Silent Discussions

Some major goals of mathematics are to have a deep understanding of the content and to be able to make connections to other concepts. I decided to look around the NCTM website for articles about such things, and I found one called “Visible Thinking in High School Mathematics.” This article is about two main methods: Chalk Talk and Claim-Support-Question. I’m going to focus on Chalk Today, because it really caught my interest.

The main idea is to have a variety of posters around the room with questions on them, generally sounding something like “What do you know about (concept).” Forever however many posters there are, say five, that many different colors of markers are distributed among the students. Students with all the same color markers are sent to a poster, and are told to write what they know about the concept. This is a totally silent activity, which is why the author called it “Silent Discussion.”Students then rotate around the room and either respond to what other students wrote or write their own new idea.

Chalk Talk gives students the opportunity to look at other students’ ideas and get their questions answered at least partially by other students. For a question such as “What is a quadrilateral,” a student may have thought of a square, but with Chalk Talk, they can get the opportunity to see a non-square rectangle, rhombus, parallelogram, or any other quadrilateral, possibly with a picture and description. It gets them thinking outside the box. If they aren’t sure about something, they can ask, and the next group at the poster won’t even know who wrote it, and they can get an answer for their question. That’s ideal.

The posters really end up looking like a mess, but the teachers can somewhat gather what the class knows and doesn’t know, as well as where the class should go next. Even if questions get answered, it still shows that students might not quite feel comfortable with a concept. On the other hand, a question asking about a possible future direction from their new knowledge can make for a great transition into the next topic. Also, students enjoy getting out of their seats, and this is a productive way to do that. It is a great idea overall, in my opinion.

Visible Thinking in High School Mathematics


10 Strategies to Highlight Strategies

Angela Watson’s Article

So often, students just want to find the answer to problems so they can get done with the assignment and do something else. I’m guilty of this myself. When I took Calculus 1-3, for online assignment, I’d look at an example problem, find where the numbers in the problem came from, switch in my answer, and move on. However, a few weeks after the test, I could not remember how to do the problem. How can we expect students to do well on standardized tests when they get homework done by taking shortcuts while gaining no real understanding of the mathematics? In our 7-12 Math Methods class, we’ve been talking about having students explain their strategies and recognize that there are multiple available strategies for many problems. We need to employ strategies to help students understand…well…strategies.

To learn how to help see students understand all strategies more effectively, I read an article by Angela Watson titled “10 Classroom Strategies to Get Students Talking (and Writing) About Math Strategies.” As the title suggests, there are ten different strategies that Ms. Watson suggests to get students to talk and write about their strategies and also just to ensure that they know.

  1. Start a lesson by talking about possible strategies, rather than with an easier problem. This strategy involves having a more difficult problem that you want them to solve, and instead of requesting an answer, the teacher has them think about how they might solve it, then get together with a partner and discuss, and finally share potential solution strategies and use them to solve the problem.
  2. Along with number 1, another strategy to use is to have students do problems without finding the answer. This has students recognize the importance of the process. The teacher can also have students discuss their strategies, and if there are multiple strategies to solve the problem, this method can help students realize it. I agree with this idea, because it will help students remember what they did to solve a problem rather than the answer to a problem they’ll never do again.
  3. Ask students about what they did throughout the process and about their work, not just how they arrive at an answer. Ms. Watson came up with 100 questions to ask to students to start conversations about their mathematics. These include questions asking about real-life examples, where certain values came from, how one could estimate answers to similar problems, and, obviously, many more. I might have to use some of her questions in my classroom.
  4. Discuss different strategies for solving problems and why people chose the strategies they did. According to Ms. Watson and probably many other teachers, students get annoyed when doing math because they can’t figure out the one single solution strategy. Showing that there is more than one strategy for solving the problem can make students more eager to do math, because they can do it in their own way. I believe in this strategy for teaching mathematics, because we do it in our methods class, and I’ve learned about connections between different types of mathematics that I never knew existed.
  5. Use math journals. In math journals, students can do their work for problems, as well as state what they were thinking at every step. They could write about what went right, where they got confused, and reasoning for it all. With math journals, we as teachers can figure out where they’re at mathematically while they have to explain their reasoning and understand it more. I might be open to using math journals, although it seems like it might take a lot of time if I were to actually read all of them.
  6. Have a smaller set of questions you can use all the time. These may be similar to the 100 questions mentioned earlier, but there are much fewer of them, and they should be used often enough that students start to think of them on their own. Some of the questions I might use are: How did you get your answer? How do you know it’s correct? What are some other strategies for solving this problem?
  7. Use effective techniques for getting students thinking. There are many ways to do this. Asking the right questions is one way. Another is to walk around and encourage or help students. Selecting the right order to address solutions in for discussions is another one. Getting to know all the ways could make me a very effective teacher.
  8. Play math games. I think that Ms. Watson might have had longer classes than the 50 minutes ones I had when I was in high school, because she suggested breaking up a class period with a 10-15 minute math game, but I think that would probably take some valuable time if done on a daily or nearly daily basis. I think I would use this strategy once a week or so to check understanding and provide an incentive for the students to learn the information.
  9. Use a program that allows you to see their work as they’re doing it. I think the main takeaway from this is that there are apps and such for which teachers can have students record their screen as well as anything they say while they’re working on homework, so they can see the whole process. This gives a good idea of where students are at with the concept.
  10. Have students create posters that contain all the strategies they’ve seen. They can start making these posters at the beginning of their learning about a concept. As time goes along, they can add the teacher’s strategies, other students’ strategies, and their own strategies.

