Make Sense of Problems and Persevere In Solving Them

"Successful problem solving does not mean that students will always conclude with the correct response to a problem, but rather that students will undertake a genuine effort to engage in the problem-solving process, drawing on learning resources described in the other practices such as appropriate tools, using their prior knowledge, engaging in math­ematical discourse with other students, and asking questions to make progress in the problem solving process. Successful problem solvers also recognize that powerful learning can be experienced even when an appropriate answer to a problem ultimately evades the student."
- Tim Kanold - Turning Vision into Action Blog


What do you do when you are presented with a difficult math problem? Try the problem below! Take note ,not so much to your solution strategy, but what you noticed about your perseverance. When was your perseverance tested? What did you do to overcome that? 

Jonathan is two years younger than his wife Hazel. Their current ages are both prime numbers. Next year Hazel's age will be a multiple of 11. Jonathan's age will be the product of two consecutive numbers. 

How old are Jonathan and Hazel?
(MATH FORUM Problem of the Week #3559)

*Want to know if you're correct? Email Sam your response and you may find a small gift in your mailbox! Correct or not, I applaud your problem solving and perseverance! 

What did you notice? Were you able to engage in the problem solving process as Tim Kanold described? 

Standard of Mathematical Practice Standard #1 states that students should be able to make sense of problems and persevere in solving them. How do we teach students to persevere and embed meaningful practice into our instruction?

When a student is presented with a challenging task, some students need help figuring out how to tackle that tough situation and need a lot of practice overcoming challenges.  Students need time experiencing productive struggle in order to build that ability to persevere. Productive struggle is when students explore tasks or problems with an appropriate level of support. Support, in this case, is asking the right questions or providing appropriate prompts to move the student forward without directly teaching them what to do.  This struggle is a balance between success and challenge that leads to increased motivation and confidence in a student's ability to be a problem solver. (See our previous blog post on Differentiation by Questioning for more information on what those prompts might look like!) 

As teachers, when our students struggle, we want to jump in and help, it's in our nature!  But, sometimes we need to avoid that temptation and provide our students with a different kind of support that builds their problem solving skills and, eventually, their confidence in math. 

Now, how do we allow for productive struggle without frustrating our students? Here are some tips for preparation and implementation:

Preparing to Implement Productive Struggle

• Prepare students for struggle
• Set classroom norms
• Share why struggle is important
• Create a safe environment for risk-taking
• Model productive struggle and thinking strategies
• Anticipate student difficulties
• Instead of preventing difficulties, provide tools to help students work through difficulties
• Differentiation
• What provides productive struggle for one student may not for another
• Ignoring this can lead to frustration for students
• Response to feedback
• Feedback on assessments and in-class work should require action from the student

Productive Struggle in Action

• Focus on the process
• Correct answers are not valid without explanation
• Incorrect answers are a pathway to learning, not an impasse
• Highlight multiple ways to reach the same conclusion
• Allow access to tools
• Do not limit the tools available for students
• Let students decide instead of directing them how and what to use
• Avoid over-helping or helping too early
• Wait to intervene; look for non-productivity
• Ensure students are doing the thinking
• Use questions to guide instead of statements to direct
• Learning when to help, when to question, when to wait
• Reflect on the activity post-implementation
• Promote student reflection and use reflection as feedback

(Closing the Achievement Gap Webinar - Edweb.net)

Questions that Support Productive Struggle

• How would you describe the problem in your own words?
• How would you describe what you are trying to find?
• What information is given in the problem?
• Describe what you have already tried. What might you change?
• Talk me through the steps you've used to this point.
• What steps in the process are you most confident about?
• What would happen if...?
• Is it possible to...?
• Is this always true...?
• Why does this work?
• What could you do next?
• Is there more than one way to think about this?
• What are some other strategies you might try?
• What did you notice?
• Could you explain what you mean by...?
• What do we need to do to clear up our confusion?
• Does your solution/conclusion make sense?

This brings us back to Standard of Mathematical Practice #1 - Make sense of Problems and Persevere in Solving Them. By allowing opportunities for our students to engage in productive struggle, we can clearly see that most, if not all, of the key behaviors of this standard are embedded into our instruction. 

Summary of Standard:

• Interpret and make meaning of the problem to find a starting point. 
• Plan a solution pathway instead of jumping to the solution. 
• Monitor their progress and change the approach if necessary. 
• See relationships between various representations. 
• Relate current situations to concepts or skills previously learned and connect mathematical ideas to one another. 
• Continually ask themselves, "does this make sense?" Can understand various approaches to solutions. 

Mathematical Language and the Common Core

As we are implementing the common core math standards and the standards of mathematical practice we want to keep in mind how those standards apply to our instruction.  One way we can do that is to model the use of precise language in our teaching.  In an article by Valerie Faulkner, Why the Common Core Changes Math Instruction, she discusses old math language habits we may have and offers a new phrase to use instead.  The following examples are all taken from her article.

Instead of  defining equality as "same as" define it as "same value as."
For example: 3 + 4 tells a different math story than 4 + 3, but they have the same value of 7.

Instead of calling digits "numbers" clearly explain the difference between numbers/numerals and digits.
For example: 73 is a numeral that represents the number 73 and has two digits 7 and 3.

Instead of saying addition makes things bigger show addition is about combining.
The "addition makes things bigger" statement becomes a problem when students are introduced to negative numbers.

Instead of saying subtraction makes things get smaller show that subtraction is about difference.
Again, "subtraction makes thing get smaller" is a problem with the introduction of negative numbers.

Instead of referring to "the answer" ask students to use "the model", "the relationships", "the structure", or to "justify their answer."
When the goal is to find an answer we forget about the most important part of the problem - How did we do that? and Why did we do that?

Just as we are asking students to be more precise in their use of vocabulary, so must teachers.  Through our modeling students will learn how to speak and write using specific and correct mathematical language.

Below are some prompts to help get your students talking about math, how they arrived at their solution, and asking others about their work.  This is an easy way to build in some of the standards of mathematical practice, particularly construct viable arguments and critique the reasoning of others.
Please let me know if you would like a hard copy of the prompts.

Using Pictures to Support ELL Students...

Learning academic concepts in a new language can be a daunting challenge. Often ELL students may have the necessary skills in their home language but aren't able to show this due to the English language demands in the classroom. As teachers there are ways that we can provide support to ELL students in order to facilitate their language development. One great method is using pictures in place of words. How could this be applied in your classroom?

• Vocabulary Crash with Pictures
• The game consists of a group of cards that use pictures to identify the necessary vocabulary, but some of the cards have the word crash. Each student picks a card and tries to guess the vocabulary word, if they get it right they keep the card if not it goes back in the pile. If they get the "crash" card they have to put back all the cards they have.
• Read to Someone using picture cards
• If students lack the necessary reading skills for this component of Daily 5 have them use picture cards and tell a story from them. Here they can incorporate all the narrative writing concepts as well as many reading concepts while they tell their story.

If you're stuck on how to differentiate a lesson for ELL students ask yourself if there is a way that they could learn the same concept through the use of pictures. Many literacy, math, and science activities can be modified easily by adding in picture cues.

Here is a link to some fun picture flashcards:

Halloween Flashcards

Instructional Strategies

From time to time, we will be adding strategies that we have used at staff meetings.  If you want to learn more about the different strategies, please click on the tab labeled Instructional Strategies.  If you have tried any of the strategies, please comment on how it worked in your room or jot down ideas of how you would like to use the strategy in your room.  The first strategy that we have shared is the Concept Attainment strategy.  Please read on to discover more about it!