Management Involvement Feedback Focus (MIFF)
Two Problem Approach
- Hand and Finger Signals
Hand and Finger signals are used to allow students a consistent way of responding. Whether it be raising your hand or waving arms across each other to show disagreement, these silent modes allow everyone to participate without distracting from the lesson.
Silent Agreement: thumbs up
Silent Disagreement: arms & hands waved across each other (Note: also known as the "I have another answer" signal)
"I don't know" or "I'm confused": hand moving back and forth over head
"I can't hear": hand behind ear
"I have an important question off the subject": raised pinkie finger
The use of space in the classroom is important for a variety of reasons. A teacher must use the full space of the classroom to facilitate the active involvement of as many children as possible. As one proceeds through a lesson, one moves to various parts of the room. The goal is to use space to bring about desired behaviors such as on task behavior (move in close), louder speech (move to the other side of the room, hand behind ear), increased participation (teacher stands in back, students present at overhead or board).
- Modes of Response
If a mode of response is given before a question is asked, students will not have to try to guess how they should respond. When there is not a mode of response given, one or more students will have called out an answer while other students are trying to guess what kind of response was expected.
I'll take a quiet hand...
Show me on your fingers...
Whisper to your neighbor...
Talk it over in your group...
Write on your paper...
- Specific Questions
In class, it would be ideal if students never had to guess at what a teacher was asking, yet often, teachers ask questions which are vague, but in the teacher's mind require a specific answer. If the question posed is not specific, but a specific answer is wanted, then this is a head trip. Teachers should try to avoid playing this game by phrasing their questions carefully. There are times when open questions should be asked, but one must be aware of what type of question has been asked. If a question has been inadvertantly open to interpretation, student thinking needs to be honored, even if the answer is not the one expected.
While students are working in groups or independently, the teacher has an excellent opportunity to move among students, looking and listening, asking questions to find out about student thinking or extend it, and give hints. This provides a quick assessment and often the chance to intervene on the spot.
- Positive Reinforcement
When specific desired behaviors are acknowledged by the teacher, it both informs and motivates students. For example, "I see that group 3 has put down their pencils and they all have their eyes on me."
It could be said that there are no wrong answers, we just have to find the question the student answered. For instance, "That would be right if I'd asked for the sum of 5 and 2, but I am looking for the product of 5 and 2."
- No Echo
Often teachers can slip into the habit of automatically repeating everything any student says. As a habit, echoing student responses can have several negative impacts in the classroom. If the teacher repeats everything, students will know that they don't have to listen to each other if it's important, the teacher will repeat it. A great deal of rich dialogue among students is lost this way, and students are likely to have a more difficult time working in groups because they are not in the habit of respecting what each person has to say.
Some examples of alternatives to echoing are:
"Would you please repeat your answer, but first let me go over to the opposite side of the room."
"Raise your hand if you heard Peter."
"Who can repeat what Sara said?"
"If you heard it, whisper it to a neighbor who didn't hear it."
"How many understood Martha's answer?"
"Who can give another explanation in their own words?"
- Wait Time
A well established management technique is to allow students time to digest a question and think through an answer, it is included here because it is so powerful in the classroom. It is recommended that 3-5 seconds are allowed to pass between a question and asking for a response.
- Deliberate Misteaks
Deliberate Misteaks can be a wonderful way to assess student understanding and increase student focus on the lesson. When students are comfortable with knowledge, they love to show what they know and catch the teacher at a mistake. Not to be used while developing concepts.
- Involvement of Visitors
In order to reduce the distraction of the arrival of a visitor in the classroom, the visitor might be asked to call on a quiet hand. Or the visitor might be invited to look at some of the students' work and find an answer or an approach that is notable. Students come to see visitors as part of the classroom experience.
About Daily Head Problems
A basic definition of a Daily Head Problem with some sample head problems for each grade level (K-6).
The Daily Head Problem is the first of 2 content strategies we recommend using as part of your regular mathematics program. Daily Head Problems are oral presentations of standards based math problems which students must compute mentally. The problems should include the appropriate math vocabulary and concepts found in the California mathematics standards and textbooks. (For second language learners please remember to use SDAIE techniques. Sheltering techniques are offered in the sample problems for each grade.)
The use of "Mathematical Number Generators" (# of sides on a triangle, # of degrees in a right angle, days in a week) conceptually enhance the Daily Head Problem and also have the added advantage of making use of math vocabulary.
