Classroom Culture: Too Focused on "The Correct Answer."

American Classroom Culture

Today we had an emergency drill and a half day, so each class was only 25 minutes long (compared to the 85 that we are used to on a block schedule). So instead of doing mathematics directly, I decided to play a game: You start out with 21 beans. Each player can take 1, 2 or 3 beans. The person who goes last, wins. It is a very simple game. I wanted the class to know that there is more to this game than just beans, so I told the students that "There is a winning strategy." and "If anyone can beat me, then I will give them a piece of candy."

As students played this game and challenged me, they were looking for the "correct strategy." Only one or two students per class figured it out, mostly by watching me and analyzing my "correct strategy." Now that I reflect on this experience, I shouldn't have said "There is a winning strategy" I should have simply said "If you can beat me, then I will give you a piece of candy." By implying that there is only one correct strategy made students focus on what I was doing instead of exploring other ways to play this game.

In addition to this, my gut instinct as a teacher is to tell everyone the "correct answer" at the end of the class period. I resisted this urge and most students left class without understanding how to win the game every time.

Japanese Classroom Culture

Several months ago I watched two classes solve a simple question. The first class was an 8th grade class and the second was a 7th grade class. The question was if you cut a cube diagonally, what is the shape of the cross section:

image from: The Electronic Journal of Mathematics and Technology, Volume 2, Number 1, issn 1933-2823. 

Retrieved from: http://lib.znate.ru/docs/index-134315.html on Sept 24, 2014

In order to solve this problem, you need to use of the Pythagorean Theorem. The 8th grade students had learned about the Pythagorean Theorem, therefore many of them figured it out. However there were groups who did not figure it out and the teacher did not give the answer at the end of the class period.

The 7th grade students haven't studied about Pythagorean Theorem. Very few students in this class solved the problems, but the teacher did not give any hints or give the answer at the end of the period. But the students still worked until the end trying to solve it.

"Correct Answer" or Process

From watching these Japanese lessons, I realized that I am too "correct answer" focused. I knew going into our bean game that I wasn't going to tell the entire class the strategy. The reason is, if students know that they will get spoon fed the answer at the end, then they loose the incentive to solve it next time. It is the process that we are more concerned about anyway, right?

Stories One Word at a Time

I know that I usually post about Mathematics Education, but I am getting a Masters Degree in Education with an emphasis on English Language Learners. I also teach a free English Conversation Class at my church once a week. This class has a 60 minute lesson and then a 30 minute game. Last week I took a game that I learned from my Drama Students (oh year, my wife and I also run the Drama program at my school).

The students sat in a circle and told a story one word at a time. One of the teachers wrote the story on the board so everyone could see it. The story didn't make much sense. It started off with one of the teachers sleeping and then her bike was flying and then a plane flew between football goal posts, etc. It was all over the place. One aspect of this game that I really enjoyed was that it required students to think about grammar and word choice. We had to corrected the grammar as we went and have short discussions about what is correct and why and it was a great way to emphasis basic grammar including articles, which my students struggle with. My students were initially taught English in a very structured setting with worksheets, this was a very open activity which stretches their abilities.

If you have an ESL class, I would encourage you to try a game like this.

Homework in a Flipped Math Class?

I spend more time assessing if my students did notes than the length of the video(s).
This morning I had a realization. I spend more time assessing whether or not my students did notes (via a 3 Question Daily Quiz and me checking notes) than it would take to watch the video in class. For example it may take around 20 or 25 minutes to hand out Daily Quiz Papers, check each student's notes, and correct the Daily Quizzes. The videos are about 10 minutes long and I assign one or more commonly two for homework. This means that I could simply show the video during class and either gain 10 minutes or not lose any time at all.

This is powerful for many reasons:

•  I have seen several students watching the video while listening to music. So they aren't hearing my explanations and listening to my think-alound as I do one or two example problems.
• I have also seen other students who fly through the notes because they move the video scroll bar to where I have writing and pause it, then move the video scroll bar to where I have writing, ... or in other words copy from the video instead of watching and listening.

