## EXEMPLARS.

## TEACHING EXEMPLARS:

## ASK.

I spent most of my time preparing for this post by researching the types of questions teachers should ask their students. The resource I found most resourceful, and what I will use most to structure this post is The Learning Center. The main point of the article was that** “Active learning extends beyond the classroom,”** and that’s what I found most intriguing and influential when reading onward. If students are introduced to asking questions and mastering concepts in school, then they can take those strategies and apply them to their studies outside of the classroom and in other areas of life. Most of the question-structuring points they brought up I agreed with but never would have come up with my own.

The first, and most obvious is:** “When planning questions, keep in mind your course goals.” **When developing questions to ask your students, you should be able to tie them into what you are learning. If they aren’t structured around the key concepts of your lesson, then chances are you aren’t asking the right questions. The questions you ask should help them develop the skills you are hoping they acquire and to aid in understanding of material you are covering. The questions you ask should make them engage and think deeper about what you are hoping they can take from that lesson and into life.

The next is: **“Avoid asking ‘leading questions.’” **I have had teachers who are really bad at this one. They ask a question knowing full-well what they want your answer to be. This is an absolute kill to creativity in the classroom. If you expect your students to think on their own, then your questions shouldn’t close off that process. Carefully think out what you ask before you say it aloud. Give your kids a chance to do some of the heavy lifting.

**“Follow a “yes-or- no” question with an additional question.” **Don’t give your students the easy way out. Maybe they were taking a guess or just spitting out an answer they heard from across the room and aren’t fully understanding any concepts themselves. Challenge them to answer “why?” or give evidence of their answer. If they can prove their answer to you, right or wrong, you know they were thinking about the concepts and problem solving a little on their own.

**“Aim for direct, clear, specific questions.” **Your goal as a teacher isn’t to trick your students out of answering a problem. Be specific and to the point about what you are asking. When posing a question, try a series of shorter questions in sequence rather than a long, complex question with multiple layers. Your aim is to see whether your students are comprehending material, not whether they can sift through a difficult question to answer what you were hoping for. If your question is aimed at the right material as simply as possible, then your students are able to give their best work and demonstrate back to you as clearly as they can.

A classic way you can avoid a quiet classroom is: **“In class discussions, do not ask more than one question at once.” **Students, especially the ones who are uncomfortable with speaking aloud or with the material you are teaching will be fearful to respond when they don’t want to be wrong. This phenomena is increased when they don’t know which question you want answered. One at a time will allow for full brain concentration on that question, and less confusion in the classroom. This way will also allow for your students to track with the class discussion better and know what topics you are covering without getting lost.

**“When you plan each class session, include notes of when you will pause to ask and answer questions.” **The more organized you are with your lesson plan, the more organized the material will be in your student’s notes and heads. Put some care into your planning and think about good placement of discussion and question time. Maybe splitting up into groups and brainstorming could be helpful. Also take into account anonymous questions you can submit by writing questions down when material is tricky and students are fearful of speaking up. Always make yourself open to questions, too. The more open of a layout you have to aid your students’ learning, the more open they will be about asking and learning during class.

The final point was worded so well by the authors that I decided to keep it in their exact words. No two students learn the same, so questions should be posed in a way helpful to each learner. This means different questions sprinkled through at different times. ”**Ask a mix of different types of questions.** You should use** “closed” questions**, or questions that have a limited number of correct answers, to test students’ comprehension and retention of important information. You should also ask **managerial questions** to ensure, for example, that your students understand an assignment or have access to necessary materials. **“Open” questions**, which prompt multiple and sometimes conflicting answers, are often the most effective in encouraging discussion and active learning in the classroom. For examples of “open” questions and the purposes they can serve, see below.”

The website also has a lot of important tips on how to allow your students to respond effectively. These include:

**“Wait for students to think and formulate responses.”** Give 5-10 seconds before rephrasing the question. Don’t simply answer your question as this will communicate to your students that they don’t have to answer. Continue posing your question until some answers are formulated.

