The Journey toward Mathematical Thinking

This has been by far one of the most exciting years in our journey to making sure all our students have access to an excellent math education which I define by classrooms of students experiencing NCTM’s Mathematical Practices on their own problem solving and explorations on a regular basis.

Only 10 percent of our students arrive in our building on track (by anyone’s definition) to pursue higher level math. Over a 3 year journey, we have successfully tripled the number of students who will have the option of taking Calculus as part of their high school career.

The most interesting part for me is that the journey has had very little to do with curriculum, lesson planning or even the quality of our math teachers. Our success has been found in growing communities of learners who walk together and encourage one another toward success. We define success as growing in their learning and being able to demonstrate their thinking. We are defining success as students reaching out to their peers to tutor and encourage. We are defining success as every student knowing that there is a community of adults ready to push them and challenge them to their best even though there will be any number of hurdles in the way. We have defined success as no teacher ever giving up on a student and no student ever thinking they can’t get started no matter how tough the journey has been for them – academically, socially and or emotionally.

I look forward to sharing the successes, challenges and lessons we seek to still learn as we have seen our students success. It has been an exciting and amazing journey with the Class of 2017!

Close Reading in Mathematics

One of the focuses for our district this year has been on close reading across subjects. As is often the case, our math department has had to figure out how to apply this ELA tool to the mathematical world. Some suggested it’s just a matter of doing word problems well, others have suggested it is helping students focus on vocabulary and still others simply don’t think it can apply to mathematics.

I continue to develop in my application of the tool but here is where I am on the journey…

First, the way our kids attack word problems has in fact changed. For a while, it was ok for students to find math words, ‘translate’ them and then solve the problem. They could look at the question being asked rewrite it and fill in the blank. However, context has become a much bigger deal for the students. The challenge for students is now to decide how much they need to understand of the problem itself and then figure out the math needed to answer the question.

Second, students have to make decisions based on their own work rather than simply move through a set series of steps to problem solving.

Third, there is a need for our students to read the problem for context, reread for the question and then read again for the information they need to answer the question and all of these require a much higher level of thinking than the students have been asked in the past.

I am in the process of developing a new set of questions and procedures for close reading with my students. I would love feedback. Here is as far as I have gotten but it needs some revision and clarity.

1. Students read the problem and look at diagrams to give a 1-2 sentence summary of what is happening in the problem, i.e. something being bought or sold, patterns beign explored, something being measured, etc.

2. Students reread the problem and identify the question(s) being asked.

3. Third reading of the problem has students looking for any detail, including units, that will help them solve each part of the problem.

4. Solve the problem and check for reasonableness.

Supporting Mathematicians in our Classrooms

Our approach to teaching mathematics (i.e. letting students explore and grow their own understanding of the material) is challenging for students who are used to being shown the pattern, practicing and then showing ‘mastery’ of the skill.

We are asking students to find patterns and learn the language of mathematics so they can share the patterns they are discovering with those around them.  Since variables are very hard for many to ‘hold’, it is helpful for them to use numbers until they are able to see the pattern in words.  As they look for patterns, use numbers that they can ‘see’ and then push them to the higher numbers which will force them to think in terms of variables.

Some things to remember as students engage in mathematics…

– Remind students continually that we learn from the mistakes we make.  Students will find patterns and see solutions differently – encourage this process.

– It is easier to ‘not care’ about the work than to admit struggling with the material.

– Seek to find a variety of entry points into every problem but do NOT get pulled in to doing the work for students.

– Encourage students to ask their peer if they agree with their answer which promotes the idea that you are not the sole authority on math correctness.

– Even you can be wrong so they should question the work you are doing as well as their own. It is a very good thing to have students challenge the answer you have provided (even if you are right).  Encourage this! For struggling learners this is a huge step in the right direction and every once in a while it wouldn’t hurt to purposely do something for them to ‘catch’.

As you work through teaching Unit 1 and 2 of the Agile Mind curriculum, I have tried to summarize the key skills students will be able to do moving forward through the activities they will do in class. While students will explore many different ideas through each of the units, these are the ones they should hold on to as they move forward.  Here is the overview I provide the tutors who work with kids through this unit to be sure the students are staying on track.

Puzzles and Pragmatics

What a joy it is to be able to teach math in an urban district!  I love the challenge of presenting puzzles to students in which they are continually looking for patterns. I enjoy the challenge of finding ways to build confidence in a set of students who have struggled through math for many years before coming to my room.  It is fun for me to figure out what are those key elements of math that they need to add to their problem solving toolbox moving forward and which ones they will continually need to remind themselves of via google or by reconstructing the process.

I love seeing student success and failure.  My favorite moments in my classroom are those when a student approaches me with confidence to tell me they are wrong but just can’t figure out why.  Their answer doesn’t make sense. They have tried a variety of things but it isn’t quite working like they thought it would – when we get to this stage I have achieved success.  Success because they haven’t given up; success because they understand something isn’t quite right; success because they have tried some things on their own before approaching me; success mostly because they actually believe they can figure out the answer and seeking out help isn’t a sign of their inability but rather a habit of learners everywhere.

Then the reality of my world sets in…I have 80 minutes a day with students who are coming in 3 years behind academically. Not one of the students I have in the 9th grade has passed the MCAS in the 8th grade, often they have not passed middle school classes, and frequently they do not see themselves of successful math students.

There is a limited amount of time to ‘prepare’ them for ‘the test’.  I love the test because it keeps me honest and doesn’t allow me to lower my standards for my kids because I somehow don’t think they can handle the same math students in other districts can.  I am not such a big fan of the test because rather than changing our structures to support the type of learning that would lead to success on ‘the test’, we panic and teach to the test.

These are the issues I would like to wrestle with in this space.  It not only takes courage to take our focus off ‘the test’ but it takes support that too often doesn’t exist.  I’d like to think that when my students gain confidence in themselves as mathematical learners this in and of itself is going to help them perform better on the test. I’d like to think that my style of very rarely giving any answers directly to students makes them think critically. I’d like to think that peppering in drills on integers and other number sense issues that are pervasive strengthens their toolbox but all these things are hard to measure and quantify.

Is there a balance that we can find between returning to the art of teaching while still maintaining expectations across communities that aren’t biased by my perspective of what students can or can’t learn?