If we watched the video in class then I would know that they were both watching the video and listening to the video.

The downside is that many students watch the video, pause it to write notes, etc. like they are supposed to. I personally love the idea of my students taking notes for homework because they can take notes at their own pace. I would hate to change the nature of the class because a few students think the notes are more important than learning from the video. I know that the videos are available online, therefore they could go back and take better notes if they want, but I don't know if they would.


After this realization ...
If I do decide to simply watch the videos during class, I realized that the video doesn't have to come at the beginning of the class period. It could come somewhere in the learning cycle. Whenever I hear Ramsey Musallam (Twitter: @ramusallam) talk about the importance of exploration, learning cycles, and his explore-flip-apply model I completely agree but I struggle knowing how to do it in Geometry in my classroom. This could be one way that I can do it.

My class could:

1. Spend some time exploring a concept using a paper activity, web activity, NCTM Illuminations activity, etc ...
2. Then watch the video and taking notes
3. Do some practice activities using problems from the textbook, or completing some task to show mastery.
4. Synthesize their learning with a detailed summary.

No homework in a math class?
This will only work if I don't assign any homework at all. No matter what I assign for homework, I will want to go around the room and assess it. This generally takes 20 to 30 minutes of valuable class time. On the other hand if I limit our math learning to 85 minutes every other day, I wonder if this will be a detriment or not. Assigning homework is a great time to have more time on task, especially for my lower students. When I am considering making a change my first thought is "How will it impact my lower students?" My top students will be fine as long as I deliver content, but my low students need support.


Explore-Flip-Apply?
If my students are only working during class, then a struggle is created between exploring and applying. I want to keep my students together. I am afraid that I might spend too much time exploring or applying and not keep up with the pace of the class.

I really struggled last year with asynchronous learning, my top students went really fast and my lowest students went really slow. If I keep them all together then the top students help the lower students. This is especially important my classes because the difference between high students and low students is really wide (at a high school with no honors math classes in a class required for every student).


Well...
This blog post isn't as enlightening as I hoped it would be. But it has helped me think through some different possibilities that I need to keep thinking about.

If you have any suggestions or comments, please leave them below. I would love to read your thoughts.

Another visit to a Japanese High School

Today I returned to Kanagawa Sohgoh High School with two math teachers from my school.

We observed three math classes and a technology class. During first period we watched two different teachers teach the same lesson. They were sophomore level math classes. We were in the first class for 45 minutes and then switched classes to the other teacher for the remaining 45 minutes. The most amazing thing is that they were teaching the same lesson on the same day at the same pace. We walked into the second class almost exactly where the first teacher left off. I don't think that this is common, but it was amazing!

Curriculum

I am continually amazed that on the surface, Japanese schools and American schools don't look much different. The teaching styles are similar enough that most people wouldn't notice huge differences between Japanese teaching and American teaching. American teachers would categorize Japanese teaching as very traditional. What is drastically different is the curriculum.

Again, we watched a sophomore level math class (Math I) and they were studying Number Theory. Specifically they talked about:

• Categorizing according to divisors and reminders. For example categorizing numbers according to 3k, 3k+1, and 3k+2, where k is an integer.
• Proved theorems that involved categorizing numbers. For example proving that the square of any odd number minus 1 is a multiple of 8. Or proving that multiplying n consecutive numbers makes a number that is a multiple of n!.
• Finally they proved and then used the Euclidean Algorithm to find the Greatest Common Divisor.

I didn't learn the content that they were learning until I was half way through my undergraduate mathematics program. Not that American students couldn't learn this material, it just isn't found anywhere in our curriculum.

The third class we attended was a senior level class. There were a total of five students in this class and they were preparing for College Entrance Exams. They played math games for the first half of the class and then reviewed one very challenging math problem during the second half of the class. This problem involved triangles, circles, inscribed angles, area formulas, trigonometric identities, and vectors. Even if the problem was written in English, I don't know if I could solve this problem. This again shows how integrated their curriculum is, and I admire them for it!

