Math Intuition

Screen Shot 2017-07-18 at 3.52.50 PM.png

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.

Photo Jul 18, 3 25 46 PM.jpg
p. 212

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.

Inservice Days

Many district are is in the midst of planning their 2016-17 inservice/institute days.  These days, sometimes called PD days, often include communicating initiatives aligned to district goals.  Sometimes school goals are included in this process.  As far as I can remember inservice days have always been part of my school year.  The content is sometimes applicable to what’s happening in a particular school, other times it’s more aligned with a district goal. Most teachers have experienced successful and unsuccessful sessions.

Last night I came across this Tweet:

David asked an important question.  I’m not an expert in the field of PD, but I’ve experienced some amazing and not-so-amazing sessions in the past.  I’ve also put together plans for PD and other sessions.  Through this experience I’ve been able to evaluate PD sessions a bit better.  Below are four questions to consider before putting together a PD session:

 


 

Are there clear expectations?

Being intentional in communicating expectations is key.  I’m not necessarily talking about listing the objectives of the session. I’m more concerned in what participants should be able to do with the information after it’s been delivered.  How will this impact teaching and learning?  Having a clear understanding of what’s expected and a timeline can help avoid confusion.

Is there an explanation of why?

I think this is sometimes missing from PD sessions.  Why are we learning about guided math, reading workshop models, grading practices, etc.?  Giving the why can help people understand the reason for a particular session.  If it’s not explained than staff may feel as though the reason is directly associated with someone not in the school, which may or may not be a good thing.

Will there be opportunities to revisit this initiative?

Educators aren’t generally fans of participating in a PD session that communicates that what’s being discussed will be fully implemented but it doesn’t happen. If the expectations is that all classrooms need to do x, y z than that should actually happen.  Starting an initiative and abandoning it halfway through the year doesn’t help with rapport or climate.  A successful PD session allow opportunities for additional help and follow up as needed.

Is there a reflection opportunity?

This may be more of a matter of personal opinion.  I tend to learn best by reflecting on what I’m learning and finding ways to practically put it into practice.  That reflection can happen after the session but embedding it in the session can be a valuable.  Sometimes a reflection opportunity can reveal itself through follow up conversations.  It also keeps the conversation going to ensure that consistently is occurring in a school/district.


When creating a PD session I tend to consider the questions above.  The questions aren’t always applicable, but it’s a place to start.  Would you add any other questions?

 

 

 

Multi-digit Multiplication Strategies

 

Multiplication strategies-01

This past week my third grade class investigated different ways to multiply numbers.  Before diving into this concept I asked the students their thoughts on multiplication.  A few students explained to the class their view on the topic of multiplication.

  • repeated addition
  • double or triple “hopping”
  • using arrays
  • “timesing”
  • Increase the number by “a lot”

Most students were able to showcase examples of the above.  Even though their vocabulary wasn’t exactly spot-on, students were able to come to the whiteboard and show their thinking.

I received different responses from the students when asking them about multi-digit multiplication.  Actually, it was more of a lack of response.  I feel like some of this is due to exposure.  A few students raised their hands and asked to show their process to multiply multi-digit numbers. These students showcased their ability to use the traditional algorithm. The class reviewed this method with a few examples.  Although students were finding the correct product they had trouble explaining the process. Students weren’t able to communicate why it worked or another method to find a solution.

On Tuesday my class started to explore the partial-products algorithm.  Students were able to decompose individual products and find the sum.  This made sense to students.  Students were able to connect an area model with the partial-products method.  They started to write number models right next to each partial-product.

Later in the week students were introduced to the lattice method.  This method seemed “fun” for the students, but didn’t make as much sense as the partial-products method.  Students were able draw the boxes and create diagonals to find the product.  Some students had trouble with laying the boxes out before multiplying.

