Unit 6: Relationships and process

I think two big ideas are highlighted during the course came out strongly in the interviews in this unit. One is the confidence to tackle complex open-ended problems. The other is developing a culture where students are focused on sharing thinking. They are comfortable learning through mistakes.

I think simply giving a problem that students don't know how to solve to a class that is not used to problem-solving leads to frustrated students and teachers. You have to first develop a culture where students become as interested in discussing the maths as in getting the correct answer. As I mentioned in my last post, I've started using dot-cards as a way to develop confidence and discussion, and help students build that all important positive relationship with maths, which is a focus in unit 6.

The unit also talks about using a mathematical process, and making this process explicit to students. This fits in with many other areas - the Writing Process, the Inquiry Cycle and Design Thinking (which is a process designed to develop innovation and problem-solving skills). It would be great if students are able to make connections and comparisons between processes in different subject areas as they grow through the school. I think the exact wording of such processes can vary, but the understanding that deep learning happens, problems are solved and new ideas are generated when we spend time immersed in the work and approach it in a variety of ways as we move through different stages is fundamental. It fits in with the problem-solving 'way of life' and a growth mindset culture.

Unit 5: Dot cards and maths talks

The unit encourages teachers to create an environment where students can find their own ways to solve problems and then share their strategies with others. A good starting point for this in Grade 1 is using dot-cards, where students explain how they know how many dots are on a card without counting them individually. This involves them seeing combinations of dots. Students share different ways they combined the dots to find the total. Some ready-made print outs of these dot cards taken from John Van de Walle's book can be found at http://www.mathcoachscorner.com/2013/07/using-dot-cards-to-build-number-sense/
Currently we're enjoying discussing one dot-card a day in 1T, and students are getting better at using different strategies such as seeing doubles e.g. 7+7 and finding ways to make ten e.g. 5+6 = 5+5+1 = 10+1 =11


Going beyond the dot cards, students can be given numerical problems to solve and then discuss their solutions and how they arrived at them.

The teacher's role is to draw out student thinking and make it visible to others, both algebrically and visually. This may involve 'rewinding' several times until they have understood the student's thinking.  Teachers should focus particularly on incorrrect answers, allowing students the opportunity to talk through their work in the hope they will see their mistake. They should also encourage students to see similaries between the different strategies.

One question I'm wondering about is the optimal time ratio between students working alone to solve problems versus time spent listening to class mates share solutions, which falls into the 'sitting and listening' category (apart from when they are sharing their own ideas.)

Jo Boaler's Latest Ted Talk

Link below. She discussed things we have not covered yet in our course. Two big take-aways for me were the impact of showing students you believe in their abilities and good ways to phrase math questions that are engaging & visual.

https://www.youcubed.org/oxford-tedx-talk/

Mistakes Are Our Friends!

“Mistakes are our friends!” Is such an important message to share with our learners. Every year, I get on my soap box and share the importance of mistakes in our classroom.  It is always nice to hear educators and researchers share information in regards to the brain. Knowing now that not just mistakes , but awareness of mistakes activates and grows the brain is fascinating. It reminds me to not only celebrate when a mistake is made, but also prepare and provide opportunities for students to make mistakes in the classroom. 

I also appreciated the importance Carol Dweck and Jo Boaler put on the word, yet. So many students and adults think if something is easy for them, then they are smart. But really they should feel cheated from an experience to grow and learn. One of my students came in the other morning and asked,”Ms. Waring, do you have any challenges for me today?” I would never have done that as a student. I was so worried about getting the right answer and scoring well on my math facts tests. So I do not think we need to educate our young learners about growth mindset. I hope the adults in schools and at home can change their mindsets from how they grew up. Parents and teachers should focus on the hard work and effort put into student work rather than the perfection and score they get. 

Sessions 3 and 4 reminded me of the importance of providing opportunities for students to make mistakes, get messy, and grow as learners. We’ll continue to make “challenge the new comfort zone."

Mistakes and the Growth Mindset

Here are some of my favourite bits and pieces from these two modules.

-When your brain is struggling that's the best time for growth.

-We want students to be struggling and getting it wrong at times.

-"I want being challenged to become the new comfort zone, not this is easy."

-"If students are afraid of mistakes, then they're afraid of trying something new, of being creative, of thinking in a different way."

