Math Intuition

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Over the past two days I’ve been reading and rereading chapter 8-9 in my summer book study. Chapter eight discusses how mathematicians connect ideas.  From what I see in classrooms, this connection of ideas is often directed by the teacher and involves some type of classroom discussion that helps students construct understanding.  Intentionally setting aside time to have math discussions and connect ideas from students is worthwhile.  The prime example of Debbie (the teacher) allowing time for Gunther (student) to put the calendar in the shape of a clock was especially a memorable portion of this chapter.  That opportunity wouldn’t have occurred if the teacher didn’t take the initiative to intentionally plan to use manipulatives to have students construct their own understanding through a math discussion.  Having these student math discussions gives educators feedback in whether students are attempting to make/create connections and whether their overgeneralizing. Creating opportunities for student to make these connections is important.

Chapter nine emphasizes the need for mathematicians to use intuition. I appreciate how the chapter indicates that math is often perceived as a very logical content area.  It’s truly not, but the perception still exists.  Tracy states in the chapter that she’s come to see “mathematics as a creative art that operatives within a logical structure.”  I had to reread this a couple times to let it sink in. I’ve heard it over and over again that someone is “not a math person.”  What I find interesting about this is that mathematical intuition is developed.  Since it’s developed over time it can change.  I tend to tackle this issue quite a bit and address it at the beginning of the school year during Open House. Providing students with opportunities to develop this personal intuition can be a game changer.  It’s up to the teacher and school to create memorable experiences for students to develop math intuition. That’s a responsibility that each teacher takes up when they open their classroom doors. By increasing their math intuition, students may also increase their math confidence. Educators need to carefully think about the different math experiences that we provide for our students.  Those meaningful experiences aren’t always found in general textbooks.

After reading these two chapters, I started to think of what perceived/real barriers stop teachers from intentionally creating these opportunities.

I think sometimes teachers feel as though they’re required to follow word-for-word the scope-and-sequence that’s provided by a district.  This can be the case when a newly adopted text is revealed and teachers are highly encouraged to follow it to a tee.  Some texts even tell teachers what to exactly say, what questions to ask, and predicted student responses.  I’ve been though many different math text rollouts and this occasionally happens.  I see it more at the elementary level though. Having common assessments with a specific timeline that everyone needs to follow can also provide pressure for teachers to fall in line with a particular lesson sequence.  Deviating from that sequence may cause issues. I find that there’s a balance between what a district curriculum office deems “non-negotiable” and room for academic freedom within a sequence.  I’ve been told in the past that a district text is a resource, but for new teachers it may be more than that.  There can be a lot of anxiety, especially if certain parts of your instruction model have to follow a pre-determined sequence and is used for evaluation purposes.

Teachers need to feel comfortable in giving themselves permission to use their own intuition.  That may be easier said than done and it depends on your circumstance.  Despite good intentions, a published text won’t meet the needs of all of your students. I believe that’s why open source resources are frequently shared within the online teacher community. Supplementing or modifying lessons/questions with resources that match the learning needs of your students happens on a daily basis.  Dan’s Ted talk hits on that point.

I believe educators have permission to do this while still meeting a strict scope-and-sequence.  Teacher confidence also plays a role with how willing someone is to try resources outside of the textbook.  Elementary math teachers need to feel empowered to be able to use resources accordingly without feeling as though it’s going to be detrimental in their evaluation.  I think that sometimes teachers don’t exercise their academic freedom to the highest potential because it’s perceived as going against a district’s plan.  Having math coaches available and supportive administration is also important in changing this perception

The work that we do is important.  Creating mathematical intuition happens through repeated experiences.

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Sometimes those experiences are beyond the textbook/worksheet and educators have the ability to make them meaningful.  I’ll be keeping this in mind as I prepare for the new school year.

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Look Who’s Talking

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I’ve been able to check off a few books on my summer reading list.  I’m now in the process of reading one book in particular.  It’s been a slow process through this book, but worthwhile as I’m actually thinking of how this applies to my practice.  That takes time. Yesterday, I was on a reading tear and made it through chapter seven.  This is where I ended up paying most of my attention. The chapter is related to asking questions in the math classroom.

In the eyes of most students, questions are often given to them, not something that they get to ask other students or even the teacher.  The ratio of questions they’re required to answer far outweighs what they ask.  I’m not arguing that there’s something wrong with that ratio, but Tracy and others in this chapter make a case to why educators should allow more opportunities for students to ask, wonder, and notice.  I think there’s value in providing these opportunities, although the management involved in that process seems challenging at times.  While reading, I came across a terrific quote by Christopher.

One of the bigger issues is the last highlighted sentence: “Quit before angering child.”  When I read this I actually laughed out loud and then started to realize how often this happens in the classroom.  Ideally, all students would be willing to make a claim, be receptive to what others have to say and then change their claim accordingly.  Some students are much more willing to engage in this type of math dialogue, while others would rather not.  There are different activities and procedures that can help move students towards being more receptive to asking questions during claim dialogues.  Notice and Wonder, 101questions, problem posing, riffing off problems and independent study options can help students ask more questions and encourage them to be a bit more curious.  That curiosity can spur students to ask more questions.  All of those are great resources, but there’s an important piece that needs to be put in place beforehand.  I believe Scott makes a great point.

