Year: 2014

Engaging students: Circumference of a circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Daniel Littleton. His topic, from Geometry: computing the circumference of a circle.

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C3: How has this topic appeared in the news?

On January 29, 2014 an internet based publisher of medical news, news-medical.net, published an article as to the link between waist circumference and health risk factors. The article is entitled Waist Circumference Measurements Help to Detect Children and Adolescents with Cardiometabolic Risk. This study was conducted in Spain and concluded that including the measurement of waist circumference in clinical practices, in conjunction with traditional height and weight measurements, will allow an easier detection of risk factors for cardiometabolic disorders in children. Waist circumference is measured by placing a tape measure at the top of the hip bone and wrapping the tape around the body level with the navel. This measurement is the circumference, and this measurement can be used to determine the radius and diameter of a human body by knowing that circumference is equivalent to 2 \pi r where r is the radius of the circle. This is certainly not the first use of waist circumference in determining health risk factors published in a medical article, however this is a very recent example. This story may be found at the following link: http://www.news-medical.net/news/20140129/Waist-circumference-measurements-help-to-detect-children-and-adolescents-with-cardiometabolic-risk.aspx.

 

 

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A1: What interesting word problems using this topic can your students do now?

One example of an engaging word problem utilizing the concept of a circumference is as follows. “Aliens have invaded earth and they are establishing colonies on Earth. You are a member of the human resistance and you need to plant explosive traps for the alien soldiers. You know that you have enough materials to build one large bomb, one mid-size bomb, and one small bomb. The large bomb has an explosive diameter of 100 feet. The blast radius of the small bomb is one-fifth the distance of the large bombs diameter. The mid-size bomb has a blast radius that is 20 feet greater than the radius of the small bomb. What is the blast circumference of each of your bombs?” This problem requires the manipulation of both forms of the formula for circumference, C=2\pi r and C= \pi d where r is equal to the radius and d is equal to the diameter. The circumference of the large bomb can be calculated directly from the information provided in the problem. The circumference of the small bomb requires manipulation of the data provided. First, the diameter of the large bomb is divided by five. This determines the blast radius of the small bomb which can be used to determine the circumference. The blast radius of the mid-size bomb is determined by adding 20 feet to the blast radius of the small bomb, and then using this radius in the formula for circumference. The solutions for the circumference are as follows. Large bomb: 100\pi or 314.16 feet, Small bomb: $\latex 40\pi$ or 125.66 feet, Mid-size bomb: 60\pi or 188.50 feet. I believe that this problem would present an intriguing challenge to the students.

 

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A2: How could you as a teacher create an activity or project that involves your topic?

An engaging activity that involves the determination of circumference will always need to include the manipulation of circles. One creative way to create circles is to form them through bubbles. The title I have chosen for this activity is “Bubblelicious Circumference.” This activity will require the following materials: bubble solution, straws, rulers, paper, and pencil. First the students will clear their desk surface, after which the instructor will pass out the straws, rulers, and pour approximately one tablespoon of bubble solution on the students’ desk. The instructor will also place a small container with bubble solution inside of it on the students’ desk. The students will first dip the end of the straw that will not go into their mouths into the container with bubble solution inside. Next, the students will place the wet end of the straw into the bubble solution on their desk and gently blow air into the bubble solution. The students will continue to blow air until a bubble forms and pops on their desk. Once the bubble pops it will leave a ring of liquid on the surface of their desk in a near perfect circle. The students will then use the ruler to determine the diameter of the circle that is on their desk. This measurement can then be used to determine both the circumference and the radius of the circle. The students will repeat this process at least 10 times, and as many times as the allotted time for the activity will allow. The circumference data will be recorded for each circle formed by each bubble blown on a piece of paper.

 

Engaging students: Midpoint formula

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Christine Gines. Her topic, from Geometry: deriving the midpoint formula.

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How has this topic appeared in pop culture?

