for writing up materials, methods and results for a study. A comprehensive notebook permits one to reproduce any part of a methodology completely and accurately. www.leapingfromthebox.com/art/rlg/labnotebook.html

Examples of Linus Pauling’s notebooks are online at http://osulibrary.oregonstate.edu/specialcollections/rnb/.
Science is a human endeavor. Discoveries stimulate discussion, re-testing, publishing and presentations at conferences. All of which can be replicated within the classroom.

Setting up the notebook
Choose the type of notebook. Paper notebooks are most familiar. An authentic professional notebook will be bound. Extra documents may be glued or taped into it.

Electronic notebooks continue to gain popularity. Some online, open scientific notebooks allow continous global scientific sharing. http://onschallenge.wikispaces.com/. For the classroom Wiki site permissions may be set to only allow wiki members (students) to work on collaborative project pages. Teacher pages may be locked.

ADVANTAGES

DISADVANTAGES

ELECTRONIC NOTEBOOK

Can be “backed up”

Easy to cut and paste

Easy to share data and documents

Easy to read

Hard to “draw” diagrams

Loss of nostalgic “art”

Loses some authenticity

Can be tampered with

Side notes may be “lost”

PAPER (BOUND) NOTEBOOK

Easy to make notes & draw

Classic format

Quick access

Easy to flip through

Tangible

More “secure” & authentic

Can’t easily be reproduced or shared

Handwriting may be hard to read

Difficult to add loose documents

Any organized notebook format can work.
Students will be comfortable with the task if a standard format or structure is used. Ideally, within the standard format, students may express their own style.

Purpose - explanation of why (importance) students are doing the activity/task

Numbered pages
Scientific integrity requires that all data, including negative data be reported. Numbering pages guards against "lost" data.

Procedure - step by step description (including materials) detailed enough for others to repeat

Chronologically organized
Notebooks tell the story of discovery over time.

Results - raw and processed data, graphs and attachments

Grading rubric
See the grading section below for a sample rubric.
Specific assignments within the notebook may be graded separately as designated by the teacher.

Conclusions - summary, reflection, further application, also referred to as line of learning

The BSCS 5E model is another way to present, interactive, inquiry-based instruction. http://www.bscs.org/curriculumdevelopment/features/bscs5es.html
The last 4 steps might serve as notebook entry subheadings: Engage, Explore, Explain, Elaborate, Evaluate Making the notebook interactive
What does it mean to make a notebook interactive?
The idea stems from the constructivist approach. It's directly tied to inquiry-based learning or problem-based learning. It challenges students to be independent, creative thinkers and writers. Inquiry or problem-based instruction ties closely with John Dewey's view of students as active learners. Students learn best by doing.
The following Wiki site provides a wealth of resources for teachers to create interactive notebooks for their classrooms. http://interactive-notebooks.wikispaces.com/

The "left page/right page" format allows the teacher to directly monitor how much information is noted by students. The right side would reflect the day’s class lesson/activity. The right side activities could include anything that is teacher led. Left side activities indicate student interaction which could include the warm-up, wrap-up and any problem solving or manipulation that students practiced in class.

http://interactive-notebooks.wikispaces.com/

"Students work out an understanding of new material by using illustrations, diagrams, flow charts, poetry, colors, matrices, cartoons and the like. Students explore their opinions and clarify their values on controversial issues, wonder about “what if ” hypothetical situations and ask questions about new ideas. And they review what they have learned and preview what they will learn...[Students] must actively do something with the information before they internalize it." http://info.teachtci.com/forum/isn.aspx

Students practice the scientific method fully when, with the teacher's guidance, they must come up with the questions, hypothesis, procedure, results and conclusion for a given assignment. This is in contrast to worksheet driven science labs where students follow along, performing the outlined steps and filling in the blanks. Click the following link for an example of a student constructed airplane lab. It is intended to familiarize students with the scientific process. http://www.middleschoolscience.com/paperairplane.pdf Click the file links for a student guide about writing a scientist's notebook and science experiment grading rubric. http://ebecri.org/content/toolkit

For the last, crucial step in the lab process, teachers must guide students to find meaning in their results. (Ruiz-Primo MA, et al., 2010) How can the results be explained? What principle is at work here? The phrase, extending the “line of learning,” has been coined to express this idea. It’s where “students develop a deeper understanding of the target concept. They apply the concept to new situations and learn new…vocabulary.” (Butler MB, Nesbit C, 2008) At a minimum, warm-ups and wrap-ups are great opportunities for students to make connections between their life and the classwork.