Let’s Talk About Math

When planning a math lesson, teachers often try to think about how to engage their students by using examples from the real world aligned with the students interests, group tasks that force students to work together to complete a specific problem, or the basic question and answer styled lesson. Rarely do they think about having a class discussion…but why? Why is it that math classes shy away from having discussions to teach content? One reason might be that it is difficult to gear questions that promote student interaction and discussion.

Students who talk about math are able to address gaps in their understanding of mathematical concepts and allow them time to express their ideas more precisely. After reading the article Talking Math: How to Engage Students in Mathematical Discourse, I have found a few tips that will help promote this type of classroom.

First, be sure that your students are aware that you expect them all to participate in the discussion. You cannot run an effective discussion if only two students do all the talking for the class. Make sure everyone feels comfortable–starting this with small groups may alleviate anxiety for some students. Second, discuss errors and solutions in detail. Be sure that each student understands the reasoning behind each solution. Third, go over definitions and introduce new mathematical vocabulary during lectures and general conversations with your students. Setting students up with the right words will make them more confident when they are talking about math in front of the group. And most importantly, let the students talk! If you hear a student say something that is wrong, let them talk it out for a bit. See if your students can help each other before you chime into the discussion. The following image shows what you might see in a classroom that has meaningful mathematical discourse.


Critical Thinking in the Mathroom

Recently I had the opportunity to attend the SDCTM/SDSTA joint conference in Huron, South Dakota. While there, I attended many different sessions that gave tips and trick about keeping students engaged while still learning the material. As society moves away from traditional teaching, I have been trying to think of ways to incorporate more lessons in a form other than lecture. I have also thought about the need to get my students to think critically and struggle productively in the process. While in one of the sessions, I was introduced to a method called 3-ACT math tasks. These were quick lessons that force the students to think and problem solve on their own and in groups. Each tasks contains 3 steps (where the name comes from):

  1. Introduce the central problem of the task with as few words as possible.
  2. Have the students determine the information they might need to solve the problem and have them guesstimate a logical answer and reasoning for their answer.
  3. With the information at hand, the students are able to solve for the solution–then set up a sequel or extension to make sure all students grasped the concept.

The 3-ACT we did in our session was over surface area. First, we watched a video of a guy who had a filing cabinet and was covering it in numbered post-it-notes. We did not get to see him finish, but were then asked to determine what he might be doing. While there were many answers, the obvious was trying to see how many post-it-notes it would take to cover the cabinet. We all guesstimated an answer for this question. After this, we were asked what information we would need to know to solve this without physically covering the cabinet. Being a room full of math teachers, it was not difficult to know that we needed the dimensions of the cabinet as well as the post-it-notes. However, in a classroom just learning about surface ares, it will take longer for the students to decide on the information they need. After waiting for the students to decide, the teacher gives the students only the information that was asked for–making them think a little extra if they get it wrong the first time. Once the class has decided on the information needed, they solve the question mathematically. Once we all had our guess, we watched a final video showing all of the sticky notes going on and revealing the final answer. We then discussed errors made (I will not give it away in case you want to try this yourself).