Finding the midpoint between two points is a fairly common situation we find ourselves in daily. Take for example cutting a sandwich into two equal halves. Here you are estimating the midpoint between the ends of the sandwich. Maybe you want the bigger half of the sandwich though. In this case you first find the middle and then move slightly away. Whether we realize it or not, finding midpoints happens all around us and bringing this to students’ attention is crucial for their development of connections.

One way to aid these connections is to demonstrate how midpoints appear in our cultures. In particular, I found a popular music video “Meet Me Half Way” by The Black Eyes Peas. The video/song is about Fergie and Will.I.Am being apart and missing each other. Fergie’s solution is “Can you meet me halfway? Right at the borderline. That’s where I’m gonna wait… for you.” Fergie and Will.I.Am’s beginning locations are the endpoints in this scenario and they will meet at their midpoint. In the video, Fergie has already reached midpoint. Here, her lyrics are “Took my heart to the limit, and this is where I’ll stay. I can’t go any further than this.” This can be interpreted as a unique midpoint. If Fergie goes any further, she will no longer be at the midpoint. Her limit is the one midpoint. At the end of the video, Fergie and Will.I.Am are reunited at their midpoint.

After this connection is made, it could be reinforced by giving students specific coordinates of Fergie and Will.I.Am and asking students to find their midpoint. For example, Fergie and Will.I.Am were shown to be on different planets in the video. So, the teacher could give them the coordinates to Jupiter and the earth. If they succeed with this problem, a follow up could be to find the endpoint when you have Will.I.Am’s endpoint and their midpoint.

green lineHow could you as a teacher create an activity or project that involves your topic?

A common issue students face regarding formulas is memorizing them without fully comprehending the formulas. They say, “give a man a fish and you feed him for a day; teach a man to fish and you feed him for a lifetime.” So, let’s not just give students a formula, but teach them how to derive the formula by letting them explore the concepts for themselves. A good activity to let students do this is as follows:

In this activity students will Investigate finding the midpoint of a line segment and derive the formula for the midpoint of two points on a coordinate plane.

Have students work in groups of 3 or 4. Each group will have a sheet of large graph paper, markers, a ruler, dice and a penny.

Procedures:

  1. Students will find two points by rolling dice and tossing penny (Dice represents number and penny represents positive or negative) and plot them.
  2. They will draw a line to connect these two points.
  3. Next, students can use the ruler to estimate where the midpoint should be.
  4. Have students investigate ways to accurately find the midpoint of the segment and challenge them to find a formula as well.

Students can create several graphs so that they can recognize the patterns. By letting them draw and plot their own graph, students will more readily realize that the midpoint is exactly in between the two x-values and the two y-values. This will then hopefully lead students to recall how to find the average of two numbers, which is essentially what the formula is. It is important that students make this connection to their previous knowledge and to guide students through this exploration, teacher can ask leading questions such as:

  • What could you use to represent the numbers so you can write a formula?
  • How did you find that midpoint?
  • Are you sure that is really the midpoint?
  • How can you find the number in between two different numbers?

 

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I don’t know about you but I’ve always thought the best educational games are the ones that actually feels like a game and not just something your teacher is making you do. This is exactly how the game “Entrapment” by The Problem Site feels like. Entrapment is actually a puzzle game. The object of the game is to create line segments such that all the given dots are midpoints to these segments.

EntrapmentMore specifically, every red dot must be the midpoint of a line segment connecting two gray dots on the playing field. In the image above, the player is one move away from finishing since there remains one red circle which is not a midpoint. This puzzle is not only addicting, but it teaches students to recognize the relationship of x and y (individually) to the midpoint. After completely only a few of these puzzles, this relationship becomes part of your strategy, which in turn pushes students further away from memorization and brings them closer to comprehension. This puzzle brings all these educational benefits, yet it just feels like you’re playing a game!

http://www.theproblemsite.com/games/entrapment.asp#.UxF5ImJdXHQ

 

Engaging students: Defining the terms acute triangle, right triangle, and obtuse triangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Brittney McCash. Her topic, from Geometry: defining the terms acute triangle, right triangle, and obtuse triangle.

