Opportunities for Extension
In any math classroom, some students are quicker to understand a concept in problem solving faster than others. In too many cases earlier finishers are given unrelated tasks to finish or the students that weren’t able to solve the problem as quickly are hurried along to finish and the solution and strategies used are revealed. Thus, those students are done engaging. That should not be the case ever, we need every student to have the opportunity to learn through problem solving. Creating opportunities for extension would be a great solution to the struggle of having such diverse learners. Also, by having opportunities for extension to problem solving continues the learning. It enforces the idea that it is about the process not the solution.
Examples of extension activities:
Students create their own math problems. Students are given parameters on what type of problems they’re creating. If you’re teaching a unit on volume, students should be writing problems that are on that topic.
Create a math game that is on the unit you’re currently working on.
Write a math story that has math problems embedded in it.
A Teacher’s Role
One of the biggest factors that plays into our students' learning through problem solving is how the teacher facilitates the activity. We all have experienced a student seeking help when they have come across the task of answering a problem based question. Many of the students that seek out help know the content that is presented in the problem, but the student is seeking knowledge about the process, the critical thinking aspect. We have to be a beacon for our students to find their own paths in breaking down a problem and making discoveries on their own. This is challenging for teachers. There are three main roles a teacher should consider when implementing problem based learning in their classroom.
Teacher as the instructor: When the teacher plays the role of the instructor, they are modeling for their students how to solve a problem, sharing the meaning of vocabulary, providing direct instructions on conventions of mathematics. In order for students to learn they need to be observing, practicing, participating, and answering questions.
Teacher as the facilitator: The role of the facilitator is to guide students to discovering the concept on their own. Facilitators do these five things:
1. Set up problems and investigations.
2. Guide their students with leading questions.
3. Refrain from delivering instruction, no procedural instruction.
4. Do show students models and encourage use of strategies.
5. Try to make themselves unneeded.
Click Here to see a small group problem solving lesson of a kindergarten teacher giving her students a task of comparing cubes. She gives two set of partners a set of blue linking cubes and a set of red linking cubes. At first, she asks them to compare with their partners, who has more cubes. You can see as you watch the video that not all the students are at the same level of understanding in their number comparisons. Some of the students are still struggling with counting. Listen to the questions that the teacher asks. She is not instructing at all, she doesn’t offer any instruction on how to compare the cubes. They are “how do you know” questions. She gives them the task to make the red cubes show three more than the blue cubes. That is as far as the instructions go, the students have to critically think about what that means. She does clarify that they don’t have to use all the cubes, which was confusing some students, but they do have to figure out on their own with only guiding questions how to make the representation of the two colored cubes the way the teacher wanted it to be.
Teacher as the coach: When students are working and discovering, they often need affirmation that they are on the right track to understanding. When a teacher is observing their students' discoveries, they are often a soundboard to their students’ thinking. This is the teacher playing the role of the coach. The students are leading the conversation, the teacher is helping students stay on an accurate path and guiding them to think as a mathematician. The coach is there to provide feedback.
Supporting Perseverance:
Often students can get frustrated when a teacher doesn’t help them when they are struggling. To them they feel like the teacher is purposefully not helping them because they don’t want to or to be mean. How can a teacher scaffold the task or give support to the students that are struggling and becoming frustrated?
Use-open ended tasks so that there isn’t one definitive solution.
Provide tools like manipulatives that will help students that may need it.
Praise effort and critical thinking, if a student is praised for trying they are more likely to continue to work hard.
Ask guiding questions, without instructing.
Provide feedback when warranted.
The “Mathematician's Chair”
The last procedure in the Share and Compare problem solving method is the part where students share their solutions and procedures. The students sit in front of the other students and the teacher. The teacher’s position in the room is important here because you don’t want the students looking to the teacher to facilitate the discussion, but the student that is sharing. The teacher is to take the same role as the other students.
What to do when a student doesn’t want to share:
Give children a chance to share with a partner.
Give praise to students during their work, even if they have an incorrect solution.
Let them share from their desks and not in the “mathematicians char”.
Assessing Students Through Observation
When we teach our students math, we are not just teaching them the content standards, but we are working hard to shape them into math thinkers. The advantage of implementing problem solving tasks such as the ones outlined in this project is that it gives teachers the chance to see students learn differently than they were before. We now have the opportunity to assess them on their math thinking skills and not just on their test scores that show their understanding of the standards on math concepts.
Math Practices:
These may vary state to state, but in Massachusetts we have eight mathematical practices. These practices are standards to teach math by. I am not talking about specific content standards, but standards that embody how students should learn and do math.
One of the mathematical practices is to Model with Mathematics. It is not our job only to teach the standards, but to make sure our students can apply models to explain their understanding even if they are able to understand math in the abstract stage.
Another math practice standard that we are able to assess during this process is the student’s ability to persevere through a problem. In order to have critical thinkers we need to give them the opportunity to critically think. Problem solving task as they way to achieve that. Students that collaborate with their peers and are trying strategies like guess and check or draw a model are persevering. It is no longer about the correct solution, but assessing their ability to work towards a solution.
How do teachers assess their students this way?
Have a checklist on hand of elements that you’re looking to assess: students are asking questions, drawing models, sharing their ideas, using manipulatives, etc.
Limit the amount of students you assess each day.
Set a certain amount of time to assess each student.
Carry notes and writing materials when you observe students
Set up interviews with each student.