Ideally the Daily Head Problem should range in length from 2-4 steps, and generally should take up no more than 3-5 minutes of class time and as the name suggests, should be done on a daily basis. Daily Head Problems are to be replayed one step at a time with the appropriate answer generated for each step. (Examples of the replay are given with the first sample problem at each grade.) The more complex math computations occur at the end of the problem. Daily Head Problems can also be cross-curricular in nature (that is they can be used in Language Arts, Science, Geography, etc.)
Making Head Problem Grids
There is a way to write three to five head problems in such a way that the parts can be mixed and matched to create many more head problems. "But my students will get bored," you might think. Surprisingly, there are so many variations available to you this way that no two head problems will be quite the same, yet students will have multiple opportunities to hear the same mathematical language. Repetition provides for increasing student success, but in an interesting way.
The grid writing shared here is just one way. If it gives you ideas of your own, all the better. What you develop for yourself will be best suited to how you think and plan.
A great place to begin thinking about your head problem grid is to gather some vocabulary you would like students to learn. You might draw words and phrases from the standards, or you might draw them from your math curriculum. In the samples available on the right, vocabulary was drawn from a whole unit.
The next thing to do is to write five head problems (I did mine from left to right). You'll notice the boxes have numbers or letters in them. That is to help keep track of different combinations to use. You can combine step 1 with A on one day, B on another and so on. The last column is the "final Mode of Response." You will want to choose that so that it works best with the final number answer of the particular combination of steps you used. So here's a math problem for you: how many possible head problems have you written once a grid is complete?
Kindergarten teachers, please note that your grids will look a little different than the other grades. The grids can still be useful for getting head problem ideas, but may not provide interchangeable parts in the same way as the grids for other grades.
Recycling Head Problems
Here is another way to get the most from your head problem creation. Take a look at a single head problem. We'll just look at the mathematical steps for now, leaving off the final Mode of Response.
||In your head, think of the number of edges on a cube. Multiply that by the number of quarts in a gallon.
Now we'll highlight words or phrases that might be changed to something else without changing the whole head problem.
In your head, think of the number of edges on a cube. Multiply that by the number of quarts in a gallon.
The last thing we need to do is make lists of words we might substitute for the highlighted words.
Here's one of my new head problems:
or for flat figures:
How many more can you think of?
- triangular prism
- rectangular prism
- square pyramid
- triangular pyramid
for quarts in a gallon:
For interest, you might even ask for the number of yards in a foot.
- pints in a quart
- pints in a gallon
- inches in a foot
- feet in a yard
What you do will depend on your grade level.
Here's one of my new head problems:
In your head, think of the number of vertices on this square pyramid.
Multiply that by the number of feet in a yard.
How many different head problems can you find?
The Daily Two Problem Approach is the second content strategy we recommend using and it involves presenting 2 standards based math problems to the total class on a daily basis. These problems should be displayed using the overhead projector, chalkboard, chart paper or some other visual that the whole class can see. It is also suggested that the 2 problems be handed out to students so they don't have to spend a lot of time recording the problem. Each of the 2 problems should be of a different nature (simple computation, multiple choice, open-ended investigation). You may choose to have students work on the problems individually, or with partners, or a combination of both.
Both problems should be debriefed with the students. Ideally the Daily Two Problem Approach should take no more than 10-15 minutes...however, sometimes the mathematical discussion generated by the problems is so worthwhile that the discussion and ensuing explorations can take up a whole math period...or more. If you choose to allow only a set amount of time for the Daily Two Problem Approach, you may not always get closure at the end of the allotted time...it's O.K. to come back to the problem(s) later in the day or later in the week. As the teacher, you make the call.
Daily Two Problem Approach is very useful for reviewing concepts, introducing concepts, assessing students, test format preparation, and developing number sense in students.
Frequently Asked Questions About the Daily Two Problem Approach
Q- How long should a two problem approach take?
A-10-15 minutes as a rule
Q- My two problems are taking at least a half hour and I just don't have time to do them every day. What should I do?
A-1 The problems you are choosing could be too difficult. You might...
1.choose or develop problems based on what students are currently learning.
2.choose or develop problems that review material covered earlier.
3.consider carrying over a problem to the next day.
4.put a problem up on the wall and invite continuing work on it.