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D2. How was this topic adopted by the mathematical community?

            As the students are walking into class, I will already have a picture of just a standard (acute) triangle on the board (be it Promethean or white). As class begins, I will pose the question of, “Who can tell me what we are looking at?” Of course, the students will tell me a triangle. I will then proceed to show two more triangles, an obtuse and then a right triangle and ask the same question. The answer will be the same for each. After I show all three, I will put a picture of all three together and ask the students what some of the differences are in each. Once we state the obvious (That there are angles of bigger and smaller sizes in each), I will then post a picture of Euclid. I will ask if anyone knows who this is. More than likely no one will. I will then proceed to tell my students that in 300 BC this man, Euclid, wrote a book called Elements. In this book. We had 4 sub books that consisted of mainly triangles. When telling this fact I will put emphasis on the word “whole” to show how insane that is. By now, the students should be in awe that someone could write so much about triangles. Then I would state that inside this book, Euclid proved that there were 3 different types of triangles. There is obtuse, right, and acute. We could then discuss as a class what we think each triangle presented at the beginning of class is just by sheer guess, and then see if they were right by then going into the actual discussion of the definition of each term. This is a fun and knowledgeable way to bring in some historical background of what they are learning. This shows that it’s just not going to go away, that it has been around for a while, and is still being thoroughly discussed in classrooms, like ours.

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A1. What interesting word problems using this topic can your students do now?

            An interesting topic that is still around today, is the Bermuda Triangle. After a brief introduction of the definition of an obtuse, right, and acute triangle, I will pose this problem: (There will be a picture of the Bermuda triangle with points labeled, (posted below).)

You are captain of the ship Euclid and are sailing straight for the Bermuda triangle. Hearing of all the bad things that can happen inside the “triangle,” you want to avoid it as best as you can. Luckily for you, you have a super power. You are able to shift one point of triangle wherever you would like. Using your super power and the knowledge of triangles we discussed previously in class, decide which point you would move, and into what triangle so that you can sail past smoothly. You will need to draw your final result with a justification of why you chose that triangle and point.

This question is not only engaging, but it makes them think abstractly. They have to use their knowledge of triangles and produce a result that fits our discussion. Then not only do they have to draw it, they will need to discuss it as well. Talking about why they chose the method they did, helps students retain and process the information better. Take into account, there are multiple ways to answer this question.
Bermuda

 

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B1. How can this topic be used in your students’ future courses in mathematics or science?

            Triangles are such a widely used topic, that it is almost guaranteed you will see them again. Not only will you use them later on in our course (Geometry), but you will be using them in pre-calculus and so forth. The main triangle you will work with in the future is the right triangle. That is why it is so important for you to learn the difference now between the triangles. Later, you will be discover the different ways you can solve for sides and angles with a right triangle, you will be discovering the different properties that come with each triangle, and how you can draw them using circles. But before you can do any of that, you have to start with the basics, like knowing which triangle is which and their definitions. I would then go in to explain that now only would triangles be used in classes, but in the real-world as well. They are everywhere we look, literally. Every time we look, we are looking at a specific angle. In the video games we play, we are always making decisions based off of the angles we can use, it’s how we build things; it’s everywhere! To have a basic understanding of something so usable in our world, would be essential to success!

Resources:

http://en.wikipedia.org/wiki/Triangle

http://www.livescience.com/23435-bermuda-triangle.html

Engaging students: Dilations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Candace Clary. Her topic, from Geometry: identifying dilations.

green lineHow could you as a teacher create an activity or project that involves your topic?

Dilations are types of transformation. One activity that I could create for my students is a matching game. I can create cards with index cards, or sheets of paper that have been cut up, that have pictures on them. Each one will be labeled and the students must classify them as dilations, why they are considered dilations, and how they were dilated. As a follow up to this activity, I could assign a topic to create their own city, or small town. They would be required to draw out their town, as well as model it using common crafts. After they do this, they will need to be able to dilate the buildings, and other such things, to make a life size city. They will not have to make the city with a model, but instead, they will need to make a blue print using their model in mind. On this blue print, they will need to inform me of the size of the dilations.

green lineHow can this topic be used in your students’ future courses in mathematics or science?

