Encouraging Conversations In Math Class.

One of the most important things in math involves conversation between students.  Unfortunately, many students arrive in high school without having the basic skills needed to share ideas or explanations of how or why they completed a certain problem.  In fact, the principal discussed that one thing they have to look for evaluations is a certain level of talk carried out during class.  

When she read out every thing the third party evaluation said students should be doing, I was surprised. We didn't get to that level of discussion in my college math classes.

In today's world it is important to encourage frequent mathematical conversations in class because it helps them deepen their own understanding while clarifying their own thoughts.  In addition, it helps turn students into a community of learners. 

Before beginning the discussion, it is important for the teacher to define exactly what they want to accomplish at the end.  With the goal in mind, the teacher has somethings to think about.  Before the discussion begins, the teacher should choose a problem which guides students towards the goal, and think about possible student responses so as to be prepared for misconceptions that might need to be addressed.

During the discussion, the teacher should monitor student responses, listening for misconceptions, and areas of concern.  Think about which students have a solution that meets the teachers original goal so the students can share with the rest of the class.  Furthermore, take time to decide what order the solutions will be presented in.  One might consider arranging the solutions from the most widely used method to one that is unique.  Finally connect solutions so students are able to discuss similarities and difference.

Finally, after the discussion, teachers need to go back through the solutions to look for misconceptions or areas requiring clarification.  As students to review their solutions to see if they seem reasonable or select a solution and have students find a method to determine if the answer is reasonable.  In addition, students should be able to justify the method they used to arrive at a solution.  Finally, take time to have students examine errors to discover where they went wrong and what they need to correct the problem. They should be given time to rework the problem until it is correct.

There are seven techniques teachers can use with students to help them learn. These can easily be used in the classroom.

1.  Revoicing or repeating - The teacher can do this or have another student do it.  The person repeats what the student said and asks if this is what the student said.  This technique can help clarify a student's thoughts or emphasize the thought.  When this technique is used frequently in class, it can help students recognize the important elements of a conversation and select these to remember.

2. Repeating - this is different than revoicing in that a student repeats or paraphrases the important ideas stated by another student.  This one helps students identify important ideas and slows down the conversation so students have time to process the information.

3. Reasoning - in this activity, the teacher asks one student to compare their reasoning with that of another student.  Students have to analyze the other student's reasoning before they can begin to compare the other student's logic with theirs.  Students end up diving deep into things, while making sense of the other person's thinking patterns.

4. Adding on - this is where once a student has finished sharing their thoughts, the teacher asks others to add on to what was said.  This encourages students to participate and is best used when only a few students are involved in the discussion.

5. Wait time - This is hard for many instructors because we don't like silence and even waiting 10 seconds is so hard as it feels like we are waiting forever.  Many students need just a bit extra time to assemble their thoughts.  When we don't give enough wait time, we discourage some students from sharing.  

6.  Turn and talk - is more of a think-pair-share. This technique allows students to talk in pairs, where they are able to express ideas in a more secure situation.  It also gives the teacher a chance to listen in to discover misconceptions, ideas that need clarification, or students who need just a bit more to be there.  This is a great assessment tool.

7. Revise - this is where students revise their thinking as a result of all this talk.  Students are able to explain how their thinking changed from where they were in understanding to where they are now.  In addition, they can explain why their thinking changed.

These are techniques to help encourage conversation in class.  Unfortunately, you cannot just implement all seven in one day.  It is important to introduce one technique to the students and have them practice it over a period of time until they have it down.  When practiced enough, students will do these automatically and their ability to discuss the material will increase accordingly.

On Monday, I'll share some conversation starters and response starters to help students who might have trouble getting started.  Let me know what you think, I'd love to hear. Have a great day.

Guided Practice

Many teachers are facing the challenge of teaching either via distance or in a hybrid model which does not allow as much time for helping students individually as much as we did before.  I worry about being able to provide enough guided practice in class so my students really learn the material.

As any teacher knows, practice is needed for students to learn to do anything, be it sports or math.  It does not mean getting them to do it perfectly the first time because more learning goes on when they make mistakes and have to correct them.

Guided practice is a way to help students practice learning a new skill or old skill at their developmental level with increasing levels of difficulty. In class, it is easy to plan for guided practice but it is not as easy when you have less time with the students in class.  Normally, it would be possible to divide students up into groups to work but with Covid-19, it is not as easy.