5.start with a very simple version of a problem, making it more challenging on following days.
6.make one problem a multiple choice question
A-2 The debrief time may be taking longer than it needs to take. You might...
1.pass out a few overhead transparencies while students are working so they can share quickly.
2.choose, as you circulate, a few students you want to have share during the debrief
3.decide not to teach during the Two Problem Approach, but to make a note to teach later.
4.make sure one of the problems requires little or no debriefing.
Q- Where can I get standards-based problems for my Two Problem Approach?
A- Your new Harcourt Math has many sources:
Q- Do two problems have to be done on the overhead?
- Harcourt Math Daily Warm-ups
Problem of the Day
Number of the Day
- Harcourt Math Mixed Review and Test Prep
- Harcourt Math Assessment Guide
- Harcourt Math Performance Assessment
A-It is important to introduce and debrief the Two Problem Approach as a whole class. Some methods might be:
Q- Is it OK to give students copies of the two problems?
- Overhead projector
- On the chalkboard
- On chart paper
A-Yes. It depends on your own objectives and resources. It can be a good way to save the time it takes for students to copy material from the board or overhead.
Q- Sometimes I do two problems, but I don't have students write anything. We just do both problems together as a class. Does this count?
A-For Kindergarten it would always count. For first grade it would often be appropriate. for second grade and above it would sometimes be appropriate. Independent work is increasingly part of the value of the Daily Two Problem Approach.
Q- What should I do if my students get stuck on one of the problems?
A-Try to discern what is causing them to be stuck. Then consider the following possibilities.
1. Continue the problem to the next day.
2. Put the problem up on the wall to be worked on for a week.
3. Do a simpler version of the problem the next day, then come back to the hard one.
4. Consider the problem an assesment and try it again after the material has been taught.
Q- I do five problems every morning and then debrief them, can this count as my Daily Two Problem Approach?
A-Yes, as long as they are not all straight computation problems.
To save time and assure depth of discussion, you might sometimes choose 2 of the 5 problems to debrief thoroughly.
Q- I have been doing a daily "2 Step" problem, is this the same as the Daily 2 Problems?
A-Not as a general rule, although there may be times when a 2 step problem works nicely as a Two Problem Approach.
Q- Do I have to share all the different approaches of solving both the problems with the whole group because it takes a long time?
A-No, not always. If one of the problems is simpler and the whole class has the correct answer, a lengthy debrief might not serve a purpose. You might have a routine of inviting a set number of approaches to be shared for each problem. A useful approach can be to note solutions you particularly want shared as you are circulating and give those students a piece of transparency to "trace" their approach while other students continue working.
Q- Can I use the standards to create my own two problems?
A-Yes! The standards are a great resource for Two Problems.
Why use them?
- To get the students' attention.
- To maximaize instructional time and minimize time wasted during transitions.
- To review important math vocabulary and concepts.
Use this list as a springboard to get started. This list is not grade level specific, but it is leveled from easier to harder examples.
Start each one by saying, "If you can hear my voice, please..."
- Clap the # of sides on a ___(geometric shape).
- Show me on your fingers (SMOYF) the # of ___(sides, vertices, edges, faces, angles) on a ___(geometric figure)
- SMOYF the # of cents in a ___(penny, nickel, dime).
- SMOYF ___ (1 more, 1 less, 2 more, 2 less) than ___(any #).
- SMOYF ___ (1,2,3...) more / less than the # I'm showing you (you hold up a # on your fingers).
- SMOYF the ___(sum, difference, product) of ___ and ___ (include negative #'s for grades 4-6).
- Make a ___(right, acute, obtuse) angle with your arm.
- When I count to 3, please tell me the # of degrees in a right angle.
- Make a polygon with your fingers.
- SMOYF the ___(1st, 2nd, 3rd...) odd number.
- SMOYF the ___(1st, 2nd, 3rd...) even number.
- SMOYF the digit in the ___ place in the # written on the board.
- SMOYF the # of feet in a yard.
- SMOYF the # inches in half a foot.
- Tell me the # of ___(in., ft., mm., cm.) in a ___(ft., yd., cm., m.).
- SMOYF ___(1/2, 1/3, 2/3, 3/6, etc) of ___(any #).
- Show me a ___(horizontal, vertical) line with your arm.
- Show me parallel lines with your arms.
