Sunday, 14 May 2017

The PYP Essential Elements and Mathematics

I was recently asked what the essential elements might look like for maths. So that got me wondering! Since the elements are the foundations of the PYP curriculum it seemed important to look at them through a maths lens.

I brainstormed ideas with this question in mind.

What should I be trying to expose/inspire/model/guide my students to be doing during my maths lessons?

Here is what I came up with. I would really appreciate any comments or feedback on this. Feel free to add or improve. 

Students will inquire into Concepts
Students will inquire into concepts within central ideas and lines of inquiry.
Students will acquire Knowledge
Students have the knowledge of number fluency/facts to enable them to solve problems efficiently, accurately and with confidence.
Students will develop Attitudes
Commitment to solving challenging problems
Confidence to share ideas with peers
Enthusiasm for mathematics
Creativity in solving problems
Integrity to admit mistakes
Students will take Action
Use strategies to solve maths in their daily lives
Observe and discuss maths at home
Make connections with how maths impacts their lives
Display an enthusiasm for mathematics
Continue to study maths at higher levels
Students will use these Skills
Thinking
  • Acquire the knowledge required to solve problems.
  • Apply knowledge of number fluency/facts to solve problems.
  • Analyse questions before solving.
  • Ask questions about maths in their everyday life.
  • Use prior knowledge to estimate answers.
Communication
  • Listen to others.
  • Present their understanding their own way
  • Ask questions when they are not sure of the answer.
  • Show their understanding using drawings, materials or written methods.
Self Management
  • Work through a problem solving process.
  • Apply known strategies independently.
  • Solve problems in their own time frame.

Research
  • Use mathematics knowledge to gain greater understanding of ideas and concepts within UOIs.
  • Observe problems.
  • Test ideas.
  • Use standard and nonstandard units to find answers to their questions
Social
  • Ability to work independently and in groups.
  • Show empathy towards others.
  • Don’t call out the answer.
  • Allow others to share their ideas.



Wednesday, 12 April 2017

Student Connections lead to Greater Learning

We were in our second week of our fraction unit. The previous week we had done a lot of visual work and using the language of fractions without really focusing too much on the symbols.

I always encourage my students to find maths connections in their lives. One student during reading came to me his books saying "Look, fractions!"


I thought, lets use this for tomorrows maths lesson!

I gave out whole cups, tablespoons and teaspoons and explained that I want them to use their imaginations to pretend that they are making these cookies. Rather than using measure cups and spoons I wanted them to practice making parts of a whole i.e. dividing a whole cup into three parts and then only using 1 of those parts or estimating out what half a tablespoon would look like.

The lesson was amazing. The students were so engaged and their imaginations were running wild mixing and stirring their cookies. Some even put them in an imaginary oven.

The language used was fantastic. I could walk around and promote the language of fractions while the students could visually see what 1/3 or 1/2 looks like.

Students even started to invent their own measures using estimation.

Students made parts in different ways. For example, some estimated 3/4 while other estimated a quarter and added it 3 times.

Students then went on to make their own recipes, some even taking it out into their recess time.

Lesson like this reminded me that I need to allow more time for my students to play with maths. So much learning happens and inquiry unfolds.






In making their own recipes students started to get creative with their fractions

 Putting the cookies in the oven.

Realising that they can use estimation to make their own measuring cups.

Thursday, 16 March 2017

Posing Multiplication and Division Problems

Multiplication and division is one of my favourite units. I find students love being able to use their knowledge of facts to help them solve problems with greater efficiency. One of the challenges to teaching multiplication and division is making sure students conceptually understand what they are doing. Can students visualise and apply their knowledge to solve problems. My students are very skilled in the facts. Where they struggle is being able to apply them and to communicate their thinking. For example, I ask my students what is 5x4?, without any hesitation they give the answer, but if I show them 5 groups of 4 or ask them to draw it amazes me how many will count them all. This is more evidence about the importance of conceptual based mathematics. 

Below are a collection of questions that I posed throughout the unit. Some worked really well, some not so good as I either posed a problem that wasn't challenging enough or I failed to extend or enable my students. 

Problems I have used:

This problem went really well. It was connected to our unit on body systems and gave the students lots to think about. Something that I want to do more of is to make my open questions more relevant to what we are doing in the classroom. I find this very difficult at times. 