The more connections a student can make between their world, real-life problems and applying knowledge, the more authentic their learning. Jon Mueller describes more about authentic learning and the end-goal of authentic assessment on his website. http://jonathan.mueller.faculty.noctrl.edu/toolbox/index.htm As students progress through the course, their notebook becomes a portfolio of their work.

Evidence shows that inquiry-based investigations significantly raise student acheivement and level racial acheivement gaps. (Wilson, CD, et al, 2010)
Writing as part of an active science program ~doubled standardize test scores for science and math. (Klentschy, M et al., 1999)

Challenges with interactive notebooks
As with all teaching activites, preparation is key. Teachers must feel comforable with their subject matter before they are ready to guide students through activities without structured student worksheets. Students may come up with multiple solutions. The teacher must hone listening and probing skills to hlep students reason and connect ideas. (Towers, J, 2010)

Case studies of teachers changing to an inquiry-based approach reveal some of the challenges as they faced. (Lebak, K, Tinsley, R, 2010 and ) All teachers wanted to change. The experience was described as a self-transforming journey. The biggest reported shift was from simplified, linear, lesson plans to guided discovery. Math and science are human endeavors. Discovery includes the history and future of a topic. Following the line of learning generated from relevant student questions can go beyond the curriculum. This is very uncomfortable for some instructors. It’s worth making the effort to become comfortable with the technique when it engages students. One high school math teacher explained, “I now treat the topic as part of an ongoing conversation of a larger theme of mathematics, i.e. equations.” (Chapman, O, Heater, B, 2010)

For students at risk or with learning disabilities, they might benefit from more directed questioning. Instead of asking these students to formulate questions, a hypothesis, and procedure, the teacher may need to give more explicit prompts, narrowing the student's options. For example to prompt the student to write a hypothesis the teacher may say, "Write this in your lab notebook. When you put a paperclip on the nose of the airplane, I think that...such and such will happen. Write what you think will happen and why." (McClearn, JA and Tindal, GA, 1999)

Assessing the notebook
The grading scheme emphasizes student understanding the concepts as well as clarity of expression and mechanics of an orderly notebook.
Below is a sample rubric modified from Mrs. Brice's online example at http://footsteps.ucsd.edu/8th_grade_support/notebook%208th%202009-10.pdf.

Formative Assessment
Student notebook entries allow both the student and teacher to see how students approach problems and what may still be confusing. Students and peers may review and modify their notebooks for clarity and mechanics before the teacher grades it.

Summative Assessment
Notebook grading is based on the expected table of contents and rubric. Each page may be marked or a summary grade given for the cumulative work since the last grading. Students may self-evaluate, peers may give feedback about notebook strengths and weaknesses, providing students an opportunity to enhance their entries, before official teacher grading. By rotating who evaluates the notebook, students get more feedback. Finally the teacher can grade the notebook.

Consider staggering the collection of notebooks for teacher grading. Writing comments on the rubric grading sheet or post-its allows students to own the content of their notebook.

21st Century Skills

21st century skills are incorporated by the nature of inquiry-based or problem-based teaching. Students must identify and solve problems. Many of them, such as in the case of the geology unit and currency conversion apply globally. Students think critically, gathering and analyzing the information that they need as seen in the disease outbreak and cell phone selection examples. Group activities require collaboration, creativity and communication. The notebook serves as a portfolio, demonstrating the student’s multifaceted literacy. http://www.p21.org/documents/P21_Framework_Definitions.pdf

Examples
Middle School Science - Microbiology - Disease Outbreak

Investigating a disease outbreak NCSCOS: 7.03 Analyze data to determine how an infectious disease may spread including:

