Japanese Mathematics Vocabulary: "We don't use this word."

I had an interesting experience several weeks ago. I recently finished my Masters Degree in Education specializing in English Language Learner Education and I am now adapting my teaching practice to help my English Language Learners. 

I realized that there was a lot of mathematics vocabulary that I didn't know in Japanese and I was limited when I was explaining something to my Japanese speaking students. So I contacted my friend who is a mathematics teacher at Kanagawa Sogo High School and has very good English. I went thought my Geometry textbook and wrote down most of the vocabulary words and a few others that I felt I would want to know in order to explain Geometry to my students in Japanese. I had somewhere around 250 words.

When we sat down and began going through the words, my friend kept saying "_________ is the word for that, but we don't use this word." I would guess that somewhere around one third of the vocabulary words that I brought, they don't use in their K-12 education. Perhaps some of this is the difference in curriculum. However my friend said that they would simply describe the word. For example we teach the word "colinear" but in Japanese schools they would simply describe the points as being on the same line or draw a picture. The same thing with coplanar, skew lines, exterior angles. and the list goes on. There isn't even a word in Japanese for a linear pair, they simply call them supplementary angles or explain that because they make a straight line, their measures would add to 180 degrees.

I walked away from our meeting wondering why we have so many vocabulary words in Geometry and mathematics in general. Because I will be teaching a Mathematics Lab next year with students from Algebra as well, I asked about some Algebra words. I again found that words like "leading coefficient" and others were not used in Japanese K-12 education. We teach students vocabulary words for concepts that we study for only one or two days. I think part of the reason is the way we were taught, so there is a culture of education that focuses on vocabulary words. 

I think Japanese education is correct when they limit the vocabulary words that they teach to the most important words that will be used throughout the course, not for a few days. Limiting the number of vocabulary words would make the mathematics more accessible to all students. However, I am afraid to do this in my practice because if I do not expose them to these words, then I will be hurting my advanced students when they go to future math classes and their teacher is using a word that I thought was unnecessary. The change has to happen in the curriculum, starting with the textbooks.

The Power of Synchronous Flipped Classrooms

Asynchronous
In education social media, it seems that teachers favor asynchronous learning over synchronous learning. citing reasons like:

• Students can go at their own pace.
• When students get stuck, they can slow down and understand that concept before they move on.
• Not all students learn at the same rate.

And all of these points are true, but my experience is that each of those points are heavily outweighed by challenges of asynchronous learning:

• Since students can go at their own pace, they put off their math homework because it isn't necessarily due tomorrow.
• When students get stuck, they lose the pace of the class and then they can't catch back up with the class.
• It is true that not all students learn at the same rate, but everyone is supposed to finish the curriculum by the end of the year. You will be constantly pushing the slower students.

Last year I tried to teach an asynchronous flipped Geometry class. The students were allowed to work at "their own pace" to watch videos and complete textbook problems. I knew that I needed more deadlines than just the end of the year, so every student was required to take the test on the same day. What I found is that the majority of my students didn't finish all the notes and practice before the end of the unit.

What I did find from the data is that there was a stronger correlation with students doing well on assessments and students doing the textbook problems and than students who watched the videos and took notes. So I decided to restructure my class so we focused on completing this practice during class and everyone would work at the same pace.


Synchronous Flipped Learning
What I found was the stronger students were more willing to help weaker students because they were doing the same problem at the same time. When I taught asynchronously, everyone was at a different place so they all wanted my help. I ran around like crazy answering questions and checking off work. The discussions between students are much richer and deeper because they aren't trying to finish their work, they are trying to understand concepts and do problems.

So the entire class follows a pacing calendar. The students are expected to come to class having taken the notes. We also take an open note daily quiz. If students come to class without notes, they don't do well on the daily quiz, but I still allow them to participate in the practice with us. Most synchronous flipped teachers don't allow students who don't have the notes to participate in the practice. They make them watch the video during the practice. I personally have chosen not to do this for two reasons. First I believe that the in-class practice is so much more powerful than doing problems out of a text book, especially for the weaker students who don't come to class prepared. Secondly, these students were in an asynchronous flipped mastery class the previous year, I am afraid that these students will simply get behind and never catch back up. Then I am back to the same problems I had last year, except I would have minimal time to work with these students individually because I am working with the rest of the class. Maybe the social motivation that other teachers state is more powerful in the younger grades. I just didn't think it would be enough of a motivation with my high school students.

I understand that this is not the ideal. In fact, the students who come to class having taken notes generally stated that they have "mastered" or are "getting the hang" of a new idea by the end of the class period. Five of the 13 students who come to class half the time or more without notes stated that they were not getting the hang of the new idea by the end of the class. But this is 5 out of my 85 students, and these are students who would probably struggle anyway.