**“Do not interrupt students’ answers.” **If you aren’t willing to allow your students to finish their responses, it may influence how or if they volunteer an answer in the future. They need that moment to find some ownership of the material and it will help you determine their understanding.

**“Show that you are interested in students’ answers, whether right or wrong.” **Look engaged and excited about what students are offering in class. Their input is important, and your encouragement can help them find value in the material and in the art of asking questions. If you are interested in them, they will likely stay interested in you and your teaching.

**“If a student gives an incorrect or weak answer, point out what is incorrect or weak about the answer, but ask the student a follow-up question that will lead that student, and the class, to the correct or stronger answer.” **Make sure they know what might have been not the strongest of answers, but give them another chance. The class can help out, but they shouldn’t feel put down or discouraged by answering incorrectly. Redemption can be a powerful thing.

Pointing out mistakes doesn’t have to be the only way– sometimes turning things back to the class to see what they think is a good way to yield the floor for some new input. If you do this every time a student is mistaken, it becomes a cue.

Also, I can find that if I am asking myself: “What do I want to know?” It will help guide my classroom in the right direction as well. Focus will remain on the topic and relevant to student learning.

Already, after reading that article and watching a few videos about the same material, I have a completely changed perspective on the importance of asking and posing questions in class. I also realized that some of my best and favorite teachers asked questions in similar ways that allowed me to gauge my comprehension and feel comfortable in the classroom. I want to make sure that I allow for full opportunity for all of my students. The use of questions in the classroom can be a major factor in the atmosphere of education and that class in particular. This information will always stick with me as I burst forth into a career in teaching.

Some feedback on this post would be how most teachers acquire this information and skills in this area. Do you find that these things reign true in a classroom setting? Any exceptions to the rule?

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## QUADRILATERALS

This week, I thought it would be helpful to merge some of the activities we did in class into one lesson plan to help introduce fourth grade level students to quadrilaterals.

I also drew some information from the Utah Education Network:

“To aid understanding, teach quadrilaterals as a whole. Define quadrilaterals as a four-sided figure and give students the opportunity to create a variety of quadrilaterals. They look for similarities and differences and sort them into several different categories according to their attributes. The sorting activity offers insight into the mathematical hierarchy used in classifying quadrilaterals. It will become clear that every quadrilateral falls into three categories:

- those with two pairs of parallel sides,
- those with only one pair of parallel sides, and
- those with no parallel sides.

This activity will set the stage for students to understand that many types of quadrilaterals exist and that these shapes have some elements in common.”

Another introduction activity I find essential would be to make poster boards with different comparisons like “rhombus vs. square,” “rectangle vs. square,” etc. over large Venn diagrams. and split the classroom into groups so they can write down what they think are different and similar between them. After the groups are finished they can present their findings to the class. Allow the rest of the class to respond to what other groups found.

Here is more from UEN that I would borrow for my classroom. Some of the activities are similar to what we did in class with geoboards, sorting, and Venn diagrams.

Invitation to Learn

Provide each student with a geoboard and geoband. Ask them to create several four-sided polygons, then choose their most unique quadrilateral to share with their group.

Instructional Procedures

- Ask the students to compare their quadrilateral with those made by other members in their group. Are all quadrilaterals different? If not, agree on how to make them look different. Record quadrilateral on
*Geodot Paper*and cut shape out for display. - Invite each group to post their quadrilaterals in one of three columns:
- those with one pair of parallel sides,
- those with two pairs of parallel sides, and
- those with no parallel sides.

Give students time to determine if all the quadrilaterals are in their appropriate columns. Discuss congruent and similar shapes and remove any duplicates.