Carrier

Vectors

Conditions conducive to disease

Examine spread of diseases

Warm-up: How does infectious disease spread? (left side of notebook)

Students write definitions of infectious disease, carrier and vector. (right side of notebook)

Students choose an example of an infectious disease from Center for Disease Control’s disease cards. In table format on the whiteboard or projected computer screen, students list the disease, carrier and vector(s). Students create a similar chart in the lab notebook and write down at least five examples. (right side of notebook) http://www.cdc.gov/museum/cards.htm

Class or group discussion(s) of what conditions make it more likely that disease will spread? Use specific disease for examples. Students summarize the top 3-5 conditions that make it easier for disease to spread in their notebook. (Examples may include: dense population, travelers, poor understanding of transmission and prevention, lack of immunization, time between exposure and symptoms, infectivity.)

Wrap-up: For your chosen disease, what are the best two ways to prevent its spread? Explain why you picked your answers. (left side of notebook)

Case study of disease outbreak

Warm-up: Imagine that 25 people show up at the hospital with a mysterious disease. You are asked to figure out what it is and how it spreads. What information do you need? (left side of notebook)

Group case studies. Paste case studies into the lab notebook. (right side of notebook)

After group discussion, answer the question, “What do I do now?” for each incident (left side). Groups present their next steps and then the teacher reads the actual outcome of the investigation. Students may be further asked to identify observational and experimental methods. http://www.cdc.gov/excite/classroom/index.htm

Wrap-up: Write the steps that epidemiologists use to solve disease out breaks (right side.) Place a check mark next to each step that you included in the group case study. http://www.cdc.gov/excite/classroom/outbreak/steps.htm

High School Science - Earth Science - Geology Expedition

Geology Expedition

NCSCOS: 3.01 Assess evidence to interpret the order and impact of events in the geologic past:

Relative and absolute dating techniques

Statistical models of radioactive decay

Fossil evidence of past life

Uniformitarianism

Stratigraphic principles

Divisions of Geologic Time

Fossil Dig

Students choose global location based on their choice of geologic time , e.g. Dinosaur National Monument, Southern Utah, Big Bend TX, Dinosaur Provincial Park Alberta, Canada, Flaming Cliffs, Mongolia, Dinosaur Cove, Australia, Karoo Basin, South Africa, Grand Canyon, Arizona

Investigate region/area to learn sedimentary rock features and geologic time periods

Learn about current species in the area and why they may be endangered or threatened to learn about extinction and population dynamics

Packing list to learn tools paleontologists use

Research previous finds in the area to learn about different fossil types (body, trace, mold, cast, amber, etc.)

Simulate dig for results

Middle School Math - Tables and Graphs - Music Delivery Investigation
Click the link to the right for specific assignment instructions.

NCSCOS 4.01 Collect, organize, analyze and display data (including box plots and histograms) to solve problems.

Music Delivery Investigation

Reflect on technologic changes in the music industry including information in tables, graphs and timelines

Compare types of graphs and their purpose (direct instruction)

Building meaning from data, using graphs (guided practice and independent practice)

Interpreting or misinterpreting data

Using data to support your position

High School Math - Statisitics - Student-chosen Data Set

Statistics

NCSCOS applies to all course goals

Analyze What's Important to You

Students search for and choose a data set that interests them. For each unit, students apply the techniques learned in class to ask and answer questions related to their data. Reflections, methods, analysis and conclusions are recorded in the notebook. Most data sets will be appropriate for generating:

Descriptive statistics

Graphs including interpretation of distributions and outliers

Forming and analyzing categorical data and continuous data

Investigating methods of data collection for their data set and sharing findings with classmates

Examples of topics might include: school/class survey, sports statistics, crime trends, driving data and student car insurance, car accidents, SADD stats, buying or building houses, global warming data, air pollution, earthquake data, weather, ocean health, census trends, global population growth, human migration, global standards of living, disease, how our school compares to the nation, AP Statistics exam scores, test scores and college admissions, likelihood of finding a job, salaries

High School Math - Geometry - Cafeteria Redesign

Geometry

NCSCOS
Competency Goal 1:
The learner will perform operations with real numbers to solve problems.