These ACT’s are a great to get kids engaged in the material. Here is a list of 3-ACT math tasks created by Dan Meyer, an officer for Desmos which is an advanced calculator application that is being implemented in current testing.

Discourse in the Classroom

The specific section that I had to read about and report about was on discourse in the classroom. This includes discourse between the students and the problems, students and the teacher, and students with other students. The article discussed how this is where real learning occurs and the importance of debating mathematics.

One of the main reasons that I believe this is so important is because it truly gives the students a chance to deepen their knowledge on the topic. Being able to debate specific steps of a problem shows a deeper understanding than knowing a procedure and reproducing it with different numbers. This gives the students a real life way of looking at the mathematics that will help them understand the concept better.

When students debate over topics and procedures, they see the importance of certain steps and really connect each step to the next.

It is important as teachers that we allow this discourse to happen and even make it happen. This happens by setting up classroom situations so that the students feel comfortable to debate and be wrong. Debating in front of the whole class makes the student vulnerable and so, as teachers, we need to be aware of how we react to this discourse and how we encourage it.

I found a source that specifically discusses the importance of discourse and how as teachers we can create this in our classrooms. The source comes from:

This article talks about the two main factors that teachers have to remember when facilitating discourse are: cognitive discourse and motivational discourse.

“Cognitive discourse refers to what the teacher says to promote conceptual understanding of the mathematics itself” (286). A common misconception that many teachers have is that they now should sit back and let the students make the connections and figure out the mathematics on their own; however, teachers still have a very important role. This being that they have to ask questions to ensure that their students are fully understanding and they have to lead them to the correct answer without giving it away.

A large problem with discourse is that it can be difficult to get all of your students to share their ideas and announce their opinions. This is why teachers need motivational discourse as well. “Motivational discourse refers not only to praise offered to students but also to supportive and non-supportive statements teachers make that encourage or discourage participation in mathematics classroom discussions” (287). A big way that teachers can provide this is by emphasizing that we learnt through our mistakes and everyone makes mistakes. However, these exchanges need to be supportive. A teacher needs to have the students explain how they got to their answer as opposed to focusing on whether it is correct or not.

I know, as a student, the best way a teacher could get me to not participate was to embarrass me in front of the class. If a teacher would explain to the class why my answer was wrong and tell me to correct it, I most certainly would not put my hand up the next time in fear that it would happen again. Teachers need to create a comfort and support in their classroom where students feel that they can be wrong and it will be okay.

Narrowing Participation Gaps

One thing that we touched upon this week in class was found in the chapters we read from ‘Strength in Numbers’ that participation in your classroom can be tricky. It can be difficult to make students who seemingly do not care answers questions and volunteer. This creates a “participation gap” where the students who participate, participate a lot and the students who do not participate, never participate. It is no surprise then that this correlates almost one-to-one with achievement as students who participate are the students who gain a deeper understanding material and know it (that is why they are volunteering answers).

The ‘Strength in Numbers’ story about the teacher who drew out participation from all of her students really intrigued me to find out more about how to get your students to participate. Especially when you have students that are all very different in terms of personality, race, backgrounds etc..

I found this article called, “Narrowing Participation Gaps” by Victoria Hand, Karmen Kirtley, and Michael Matassa that does just that. The url is:

The article discusses three specific ways to encourage and increase participation by all students which, in turn, will narrow the gap. First,

  1. Organize Mathematical “Contributions”
  • Participation is not only talking or answering questions
  • More than one way to get math answers
  • Prompt students unclear answers with directing questions

The first bullet point makes the point that teachers too often ask a question and evaluate the student’s knowledge based off of who answers and how they answer the question. It is important to remember that math is all about process and teachers need to evaluate how the students got to their answers. There are other ways of participating in class other than speaking as well. An important quote that goes along with this explanation is, “This orientation also prioritizes correct uses of academic language over students’ sense making”.

The third bullet point is one that was not said, but I observed from the example in the article. These directing questions can be uber-focused on the words that students use such as “length” and “width” as it may have a different meaning to the student than it does to the teacher. Thus, these clarifying questions help the teacher know what the student is saying and it helps the student understand what he/she is saying. This can also help narrow the participation gap when students aren’t afraid that they will be “wrong”, but rather they will get guided to the right answer. It is important not to simply give the answers to the students as they are learning absolutely nothing there. Confusion is the best way to learn.

