Mathematics, Inquiry and Neuroscience

​I’m currently reading two books to improve my professional understanding and practice of Inquiry and Mathematics. I am so excited for Jo Boaler’s Mathematical Mindsets to meet Kath Murdoch’s The Power of Inquiry. What delicious learning! I will also be keen to observe and collect feedback from my students to see how our attitudes, skills, knowledge, understanding and action improves in Math.

As we begin the school year, our scope and sequence outlines that we address whole number. This will be a great place to start, however, I can be one to get bogged down in detail. I sometimes worry that I may just strangle the life out of an inquiry before it even leaves the pages of our planner. So I pull myself back and feel encouraged to read Kath write, “The challenge, then, is to acknowledge the way we can scaffold our planning and teaching by referring to a process without the process becoming overly prescriptive.” (p.77) That was all the affirmation I needed to open the inquiry up and ensure learner ownership in the process. Let the messy Maths begin!

Before we had even launched our inquiry into number I heard an all too familiar comment when a student announced, “I am NOT good at Maths!” Something Jo Boaler warms us is a damaging message that we need to challenge and change because science and research proves otherwise. I smile knowing that this student will feel and think differently by the end of this inquiry.

In an attempt to stay fluid I run through a quick checklist in my head of the essentials to frame a successful open inquiry. A central idea, personal connections to the concept, a guiding question, hands on materials, student choice and differentiation for content, process, product and environment. I read carefully through the outcomes and content points in the curriculum, gather my materials and dive into the Constructing Meaning stage of our three-part Math inquiry with Transferring Meaning and Applying With Understanding to follow.

We begin by anchoring our inquiry around the central idea, “The base 10 place value system extends infinitely in two directions.” To prepare for how Maths is learnt, we will proceed through the inquiry by students “doing” the following types of learning engagements at each stage of the inquiry:​

​1. Constructing Meaning: modelling

2. Transferring Meaning: ordering, reading and representing

​3. Applying With Understanding: Where do I use integers?

We immediately follow with a question that invites personal connection with the content, “Where have I seen negative numbers in my life?” I call this brain rain because it reminds me of the sound of the first few slow drops of rain before the heavens open up and it gushes down in a loud outpour. One student bravely but quietly offers one idea and the whoosh the discussion erupts and I can’t keep up the recording of their wonderful personal connections.

Temperature
Dials
Number lines
Ruler
Dice
Receipts
News
Finance
Sat Nav
Boating
​Movies
TV Shows

Sea Level
Swimming
Diving
Submarine
Calculators
Graphs
Compass
Degrees
Thermometer
Decimals
Percentages
Fish Finders
​

What is interesting to observe is that this lively loud discussion seems to be constructing a shared visual in our collective mind’s eye or a common narrative between us about how we see positive and negative integers. I hear a lot of students affirming peers’ descriptions with an excited, “Yeah! Yeah!” We are moving so fast that many words have flown around which we need to capture, clarify and begin using. One student suggests it’s already time to begin our Math Word Wall which is student created and maintained paper or e-space to use as a reference chart along with our on-line and paperback Math dictionaries. We set a culture that encourages, “If there is a precise and more accurate vocabulary word then find it, write it and begin using it. Math words are yours!”

We then dove straight into modelling using hands on materials. This stage of the inquiry is driven by student choice. The purpose of student choice here is that each student is in control of entering the learning engagement at their own level of knowledge, skill and understanding. This is differentiation. I explain that we are going to explore integers using a range of dice and I suggested a few learning engagement frameworks such as ordering, repetition and comparison but everything else was their choice: ​

What dice they choose either single digit dice, double digit dice, large integers formed from dice in expanded form or moving beyond whole numbers to work with decimals formed from dice in expanded form (Content)

What tools they use choosing finger counting (read Jo Boaler’s article Why Kids Should Use Their Fingers in Math Class about what neuroscience research supports), calculators, regular or  e-whiteboards  (Process)
How they learn such as co-discovery or a competitive game (Process)

How they model and record by either physically placing items on created number lines, using a paper/digital record or through conversation (Product)

​Where they work in the room whether it is the table, soft furnishing or on the floor (Environment)
Who they work with choosing individual, partnerships or a group (Environment)

And then our classroom was a beautiful mess (just as Kath Murdoch warns) as students expressed themselves through Math while learning Math but most importantly in a way that made sense to them while they Constructed Meaning. The following pictures and captions share the range of explorations they designed and were engaged in.

​Students selected and rolled 20-sided die to find a starting positive integer. They rolled a 6-sided die to subtract until they passed zero. They used a number line and calculator for support and checking of accuracy.

​Students selected and rolled a 20-sided die to establish a starting positive integer and an ending negative integer. They rolled a 6-sided die to subtract and kept score with agreed points for first past zero, landing on zero and reaching the target negative integer. They agreed on jobs and took turns using calculators if they needed to check mental visualisations.

Students determine which integer will be negative or positive by rolling a +- die or flipping a two-coloured counter.

Students order their positive and negative integers on a number line where zero is marked by a pencil or piece of paper.

The students on the right chose 12-sided dice and the students above chose expanded-form dice that produce large integers.

These students used a Maths dictionary to revise how a number line worked. They created a number line out of two rulers, selected 20-sided dice, used a two-coloured counter to determine which die was positive or negative and marked zero with a handy paint brush.

They have placed the negative dice accurately aligning the numbers of the ruler with each die value, however, they've faced a creative problem with using a second ruler with positive numbers descending from 30 after zero. They are ready to design an accurate and useful number line for future tasks.

The students above chose to move beyond whole numbers to work with decimals that were formed from expanded-form-decimal dice.

They determined if the decimal was positive or negative by rolling the +- die.

They decided to go further and find the difference between their two decimal numbers and made it a competitive game where the winner had the greatest difference.

At this point I had a good handle on where learners are at from formative assessment. I have observed their prior experiences, vocabulary, level of skill, conceptual understanding and mathematical notation. But I have also gathered a lot of other qualitative data too. I know how they feel, their attitude, comfort level, willingness, open-mindedness and interest. This is a rich bank of information to now co-plan (with the learners themselves) the next stage of our inquiry Transferring Meaning Into Symbols. And I’m pleased to report that the student who announced, “I am NOT good at Maths!” was fist pumping the air in Math excitement and celebration during her inquiry and she asked to do it again. I'll have to ask her if she means what we were learning, how we were learning or both.

I hope that Kath Murdoch and Jo Boaler enjoyed their first meet as much as I did. Thank you for the springboard into learning more about Math and Inquiry...stay tuned to find more as my learning unfolds.