During the last day of the week students were asked to explain in written form how to multiply multi-digit numbers.  Even though all of the students could use the traditional, partial-products and lattice methods, they were stuck for a bit.  Soon, most students started to lean towards using the partial-products method to explain how and why this method works.  I asked one student in particular why it made sense and she said “I can see it visually and in number form.”  Although most students were able to use the other methods effectively they didn’t seem confident enough to explain why the strategies worked.

Students will be expected to multiply multi-digit numbers on the next unit assessment.  The method to multiply these numbers will be determined by the student, but I’m wondering how many will gravitate towards the strategy (not just the process) that they understand.

 

 

Mindframes and Teaching

Screen Shot 2015-11-07 at 9.27.27 PM

This last Thursday I was fortunate enough to attend a conference around the topic of visible learning in schools. The conference had so much information.  I had to filter and compartmentalize the discussions and presentation.  One of the discussions revolved around John Hattie’s effect size and how schools can make learning visible in classrooms. The last day was dedicated to organizing a school plan that’ll be carried out through the remainder of the school year.

One of the more memorable pieces of the conference were the discussions that happened between the school teams.  My school sent a team of four teachers and two administrators to this particular conference. Discussing our views on teaching and learning was a powerful experience. Many members of the team don’t regularly work with one another, so meeting to discuss these issues brought about other views as we’re all in different roles . Not everyone thinks the same and each member of the team was willing to hear out different perspectives. As a team, we agreed that our school has some great initiatives happening right now. That affirmation was great to hear, but at the same time, we felt that there are steps we need to make to become better. In order to put these initiatives in place the school has to communicate the importance and reasoning behind these proposed changes.

This brought up another discussion about how change will not happen unless stakeholders are truly committed to the cause. Even if they’re committed, the initiative doesn’t reach its full potential unless the organization and individuals have mindsets that are aligned with the initiative. This type of thinking falls in line with Hattie’s Mindframes for Teaching. Teachers have beliefs that impact their teaching. That belief often stems from a self-developed mindframe. Understanding your own mindframe can help stakeholders better define their own role.  The mindframes are explained in the video below.

All of these mindframes are discussed in Hattie’s Visible Learning book.

Early in September my school was introduced to the idea of teacher mindframes. A staff meeting was designed to have educators analyze Hattie’s mindframes and reflect on their own. We plan on revisiting this topic throughout the school year. Understanding deep-seeded beliefs about our role in education can help bring awareness to how we think.  I believe that thinking impacts instructional decisions that influence student learning.

Computation and a Growth Mindset

growth
Encouraging growth

This past week my third grade class started to use multiplication and division strategies to solve world problems.  They’ve used arrays before and are now applying their understanding of multiplication and division.  That practical application can be a challenge for some and I feel like it’s partially because students aren’t yet fluent with their facts. In an effort to collect a bit more data on what particular facts students were struggling with I gave the class a short 17 question Kahoot! quiz. The quiz was related to multiplication and division facts.

Screen Shot 2015-10-17 at 12.48.51 PM
Click picture for actual quiz

In the past I’ve used Kahoot to review concepts and skills in a game-based format.  I’d estimate that the majority of Kahoot quizzes have a limited amount of time and points are scored.  This is fine and I’m not against using this format, but it didn’t work for my purpose.  I wanted students to take their time and diligently pick an answer.  So, each student grabbed an iPad and completed the quiz on Wednesday.  It took about five minutes or so and students reflected on how they thought they did on the quiz.  The class then reviewed multiplications strategies and connected how multiplication and division are connected.  The homework for that evening also reinforced some of the computation strategies that we’ve been practicing in class.

The next day students were given the same Kahoot quiz.  The question order was changed and students were allowed to take as much time as needed.  I printed out both the first and second quiz results for the students to see the difference between the scores.  Students glued both sheets in their math journal and were asked to respond to the journal prompt below.

“Was there a difference between your first and second scores?  If so, why do you think the results changed?”

Some of the responses are below.

This slideshow requires JavaScript.