- Identify the columns with the appropriate headings:
*trapezoids*(one pair of parallel sides),*parallelograms*(two pair of parallel sides), and*trapeziums*(no parallel sides). - Use the
*Quadrilateral Family Tree*handout to discuss the properties, attributes, and characteristics, as well as the interconnective and hierarchical commonalities and differences, between and among quadrilateral shapes.- Have the students look at the relationship between squares and rectangles. What are the characteristics of each? Is a square a rectangle? (Yes, it has four equal angles.) Are all rectangles squares? (No, many rectangles do not have four equal angles and four equal sides.)
- Have the students look at the relationship between squares and rhombuses. What are the characteristics of each? Is a square a rhombus? (Yes, it has four equal sides.) Are all rhombuses squares? (No, many rhombuses do not have four equal sides and four equal angles.)
- A Venn Diagram is a good visual aid to illustrate that a square is both a rectangle and a rhombus.

- Further explore the relationships between quadrilaterals by having the students work with roping quadrilaterals. Provide each pair of students a set
*Quadrilateral Pieces*and two or three pieces of string to make a*Quadrilateral Venn Diagram.*Ask them to place the appropriate quadrilateral pieces in each ring according to the following labels:

Ring 1 (Left side): At least one pair of parallel sides

Ring 2 (Right side) No sides parallelAsk students to justify their placement of different pieces. What do all the shapes in one ring have in common? How might the shapes in one ring be different? (Some shapes in Ring 1 are trapezoids, and some are parallelograms.) What different label would eliminate one or more of the shapes from a ring? (Only one pair of parallel sides.) If we drew a giant circle around everything, including any shapes that are outside the rings, what might the label for this new ring be? (Quadrilaterals) Try further explorations using the following labels:

Ring 1 (Inner ring): All sides of equal length

Ring 2 (Outer ring): At least one pair of parallel sides

Ring 1 (Left side): At least one right angle

Ring 2 (Right side): No right angles

Ring 1 (Left side): All sides the same length

Ring 2 (Right side): At least one acute angle

Ring 1 (Left side): At least one set of parallel sides

Ring 2 (Right side): At least one obtuse angle

To test for understanding I could have my students write reflections explaining where they would place quadrilaterals or have them place them on a Venn diagram. A good relationship to have students explain is between the rectangle, rhombus, and square.

As I reflect on this activity, I wonder how I would choose to break up these activities in a unit and what things would be important to include that I may have forgotten. I also wonder if there is anything that I could cut out or if I need to take a different direction with some activities to aid learning of students with different abilities. Overall, I enjoyed looking through our work over this unit, and learned a lot when reflecting on what was helpful for me as a learner.

I think this lesson adds a sense of ownership to the material for the students. They can work in a small group setting and share some ideas that they might not have in front of the whole class. They also have the opportunity to share what they have done- to feel some pride over their work.

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## DOING MATH EXEMPLARS:

## BRAILLE.

So I spent a lot of my time preparing for this post by looking at Braille and how that system could apply to teaching math in elementary schools. I read the article with lesson plan ideas for elementary students and learned a lot about the history of Braille and who invented it. I found it really interesting how there was a system before Louis Braille invented his system. He took something that wasn’t practical for his lifestyle and changed it to suit him better. The assignment of different patters to letters and numbers was intriguing and I don’t know how hard or easy it would be to be blind and learn how to read it. I loved the options for the activities to do with students especially the younger ones using colors and then moving forward with the older students and teaching them more about the background of Braille and diving deeper into the idea behind our projects and assignments.

I furthered this look at Braille by doing the worksheet that we received called “Pixel Patterns.” I basically began by making my own mock Braille system which looks like this:

You can see that I could use my system to write my name as well. I didn’t plan it out in any specific way, so I figured there was probably a more patterned way of arranging the dots, hence Braille. The Braille alphabet has a more systematic way of arranging the dots going from one letter to the next, and it even has a symbol to motion for a capital letter– something I hadn’t even thought of! Here’s what my name looks like in Braille.

For further confusion and curiosity I went ahead and shaded in my own 5 pixel alphabet. It would be hard to memorize and get used to, but here it is!

It’s hard for me to pinpoint a pattern to the Braille alphabet, but looking at it I can tell it sequences well. I would be curious to hear how my students would describe the pattern, and I would also like to give them the opportunity to make their own alphabets and explain why they chose to do it that way.

