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# Week 1: Blooms digitally

Let us look at Bloom's Taxonomy, revised for modern technologies. You can find the article that goes with this diagram here: http://edorigami.wikispaces.com/Bloom's+Digital+Taxonomy

None of the activities in the above diagram are specifically mathematical. Your task is to name two math activities for each level in the taxonomy, and to explain (in one or two sentences) why they belong to the level, and what technology (if any) you would use.

For example...

Creating. Design your own "function machine" that turns one number into another number. For example, the machine can double the number and then add 5 to it. This is Creating because you design your own math formulas.

Applying. Program the rule for your function machine using some computer software, so that it gives results for the numbers you type in. This is Applying because you need to execute the math design and programming. I would use spreadsheet software or Scratch for this task with students.

• Creating
1) Scholars could create mathematical music videos using instrumentals from songs that relate to their culture or peers. The songs/videos would be based around mathematical concepts learned in class. This would help scholars remember certain concepts in a create, fun, and engaging way. Here is an example of a high school which uses this technique. https://www.youtube.com/watch?v=OFSrINhfNsQ

2) Scholars could create original posters that explain mathematical concepts covered in class. This empowers the scholars because their work would be displayed around the school and in classrooms. It also gives the scholars a sense of ownership in the classroom because their posters can be used instead of purchasing pre-made posters that teacher supply stores sell. The poster could cover concepts covered in their class or concepts for taught in lower grades.

Evaluating
1) Scholars could score or critique other scholars work when presenting a class project. The scoring of the other scholars work could come by way of a rubric, which would also be factored as a part of their grade. Their feedback would also provide positive critiques of each other's work and allow them to improve as scholars and presenters.

2) Scholars could use a graphing scientific calculator to explore what happens when they change the slope or y-intercept value and how it effects their linear equation. This would allow for them to develop their own idea for these concepts before being formally introduced. They would also be able to articulate what happens in their own words.

Analysing
1) Scholars could create shake colored M&Ms in a cup and dump them out, eating all the one's that are face down. They would record the number of M&Ms face down for each roll until they ate them all. They would then plot that data and analyze it to see if creating an exponentially decaying graph. The data could be entered into a graphing scientific calculator and a they could determine the equation for a best fit line. The software on the calculator could assist them in there analysis.

2) You could use movies or cartoons to teach students Newton's laws of motion. For example, a teacher could use the movie Transformers to demonstrate what laws of motion the movie follows or breaks. That movie is pretty popular, so the teacher wouldn't have to worry about students paying attention.

When the students figure out which laws of physics are being broken, they could write down what would actually happen if the law hadn't been broken. This would make class a little more interesting because, for example, if Optimus Prime made a giant leap that is impossible, the students would probably have fun imagining the character falling.

Applying
1) Scholars could create math riddles, word problems, brain teasers for other scholars to complete in the classroom. The exercises could also be shared on the school web site, newspaper or newsletter, and other math websites for scholars around the world to solve. The scholars would be creating math exercises ranging in difficulty due to their different learning styles and abilities. As the teacher, you can assess their understadning of each topic and if they successfully applied it in their self-created exercises.

2) Scholars would create a shopping spree within a budget of \$1000. The goal of the assignment would be to shop at 5 different stores (preferably that sell their favorite things) while spending the most amount of money, yet walking away with the most articles of clothing. An article of clothing would consist of a top, a bottom, and footware. There would be a minimum and maximum amount of items they could purchase from each store. The scholars would be applying to apply percentage discounts and calculate taxes if applicalble. It would also teach them the skill of bargain hunting, living within your means, and getting the best value for you dollar.

Understanding
1) Scholars could find real world examples that correlate to concepts taught in class. This is important because scholars have to make connections to the work they learn in class. It is more important for them to relate to what they are learning in terms that they may understand. Some examples may be as simple as figuring out the tax when they go shopping or how calculus relates to the construction of a roller coaster. Math without a real world or deeper connection can often become confusing and pointless to many scholars.