Dilations can be used in many different subjects. Dilations can be used to find sides of a triangle when learning about the triangle congruence theorems. These can be useful in algebra when finding side lengths of figures. This may not happen very often, but it is displayed in algebra. Another place that it will help, although it may not be math, it will help in math classes for architectural students, as well as help people in construction. Many science classes require science projects that work and simulate something real. Dilations can be used when making these projects because you can’t make a real river, but you can structure something that is a smaller figure to the real thing, same thing as a volcano. With architecture, dilations can help with making blue prints and can help in building these blue prints with dilations in mind. With construction, those are blue prints too. I’m not saying in order to build something you must know how to dilate something, but it will help tremendously.

 

green lineHow has this topic appeared in pop culture?

To get the students engaged in the topic, I could bring up the Disney channel movie ‘Honey I Shrunk The Kids.’ This will bring up a discussion with the kids when I ask them what the dad did with his shrink ray. Some ideas that may come up will be that he made them smaller, and then at the end of the movie he made then bigger, back to normal. But in the people were still the same people, they didn’t change, only the size did. At least I hope that is what happens in the discussion. I could then instruct the students into pretending that they had a shrink ray and ‘shrink’ some shapes, as well as other students. This activity, and their answers will be recorded on a chart that they will turn in at the end of class. They, themselves, can decide what size they want to shrink to, but they have to remember to bring the student back to normal at the end of class. I think this activity will be fun for the kids because they will never forget what a dilation is, since they have been ‘dilated’.

Engaging students: Finding the area of a right triangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Andrew Wignall. His topic, from Geometry: finding the area of a right triangle.

green lineWhat interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

To introduce the topic of the area of a right triangle early in a lesson, we can first examine the area of a rectangle, which students should already know how to do.

Say you have a large rectangular garden, 60 feet wide and 10 feet long. Home Depot sells sod (which is a pre-grown grass on a net that can be spread on the ground) at a rate of $3/square foot. What is the area of the garden, in square feet? How much sod should you order? How much would it cost to cover the entire garden with sod?

Instead of having the entire garden covered with sod, suppose you wanted to cover part of the garden with sod and leave the rest as soil for planting flowers. To make it more visually interesting, you decide to set the sod as a triangle? The sod triangle will have a base of 60 feet and a height of 10 feet. What is the area of this triangle in relation to the area of the entire garden? What is the area of this triangle? How much sod should you order? How much would it cost to cover the triangular area with sod?

Through this activity, we can investigate a relationship between right triangles and rectangles, and also the relation of the area of a triangle compared to the angle of a rectangle.

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website

One tool to show the area of a right triangle quickly and easily is the Area Tool on Illuminations (http://illuminations.nctm.org/Activity.aspx?id=3567). With trapezoids, parallelograms, and triangles available, you can click and drag the three vertices of a triangle and instantly see how the area is affected. You can create a quick table and keep a running tally of the base, height, and area, so you can recalculate in front of the class.

Illuminations has a sample lesson plan available online for discovering the area of triangles, and integrates this tool into the plan. If not using this tool as part of a similar plan, we must understand that this tool will not be great for introducing the lesson, as there is no button to lock onto a right triangle. However, there is a button to lock the height, so when you move the vertex opposite the base, you can see how the area does not change, see how the height can be outside the triangle, and extend the formula for the area of a right triangle to the area of any triangle. This tool can then be used in further lessons when discussing the area of parallelograms and trapezoids.

 

green lineHow can this topic be used in your students’ future courses in mathematics or science?

