1. Make sense of problems and persevere in solving them.
How are middle school teachers helping their students make sense of problems? How are they helping them learn to persevere?
How can students be helped to:
- explain to themselves the meaning of a problem?
- look for entry points to a problem’s solution?
- analyze givens, constraints, relationships, and goals?
- make conjectures about the form and meaning of the solution?
- plan a solution pathway rather than simply jumping into a solution attempt?
- consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution?
- monitor and evaluate their progress and change course if necessary?
The CCSS states:
Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
What are you doing to help students develop this practice? What makes it hard? What challenges are you encountering?
To develop perseverance, I am very hands-off. I tell the students it is okay to struggle a little bit. I place 3 squares stacked up on the table when they are working in groups. One green, one red, one yellow. They work like traffic lights. If they have green on top, I know they are going strong, lots of ideas are being tossed around, etc. If it is yellow, I know they are sweating, but they are working through the challenges together. If red, the group is at a standstill and I know they can’t move forward without my help.
It is very hard not to help too much because students don’t like to be uncomfortable.
I love the traffic light idea! I’m curious, Amy, what “too much” is. Can you describe that? Also, do you have some ways that you keep yourself from helping too much?
Encouraging students to persevere in times of difficulty requires an understanding of the individual student, his/her past educational experiences, and the strengths and weaknesses in the student’s ability in a variety of areas. In particular, I work with alternative education students and need to be hyper aware of past issues around school. So, with that being said, I help students challenge their minds by working to eliminate parts of the activity that are NOT crucial to problem-solving. For example, I may be the group’s scribe for my students who are resistant to writing yet another thing in their day or whose hand-writing will not be legible to the group. In addition, I will help structure the physical space for easy access to needed supplies. On a different note, we talk a fair amount about what type of answer makes “logical” sense before starting the problem. This can be estimating, but can take a variety of forms depending on the question. It helps weed out potential answers that are not possible through reasoning. That serves to help with student frustration level, and helps them to “save face” in front of peers.
Evelyn, most of my classroom experience was at a schoolwide Title 1 middle school. Before I left the classroom to join the Math Forum staff full time my group of seventh graders had an average reading level of 4th grade and they had similar challenges in mathematics. There were two things that I did that were a little different (at the time) that I think helped develop perseverance:
1. we approached problems from different perspectives — hands-on manipulatives (or concrete manipulatives), virtual manipulatives, and paper/pencil were may favorite mixtures
[example: Traffic Jam Activity: http://mathforum.org/alejandre/frisbie/jam.html and also here: http://mathforum.org/clime/page1.html ]
2. I would just do a little bit at a time each day — I wrote that up later and it’s here:
Think You Don’t Have Time to Use the PoWs
http://mathforum.org/pow/teacher/PoWsDontHaveTime.pdf
The pdf refers to the Math Forum’s Problems of the Week (PoW) but I think the general ideas could be applied to any problem solving activities.
I am a math coach at a middle school in New England. We have students from 26 different countries and 50% of our students receive free or reduced lunch. I love teaching in our bustling, exciting school. To encourage perseverance this year, I really wanted to emphasize discourse in my work with teachers. I began with Mary’s class. My schedule overlapped with Mary’s and so I joined her class 15 minutes after it began everyday. Mary and I met and decided that we would teach several Connected Math investigations over a two week period. We looked over the investigations and choose the parts that she would do and the parts I would do.
When I walked in everyday Mary was finishing warm up exercises and going over homework. This took longer than I thought it would, and I was often left with 30 minutes or less to teach. Once it was my time to work with the class,I had students work on an investigation. When we reflected as a class, I reflected questions back to the students. “Can anyone answer Katie’s question? What do people think about what Tony said?” Sometimes there were uncomfortable silences as I waited for students to think about how to explain something. Mary had a hard time with this. She would jump in and say, “You know I think I can explain that really well, if you can’t.” I met with Mary after class and said, “I am trying to get discourse going. I want students to explain things to students. They can do it. We just need to give them time.” Unfortunately, because of the short amount of time I had, the class never made the leap to the regular discourse that I wanted them too. I left frustrated and slightly defeated.
But I also left ready to do better with the next teacher. I also met will Sally ahead of time. I explained that for the type of investigations I wanted to do I needed the warm up to be only 5 minutes. Also, my schedule didn’t overlap with hers, so I wasn’t late everyday. I also asked that we trade off for longer periods of time and that we didn’t interrupt each other’s lessons. I asked Sally if she would write up notes for me everyday on how things are going. Also, I would do the same for her later. She decided to also make a videotape of me everyday in addition to the notes she would give me. I began the first investigation and my heart pounded. We reflected as a class and I asked a student to present. The student made many mistakes, but I resisted the urge to explain it myself. I asked students if they had questions or comments for the student at the board. Students were unsure what to do at first, but slowly, students began to correct the struggling student. I kept going with the idea that students could answer students questions. By the end of the first week, a student raised their hand and said, “I have a problem I am struggling with. I’d like to come up to the board and see if the class can help me.” Sally lit up. “This is really neat,” she said. “I’m going to keep doing this.”
I walked away smiling and knowing that next year, I’ll try again with Mary.
Thank you, Fred! This is a lovely example of perseverance at several levels.
The only teaching i have done in my past is my student teaching. Being a new teacher a lot of people give me the advice to set the tone early. Which in my mind means set a model for them. I taught in many special education classrooms with behavioral problems and emotional difficulties. I never thought i would encounter such students. There is a difference in telling kids they can do it and believing they can do it. I always try to look at the positive, nothing negative comes out of my mouth or theirs. I also praise them for their efforts and try to get them to keep going by asking engaging questions. If i don’t give up, they wont give up. They don’t have to b perfect, they don’t have to be straight A students they just need to work hard and put in effort.
Activities consist of as many different things i can do, this way i see different students strengths and can focus on their strengths. Using m&m’s to find volume, having them make posters, working in pairs. The “a mile wide, and inch deep” concept is so brand new for me and i am excited to have students discussing with each other and having hem come up with rules and theories based on their investigations. It almost feels like science class
Creating a respectful, safe classroom environment with the understanding that we all learn from mistakes. Struggling with math problems creates learning opportunities. Also private think time allows students to think on their own about a problem before working with others. Without think time some students might not have enough time or opportunity within a group to express their ideas. As students work through strategies toward a solution, I would check in with them and take note of students that could explain their solution. If students finish early, I think it would be beneficial to explain their methods to each other. If students are having trouble, I would ask questions such as what is being asked, what information is given that is needed and does this problem resemble previous problems?