2) Scholars could simulate life as an independent. I find this to be successful and fun in many classes because you always have those scholars who believe they can live on their own or constantly making comments like "My parents get on my nerves or I wish I could move out." To make the experiment fair, I would assign all the scholars a minimum wage job being that is the only job they could obtain without a high school diploma. The scholars would have to find a living arrangement and calculate their utilities (gas, electric, water) and food expenses. Those are the bare necessities. They may also calculate car, phone, insurance, internet, clothing, washing clothes, and various expenses that may occur in life. Their goal would be to save more money than they spend, realize the importance of higher education, and developing their successful business if they choose not to complete high school or attend college. They would know the importance of budgeting and how things fluxuate in the real world.

Remembering
1) Scholars could complete a series of minute math drills three days out of the week. They would start with addition and eventually end with fractions, decimals, and percents. Their goals would be to complete fifty problems in a minute. Each scholar would be on different levels because everyone has their own goals and levels of intelligence. They should be able to track their growth through each week and begin getting more problems right eacht time. This improves their muscle memory and ability to recall basic mathematical facts as they progress in the realm of mathematics.

2) Scholars could explain their steps when solving mathematical word problems. This assists them in recalling important facts, explainig their logic as to how they solved the problem, allows them to notice their own or other scholar's mistakes, and it improves their critical thinking. It also infuses the use of literature skills along with their math explanations.

• Nice rap there, Green Machine!

My favorite project of that sort is "I will derive":

I like the word you use for people who learn, "scholars" - it defines a nice role for them!

Some people could argue that many (most?) tasks on your list are at the Creating level. What I see there is lower-order tasks embedded into more interesting, creative higher-order tasks. This is a strong education design principle, overall. You may find some scholars resisting it, though. Have you had that problem with your designs?

•

I chose all of my activities according to the verbs associated with each of the key terms, and realize that many of my activities can apply to more than one key term.  Any comments, suggestions, etc. are welcome!

Remembering:

1.  Student has memorized times tables by using computer flash cards (such as those athttp://www.math-drills.com/flashcards/flashcards.shtml,http://www.aplusmath.com/flashcards/index.html,

http://www.apples4theteacher.com/flash-cards.html).  Some sites allow the user to provide their own content for the flash cards, making this applicable for multiple grade levels (such as older students who need to learn the Greek letters often used in mathematics notation).

2.  Student remembers (after teacher demonstration) that, when using a calculator, sometimes you need to insert extra sets of parentheses.

Understanding:

1.  Students correctly interpret word problems to decide what techniques are necessary to solve.

2.  Students understand the concept of 'Garbage In, Garbage Out" and limitations of available technology.  I've seen a lot of tech users (of all ages, all kinds of technology) forget this at one time or another (myself included).

Applying

1.  Students independently (and correctly) insert extra parentheses (if needed) when using a calculator.

2.  Students use mathematical formulas (or appropriate technology, whether it's a calculator or software) to solve problems.

Analyzing:

1.  Students provide others with an outline of how to use mathematics software for specific tasks (such as my son outlining the steps necessary to find lines of symmetry using the Geometer's Sketchpad software).

2.  Students (and parents) use an Excel spreadsheet to compare the costs of attending the colleges they've been accepted to (we've been doing a lot of this lately; my oldest son will be starting in the fall).  Students can also use an Excel spreadsheet to keep track of their personal income and expenses.

Evaluating:

1.  Students use their evaluation of mathematical situations to achieve particular outcomes.  One example is using probability when playing computer games (like backgammon or cards) to evaluate which move is best.

2.  Students critique their personal financial spreadsheets to decide if any of their spending habits need to be modified.

Creating:

1.  Students use Excel to create a financial spreadsheet template that automatically includes fixed, recurring costs (such as a monthly cell phone bill, food, and transportation costs for commuting to their job).

2.  Students use software to create an interdisciplinary presentation that teaches the rest of the class about how a particular math concept relates to other areas.  For example, a student would use Adobe After Effects to demonstrate how trig functions control visual media parameters such as scale and opacity.