Fortunately, there are some methods of guided practice we can use in this time of unique instruction.  One is called the "Gradual Release of Responsibility Method" which is also known as the "I do, We do, You do" model.  This method moves from teacher focused to student focused by transferring responsibility.

When the teacher models the action, it is the "I do" part.  With the teacher modeling, students get a chance to see the task and how it is done.  It introduces them to it.  It is suggested that teachers break the material into small, clear, steps by using some sort of visual chart to help students understand the process.  Include mnemonic and acronyms to help facilitate student recall while encouraging students to take notes of things they might have trouble remembering in the future.  Furthermore, it is important to give students time to ask questions at the end of each step for clarification and to allow them time to process.

The next step is the "We do" stage where it is done together.  I've done this stage by having students do the same process I modeled in the "I do" stage.  This can also be done by working in small groups or in pairs. In terms of zoom, it would be done via the use of breakout room but it is much harder with the 6 foot distancing which is why one can create worksheets with missing information that students fill in.  

In addition, one might ask what the next step is?  This helps students work on recall to move the information from short term to long term memory.  One might also ask students to debate whether a certain step is better than another such as would you divide by the number outside the parenthesis in a distributive property problem rather than multiplying?

The "You do" stage is actually divided into two parts.   In the first part the teacher is facilitating responsibility from herself to the students and in the second stage, students are given full responsibility for completing the problems.  In the first part, students are encouraged to complete as much of the task as possible with the teacher stepping in to gently guide the student towards completion. In this stage, teachers should ask students to explain what they've done and why they did it that way.  Students should be encouraged to ask questions when they stall and teachers should provide direction via open-ended questions, prompts, or nudges to help the student remember the next step.

In the final stage for "You do", the student is expected to complete a task from start to finish and be assigned a number of problems to complete to reinforce learning.  One can also have students do similar problems that have a small twist or are just a bit different to help with transference of material. 

Sometimes having group or pair work is difficult in class due to the coronavirus. I've found that if I create a worksheet with an example complete with explanation and then write the following problems so students fill in the missing material, it can do the same thing.  What I mean about following problems is that I might have two or three that are missing the final step so students have to write that in.  Then I might cut out the final two steps so students have to fill those in to complete the problem.  Eventually, the only thing students have is the problem it's self.  This has been especially helpful for students who struggle with math.  

Let me know what you think, I'd love to hear.  Have a great day.

Happy Labor Day

Warm-up

I will tell you the number you get at the end of this problem.

Write a three-digit number (with three different digits).
Mix up the digits to get another three-digit number.
Subtract the smaller number from the larger.
Add the digits in the difference. (If you get a two-digit answer, add the two digits to get a single digit.)
Subtract 5 to get a final number.

Your answer is 412.

Warm-up

I will tell you your answer at the end of this math problem.

Think of a number between 1 and 100.
Multiply your number by 4.
Add 12.
Multiply this number by 2.
Add 16.
Divide this number by 8.
Subtract your original number.  

Your answer is 5!

Math Games for the Social Distanced Classroom Part 2.

There are more games and more games that can be modified to play in the classroom during times of social distancing.  Games we can play during social distancing, or via distance learning are important.  No more sending problems down the rows.  No more moving people up or down the rows.  So we have to make changes.

1.  Match Game - This is where students are given a worksheet with three columns.  The first column has the problems while the second column has the answer but they are not lined up correctly.  In other words the answer that is next to the problem do not match up.  In the third column, students work out the problems and once they have the answer, they write the letter of the answer next to the problem.  

This can be done with any math topic from solving simple equations to trigonometric functions to multiplying binomials.  It takes just a bit of work to prepare the sheets a head of time but well worth it.  Furthermore, this game could be done using cards on the desk with the problem worked out on the back of the card with the problem.  

2. Round Trip - Although this is normally done with students standing next to each other, it can be played with 6 feet of distance.  This can be played with students sitting at their desks while using white boards. First step is to designate the the first two players but have the other students ready to work the problem to check the two designated players work.  Show the first problem and let the two designated students work out the problem.  The first one with the correct answer as checked by the other students get to play against another student.  The idea is if one student can work their way around the room completely, they are the total winner otherwise, go with the one who answered the most problems.

Any math topic can be used for this game which makes is possible to use it in everything from basic math to calculus and anything in between.