Since triangles are one of the most basic shapes, the area of triangles comes up time and time again. Triangles will also be used to find the area of more complex polygons, such as hexagons and irregular polygons, by breaking down complex shapes into simple triangles and quadrelaterals. Trigonometry uses right (and non-right) triangles extensively; in Precalculus, we will revisit the area of triangles, and learn how to find the area of triangles without explicitly being given the base and height.

Outside the classroom, the area of a triangle is used extensively in architecture, as triangles are strong, and triangular trusses and frames are used in many steel structures. As the inside empty area of the triangle increases, then the stress on the triangle increases, and architects must take this into consideration.

Triangles are also used in 3d computer graphics, as the 3d shapes they design actually consist of lots of little triangles, and they have to fit textures of a certain size (say 512 pixels x 512 pixels) onto a few triangles, so it is important that they know how and where for these textures to lie.

 

References

Math is Fun, “Activity: Garden Area”. http://www.mathsisfun.com/activity/garden-area.html

Illuminations: Resources for Teaching Math, “Discovering the Area Formula for Triangles”. http://illuminations.nctm.org/Lesson.aspx?id=1874

Illuminations: Resources for Teaching Math, “Area Tool”. http://illuminations.nctm.org/Activity.aspx?id=3567

Home Depot, http://www.homedepot.com/p/StarPro-Greens-Centipede-Southwest-Synthetic-Lawn-Grass-Turf-Sold-by-15-ft-W-rolls-x-Your-L-2-97-Sq-Ft-Equivalent-RGB7/202025213

Math is Fun, “Heron’s Formula”. http://www.mathsisfun.com/geometry/herons-formula.html

Maths in the City, “Most stable shape – triangle”. http://www.mathsinthecity.com/sites/most-stable-shape-triangle

Andre LaMothe, “Texture Mapping Mania”. http://archive.gamedev.net/archive/reference/articles/article852.html

 

Engaging students: Defining the terms perpendicular and parallel

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Allison Metlzer. Her topic, from Geometry: defining the terms perpendicular and parallel.

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B1. How can this topic be used in your students’ future courses in mathematics or science?

The concepts of perpendicular and parallel will be implemented in many of my students’ future mathematics courses not only in high school, but also in college. In algebra, the students are asked to find the slope or the rate of change. In looking at the slope, students are asked to find if it’s parallel or perpendicular to another function’s slope.

In geometry, many shapes have properties that define them as having parallel or perpendicular sides (i.e. squares, rectangles, parallelograms, etc.). Also, in order to decide if triangles are similar, their corresponding sides must be parallel. In order to use the Pythagorean Theorem, the triangle must be right angled or have the two legs perpendicular to one another.

In calculus, students are asked to find orthogonal vectors which are also defined as perpendicular vectors. Also, calculus incorporates concepts from algebra and geometry which in turn, include parallel and perpendicular lines.

Therefore, many, if not all of my students’ future math courses will use the topics parallel and perpendicular. Thus, it would be important for me to teach them the two concepts correctly now so that there wouldn’t be any misconceptions in the future.

 

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C3. How has this topic appeared in the news?

One big thing the news talks about every two years is the Olympics. Using the concept of parallel and perpendicular, the constructions are made for all of the different events. Apparent examples of events incorporating parallel lines are track, speed skating, and swimming. The one I will focus on is swimming, namely because it is a very popular Olympic event and one of my favorites. Pictured below is an Olympic swimming pool of 8 lanes. Do the lanes appear to be parallel? Two things that are parallel are defined as never intersecting while also being continuously equidistant apart. One can clearly see the lanes of the pool never intersect. If they did, then the contestants could interfere with one another. Also, because the Olympics is a fair competition, the lanes are equidistant in order to give each contestant a fair and equal amount of room.