• I am watching "The Wire" PBS series, specifically Season IV that deals with schools. Yesterday's episode dealt with a math teacher who finally was able to reach "street kids" - by relating math to gambling games they see all around them. They greatly appreciated "the secret knowledge" that there are more chances to get 6 than to get 4 out of two dice, etc. You wrote in your sign-up about students from different backgrounds, too.

The spreadsheet task for families is excellent design. As you said, many tasks can be considered to belong to many levels - this one, in particular, spans everything, because it will incorporate remembering (formulas, UI) and up. You will find that projects tend to span many levels on any taxonomy!

"Garbage in, garbage out" made me chuckle. Hear, hear! And there are so many neat math things you can do with it... To use trig functions you mentioned, a graph that reaches up to 2 should be a red flag for sin or cos functions with any argument in them...

• Remembering

1. Students will search online for situations involving percent such as a 25% off sale, 15% down, 7% interest, etc. This is remembering because they have to identify examples of what they’re learning in class and recognize real-world math problems involving percentages.

2. Using graphical representation of geometric shapes, students will identify which are reflections, rotations, or translations in the plane. This is remembering because they have to recognize and identify the different situations. I would use geometric graphing software to create different situations using a variety of shapes.

Understanding

1. Working in pairs, students will use the internet to research a mathematician and a topic from geometry class (for example, Archimedes’ method of approximating pi) and write up a short summary. This is understanding because students have to search the web, sift through the information to find what relates to their topic, and summarize the results.

2. Students will use what they have learned about triangle congruence to write up a step-by-step method for determining if triangles are congruent. This is understanding because students must interpret triangle congruence theorems and explain how to use them in practice.

Applying

1. Following #1 from “Understanding”, each pair of students will present their mathematician and describe his or her contributions to mathematics and how it relates to what they are learning in geometry. This is applying because they have to link the information from their research to class and share the results.

2. Students will take on the role of air traffic controllers and direct incoming airplanes by solving problems involving the distance formula and linear equations. I would use NASA’s FlyBy Math software so students can work on these problems interactively. This is applying because students have to use their algebra skills to carry out a plan for directing airplanes.

Analyzing

1. As a class, students will create a series of bar graphs of class favorites, such as favorite lunch in the cafeteria, favorite TV show, etc. Everyone will contribute their answers, and then students will work in groups to organize the data to create a bar graph. This is analyzing because the class will organize and compare the results.

2. Students will use probability experiments involving dice to calculate theoretical and empirical probabilities. For a probability experiment (such as tossing two dice) students will write out the sample space and calculate the theoretical probabilities. Next they will perform the experiment and calculate empirical probabilities. This is analyzing because students compare theoretical and empirical probabilities and organize their results.

Evaluating

1. Using graphing software, students will graph a variety of specified equations which are horizontal and vertical shifts of familiar functions, such as y = x^2 + 2 and y = x^2 – 5; y = (x + 4)^3 and y = (x – 3)^3. Using the results, they will try to come up with a method for describing what causes horizontal and vertical shifts, and they will use graphing software to test their prediction. This is evaluating because students have to come up with a hypothesis and then test it.

2. Using scatter plots, students will use their judgment to graph a line that fits the data and determine the equation of that line. Students will then use the linear regression feature of graphing calculators to determine the line of best fit and the regression coefficient. This is evaluating because students have to use what they know to make a judgment and then test their guess.

Creating

1. In groups, students will design a game of chance using the concepts of probability distributions and expected value. The groups will write up the rules of their game and decide on a price to charge. They will also write a short summary that includes the probability distribution and an explanation of why they chose the price that they did. This is creating because they have to use what they know about probability and expected value to design a game.

2. Students will work in groups to plan a trip to another planet. Each group will be given a planet and the speed of their spacecraft, and they have to plan the trip. They will be responsible for determining how much food and fuel they need. This is creating because students are not given a specific method for answering the questions so they must devise their own problem-solving method to researching the distance to the planet, use the speed to calculate length of time, and determine amount of fuel and number of meals needed for the trip.