"What does length of time mean?"
"What's a number between 60-80?"
"How many days are there in a week?" 
(this also gave me a great formative assessment tool and informed me that I need to teach this)





This was linked to our HWEO unit. The students really enjoyed this task and it worked out to be a great visual tool too.

This open question was great for getting students to practice their knowledge of facts in an open setting. 


A student took this photo on a multiplication walk and I used it in our next lesson. Students love it when you use their ideas. It gives them ownership. 


Taking photos from when you're out and about. 




Using provocations are also a way to engage students.


Not really an open question, but asking students to find different arrays and letting them see that they can use a variety of facts to solve problems is also important. 




A few more examples that I have used. 









Feel free to share any questions you have used or if you think these are good questions. 

Sunday, 22 January 2017

Posing Open Questions

Below are examples of open questions that I posed to my students during our Addition and Subtraction Unit. While some worked well, some bombed. I would like to blame the students, but like always, it was because the questions or tasks didn't reflected the students needs, I used the wrong numbers or I didn't word the problem correctly. Regardless, writing open questions is a learning process. They are not easy to write, but when they work maths lessons come alive and so too student learning. The important thing is that I reflected on why they worked or why they did not. I can then make changes for next year and makes maths lessons even better.

Here are some of them


Students really struggled with this as there was a lot of information missing. What I learnt was that some students would take risks and trial different numbers. Others waited for me to give them temperatures and had a hard time thinking about how to start. Maths to me is about making mistakes and having a go. This ended up being a good lesson as it told me that I needed to teach my kids to be risk takers. After observing their maths journals, I also found out that I we need to work on what 'difference' means and how we can use subtraction to find the differences between numbers.


This worked well, but I forget to tell them they can only use the numbers in the box... next time!



These questions are always good. It is important though that you ask students to show their thinking.


These kinds of questions will tell you if students understand the concept of 'equals'.


I have a pair of dotty socks, hence the question. This was ok. My students struggled with the meaning of 'pair of' so we had to talk about that first. Always, good to remember that don't assume students always know what your talking about... especially in an ESL setting.



This worked well. You can make these receipts online. It was close to Xmas time so kids enjoyed this. They then told my kids what I bought them. Again, lots to comprehend here. I enjoyed this as it made them think in context and what could possible answers be.


This was linked to our weather and data handling unit. It worked really well. Students liked having the choice of country to choose from. Reflecting on their answers, studdents learnt and gained a better understanding about how can use subtraction to find the difference between two numbers.

 ¨
This again was a problem that students found difficult. I think the wording was not great and maybe too complex for ESL students. We did talk about it first. Most struggled with thinking about what prices to use.



These work well and students love to have choice and to work with others.



Another problem linked to our unit.



I really liked this problem. I love finding maths problems when I am out and about.



Given choice is always good.


These questions or task above are a few examples of things I have posed to my students. I hope you find them useful. If you have any advice please comment below. Always great to share planning.


Tuesday, 10 January 2017

The Right Strategy for the Right Problem

For some time now an expectation for my students has been to solve problems in a variety of ways. Many based on outcomes from we have set. I have constantly been asking them, "How could you solve that another way? 

Recently, I have been thinking "Why?" Why do they need to solve a problem in 2 or 3 different ways. In real life situations we don't solve a problem and then think of another way to solve it. I understand that being able to have different strategies teaches students to look at things in different ways. But wouldn't be better if the could analyse a problem and then reach into their tool box of strategies to find the right tool for the right problem!

So I have started asking my students, "What strategy would be helpful in solving this problem?" Not all problems lend themselves to certain strategies. Sometimes place value lends itself well, but other times making tens can be more efficient.

This made me think that rather than teaching or guiding my students to a specific strategy I should be getting them to think critically about a problem and then ask them what strategies would be most helpful and why?

Getting students to analyse a problem and then think about the strategy that they will use also provides great evidence of their understanding. 

What I was finding was that my students were simply taking a problem and trying to apply any strategy that we had shared without really thinking first about what strategy would be most useful. 

I have now changed how we approach problems. I have start to allow more time for students to think about the problem and the strategy that might make the problem easier and more efficient for them to solve.

Wednesday, 30 November 2016

An Inquiry into how to plan for inquiry maths

I am passionate about the benefits of inquiry maths and inquiry in general, but recently, after completing a workshop with wonderfully passionate teachers, our discussions got me thinking about the difficulty teachers face in implementing inquiry maths into their planning. I felt that the confusion of when and how to teach was making teachers despondent before they had even started, with this then resulting in teachers falling back to what the know i.e. text books or worksheets. There seems to be a lack of information or resources out there about how to create a structure inquiry maths. Now, I know that the word "structure" and inquiry might sound conflicting, but my intention for creating structure is to provide teachers with the when and the how to teach and for the students to determine what we teach. I also really wanted to give both myself and my students more time.