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Each child has their own tolerance for struggle.  That struggle can turn into frustration quicker for some more than others.  This happens with children and adults. I think most educators have been in situations where a student makes a claim and then retracts it after its been shown that their response wasn’t quite right.  That student then disengages and it’s challenging to get them to be assertive afterwards.  How can this be avoided or is it possible to avoid these types of situations?  I don’t know the exact answer to this, but understanding the level in which a student can struggle without frustration is important.  Struggle is part of what happens in any math class. That productive struggle is what’s often needed before students construct their own mathematical understanding.

Enabling students with tools and models can help in these struggling situations.  I’ve also seen this struggle occur during whole class guided math conversations. Some students shut down when they are called out by another student.  They think that disagreement means that they’re being challenged or attacked. That’s not the intention, but it may be perceived that way by other students. It may be helpful to model what appropriate math dialogue looks like.  After the modeling, practicing that type of math claim dialogue and providing opportunities for questions can help smooth out the process.

I also believe some students are not used to making a claim in a verbal format.  Students are definitely used to talking.  Ask any teacher.  Also, they’re probably familiar with providing reasons why they agree/disagree on paper, but communicating it in a verbal format can cause some issues. Providing these students with sentence starters, using technology that can be shared with the class, or using other appropriate means can help students engage respectfully in a productive math dialogue.

I’ll be keeping these ideas in mind during my planning process.

 

Making Math Mistakes

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This summer I’ve been reading a few different books.  One of them is Becoming the Math Teacher You Wish You’d Had.  It’s part of a book study that started a few weeks ago.  Kudos to Anthony for helping start the study.  I’m slowly making my way through the book, following the tag and listening to people’s comments on Voxer.  My highlighter has been busy.  I appreciate all the different teachers that Tracy showcases.  I’m currently in chapter four, which is related to making mistakes in the math classroom.

I believe making mistakes is part of the math learning process.  I don’t think I’ve always communicated that enough.  Some students that I see come into the classroom with an understanding that mistakes are evil.  They’re not only evil, but I’ve seen them used to humiliate and discourage students and peers.  I believe these types of behaviors tend to crop up when the culture of a classroom isn’t solid.  Of course there are  many other variables at play, but a classroom culture that doesn’t promote risk-taking isn’t reaching its potential.

Tracy showcases different teachers in chapter four.  All the educators highlighted seem to be able to communicate why it’s important to look at mistakes as part of the math journey.  This chapter is full of gems.  A couple takeaways that I found are found below.

  • The math teachers that are highlighted seem to understand that mistakes are opportunities.  When they happen, teachers have a choice to make.  Modeling and showing students different ways to react to mistakes is important.  Students need to be able to understand and be accustomed to making mistakes in stride.  This can be a challenge since some students stall or immediately stop when they run into a mistake. Mistakes shouldn’t be perceived as failure. If a student makes a mistake they need be able to have tools and strategies to move forward.  They need to also find the underlying reason to why the mistake or misconception happened.  Having a misconception investigation procedure in place for these instances is helpful.
  • Using classroom language that creates safety is key.  Teachers need to be able to have phrases in the bank that empower students to participate and take risks.  I found that the teachers highlighted in the book often ask questions related to students explaining their reasoning.  They also set up the classroom conversation so that students build upon each others’ responses.   Students speak their mind about math in these classrooms.  They’re not afraid to respectfully agree or disagree with their peers and explain their mathematically thinking.
  • I noticed that the teachers played multiple roles during the observation.  Teachers often gave students time to work with partners/groups to discuss their mathematical thinking.  This time of group thinking and reporting happened throughout the lessons.  Teachers often anticipated possible misconceptions and guided the classroom discussion through students’ thinking.  The teachers asked probing questions that required students to give answers that displayed their mathematical thinking.  Teachers didn’t indicate whether an answer was correct or incorrect.  Instead, educators asked students to build upon each others’ answers and referred to them as the lesson progressed.

I can take a number of the strategies identified in the observations and apply them to my own setting.  I see benefits in having a classroom conversations where students explain their math thinking.  That productive dialogue isn’t possible unless the culture of the classroom is continually supported so that students feel willing to speak about their thinking.  Students aren’t willing to take risks and explain their thinking to the class unless a positive culture exists.  That type of classroom needs to have a strong foundation.  That doesn’t take a day, or a week.  Instead, this is something that is continually supported throughout the year.  Next year I’m planning to have students use the NY/M tool again.  I’d like to add additional pieces to this tool.  I’m also planning on using more math dialogue in the classroom.  I believe students, especially those at the elementary level, need practice in verbally explaining their mathematical thinking to others.  That verbal explanation gives educators a glimpse into a student’s current understanding.  I also believe that giving students more opportunities to speak with one another about their math thinking will help them develop better explanations when they’re asked to write down their math thinking.

I’m looking forward to starting chapter five on Monday.