Cafeteria Redesign

Analyze and design the school cafeteria to decrease congestion and increase capacity

Students define the problem

Students come up with lists of what they need to know and then get data. E.g. measure and calculate area, furniture dimensions, space required around furniture for movement, peak number of students using the cafeteria, rate of flow into cafeteria and in lunch line.

Use geometry concepts to pose and defend solutions, including any implementation costs

High School Math - Algegra I - Equations for the Real World, currency conversion, cell phone plan comparison, careers in banking (working with exponential growth)

Algebra I

NCSCOS 1.02: Use formulas and algebraic expressions, including iterative and recursive forms, to model and solve problems.

Equations for the Real World

Choose one of the examples.

Create an equation to model currency conversion, including an exchange fee.

Investigate the cost of three cell phone plans. Build equations to fit each plan’s fees. Use the equations to select the best plan for a low, medium and high level of cell phone usage.

Science, Technology, Engineering and Math (STEM)
STEM initiatives offer a wealth of real-world learning ideas. http://www.pbs.org/teachers/stem/

Science and Math News
Breaking news may provide ideas for more project-based inquiry.
The following websites contain a wealth of science and math-related current events.

http://www.nytimes.com/pages/science/index.html More than just news the NYTimes science website includes pertinent slide shows, interactive multimedia and games. Check out the companion site, http://learning.blogs.nytimes.com/ for math puzzles and quizzes related to articles and a section for student opinions. http://www.newsobserver.com/news/health_science/ Science of local (The Triangle in NC) and national interest may be found here. Given our biotech/technology focus, stories often relate to jobs and local services. http://discovermagazine.com/ The online Discover magazine separates science disciplines, including a math/physics section. For the most popular stories and the latest stories click on the appropriate tab near the top of the page. http://www.sciencedaily.com/ Find comprehensive, up-to-date coverage of science stories and check out the math/computer link.

The Inquiry-based Math and Science NotebookInquiry-based learningoccurs when students pose their own relevant questions and seek their own solutions.An interactive notebookbecomes a portfolio of student work through the school year. It serves as both a formative and summative assessment. Keeping a notebook promotes writing, thinking and organization. Follow this link to see a summary slide show about interactive notebooks."Writing is how we think our way into a subject

and make it our own."William Zinsser, author of

Writing To LearnLearning is a hands-on, minds-on experience. National Science Education Standards,1996Math & science notebooks in school & the professional worldhttp://www.sciencenotebooks.org/(examples by grade and subject)www.physicsclassroom.com/lab/nbookideas.cfm

www.ruf.rice.edu/~bioslabs/tools/notebook/notebook.html

www.benchfly.com/blog/how-to-keep-a-lab-notebook/

Science is a human endeavor. Discoveries stimulate discussion, re-testing, publishing and presentations at conferences. All of which can be replicated within the classroom.

Setting up the notebookChoose the type of notebook. Paper notebooks are most familiar. An authentic professional notebook will be bound. Extra documents may be glued or taped into it.

Electronic notebooks continue to gain popularity. Some online, open scientific notebooks allow continous global scientific sharing. http://onschallenge.wikispaces.com/. For the classroom Wiki site permissions may be set to only allow wiki members (students) to work on collaborative project pages. Teacher pages may be locked.

NOTEBOOK

NOTEBOOK

Any organized notebook format can work.

Students will be comfortable with the task if a standard format or structure is used. Ideally, within the standard format, students may express their own style.

InstructionsTitle and dateTable of contentsWith an online, current table of contents students can check that their notebooks are complete.

Purpose- explanation of why (importance) students are doing the activity/taskNumbered pagesScientific integrity requires that all data, including negative data be reported. Numbering pages guards against "lost" data.

Procedure- step by step description (including materials) detailed enough for others to repeatChronologically organizedNotebooks tell the story of discovery over time.

Results- raw and processed data, graphs and attachmentsGrading rubricSee the grading section below for a sample rubric.

Specific assignments within the notebook may be graded separately as designated by the teacher.