I also watched the Domino Chain Reaction video, pretty neat!

The creativity quotes page was a great read as well.. here’s one that stuck out:

“The trick to creativity, if there is a single useful thing to say about it, is to identify your own peculiar talent and then to settle down to work with it for a good long time.”

We as teachers need to remember that our students have unique talents and can be creative in different ways.* I hope to structure my classroom to allow for that creative space for all of my students one day.*

I could do this by allowing for more creativity on assignments- like this braille worksheet, no two students would create the exact same alphabet. The same should go for most projects. Creativity shouldn’t be shut out just because a child is in math class. Instead, it should be encouraged!

How do you feel my blog post went this week? I sure learned a lot while creating it, and felt my own creative juices flowing. I really loved learning about Braille more. I now have a better understanding after only a few hours of looking into it’s history.

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## EN CROIX.

Working on furthering our study of patterns this week, I joined up with the “dance crew” to try doing math with our feet. While working on this task, I immediately turned to my ballet instincts. I dance, and a lot of things in ballet are structured around a square. At the beginning of ballet class, you start of at the barre and warm up. Most every combination is focused on your own little square that you point and extend your feet and legs to. At the barre, it is most common to have combinations that work directly to your front, side, and back. If a combination makes it’s way around repeating the same movement to the front, side, back, and then back to the side it is referred to as “en croix” which translates to on the cross. Your feet are moving in a pattern that resembles the shape of a cross. Not only can patterns be found in the repetition of your exercise at every position, but also the consistency of using “en croix” between every new exercise introduced. Yeah, ballet patterns!

For my weekly work I took pictures of different ballet positions in centre- usually what comes after barre in class. These positions are also centered around a square and are consistent throughout most ballet classes. Therefore you can find a pattern working within the class to see everyone using the same position, but you can also see the pattern between different classes that use the same positions.

1. Croisé devant

2. En face

3. Écarté devant

4. Efaccé devant

5. A la seconde

6. Croisé derrière

7. En face

8. Écarté derrière

9. Efaccé derrière

10. Á la seconde (opposite of above)

Someday I might want to teach ballet. If I do, and if I work with young children, incorporating pattern work would be a part of my class. Maybe allowing those kids to see patterns in something they love with help them excel in school.

I loved working and thinking about my work this week. Ballet is my life and incorporating into the classroom and even my homework is awesome. It’s important to recognize what you love, so that once you manage a classroom you appreciate those things and can let the students have some of their favorite things incorporated as well.

A few questions I’d like to answer from feedback:

Are there any typical patterns or sequences in ballet that get reused a lot?

Usually in class, we will work on a sequence or a pattern of steps each class period and add on to it as we go. Sometimes these sequences occur in “barre” which is the first warm up we do in class or in “centre” which is the next part of class using the whole floor. One sequence I can think of is “tombe, pas de borre, glissade, grand jete.” It’s a very common and flowing set of movement I’ve had in nearly every class I’ve taken. Here’s a link to a dancer doing just that! http://www.youtube.com/watch?v=N872wtRZANE

It would be interesting to see these abstracted to positions on a square.

here’s an image I found that puts all the positions on a square- it’d be hard to follow if you weren’t already familiar with the positions: http://www.laban-analyses.org/jeffrey/1996-jeffrey-scott-longstaff-phd-thesis/IVA60_ballet_prototypes1.jpg

Is there a way/system ballet choreographers use to write down a dance? Or is it just visual record of the dancers’ movements?

Some teachers will write down combinations to help you remember if need be, I’ve done it a handful of times myself, but it’s usually just a jumble of words that you wouldn’t understand unless you are familiar with the way that teacher speaks or have a little understanding of how the dance should go. One of the best ways to record movement is by video. When I collaborate with a friend of mine while working on pieces, we often record entire class sessions when we are creating so we can reference anything we might have forgotten.