2. Expand “Smart-ness”

  • Expand perception of who is “good” at math
  • Complex Instruction in groups
  • Assigning roles
  • Reward different ways of thinking

Group work, in general, causes more participation, as each member is responsible for their own work. This is very true for what roles are assigned and each student has a responsibility. In my opinion, group work sometimes creates less participation as the “smart student” does the work. Assigning roles changes this and allows students to see the way others think and it will expand what they deem as smart as well.

3. Engage Instead of Motivate

  • Takes away the blame on students
  • Change the classroom as opposed to the student
  • Don’t label students

While I agree it is important not to label students, I do not necessarily agree with “engage instead of motivate”. Yes, there is a time and place where you do the work along with the students, but giving motivating problems is good as well. I believe there must be a balance of both to really get the students to participate. The article talked about how teachers should allow foreign language speaking students to solve the problems in their first language and then to explain it in english and I think this is a great method. Bottom- line I believe you still have to motivate your students and it is not all about doing the work with your students because then they will start to bank on you for the answer and that will only hurt their learning.

Increasing Student Participation

Student discussion and input can greatly impact learning in a positive way. The issue is that there are always students who are more vocal and have higher status. A few student will throw their hand up every time a question is asked to the class. Meanwhile, there are students who are constantly concerned that their question or answer is wrong. A major question we have to answer is: How do we get all our students to provide their input?

The article I read, “Increasing Student Participation” from The Teaching Center, which is part of the University of Washington in St. Louis. The article gave several ideas, but the ones I found noteworthy, along with some of my own ideas, are the following:

  • Set up ideal physical setup for discussion
  • Set expectations
  • Establish environment of caring and respect
  • Have class schedule set up for time for discussion
  • Respond positively to student ideas

Set Up Ideal Physical Set-Up

According to the article, there are many things that a teacher can do for the class to prepare them to participate in discussions. One thing is just the way the classroom is set up. The article suggests creating a U shape with the desks, which helps, since the students are facing each other and not just the teacher.

Set Expectations

Also, set expectations for how much each student should participate. If the intent is to have every student contribute, let them know that every one of them has to contribute. One option is to have participation be part of the grade, but if we do our job as educators, getting students to contribute to a discussion should not have to be something that we have to force them to do, but eventually something that they will do on their own with our support.

Establish Environment of Caring and Respect

One idea that the article did not address but I’d like to is establishing an environment of respect and caring. If we as teachers make it clear that every student’s input is valued, no matter their status in the class, then students will respect what other students are saying and not degrade them for it. As a result, students will not be afraid to speak their minds as much as they would otherwise.

Have Class Schedule Set Up for Time for Discussion

One way to promote discussion in a classroom is by allowing plenty of time for it in the proper way. If a teacher only asks for questions and students’ thoughts at the end of a lecture, they may not remember what they thought throughout the lecture. On the other hand, if a teacher stops after every fifteen minutes of a lesson and asks a question to the students and asks for their questions and comments, the students may better remember what they just learned. Also, asking a question and taking responses immediately is not always the most effective way for students to formulate ideas and give a response. Giving students 10-15 seconds to think allows everyone to come up with an idea before anyone else responds. Mixing up discussion methods can make it so every student can give a response. Moving from partner groups to small groups to large groups to a whole class discussion can make it so every student feels comfortable sharing at some level.

Respond Positively to Student Ideas

For students that lack confidence, pointing out their good ideas is very important. Even if their answer is not totally correct, pointing out what is correct or paraphrasing so it is correct can give them a confidence boost and think of their response as valuable. As stated in the last section, having students submit responses online or in some other private way and responding positively to that will give them the confidence to participate more in partner groups, where positive feedback will give them the confidence to participate in small groups, and so on.

The article “Increasing Student Participation” gave some intriguing ideas for how to get more students involved in discussions. I think that this is a very important subject to bring up, because all through my education, there were always people who would contribute every day and people that wouldn’t contribute at all. Sharing ideas can be a great learning experience for everyone. Hopefully we can use these ideas to have all our students contribute in meaningful ways.