More time to process questions.
More time to play with materials.
More time to explain their thinking.
More time to actually practice what I teach.
More time for students to share their understanding.
More time for conversations.
More time for students to pose problems.
More time for students to learn number fluency.
More time to assess authentically and in a manageable way.


So I started my own inquiry... 

Step 1: Tuning In (what do I already know?)

I started to think about what a week of inquiry maths might look like. I took all the components of what I thought would make a great maths environment and started including them into a time table. I took ideas from the likes of Jo Boaler, Ann Baker, and Peter Sullivan and used resources such as Nrich, nzmaths.co.nz, and the Teaching Student Centered Mathematics text by Van De Walle to help me create a "structure".

Plan 1:



Step 2: Finding Out

I researched to find out information to help support teachers with resources and information on how to carry out each activity or task on a particular day. I linked in readings and videos from Youcubed.org and Ann Bakers Natural Maths blog. I searched articles on Peter Sullivan's phases and prompts. I included links to TSCM book and the Polya problem solving approach. 

The information was there, but would it work? Was it practical? Would it provide more time? Would it promote inquiry? What anyone be interested? Would it make my students better mathematicians?

Step 3: Going Further

I took this idea to my colleagues who were, for the most part, on board with the idea of having a structure in place, but how did I know if the plan was good or not? Was it actually good for students? Was it good for teachers? 

I then turned to some "experts". I sent them my ideas and got there feedback. I was very appreciative for their help. It's great to be able to bounce ideas and reflect on their advice. 

Step 4: Sorting Out

Their advice...

very comprehensive but is too much there?

Do students developing a deep understanding of important mathematical concepts?

How are we engaging the students in this process?

How are you identifying what they know... yet to know...choice... developing theories...

Are students enjoy mathematics when they struggle and they want to find a solution and they are given time to work through possible strategies.

It is flexible?

Is it focused on the student?

There is danger in that .... as mathematics can then become a series of activities.

Step 5: Back to Finding Out

I then went and looked a Ann Bakers idea for a what a week of maths might look like. I included the idea of a "mental routine" to begin the lesson. I like this because if give me time to teach skills in a conceptual, collaborative and informative setting. I also then took the idea of posing problematized questions (open-questions tasks) and then using the outcomes from that lesson to pose a strategy based lesson.  This formed the basis to plan number 2.

Step 6: Taking Action

I have implemented the plan into my classroom and will give it time to see if works. 

Plan 2:




Step 7: Reflecting

So far I am enjoying this approach. The number talk (mental routine) at the start is a great way to reinforce skills and to build students understanding of mathematical language and concepts. I really like the Poyla problem solving approach. I think it scaffolds students nicely through the process of solving a problem. The difficult part I am finding is posing good challenging tasks. I am getting better at creating them and I reflect a lot on the ones that work and ones that bomb ... and there have been a few, but when you pose a question that the students are engaged and challenged you can literally see the thinking happen and this is what I believe in the most. 

I will blog some of my open problems soon as I would be keen to get feedback or advice.



Thursday, 17 November 2016

Using Formative Self-Assessment to the Students and Teachers Advantage

I have often struggled with how best assess and then teach the skills that are essential for helping students solve problems more easily. 

My latest attempt has been to make up self assessment checklists where students first assess themselves as either, I don't know yet or I know this. I then conference 1-1 with each student as a check to see if they know or have they rote learnt it. Understanding these skills conceptually is super important. 

I found that students were really honest and principled when filling them out and that I didn't have to change too many.

This next step is how do I get students to learn these skills so that they are targeting what they need to know. One way I approach this is by setting them as goals for home learning activities. Another way is to give them time during the week to practice. Finally, I am also starting to bring them into my number talks or maths routines as quick mental warm ups. 

I like this method of assessing. It gives students ownership of their learning and promotes the learner profile attribute of being knowledgeable. 

People often think that inquiry maths has nothing to do with skills! I completely disagree. If we want students to inquire mathematically it is essential we give them to tools to do so. 



If you have any other ways of assessing and teaching knowledge skills then please feel free to share!