I found out today that there are many different levels of textbooks. The mathematics textbooks that are used by Kanagawa Sohgoh High School are some of the hardest textbooks offered. This may explain why the content was so challenging. There is a lot of talk in American schools about Standards and about the Common Core. The Japanese school system does have content standards, but they value the textbook over the standards. This works very well because their textbooks have a very demanding curriculum.

 

Games

This third class played a number of games during the first part of the class. They were very interesting. I will share them with you.

Game 1: Police and Thief

One player starts as the Police and the other player as the Thief. The Police get to go first and they can move one space. The Thief can then move one space. Through playing the game the players learn that the triangle at the top is a very important part of this game. If the thief goes in there, then they will be caught. Otherwise they won't be caught.

Game 2: Two Groups of Stones

The game is played by drawing two groups of stones one with 25 stones and the other with 18 stones. The first player can remove as many stones of the same color as they want, the second player can remove as many stones of the same color as they want, and it returns to the first player. The goal is to be the last person to take a stone.

I won't tell you the strategies of how to win this game. The trick is to play it several times and begin to notice patterns. What I will tell you is that one of the players (player 1 or player 2) has the advantage. But I won't tell you which. If you have any guesses, please leave them in the comments below.

Game 3: Three Groups of Stones

A third game is a similar game about removing stones. The rules are the same, two players alternate turns by removing stones of the same color and the player to remove the last stone wins. But now there are three groups. The first version has 3 stones, 2 stones, and 1 stone. Like this:

Another is with 7 stones, 5 stones, and 3 stones. Just like the other stone game one of the players has an advantage and will win, unless they make a mistake. I won't tell you which player has the advantage. But if you think you know, please leave your thoughts in the comments below.

Games

I was so glad that I watched this class play games. These games are very different than the games I have ever played in American schools. These games still required critical thinking and problem solving, and it took some mathematics understanding to understand the strategy.

Student Presentations

I wanted to mention some of the things that I saw in the technology class. The students were giving presentations from PowerPoint that they prepared on technology in the future. For example they talked about designing Smart Phones so they can be used by blind individuals, 4D televisions (including things like smells), food that can be stored in data and then reproduced at will, and a way of finding lost items using a Smart Phone. The students had three minutes to present their ideas and they were graded by their peers by answering 13 yes or no questions, and overall score, and comments. These kinds of things are not uncommon in America, but I was very impressed by these presentations.

Teachers

A learned a few more things about Japanese teachers. First of all many high school teachers don't have degrees in education, they have degrees in their content area and then get a teaching certificate later. They also teach about two out of the four 90 minute periods each day (not three like is common in America). So the teachers have much more time to plan and collaborate.

Finally their school is divided up into committees. We have this at my school as well, and I am guessing this is not uncommon. But in case you are interested here are the committees at this school:

• Research (look at how they can prepare their students for entrance exams, decides how many home room classes to have, and fills out reports).
• School Management
• Career Advisory
• School Activities
• School Behavior (discipline)
• International (this school has several sister schools around the world and their students travel all over the world)

I just thought it would be interesting to compare the kinds of committees at this school and your school. Also there are between 5 and 10 teachers on each committee. Committees between 8 and 10 are the most common.

Differences between Japanese Education and American Education

I am continually fascinated by Japanese education. It is just interesting to see another education system, especially a successful one, and look at the similarities and differences. The more I talk about my experiences and things that I am learning, people want to know "Why is Japanese Education better?" and my answer is "I don't know." Anyone who is involved in education understand that it is very complex and connected to the students that we serve, so it would be impossible to research and find the that make Japanese education work that American education is missing. That is the wrong question to ask. But I do want to point out some differences:

1. Children are left behind

The responsibility of learning information is set squarely on the student's shoulders. Teachers come to class, they lecture, the students take notes, and speak only when invited to by their teacher. In fact, Japanese high school is more like American college than American high School. The students know that if they don't study well, they won't pass their next entrance exam and therefore won't be on a track to get into a good university. But it is their choice to learn or not. I have seen multiple students with their head on their desks sleeping during class. The teacher doesn't make any comments about it, as long as they aren't disturbing the class, it is fine.