Another reason that I prefer the synchronous flipped classroom is that parents and students seem to understand it better. Meaning that I have to do far less explaining about what the model is and how it is used.

My Situation
I understand that every teaching situation is different. Maybe my bias is a result of trying to use Flipped Mastery from my first year of flipping. Maybe if a did a traditional flip first and then moved to Flipped Mastery then I would have had the opposite experience. But from talking to other flipped teachers and students at my school, I don't think so. I still think there is more power  the synchronous flipped classroom than in the asynchronous flipped classroom.


Comments?
Are you a flipped class teacher who has had the opposite experience? Do you disagree with my assessment? Please comment below.

Flipped Geometry Survey Results

When I was a student teacher, my cooperating teacher taught me that I need to embrace my students and allow them to teach me how to be better. Because of that, every year of my teaching I have given my students at least one survey, if not more than one.

At the beginning of the fourth quarter in school year 2013-2014, I gave my Geometry students the following survey using Google Forms. I love Google Forms!! The survey is anonymous, but I told the students that I will be using these results to make decisions about next year so please be honest.

If you are curious here is the Survey:

Many of the results were not surprising, I will show them in the tables below.

I was curious how long it takes students to complete the notes. I commonly assign one or two videos as homework. So students can expect to have an average of 15 to 20 minutes per video. 

The amount of time students spent on one video seemed independent of their overall success in the class. Some students who struggle spend more time on the videos and some students spend more time, which help them do very well. The opposite is also true. Some strong students don't need to take a lot of time on notes and some weak students should spend more time.

From time to time I see students who are watching the video with earbuds in, but I notice that their earbuds are plugged into their music instead of the computer. So I was curious if this makes a difference.

This relationship wasn't very strong, but in general, students who get As and Bs in my class listen to the videos. It was more common among C, D, and F students that they listened to me less often.

Next I will move to one of the most important relationships to overall success in my class.

The relationship between watching videos before class and overall success was much stronger than I expected it to be. Every student who got an A or B in my class either Always watches the video or usually watches the video. Students who got Cs, Ds, or Fs came to class less frequently having watched the video and taken notes.

Students can ask themselves, do I want to act like an A or B student, or a C, D, or F student? Students who act like A or B students usually get As or Bs. And it starts with consistently coming to class prepared.

The next question is about our Daily Quiz. The quizzes I give are only 3 questions and students can use their notes. The questions are written so that if you watched the video and took good notes, all three questions should be easy, but if you haven't watched the video you may only get one correct.

I was very surprised to learn that 77% of my students felt that having a Daily Quiz is a motivation to take better notes. I realized that I spent longer assessing whether or not students watched the videos and took notes than the length of the notes. Meaning I was considering doing away with Daily Quizzes and just showing the video at the beginning of class. This survey taught me that students need more time than the length of the video to process it, and the daily quiz is a motivation to take better notes with more examples.

This is what I hoped would happen when I decided to give daily quizzes and I am very glad that this is what is happening.

We spend about 60 minutes each class period going over problems. I present a problem on the board and the students do the problem on individual white boards or on paper. After doing this for 60 most of the students feel that they have mastered the idea or are getting the hang of it.

There were two students who said "I usually feel just as confused as before we started." I looked very closely at the responses of these two students. These are also students who frequently didn't come to class having taken notes and these are students who "Only listen to Mr. Lewis if I don't understand." This is probably why they are confused.

Now if I separate the above data by whether or not they come to class having taken notes we see some powerful results:

The students who almost always have notes understand the material when they leave, while those who usually don't have notes may be getting the hang of it, but may be confused.

I was surprised that so many students were interested in optional problems. I don't know what I am going to do about this. The idea of it being optional makes me wonder if students would do it. Maybe make it extra credit or something. I need to think about this more.


Finally their third quarter grade separated by how frequently they come to class having taken notes.

If you almost always come to class having taken notes, you will probably get an A or a B. If you usually come to class having taken notes, then you will probably get a B or a C. If you usually don't come to class having taken notes, you will probably get a C or a D. I was shocked at how clear this data was.

One incredible thing about this relationship is whether of not you always come to class having taken notes is something that each student has control over. I think students feel that their grade is determined by how "good they are at math" and although this is true, they have control over how good they get at Geometry, by whether or not they always come to class having taken notes.


Statistics
The AP Statistics teacher in me needs to put a disclaimer. There is a "confounding variable" of each student's dedication to school and ability. Does taking notes make you get better grades? Or do students who get better grades always take notes because that is what they do as a student?

I don't want to imply any cause and effect relationships with this data because it was an observational study. But I think students will understand that if you come to class having watched the video and taken notes, you will be prepare for the practice, you will understand the practice, and then do better on assessments and therefore do better on the class overall.