-Mathematical Practice 1 - Make sense of problems and persevere in solving them.

-Encourage students to be sense makers.

-Reward students for experimentation and trying things out, not for correct answers, but for having ideas and being willing to test them out.

-Effort is the secret of life.

-In this class mistakes are expected, inspected and respected.

-highlight mistakes positively in feedback and then get them to correct their own work.

-Don't teach to tiny curriculum standards but to big ideas in maths.

-Covering everything doesn't mean students learn everything.

-When students finish work have them think deeper, ask them to think of other problems similar to those they have been working on and work on those.

Reflection

"Making mistakes is the most useful thing to be doing"  reaffirms to me the importance of the children showing their thinking when they are solving a problem and not just focusing on the answer.  Being able to describe the strategies that they are using confidently to their peers/teachers is an important skill for children to learn.  Being able to work through their strategy and see where they went wrong is also very important.  Continually encouraging the children to understand that getting a question wrong doesn't matter so long as they are confident with the strategy they are using and are able to see where they made the mistake.

It has made me think about the terminology that I use with the children. "Not quite yet" is an interesting term that can be used with the children to encourage them to keep trying and is very positive. The need for me to give children time to correct their work is very important.

Knowing that by developing children's ability to be confident and persistent problem solvers will really help then in later life.  These are skills that they will then go on and use to make them successful in their careers. 

Jo has really made me think about the type of problems and investigations that I should be doing with the children.  I think it is really important to teach them the skills they need for good problem solving: Team Work, Communication, Flexibility and Persistence.  Nrich is a great resource for investigations for the children to work on to develop their skills of problem solving.

Finally SLOW DOWN.

https://nrich.maths.org/

Feeling positive about changes that I am going to make to the way I teach maths in grade 4.

Making Mistakes - Pushing Our Thinking

As Carol Dweck said, math is something you learn. It’s a set of skills. Everyone can get better at math. I’ve always encouraged students to value their mistakes and to understand that this is part of the learning process. I didn’t realize how mistakes can actually make the brain grow. After participating in this session, I would like to go further with my students about how we can look at the mistakes we make - I could share more literature with the children such as, 40 Mistakes that Worked or 11 Science Experiments that Failed to prompt discussion. I would also like to give time for students to share their responses to how they feel when they make mistakes. We can further the discussion by talking about the attitudes that develop.

I’d also like to make sure I give plenty of opportunities for everyone in the class to engage in rich, challenging open tasks to push their thinking and allow them to experience the ‘complexity and messiness’ of real world mathematics. Also - I will reevaluate ‘timed’ fact quizzes or tasks as ‘faster isn’t smarter’ and I do not want students to associate feeling nervous or anxious with completing math.

Mistakes: The Struggle Is Real

Mistakes are powerful.  We know this already.  Anyone who has ever failed at something in their life would probably agree with this statement.  Unfortunately, mistakes also have a way of hanging around for awhile.  Depending on the mistake, an individual could feel slightly embarrassed or ashamed, which makes the mistake incredibly hard to forget or overcome.

Furthermore, the impact a mistake has on an individual is directly linked to their mindset and the way they perceive certain events.  A person with a fixed mindset may view a mistake as a roadblock.... a warning sign reading 'PROCEED WITH CAUTION'.  On the flip side, a person with a growth mindset might view that same mistake as a simple detour; an opportunity to learn something new.

Jo's mention of the 'didactic contract'- the feeling that you want to spoon feed math to students who struggle- was also really interesting to read about.  It takes time for a student to work through a challenging problem, and it takes time for them to develop their own understanding of the concept at hand.  However, teachers know that time is precious and in order to 'get through' what needs to be done, sometimes it is easier just to show them how, or do it for them.  

On a different note.... Jo mentioned that challenging, problem- based mathematics helps to create future entrepreneurs and innovators.  This statement made me think about a group of students who I am currently teaching. All very strong mathematicians, and all very interesting in inquiring into industry, innovation, and infrastructure! An interesting personal connection to the session... and I look forward to sharing this information with my students.