Conclusions -summary, reflection, further application, also referred to as line of learningThe last 4 steps might serve as notebook entry subheadings: Engage,

Explore, Explain, Elaborate, EvaluateMaking the notebook interactiveWhat does it mean to make a notebook interactive?

The idea stems from the constructivist approach. It's directly tied to inquiry-based learning or problem-based learning. It challenges students to be independent, creative thinkers and writers. Inquiry or problem-based instruction ties closely with John Dewey's view of students as active learners. Students learn best by doing.

The following Wiki site provides a wealth of resources for teachers to create interactive notebooks for their classrooms.

http://interactive-notebooks.wikispaces.com/

The "left page/right page" format allows the teacher to directly monitor how much information is noted by students. The right side would reflect the day’s class lesson/activity. The right side activities could include anything that is teacher led. Left side activities indicate student interaction which could include the warm-up, wrap-up and any problem solving or manipulation that students practiced in class.

"Students work out an understanding of new material by using illustrations, diagrams, flow charts, poetry, colors, matrices, cartoons and the like. Students explore their opinions and clarify their values on controversial issues, wonder about “what if ” hypothetical situations and ask questions about new ideas. And they review what they have learned and preview what they will learn...[Students] must actively do something with the information before they internalize it." http://info.teachtci.com/forum/isn.aspx

Students practice the scientific method fully when, with the teacher's guidance, they must come up with the questions, hypothesis, procedure, results and conclusion for a given assignment. This is in contrast to worksheet driven science labs where students follow along, performing the outlined steps and filling in the blanks. Click the following link for an example of a student constructed airplane lab. It is intended to familiarize students with the scientific process. http://www.middleschoolscience.com/paperairplane.pdf Click the file links for a student guide about writing a scientist's notebook and science experiment grading rubric. http://ebecri.org/content/toolkit

For the last, crucial step in the lab process, teachers must guide students to find meaning in their results. (Ruiz-Primo MA, et al., 2010) How can the results be explained? What principle is at work here? The phrase, extending the “line of learning,” has been coined to express this idea. It’s where “students develop a deeper understanding of the target concept. They apply the concept to new situations and learn new…vocabulary.” (Butler MB, Nesbit C, 2008) At a minimum, warm-ups and wrap-ups are great opportunities for students to make connections between their life and the classwork.

The more connections a student can make between their world, real-life problems and applying knowledge, the more authentic their learning. Jon Mueller describes more about authentic learning and the end-goal of authentic assessment on his website. http://jonathan.mueller.faculty.noctrl.edu/toolbox/index.htm As students progress through the course, their notebook becomes a portfolio of their work.

Evidence shows that inquiry-based investigations significantly raise student acheivement and level racial acheivement gaps. (Wilson, CD, et al, 2010)

Writing as part of an active science program ~doubled standardize test scores for science and math. (Klentschy, M et al., 1999)

Challenges with interactive notebooksAs with all teaching activites, preparation is key. Teachers must feel comforable with their subject matter before they are ready to guide students through activities without structured student worksheets. Students may come up with multiple solutions. The teacher must hone listening and probing skills to hlep students reason and connect ideas. (Towers, J, 2010)

Case studies of teachers changing to an inquiry-based approach reveal some of the challenges as they faced. (Lebak, K, Tinsley, R, 2010 and ) All teachers wanted to change. The experience was described as a self-transforming journey. The biggest reported shift was from simplified, linear, lesson plans to guided discovery. Math and science are human endeavors. Discovery includes the history and future of a topic. Following the line of learning generated from relevant student questions can go beyond the curriculum. This is very uncomfortable for some instructors. It’s worth making the effort to become comfortable with the technique when it engages students. One high school math teacher explained, “I now treat the topic as part of an ongoing conversation of a larger theme of mathematics, i.e. equations.” (Chapman, O, Heater, B, 2010)

For students at risk or with learning disabilities, they might benefit from more directed questioning. Instead of asking these students to formulate questions, a hypothesis, and procedure, the teacher may need to give more explicit prompts, narrowing the student's options. For example to prompt the student to write a hypothesis the teacher may say, "Write this in your lab notebook. When you put a paperclip on the nose of the airplane, I think that...such and such will happen. Write what you think will happen and why." (McClearn, JA and Tindal, GA, 1999)

Assessing the notebookThe grading scheme emphasizes student understanding the concepts as well as clarity of expression and mechanics of an orderly notebook.