2. No Differentiating

Differentiated Instruction is a hot topic and a discussion among American teachers, but not Japanese teachers. American teachers strive to adapt their teaching to meet their kinesthetic, auditory, and visual learners as well as meeting Garners Intelligences. Japanese teachers stand at the front of the room and lecture and draw diagrams on the chalk board. From what I can tell, they may not even know their student's names, let alone their learning styles. When they call on a student they are usually looking at a class list.

3. Not Standards, Curriculum

I have limited working knowledge of Japanese curriculum, but it is an area of interest. There are national education standards approved by the Ministry of Education. But in talking with Japanese teachers, I have never heard them say anything about standards, but they do talk a lot about textbooks. There are only a few textbooks that are used in Japan and they are all approved by the Ministry of Education and therefore you don't deviate from them. The examples that the teacher gives are the examples that are printed in the textbooks. Very little is done that isn't directly from the textbook.

4. Teacher Observations

American teachers are not used to being observed. In my entire teaching career of about four years, I have only been observed about three or four times. I have only had two individuals come into my class and watch me teach. This is very different from Japanese teachers. During a few months of the year, there are open lessons where anyone can come and observe a teacher. So each teacher may have one open lesson a year where the principal, other teachers, or parents are welcome to come and watch them teach. I have also heard that the first full year of teaching is done with an experienced teacher assisting them. Usually in the back, but providing feedback.

Conclusion

I don't know what will fix the problems that we are seeing in American Education. It is my opinion that the American Education system is too complex for simple solutions. Slow and gradual progress is the only way that we will improve our schools. I have highlighted some areas that are very different between Japanese Education and American Education. I find them interesting, I hope you do too.

Using Requirements on Teachers to Inspire Creativity

One of the most liberating and powerful attributes in education is freedom. Sometimes as teachers we don't feel that we have a lot of freedom. We are required to teach Common Core, State or other Standards. At the same time we are required to meet other school or district improvement plans. This is on top of lesson planning, grading, and differentiating for a wide variety of learners. But the truth is, in all of this, there is still room for freedom. In fact, the demands on you as a teacher can inspire creativity.

One TED talk that I think about frequently is Phil Hansen's TED talk entitled Embrace the shake.

He is an artist who developed a tremor in his hand which kept him from drawing pointillist drawings. So instead of giving up, he decided to embrace the shake. I love the moment when he goes to the art store and can buy anything, he doesn't know what to do. He then sets limitations that inspire the creativity.

My goal as a teacher is to use standards, improvement programs, and other demands on me and my time to inspire creativity. I am always willing to try new things in my classroom. Some of them fail, but most of them work and can then be improved upon.

This same idea is passed to my students. I love to assign open ended projects. For example

• Create a game that reviews the ideas from this chapter, the game must include content questions from the chapter that you wrote. This can be:
• A board game
• A card game
• A video game
• Present a summary of all the important ideas from this chapter. Be creative, for example:
• Make a video: a movie trailer, news broadcast, or video of toys discussing the ideas.
• Write a story: a children's story that incorporates the ideas from this chapter.
• Create a PowerPoint, Prezi, etc.
• For more ideas go to:
• http://khsgeometry.weebly.com/quarter-1-projects.html
• http://khsgeometry.weebly.com/quarter-2-projects.html

I have been amazed at the quality of work that I get when I allow the students freedom with specific parameters. Last year when I detailed exactly what I wanted, I received exactly what my rubric stated, but it lacked passion and enjoyment.

I hope that educators will embrace the demands on us, and use it to inspire creative solutions to the daily struggles of being a great teacher.

Creative Constructions

Recently my Geometry class were studying parallel and perpendicular lines. We discussed how to construct parallel lines (using a transversal and constructing congruent corresponding angles). Then on a quiz I showed my students this picture and asked them to construct a line parallel to line AB through point C.