As you can see, some of the students are connecting the idea that improvement, effort, and growth is important.  I’d say this is a move in the right direction.  This year my school is emphasizing the idea of Dweck’s growth mindset.  Teachers are encouraged to use terms like persevere, not yet, and effort fairly frequently.  Students are hearing this type of speak and even being asked by administrators questions related to having a growth mindset.  By doing this activity I feel like students are starting to internalize that effective effort helps produce better results.  Instead of just talking about growth mindset and the benefits, students need to be able to make a meaningful connection between effort and achievement. I feel like preaching that effort alone will reap success isn’t the whole story.  I feel like students need to be able to document their journey and internalize the connections. I’m hoping to continue to use these types types of reflection activities throughout the year.

Students and E-portfolios

Student E-porfolios
Digital Portfolios

Last week my math students wrote in their math journals about their experience in math class so far.  Their entries were fascinating and many students documented their learning that took place since the beginning of the school year.  Some students drew pictures and wrote lengthy paragraphs indicating skills learned.  At the end of the class the journals were put back in their designated place in the classroom.  I looked over the journals and made comments.  Afterwards, I starting to think about what happens to these types of journals after they’re sent home at the end of the year.

What happens after a student receives back their classwork? The work is often presented in a number of ways: hanging up the assignment, placing it on bulletin boards, showcasing it around the school, or sending it home for refrigerator placement.  I’m not sure what happens after the assignment heads home. Optimistically, I assume that they’re kept forever, but most likely the assignment moves towards a recycling bin at some point.

I’m finding that the work that students complete is becoming increasingly digital. Regardless of how the work is created, it’s often captured and presented in a digital form. Student work that’s completed and presented digitally lives on.  Not only does it live on, but it can be seen by people outside of the school, state, or even nation.  For example, students might use base-ten blocks to show their understanding of how to add numbers together.  The end product, although it may be a physical representation, has an opportunity to be captured digitally and communicated to stakeholders.  Some school districts are finding that they can help showcase student understanding through digital means.

I’ve found that some of these same school districts have moved towards a student e-portfolio model.  This is much more prevalent at the middle and high school level, but exists in small pockets at the elementary level. In some cases, students have access to their own e-portfolio and they submit their work digitally. Over the past couple of years I’ve seen elementary teachers use Weebly, Google, Seesaw, and Showbie to have students submit their work digitally.  In turn, student receive feedback and document their learning experiences in the process.

A few teachers in my school are currently using Seesaw to have students’ submit their assignments.  Teachers need to approve the submissions and parents are notified that items are located in their child’s portfolio.  Teachers and parents can provide feedback to the students.  Students can even take that feedback and resubmit their projects as needed.

Silicon Valley has also paid close attention to how this is playing out. Learning management systems (LMS) are starting to become more of the norm as students and teachers become more familiar with how they work. As districts become more familiar with LMS, questions about student privacy and data collection should be addressed. Having an online student portfolio gives teachers, students, and parents opportunities to be transparent in communicating what’s happening in class. This type of student work evidence goes far beyond a classroom newsletter.  Being able to submit assignments and receive feedback digitally encourages learning beyond the school walls.  Submitting projects digitally also allows teachers to give feedback a bit differently.  Instead of writing feedback on papers, teachers can record comments verbally or record a brief video with examples.  Although I prefer to give feedback 1:1 in person, giving feedback digitally has its advantages. Ideally, the student e-portfolio would follow the student throughout a school district.

Back to my students’ math journals … so the next day I had students submit their work to their e-portfolios.  Through this action, students were taking their physical work and making a digital copy.  Parents were able to immediately check out their child’s work and make comments.  Some parents made comments, while others just view the work.  I’m not looking for interaction on everything submitted, but I feel like having that opportunity to communicate and the transparency involved is important.  It also can help initiate the “how was school” talk that happens when children come home from school.  Through the years the physical journals may stay intact, but the digital copy will always be accessible.  Having access to past entries can help students see the growth that they’ve experienced during their journey.

How do your students document their learning journey?