Below is a sample rubric modified from Mrs. Brice's online example at http://footsteps.ucsd.edu/8th_grade_support/notebook%208th%202009-10.pdf.

Formative AssessmentStudent notebook entries allow both the student and teacher to see how students approach problems and what may still be confusing. Students and peers may review and modify their notebooks for clarity and mechanics before the teacher grades it.

Summative AssessmentNotebook grading is based on the expected table of contents and rubric. Each page may be marked or a summary grade given for the cumulative work since the last grading. Students may self-evaluate, peers may give feedback about notebook strengths and weaknesses, providing students an opportunity to enhance their entries, before official teacher grading. By rotating who evaluates the notebook, students get more feedback. Finally the teacher can grade the notebook.

Consider staggering the collection of notebooks for teacher grading. Writing comments on the rubric grading sheet or post-its allows students to own the content of their notebook.

21st Century Skillsidentify and solve problems. Many of them, such as in the case of the geology unit and currency conversion applyglobally. Studentsthink critically,gathering and analyzing the informationthat they need as seen in the disease outbreak and cell phone selection examples.Groupactivities requirecollaboration, creativity and communication.The notebook serves as a portfolio, demonstrating the student’s multifacetedliteracy.http://www.p21.org/documents/P21_Framework_Definitions.pdf

ExamplesMiddle School Science - Microbiology -

Disease OutbreakNCSCOS: 7.03 Analyze data to determine how an infectious disease may spread including:

High School Science - Earth Science -

Geology ExpeditionNCSCOS: 3.01 Assess evidence to interpret the order and impact of events in the geologic past:

Middle School Math - Tables and Graphs -

Music Delivery InvestigationClick the link to the right for specific assignment instructions.

NCSCOS 4.01 Collect, organize, analyze and display data (including box plots and histograms) to solve problems.

Student-chosen Data SetNCSCOS applies to all course goals

- Students search for and choose a data set that interests them. For each unit, students apply the techniques learned in class to ask and answer questions related to their data. Reflections, methods, analysis and conclusions are recorded in the notebook. Most data sets will be appropriate for generating:
- Descriptive statistics
- Graphs including interpretation of distributions and outliers
- Forming and analyzing categorical data and continuous data
- Investigating methods of data collection for their data set and sharing findings with classmates
- Appropriate generalizations based on results

- Good links to online data include:

http://serc.carleton.edu/introgeo/teachingwdata/PublishedData.htmlhttp://www.statsci.org/datasets.html

http://mathforum.org/workshops/sum96/data.collections/datalibrary/index.html

High School Math - Geometry -

Cafeteria RedesignNCSCOS

Competency Goal 1:

The learner will perform operations with real numbers to solve problems.

High School Math - Algegra I -

Equations for the Real World, currency conversion, cell phone plan comparison, careers in banking (working with exponential growth)NCSCOS 1.02: Use formulas and algebraic expressions, including iterative and recursive forms, to model and solve problems.

Science, Technology, Engineering and Math (STEM)STEM initiatives offer a wealth of real-world learning ideas.

http://www.pbs.org/teachers/stem/

Science and Math NewsBreaking news may provide ideas for more project-based inquiry.

The following websites contain a wealth of science and math-related current events.

http://www.nytimes.com/pages/science/index.html More than just news the NYTimes science website includes pertinent slide shows, interactive multimedia and games. Check out the companion site, http://learning.blogs.nytimes.com/ for math puzzles and quizzes related to articles and a section for student opinions.

http://www.newsobserver.com/news/health_science/ Science of local (The Triangle in NC) and national interest may be found here. Given our biotech/technology focus, stories often relate to jobs and local services.

http://discovermagazine.com/ The online Discover magazine separates science disciplines, including a math/physics section. For the most popular stories and the latest stories click on the appropriate tab near the top of the page.

http://www.sciencedaily.com/ Find comprehensive, up-to-date coverage of science stories and check out the math/computer link.