This data was collected in my class, with my students, within my teaching model. These results can't be used in other Flipped math classes.


Comments or Questions?
If you have any comments about this data or the analysis, please comment below.

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.

Japanese Research Lesson

On March 21st I was invited to visit a junior high school in Wakoshi of Saitama-ken. A "research lesson" was being taught by two different teachers. So we saw the same lesson twice. Once with eighth grade students and then with seventh grade students.

Previous Lesson
The lesson prior to this lesson the students were given this picture. It is a cube. Points P and Q are the midpoints of the sides.

The students were asked "What kind of quadrilateral is PQGF?" The students used nets to think it through. I wasn't present for this lesson so I don't know the details of this lesson. It turns out that PQGF is a rectangle.

This Lesson
Then in this class, the students were given this picture. Points P and Q are the midpoints of the sides.

The question that was asked to the students is "What kind of quadrilateral is DQFP?" The students were given time individually to think about it, and then the teacher asked students to raise their hand based on what shape they thought it was. The common answers were square and rhombus, but there were students who thought it was a rectangle, parallelogram. I was impressed that no student was concerned of the social implications of their initial guess. I think that if I asked this question to my students, many students would vote how the few "smart" students vote. I was impressed that students felt safe enough to voice their thoughts, even if they were the only ones.

After students made their initial guess, the students got in groups of 3 in their han. (Their han is a grouping of about 6 students. These groups are set at the beginning of the year and are the same throughout the entire year). Since students had different opinions, they were to discuss their reasoning and as a group decide on what shape they thought it was and why. The students were also given the following net:

As they were reasoning, they were invited to use that net. Some students drew the shape they thought the lid was and then cut it out to see if it matched. For example the groups that thought it was a square constructed a square on the net and then cut it out. Or I saw one group who thought it was a rhombus, construct a rhombus of two equilateral triangles and see if it matched. Neither of these worked. I saw another group cut out the net as shown, then place the lid face down on another piece of paper and trace the top.

At the end of the lesson the teacher didn't announce what the shape is. I was really impressed by this. If I was teaching this lesson I would want to wrap it up and say what the answer is. The truth is, if the teacher did give the answer at the end a two things would happen:

• The students would stop thinking about the problem. If the teacher doesn't announce it, there will be students to continue thinking about the problem on their own and may come up with an answer or other connections.
• The next task that the teacher presented, the students would be significantly less motivated to attempt it because they would know that if they wait until the end of the lesson, the teacher will tell them the answer anyway.

My Reflection
I loved this task! It was very simple to understand, but very complex to solve. I loved that it was open on how to answer the question.

I love research lessons! I wish that I could find more lessons like this to use in my class, with my students.

Japanese Feedback on My Teaching

On Friday, February 21, 2014 I was invited by Dr. Doug Corey of Brigham Young University (when I was an undergraduate at BYU, I was one of Dr. Corey's research assistants) and Dr. Ninomiya of Saitama University and several other graduate and undergraduate students. Please see my blog post about that "research lesson." (coming soon!)

Then on Tuesday, February 25, 2014 Dr. Corey, Dr. Ninomiya, and a graduate assistant came to my school and observed three lessons. Two that were using the Flipped Mastery model, and a Geometry lesson about Translations and Reflections. After that 85 minute lesson we sat down and talked about the lesson. Since graduating from BYU about five years ago I have never received such though provoking feedback a single lesson.

Kadai (underlying deeper mathematical idea)

The first comment that Dr. Ninomiya made to me was describing mondai (pronounced moan-die) and kadai (pronounced ka-die). The mondai are the kind of problems or questions that are asked to the students. The kadai are the underlying deeper mathematical ideas. The goal of the lesson is not for students to solve the mondai, but to think and learn about the kadai. The kadai is sometimes a phased as "Let's think about ________". The kadai isn't specifically shared with students, but teachers will talk about the kadai as the goal of the lesson with the students. This is especially true at the end of the lesson, with a summary of what students have done by solving these problems.

Dr. Ninomiya didn't understand the kadai of my lesson... because there wasn't one. American curriculum is a collection of separate topics and problem sets, it doesn't lend itself easily to underlying big mathematical ideas. On the other hand, Japanese curriculum is more focused and builds upon a single mathematical idea over several lessons. As I progress as teacher through my career, I have been thinking of pulling away from the textbook (not from the Standards). I don't know how valuable remember types of problems are in the long term. This seems like a topic for another blog post. At the very least, I can work on framing the lesson within a kadai.

Notes and Summaries

Another topic that we discussed was students effectively using their notebook. In my class, the students watch a 10 minute video, take notes, and then write a summary. Dr. Ninomiya liked the idea that I had my students write a summary. He said that many teachers want their students to do lots of problems, but it is better that students are asked to summarize what they learned. It becomes a synthases exercise.