Because the Olympics is a well-known event featured in newspapers, articles, and on TV, the students will be able to understand this real world application of parallel and perpendicular.

pool

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E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Before I would play the video, https://www.youtube.com/watch?v=vnnwfcDcNlY, I would first ask the students to think of as many examples they can of parallel and perpendicular in the real world. After about a couple of minutes, I would tell them to keep those in mind and see if the video included any they didn’t think of. I would play the video from 1:25 to 3:05 which is the portion that displays all of the examples. It has clear pictures of recognizable objects which incorporate parallel or perpendicular lines. Also, the video has labels on the pictures to even more clearly describe where the components of parallel and perpendicular lines are. I believe that the initial brainstorm along with this video would get the students thinking about the importance of parallel and perpendicular lines. Also, I would make the connection that those examples would not be considered parallel or perpendicular unless they met the following definitions. Then I could explicitly define both parallel and perpendicular.

Thinking of real world examples, and seeing pictures of them will help the students understand what parallel and perpendicular lines should look like. After they have this initial understanding, they then could get a better grasp of the definitions. Also, they would recognize the importance of following the definitions to correctly construct objects involving parallel and perpendicular lines.

References:

Detwiler, dir. Intro to Parallel and Perpendicular Line. YouTube, 2010. Web. <https://www.youtube.com/watch?v=vnnwfcDcNlY >.

Medicine’s Uncomfortable Relationship With Math: Calculating Positive Predictive Value

Taken from: http://archinte.jamanetwork.com/article.aspx?articleid=1861033&utm_source=silverchair+information+systems&utm_medium=email&utm_campaign=archivesofinternalmedicine%3aonlinefirst04%2f21%2f2014

In 1978, Casscells et al1 published a small but important study showing that the majority of physicians, house officers, and students overestimated the positive predictive value (PPV) of a laboratory test result using prevalence and false positive rate. Today, interpretation of diagnostic tests is even more critical with the increasing use of medical technology in health care. Accordingly, we replicated the study by Casscells et al1 by asking a convenience sample of physicians, house officers, and students the same question: “If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about the person’s symptoms or signs?”

Approximately three-quarters of respondents answered the question incorrectly (95% CI, 65% to 87%). In our study, 14 of 61 respondents (23%) gave a correct response, not significantly different from the 11 of 60 correct responses (18%) in the Casscells study (difference, 5%; 95% CI, −11% to 21%). In both studies the most common answer was “95%,” given by 27 of 61 respondents (44%) in our study and 27 of 60 (45%) in the study by Casscells et al1 (Figure).

Encouraging Students to Tinker

A recent blog post from Math Ed Matters had the following pedagogical insight:

How do we encourage students to tinker with mathematics? As a culture, it seems we are afraid of making mistakes. This seems especially bad when it comes to how most students approach mathematics. But making and then reflecting on mistakes is a huge part of learning. Just think about learning to walk or riding a bike. Babies are brave enough to take a first step even though they have no idea what will happen. My kids fell down a lot while learning to walk. But they kept trying.

I want my students to approach mathematics in the same way. Try stuff, see what happens, and if necessary, try again. But many of them resist tinkering. Too many students have been programmed to think that all problems are solvable, that there is exactly one way to approach each problem, and that if they can’t solve a problem in five minutes or less, they must be doing something wrong. But these are myths, and we must find ways to remove the misconceptions. The first step is to encourage risk taking.

A few months ago, Stan Yoshinobu addressed this topic over on The IBL Blog in a post titled “Destigmatizing Mistakes.” I encourage you to read his whole post, but here is a highlight:

Productive mistakes and experimentation are necessary ingredients of curiosity and creativity. A person cannot develop dispositions to seek new ideas and create new ways of thinking without being willing to make mistakes and experiment. Instructors can provide frequent, engaging in-class activities that dispel negative connotations of mistakes, and simultaneously elevate them to their rightful place as a necessary component in the process of learning.

Here are a few related questions I have:

  • How do we encourage students to tinker with mathematics?
  • How do we destigmatize mistakes in the mathematics classroom?
  • How do we encourage and/or reward risk taking?
  • What are the obstacles to addressing the items above and how do we remove these obstacles?

Source: http://maamathedmatters.blogspot.com/2014/04/encouraging-students-to-tinker.html