I had one student who turned in this as an answer. 

  

Now I am the kind of teacher who is fine with my students doing math in multiple ways. I will not require my students to do it my way, but it is important to me that the methods that are used will always work. I asked this student to explain what he did and he couldn't. All he would say is "But they're parallel" and I asked "Why?" Finally I said "Unless you can explain to me why it will always work, I won't give you full credit.

But I was really intregued by this problem. So I sat down and sketched it using Geometer Sketchpad and I found that, in fact, it did always work. It wasn't until I saw the shape moving that I realized what he had constructed and why it always works. Here is a picture.

This student constructed a rhombus, which is a parallelogram. So the line he constructed actually was parallel. So we discussed why it worked and I gave him full credit.

Then later, on a test. I gave my students this picture and asked them to construct a line perpendicular to l through point A.

The method we covered is by drawing a large arch through point l and then bisecting the segment. But this is what that same student constructed.

This was much easier for me to see why it works and I gave him full credit. I am still not exactly sure how to describe why this works. I think the best explanation is by using an isosceles trapezoid.

These creative constructions have made me consider coming back to constructions when we cover congruent triangles and quadrilaterals. There are some great connections to be made here.

Triangle Congruence Card Game

I had an idea today about a card came to practice Triangle Congruence. There would be a game board would look something like this:

Students would have three cards in their hand. Then they would pick up a fourth card, and then discard one card. The first student to have three cards that make a pair of congruent triangles must show their cards to the group and tell them which congruence postulate or theorem they used.

This idea can probably be improved upon, for example adjustments to the number of cards to make the game a little more difficult. For example in a few trial runs I found that it was not uncommon to finish your hand in 2 or 3 turns.

Download the pdf

Geometry In-Class Activities

Here is my brainstorm of activities that I might use next school year. I want to make every class period different so my students don't get bored.
 

“Variety’s the very spice of life 

That gives it all its flavour.”
-William Cowper (1731-1800)

Starter Activities
1. Summary and Question
Teacher Preparation:
· None

In Class:
· Students write a summary of what they learned the previous night and either ask a question or write something they found interesting
Pros:
· Quick summary/synthesizing activity.
· Writing activity
Cons:
· It will be difficult for some students.
2. Flash Cards
(This activity can be used as a short opener activity in sections that have lots of vocabulary.)
Teacher Preparation:
· Teacher creates flash cards that students can use to drill material. Name on one side, picture on the other.
In class:
· Done after the daily notes quiz.
· Students drill themselves going from name to picture and then from picture to name.
· Then students get in pairs and drill their partner.
Pros:
· Great for sections that have lots of vocabulary.
· Lots of quick practice.
· Solidifies important vocabulary.
Cons:
· Lots of teacher preparation, unless these flash cards can be found online.

Practice Activities
1. Textbook Problems
Teacher Preparation:
· None

In Class:
· Have students do the assignment that we assigned last year in their textbook.
· They should do it by themselves, but they are encouraged to ask their partner or other students for help before they ask their teacher.
· When they are finished they should check their answers in a different color pen or pencil, and then fix their mistakes in that same pen or pencil.
· Then students should take their “Mastery Check” by themselves to see if they can do about 3 problems without any help.

Pros:
· No preparation for the teacher.
· Students can work at their own pace through the problems.
· Students can help each other.

Cons:
· Student engagement might be low.
· Deeper understanding opportunities will be missed.
· Students who get too much help from friends will think they understand, but they don’t.