- Show me intersecting lines with your arms.
- Show me the direction of the ___(x or y) axis with your arm.
- Show me the square root of ___ (4, 9, 16, 25, 36. 49. 64. 81, 100) on your fingers.
- SMOYF the # of factors a prime # has.
- SMOYF the ___(1st, 2nd, etc) prime #.
- SMOYF ___10%, 20%, etc.) of ___(any #)
- (Have data on the board or overhead.) SMOYF the ___(range, mode, median, mean) of the data.
The National Library of Virtual Manipulatives
Where to start:
Click on "Virtual Library" to explore a large number of very well done interactive visual representations. A nice place to start is with any of the base blocks activities. There are several different levels of base blocks focusing on different operations.
Be sure to try, under Geometry, Platonic Solids. There you can see solid shapes, change the size, rotate them, change colors of faces (helps keep track if you're counting!).
As you explore, be sure to check higher grade levels than your own. This is from Utah. You will find some activities in 9-12 that are in our California standards in elementary grades.
I had trouble loading the activities in two versions of Netscape on my iBook. In Internet Explorer they sometimes required two attempts before they would load. I tried closing the activity window and clicking the button for the same activity. It worked. The activities are worth the effort.
The National Council of Teachers of Mathematics (NCTM)
Where to start:
From the home page, Click "Principles and Standards Resources" on the left of the screen. This will take you to a page where you have two choices: Principles and Standards and Illuminations.
Principles and Standards
These standards make great professional reading. If you are looking for the Big Ideas, this might be a good place to start.
You might choose one strand for your grade in the Principles and Standards to read carefully. How is it different from other standards you are required to work with? How might it help clarify and support your implementation of those standards?
There is a real wealth of resources to support great mathematics teaching here.
I-Math Investigations and Interactive Math•lets are interactive activities for your students that support conceptual understanding.
Inquiry on Practice offers video vignettes, research reports and articles for teachers.
Links to lesson plans and web resources are also available here.
The Math Forum
Where to start?
Anywhere, depending on what you are looking for. Here are some of the kinds of things you can find here.
You can find answers to your own math questions. Dr. Math has a library of questions and answers. And if you don't find your answer there, you can submit your question.
You can find lots of grade- and strand-specific links to many kinds of resources as well as access to lots of teacher created lesson plans.
You can find interesting and challenging Problems of the Week from different strands, and with different levels of difficulty.Try some yourself!
There are lesson plans and information about any area of mathematics you can think of. Want to know what an eleven-sided figure is called? You can find it here. Want to know why a negative times a negative equals a positive? You can find it here.
One of the most powerful aspects of this site is that you can search it for specific areas of math. You will get all kinds of possibilities to explore.
Where to start:
This site offers a number of ways to do facts practice, including printable worksheets, at different grade levels.
Try the interactive areas before letting kids do them. I found the java flashcards didn't work properly on my computer. The worksheets seemed to set up nicely.
The Maths File Game Show
Where to start:
I would recommend the games. There is quite a selection of them. They are for intermediate students, but great for adults too. Mathematical information and tips for playing the game are easily available. There are printable puzzles. The designer of this site clearly has a sense of humor as well as a love of math.
I was prompted to obtain and install the latest version of Shockwave. I don't tend to stay up with such things. But it installed easily and when I returned to the site everything worked very well.
Eisenhower National Clearinghouse for Math & Science Education (ENC)
Where to start:
This is a very rich site. I would recommend the tour, just to get the idea of how the site works. It is very well organized. You will find every imaginable kind of teacher support for the teaching of mathematics there. I found the Maths File Game above from the ENC site.
This site is a very good resource for teachers who want to grow in their mathematics teaching, and who are interested in the enjoyment and understanding of math for their students.
Everything seemed to work just fine, but this site is so big and rich, I have a tendency to get overwhelmed. It's important to remember you can bookmark it and come back. Explore it a little at a time.
California Department of Education
The Mathematics Standards and its accompanying framework are on this site and can be downloaded as a PDF file. Test score and API information can also be found here.
The International Study Center (TIMSS)
Lawrence Hall of Science, University of California
GEM - The Gateway to Educational Materials
Internet Resources for Elementary Mathematic
An interesting site filled with lessons and projects that are worth considering.
Kathy Schrock's Guide for Educators
This site has more resources than you probably have time to access. It's a great site.