• Ooh, all the sci-fi and space exploration! Can I be in your class?!

"Students will use what they have learned about triangle congruence to write up a step-by-step method for determining if triangles are congruent" - this is a very strong task for Understanding. It is routinely used in Eastern European mathematics pedagogy, and also in Asian countries. Sometimes, students are also paired later (especially popular in Japan) and explain their methods to one another, which would be Evaluation because they also debug the procedures. Teachers in Japan and China do this for lesson plans, too (step-by-step planning, then sharing/debugging in pairs).

I think your graphing Evaluating task is one of the best overall methods for undersanding the behavior of functions. This is the example of what people call "computer-based mathematics" (in contrast with 'computer-delivered").

• Applying Bloom’s Taxonomy to Math Technology

Remembering

1.       Have students research online to find an example of a real-life scale drawing or model. Students will use any research engines or online sites to access a map or model. This is Remembering since the students are locating, finding, and identifying examples of scale drawings.

2.       Students will create a short PowerPoint presentation (3-5 slides) of how to multiply and divide fractions, including step-by-step examples, intended for a student who was absent for the lesson that day. This is Remembering since the students are identifying and describing important steps from the process of multiplying and dividing fractions.

Understanding

3.       Have students write a letter using a word processor to Fred Maseres to explain the existence and importance of negative numbers, using research to understand the time period of Maseres. The students will first read the following excerpt from an online site on the history of negative numbers. The History of Negative Numbers. Back in Ancient history, these mathematicians were convinced that negative numbers did not exist and were very against the idea, especially Fred Maseres! This is Understanding because it requires the students to use their knowledge of negative numbers and interpret it in a different time period.

4.       Students will create a tweet on what they learned in class that day. If the students do not have a twitter account, they are to create a tweet using a word processor and email it to the teacher instead.  This is Understanding since the students are paraphrasing what they learned in class that day in their own words.

Applying

5.       Students will create a short skit that represents an arithmetic property. Example would be a scene of a student putting on his shoes and then his coat and then a scene of putting on his coat and then his shoes (commutative property).Working in groups, the students will either perform the skit in class or create a video recording of the presentation. The video option will also allow more tech-savvy students the opportunity to play with editing. This is Applying because the students are implementing the definitions of the properties into real-life situations.

6.       Students will research online the distance from the Sun to each planet and convert it to scientific notation. This is Applying since the students are carrying out the process of converting to scientific notation using real-life data that they have researched.

Analyzing

7.       Have students break up into groups of three students each and each group fill in a two-circle Venn  Diagram labeled equations and inequalities. This activity would be done before a lesson on solving inequalities after the students have solved equations. I would use an online program for creating Venn Diagrams (Example: http://creately.com/Draw-Venn-Diagrams-Online) for the students to complete this activity. This is Analyzing because it requires the students to compare properties of equations and inequalities and organize the information.

8.       Students will summarize the information in a chapter by creating their own study guide. Students will use the outlining tools in a word processor. In class, students will be able to share their study guide with other students and see what important information they missed or can expand on in order to study for the test. This is Analyzing since the students are organizing and outlining the important information in the chapter as well as sharing and comparing their ideas with other students.

Evaluating

9.       Students will hypothesize what happens to a graph when the value of the slope increases, decreases, or changes sign. Then students will confirm or revise their hypothesis based on the results they see when they plug in different equations in their graphing calculator. This is Evaluating since the students hypothesizing and experimenting with different linear equations to understand how the slope of the line affects the graph of a linear equation.

10.   Students will use calculators to find a pattern with exponents. They start with any base and then keep increasing the exponent on the base. Once they discover the pattern, the students will hypothesize what a zero and negative exponent will equal based on the pattern that they found. They will then use their calculators to verify or revise their hypothesis. This is Evaluating since the students are experimenting, hypothesizing, and using resources to find out what a zero and negative exponent is based on their previous knowledge of exponents.