2. Peer Instruction http://blog.peerinstruction.net/
Teacher Preparation:
· Print off Multiple Choice Questions
· Teacher complete problems to know which of the problems are good “conceptual” questions. (Hard questions that get at the big idea of the mathematics).
· Cut the Standardized Test Prep paper into strips with 1 question and answers on each paper.
· The first time you do this, you will need to prepare (or have your first class prepare) Four cards for each student that say A, B, C, and D on them.
In Class:
· Present a problem and have the students complete it individually and commit to an answer.
· When all students have finished, have them all show their answer at the same time.
· The teacher counts how many “correct answers” there were but does not announce it to the class.
· The teacher then asks the students to find someone who has a different answer than you and discuss your thinking.
· After a few minutes the teacher asks the students to show their answer at the same time.
· Again the teacher counts how many “correct answers.” (This is used to see which questions are good questions. Good questions will have a big increase from first and second attempts).
· The teacher announces the correct answer. They can have a student present the answer, or complete it themselves, or if there are lots of students with the correct answer then they can move to the next question.
Pros:
· Students discuss their thinking and listen to each other’s thinking.
· Deeper understanding takes place.
Cons:
· A little bit of preparation for the teacher.
· Only a few problems will be attempted during a 45 or 60 minute time period. (Which is why the selection of quality problems is so important.)
· I fear about students changing their answer to the answer that their “smarter” friend’s answer.

3. Group Games
Teacher Preparation:
· (Same as Peer Instruction)
· Print off multiple choice questions
· Teacher complete problems to know which of the problems are good “conceptual” questions. (Hard questions that get at the big idea of the mathematic s).
· Cut the Standardized Test Prep paper into strips with 1 question and answers on each paper.
· White boards with markers and erasers – If available

In Class:
· Students get in teams of 3 or 4. – Pick a team name and decide on their ordering (1, 2, 3, 4)
· A question is presented.
· All the students complete the problem on their white board with work and an answer.
· They may help each other in their groups.
· At the end of the time limit. A number between 1 and 4 is randomly selected and that student in each group’s answers are compared and the teams with correct answers get points and teams with incorrect answers don’t get points.
· If multiple groups get the problem wrong, invite a student up to the board to complete the problem (for an additional point) or do the problem yourself. Answer any questions and probe for deeper questions.
Pros:
· Student engagement and enjoyment is high.
Cons:
· Weaker students rely on stronger group members for work and answers. Weaker students don’t get effective practice.
· Deeper understanding opportunities are missed.
4. Student Created Word Problems
(This will not take an entire class period. Should be paired with another practice activity. Maybe be used after the other activity)

Teacher Preparation: 
· None

In Class: 
· Students get in groups of 3 or 4. 
· Students write word problems that are difficult but clear. 
· Switch problems with another group.
Pros: 
· No Teacher Preparation 

Cons: 
· Quality of problems can very.

5. Student Transcriptions
Rubenstein, R. N., & Thompson, D. R. (2001). Learning mathematical symbolism: Challenges and instructional strategies. The Mathematics Teacher, 94(4), 265–271.
(Best used when notation in emphasized or when diagrams are emphasized)
Teacher Preparation: 
· Prepare cards of notation or diagrams.

In Class: 
· One partner reads a symbolic expression or sentence while the other writes what he or she hears using symbols. 
· In another variation of the transcription activity, students can practice using mathematical language by directing their partners to draw figures given to students on index cards.
Pros: 
· Prepare cards of notation or diagrams. Cons: 
· Drawing diagrams may not be the most effective way of practicing the mathematics of the day, but would be good for critical reading and thinking activities.
Content Review Activities
1. Student Created Problems
(I imagine this best being used as a review activity)
Teacher Preparation:
· None

In Class: 
· Students create (word) problems over the material that will be covered or assessed. They may use notes, textbook, etc. 
· The students either give them to the teacher and a group game is played.
Pros: 
· No teacher preparation Cons: 
· Quality of student created questions can very.
2. Student Section Summaries
Teacher Preparation:
· Prior to the review tell groups of students which section they will be assigned to review for the class the following class period. Highly recommend that they review that that section/topic.

In Class:
· Give the students time in class to finalize their presentations and pick three problems for the class to do.
· Give each group a chance to review their section and ask their questions.· The other groups complete the answers. Group points can be awarded.

Pros:
· No teacher preparation
· Students review one section in depth.

Cons:
· Students only review on section in depth.
· Fewer problems are presented.