In my class, for about an hour, we do problems from the textbook. The students write down the answer on individual white boards. I do this because:

1. Students enjoy it. At the beginning, the students loved the idea of using white boards. Even after using them everyday for months, they still enjoy it. Maybe after using paper and pencil so much, they like the change.
2. Because students are writing with a large white board marker, it is very easy for me to scan the room and see what all of their answers are. I then know what mistakes have been made and the kind of thinking my students do.

Dr. Ninomiya commented about the fact that students don't have a record of what they did in class. Once they erase their white board, then it is gone. They can never go back to that problem and look at it. Notes have several purposes:

• Record of what they, the student, did: The textbook isn't always what they did, but their notebook is a record of what they did. (By the way, because textbooks are so small and thin, students are encouraged to bring, not only their textbook to class, but all of their math textbooks from junior high school or high school to class.)
• Notes remind students of what they did last class: Teachers ask students to review their notes from last class, so they can continue the ideas from last class and build onto the previous class period's discussion.
• Notes are used to study: Studying for semester exams are very important, and a student's notes are very useful when preparing for these exams. These notes are for their future self.

I don't think I am ready to give up white boards, because of the above reasons, but I think students could add things to their notes as we are doing problems on white boards.

Whys that I might implement these ideas in the future:

• Require more from the summary: Right now my student's summaries are very simple and are only two or three sentences long. I want to look into Cornell Notes and other AVID materials and see what they say about summaries.
• Use their summaries at the beginning of the lesson: Occasionally have them switch notebooks with someone around them and have them read each other summaries. Are there any important ideas that this person left out?
• Add to their notes during the lesson: My students usually don't write down the example problems that are in the videos in their notes. They just don't see value in them. (This is one reason why I been shortening the videos by removing about half of the example.) I want to change this perspective. I want my students to work out a problem or two, in their notes, show their work and write down explanations.
• Add and clarify their summaries at the end of the lesson: At the end of the lesson, the students need the teacher to remind them what they did that day. This also gives a teacher a chance to summarize the lesson and, by example, the importance of summarizing the lesson. By the way this is called the matome (pronounced mah-toe-mei) in Japanese education. The students will then add or clarify what they have written from the summary.

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

Proof in Geometry

When I was a high school student I hated the sections in Geometry on proof. My teacher frequently said that "implied" too much. It was hard for me to wrap my brain around the idea of proof and how to prove something. After high school I went to college where I completed a bachelor program in Mathematics and Mathematics Education. My coursework in Mathematics has taught me that higher math is more about logic and proof than solving equations. So even though I hated proof as a student, I feel that it is really important.

Previous Status Quo
I am about to start my fourth year of teaching Geometry. In the past my students have always complained about proof (like I did when I was in high school). Because I feel that proof is important, I want to find a better way to introduce it and teach it. I have an idea, and I am interested to see what will happen this year.

The book we use (Prentice Hall (C) 2011) introduces proof in Chapter 2 and immediately has the students do Algebraic Proof and then proofs about angles. As if solving an equation on one side of a two-column proof and the reasons on the other is enough of an introduction to proof that students can jump into some complicated Geometric proofs in the next section. For example the very first proof in the book is The Vertical Angles Theorem, which is not simple. It requires students to think outside the box and use an angle other than the two vertical angles, notice that they are supplementary, and then do some pretty complicated Algebra.

It is during and after these first two sections of proof that my students started complaining about proof. I would then tell them that we have at least two more chapters of proof and they would grumble.

In Chapter 3 we would do proofs about proving lines to be parallel. Students got pretty good about recognizing the angle pairs and knowing which Theorem or Postulate needed to be used (ex: Corresponding Angles Postulate, or the Converse of the Alternate Interior Angles Theorem). But when it came time to put those ideas into a two-column proof, they would get so confused about where to put statements and reasons that they couldn't write the proof. The same thing happened in Chapter 4 when we would prove that triangles are congruent.

My Idea
My current thought is to not introduce two-column proofs until Chapter 4 when we prove triangles to be congruent. Before then, require that the students give a well thought out justification that includes Theorems when applicable. They could do this for triangles in Chapter 2 and parallel lines in Chapter 3. Then after students have been working on written justifications for two months, they will then be introduced to "formal proof" meaning, two-column proofs. We would only do two-column proofs for triangle congruence, CPCTC, and later in the second semester, proving that triangles are similar.

In essence, I am trying to encourage students to understand the concepts behind proof without getting confused by the formality of a two-column proof. I don't want the students to get lazy, but I do want students to think through why something is happening instead of guessing at statements and reasons.

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.