Creating

11.   Students will create a scale drawing with a new arrangement for my classroom using the measurements gathered in class. This will be given after the students have been taught about scale ratios and how to construct scale maps. The technology I would recommend the students use is Geometer Sketchpad. This is Creating because the students are designing and constructing their own scale drawings.

12.   Students will create a survey question to gather data on a question with at least four possible choices. They will then type a chart using a table feature in Microsoft Word or an Excel spreadsheet displaying their data with the question that they asked as the title for the spreadsheet. The columns in the chart will include the tallied data under each category, the fraction of the whole that each category represents, the decimal equivalent, and the percentage it represents. Students will use the following link to make a nice computer-generate graph to really showcase their data, refer to the following link: Create a Pie Chart. Students will also write a brief report (at least one paragraph) to summarize the conclusions and observations of their research. This is Creating because the students are designing their own survey and analyzing the results.

• Impressive task design, Gina! What is interesting is that you made your low-level tasks (remembering, understanding) very creative. The letter task, wow. I gather you like history?

Some people would argue that such tasks should be placed at the Creating level since students are designing, building and composing novel identities. I am not so sure. What do you think?

• To answer your question about whether I like history or not, I'm actually not a huge fan. I appreciate the subject, but never really got into it. It's my students who are the history buffs. So I try to incorporate historical accounts into my lessons of the history of numbers and math formulas. They love it!

I do think that you may be right with the fact that the students are creating for the letter assignment. Most of my activities and projects are geared towards gifted students and their interest and talent levels, so I think that is why my tasks involve higher level thinking. Could these taxonomies be differentiated based on student ability levels?

• I see what you did there, Gina! Now, the relationship between giftedness and low-order tasks is somewhat stormy. Some people define giftedness as turning EVERY task into a high-order task: "Seeing the ocean in a drop of water." Some gifted learners can control this feature of their mind and, upon request, turn it off and just do those primitive exercises at low orders of Bloom's. Others have, to use another quote, "an addiction to complexity" and suffer if you try to make them stay at low-order tasks without transforming the tasks creatively.

Math educators of gifted students are faced with a huge problem. Higher-order tasks are very hard to assess. As a result, take a math test, any test... You will see most or all tasks on the test sitting at Applying or lower. How do we help gifted students cope with this?

One of my favorite design appropaches to that problem is what you did, or what Katherine in the response above did for her Remembering task (a scavenger hunt). You make low-order tasks like Remembering a part of higher-order projects. For the purposes of classification, the project as a whole is probably still at Creating or so level on Bloom's, so that gifted students are okay with it. Still, they get those low-order tasks in as their practice...

•

From Lisa Rittler:

Examples from Bloom’s Digital Taxonomy:

CREATING:

1-students create a large square with 100 smaller squares of equal size inside of the large square. This will help teach them decimals by using shading of some of the 100 squares for different decimals. (http://www.ixl.com/math/grade-6/what-decimal-number-is-illustrated)

2-students cut circle out with 8 pieces (like a pizza) & then show proportions of PIZZA that we talk about in class… for example 2 pieces = ¼

Creating because they create the squares & circles themselves

EVALUATING:

1-Students work in pairs with calculators & they do say 5 different math equations .The equations are not the same. for example, subtracting .. & then the other student evaluates whether the other student is correct by doing the opposite math…so they would do addition.

2-Teacher creates bar graph at beginning of year to see how quickly for example students can do multiplication correctly. Each month, the teacher updates it & the students can see their progress. She puts the progress on her website so kids can log on when they are at home to see how they did.

Evaluating ..students and teachers assessing what makes sense to them.

ANALYSIS:

1-students read word problems & analyze what type of math equation they can make to figure out a word problem.