Reflection

I feel that the evidence about brain growth and growth mindset is a very powerful message since it allows children the freedom of making mistakes and encourages them to learn from it. I discussed this in class and they loved the idea of ‘ Maths neutrons’. We always talked about  ‘stretch mistakes’ in our class room but I feel that the growth mindset is a more positive idea on learning as it gives all children a level playing field since they can shift their focus from a perfect final product to process-oriented learning. Luckily, we are at a school where this is part of the learning culture. It was encouraging to get confirmation that when Maths methods are discussed in the context of a problem, it leads to students developing persistance and confidence in later life. They loved hearing the story about how Laurent Schwartz, felt stupid because he was one of the slowest thinkers!

I did the activity that was mentioned in this session where children took a piece of paper  and crumple it up with the feelings that they have when they make a mistake, and then throw it at the board, expressing those feelings. Then we took the paper back, unfolded it,and colored in all the lines that have formed to represent the brain growth they had when they made errors. Children loved this activity!!

I also showed the 2 videos from youcube in class and the discussions that followed were great!

https://www.youcubed.org/what-is-number-sense/

https://www.youcubed.org/four-boosting-math-messages-from-jo-and-her-students/

I agree that often children are the “victims of excellence” and the pressure is not just from the teaching community but parents who have inflexible notions of excellence and push their children to achieve it.

I love workshops where you can take the ideas straight back to the classroom!

These sessions absorbed a lot of my thinking time. It took me a long time!  I too, am a slow reader.  I am not particularly quick as a mathematical problem solver either.  But I enjoy them both equally.

Sessions 3 and 4 raise important questions for me concerning attitudes to mathematics amongst students and teachers.  I meet many teachers and parents who admit to a residual anxiety about maths, mostly born out of their school experience.  The testing culture discussed in this session is an obvious contender for the source of this maths anxiety.  Similarly, the association of speed with recall tests or other mathematical tasks is another cause of stress in the learner. The session highlights the negative effects of a testing culture, in particular, the learner's association of mathematics with testing. To the learner, mathematics and testing have become synonymous.  In test-driven environments, students become preoccupied with test scores/grades and the net result of this is often raised anxiety.  

In our school we are not constrained by such a testing culture and have worked hard to develop positive attitudes to mathematics with our students, however, negativity concerning errors still persists amongst some children and finding ways of changing this is important.  The references to the brain research were very thought-provoking; the realisation that your brain is actually growing when confused! (I need to read more about this).  An environment with a growth mindset accepts that mistakes are inevitable, especially when confronting difficult problems, and they can pave the way to conceptual leaps.

  We need to move away from maths problems that are always resolved in one lesson, or a set time, and lead students towards greater complexity and challenge.  Persistence was a theme here.  I was reminded of 'Thinking things through', by Leone Burton, written in 1984 and still a very relevant book on, in my view, dismantling the didactic contract.  The problem solving work of mathematicians working in Silicon Valley referenced in the session had to deal with 'messiness' and complexity, persist and be resilient ...  We need to continue to engage students in rich problem solving that resonates with the real world.

Thoughts on Sessions 3 & 4

Many interesting and often new (for me) ideas were raised in these sessions. First session 3...

I liked what was said about the need for greater resilience and persistence. It was important to note that successful entrepreneurs made MORE mistakes than others. We shouldn't be designing work so that students get most things correct-make it more challenging. Unsurprisingly, there are many benefits to problem based learning approach. We need to show students how to use the feedback they are given to get better.  Loved the idea of using "yet" in you don't know this yet.  There is an over emphasis on speed.

Looking at this from an LS lens I can attest to the stress caused to students when they feel under pressure to quickly produce an answer. The didactic contract also resonated with me as students often try this approach. PBL sounds great in theory but hard to do with younger kids. Good stuff in this session!

Session 4 head some good points as well.  It is useful to have our students make sense of maths. In the video we saw many students thought the answer was six but how can the quotient be bigger than the dividend? that didn't make sense so was a clue to students to try something else.  There is value in explaining what someone else said-it is a mark of understanding.  It is important to select tasks that promote engagement-not just the search for an answer.  Formative assessment is much more useful than summative--students need to know where they are when they are able to do something about it!  The self-esteem movement was wrong-it has diminished children's growth mindsets.  The Growth Mindset Task Framework consists of: openness; different ways of seeing; multiple entry points; multiple paths / strategies; and clear learning goals and opportunities for feedback. Tracking sends a message that damages students (high and low), impacts teacher expectations, defines the type of work students are given and results in teaching to the middle.

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