3.  Loopy numbers - This game has students follow a specific rule such as one follows for a function machine.  The rule is "Multiply the ones digit by four and add 1 to the product.  This rule is applied to the result of the previous calculation where you multiply the ones digit by four and add one to the product and continue.  An example might be begin with 15 so you do 4 x 5 + 1 = 21.  Then you take the 21 and multiply the 4 x 1 + 2  = 6. The next round would be 4 x 6 + 0 = 24.  Then 4 x 4 + 2 = 18 followed by 4 x 8 + 1 = 33.  Then we have 4 x 3 + 3 = 15.  At the end, the student should end up with the number they started with.  No matter what two digit number they choose, if they follow this rule, they will end up where they started.

4.  Coordinate Battle Ship - This is a version of battle ship that uses two squares of graph paper on one sheet.  The left side is filled with the players ships while the right one is where they mark down hits, etc.  To start, each player marks down their ships on their coordinate plane with one carrier, two destroyers and three submarines using dots. When both players have placed their ships, one person calls out a coordinate of where they think the other person has a ship.  The other person says hit or miss and then calls out a coordinate for them.  With social distancing, it is easy for people not to sneak a look, especially if you have students use a folder to create a wall.  If you'd like, this site has a sheet all ready to go along with the directions.  Students have a fun way to practice or review the use of coordinate planes. 

Here are four more games that can be played in a socially distanced classroom.  Let me know what you think, I'd love to hear.  Have a great day.

Math Games For The Social Distanced Class Room, Part 1.

 

My school year has started with about three weeks of inservice and at this moment the students are due to start on September 14th with masks and desks spread out as close to 6 feet apart as possible.  Furthermore, only half the students will be in class at anyone time.  I realize I could assign students to play games on digital devices but those do not usually allow as much interaction among class members so I decided to find games I could have students use in class.

I'll be sharing several in today's column and I'll share several more in Friday's column.  The games are fairly easy and won't need a ton of preparation.

1.  I have...... Who has....  game.  This game uses cards with an answer and a question but the answer does not go with the question on the card.  For instance the card might read "I have x = 3, who has the answer to x - 7 = 4."  The answer of x = 11 is on someone else's card with a new question.  To play, you make sure you have enough cards for one per student and that the answers and questions line up so the last person to have the answer is the same one who read the first question.  After I pass out the cards, I usually give students a chance to read their cards so they are ready for the game.

To start have the teacher or a student read the first question such as " Who has the answer to x - 7 = 4?" Give students a chance to find out the answer to it so they see if they have the answer.  If students are struggling , encourage students to talk about solving the problem so they can help each other.  Once the answer is figured out, let the students see who has the phrase " I have x = 11".  The student identifies themselves and reads the next "Who has problem?" It continues until everyone has had a chance to answer  a question and ask a question.  

This game is a great way to have students practice previous and new skills.  It can be used for any grade and almost any topic from basic addition, to multiplying binomials, to factoring, to practicing multiplication facts.  I've used it to help scaffold missing skills and used it to help students learn a new skill.  I also admit, I have messed things up so I didn't manage to go all around the classroom and I've also managed to accidentally come up with equations that ended up with the same number.  On the other hand, when I've gotten all the cards set up correctly, it has been lots of fun.

2.  Math Bingo.  I pass out empty Bingo cards to my students.  On the board, I have a huge number of possible answers.  I ask students to fill out their bingo cards with answers from the board.  For instance, if I wanted to have students practice order of operations, I might say, select any numbers between 1 and 99 and write them down on your card in any order.  

Once everyone had their cards filled out, I might write the first equation on the board such as 2 x 3 + sqrt 25 x sqrt100 =    .  I let the students figure out the answers by doing the calculations and usually someone calls out an answer but no one writes the answer down until a second or third student confirms the answer is correct and then students cross it out if they have it on their card.  I'll put another equation and wait till they come up with the answer.  Usually, students want to play blackout because they are having so much fun.

I've used math bingo with fractions, order of operations, one step, two step and multistep equations, factoring trinomials, area, perimeter, trig functions, and any other math topic I can think of.  Again, it allows students to practice their already learned skills and newly acquired skills.  

3. The Great Escape - normally requires students to line up but in the time of social distancing, it won't work.  Instead have students stay seated at their desk.  Go through and ask each student three questions.  If the student gets the first one right, ask the second, but if they don't get all three correct, move on.  If the student manages to get all three correct, they can leave a minute early from class, or get to do fewer homework problems that night.  The reward is up to the teacher.

On Friday I'll be sharing a few more games that can be done in a classroom where the desks are 6 feet apart due to social distancing.  Let me know what you think, I'd love to hear.  Have a great day.