2-teacher shows different ways to do multiplication. …like lattice, partial products, etcThen, she tells the students to do the problem however they like to figure it out & then analyze which way is more successful for her class

Analyzing because you are figuring out what works best

APPLYING:

1-After doing work with angles, help students pretend to build a house on a piece of paper & create walls that are, complimentary, supplementary, etc

2-After a proportion lesson with fractions, have students on stand in the classroom & have them take turns moving other students back n forth to create appropriate fractions to the whole amount of students in the classroom

Applying because you are putting math into real life situations

UNDERSTANDING:

1-Using a smart boards Have students be on teams & work off a website practice test from study island & see who gets more correct…which team wins gets a REWARD J

Showing how they understand the concepts taught

2-Students take turns being the teacher & create a short lesson of how to explain how to do division for example

Understanding because you are able to give examples of lessons taught.

REMEMBERING:

1-Using study island to have students pick a grade lower math than they are & perform these on teams in class

2-Have students create a MATH GAME to remember different geometric figures… they can make teams & have pictures of different thngs in the world & the other teams have to name the geometric shape

Remembering because you are showing that you have remembered something that was learned earlier.

• Lisa, I think it's a great idea to make examples of TEACHER actions - after all, teacher tasks come in different levels too.

I would say that PLAYING a math game for remembering figures and shape is a Remembering task, but developing a game can be a Creating task (depending on how you arrange that). So if students design rules of the game themselves, it will probably go into Creating levels.

I really like that 100-square respresentation for decimals. We could take that same task further by inviting students on a scavenger hunt for hundreds, such as hundred pennies in a dollar, or hundred millimeters in a meter, and do the same task with those objects. Scavenger hunts is one way of making tasks a bit more OPEN for student interpretation.

I want to share a piece of technology that goes with that task, in my mind - and is very beautiful:

• Remembering

Have the students creat flash cards and then work in pairs to match words to their definitions. This is a LOTS. This could be done the good old fashioned way with note cards, or using a smart board. This is remembering because the students must recall information.

Have the students find pictures of the Golden Rule. This can be done on any search engine. This is remembering because the students are searching and finding.

Understanding

Students are responsible to teach a topic to the class. They must understand the material in order to teach it to others. They need to use a power point presentation to present the material to the class. This is understanding because the students must interpret the information into a presentation.

The students must read a chapter before it is taught and put the information into their own words. Paraphrasing is a way to show that the material is understood. This is understanding because the material is being summarized.

Applying

Use different operations to solve equations. The student will use the procedures they have learned in order to solve equations. This is applying because the students must carry out the different operations.

Use a graphing calculator to graph functions. They must know how to operate the calculator in order to ensure the graph is correct. This is applying because they are executing commands in the calculator.

Analysing

Have the students generate a list of random numbers in the calculator. They must then organize their findings into graphs and pie charts using power point. Try to see if there are any patterns or if the numbers are truly random. This is AnalysingHave because they are organizing data into charts.

Look at different functions and their graphs, this can be done on several math sites online or on a graphing calculator. Compare the graphs to see how the changes in the functions affect the 'parent' graphs. This is AnalysingHave because the students are comparing the changes in graphs.

Evaluating

Students work in group to collaborate on a project. They must create a presentation as well as write a paper on the topic they do. Working with others is a way to evaluate material. This is evaluating because the students are collaborating.

Work in Geometers Sketchpad to see what happens to triangles that are inscribed on a circle. Move the vertices around the circle and also change the size of the circle itself to see what happens. Put the results into a poster for classmates to see. This is evaluating because they are checking to see what happens to the triangles.

Creating

Create a flow chart for 4 sided geometric figures. Show the similarities as well as the differences. This can be done on poster board or as a power point. This is creating because students are making flow charts.

Design a floor plan for a house. This can be done on Geometers Sketchpad. They build a scale model of the house designed. Try to make the house as realistic as possible ( no one room houses). Be creative and elaborate. This is creating because the students must design and construct the models.

• Excellent task design! There are quite a few nice projects here. You seem to like charts and diagrams, and yes, complex diagrams make great group tasks because there are many steps to them, meaning work for many people.