Technolandy

Day 2 (of 2024/25) Book Share: Math-ish by @joboaler (yes, I’ll get a copy for any of my staff who want one) #trueBoaliever

Web resources: mathish.org youcubed.org

Haters gonna hate – but I’m gonna shake it off and say that I love the thinking Jo Boaler (and others – looking at you Sunil Singh @mathgarden) are encouraging regarding fresh eyes at math. I was reminded of this importance with a recent twitter/x post that talked about a teacher (and principal) getting mad at a parent (and their child) when they insisted that 2 divided by 0 = 0. It’s not that simple. And that’s not a correct answer(s). Yes – there can be multiple answers to this, and it’s the debate and discussion in math that has been definitely missing (and making people hate this subject) for generations.

Love her opening example that helps show that not everyone sees maths the same – much like everyone can interpret a reading differently or indeed write in unique ways (both in pen grip and word choice). Similar to Ted Lassos pointing out “be curious not judgemental” (thanks uncle Walt)- we need to ask more back and forth questions about the math! Be curious and seek more than ‘a’ right answer (yes – there are times when ‘an’ answer is required). 

Making me feel right about some of my earlier thinkings about math – that we need to do what we have been successful with in reading and writing – talking and sharing more; collaborating when appropriate; taking time to get to ‘why’; working independently (in comparison to the work) a lot, instructional some of the time and into frustration when it is fun (or challenging – like puzzles) – just gotta be mindful not to get stuck in frustration. 

I can also hear many teachers I have worked with laughing at me because of how frequently I embrace the -ish mindset in much of my learning and teaching and doing. And being. 

Loving the concept of “narrow math” and how that is part of the death spiral that affects too many students… this is an impoverished mindset which is why so many end up hating maths so quickly. Diversity matters and helps bring joy and understanding to more students working with math. 

Appreciate the reminder of how the maths tend to also be the most tested – and where speed is also a key factor – and I know there have been times where wpm were factored into literacy, but sometimes , maybe more times , we need to allow time to savour/percolate/ponder the question to better answer… my ongoing reminder to my teaching staff: quality over quantity. 

The three headed monster that limits love of maths: over-testing; believing math is just a set of procedures to follow; and that there is a ‘math brain’ that you either got… or not…

Learning to Learn

The importance of metacognitive thinking – learning how to learn – and strategies we have (as I have noted in the past) used in many other subject areas but not in math (and why I love the popularity of Peter 

Liljedahls ‘Thinking Classrooms’ for math. More talk and collaboration!

Good breakdown of Hattie’s metadata analysis too… useful explanation as to how some strategies/approaches have more of an overall impact on learning (ability grouping is low low low – just like with reading/writing). There is power in Metacognitive. Especially with maths. (Sidebar… the number of people in North America who have messaged me about my use of the ‘s’ on maths… it’s not a small number…) Metacognitive: 1 – self-awareness/reflection; 2 – deconstruct the task at hand; 3 – assessment. 

Love the examples give  to begin ‘talking’ during math class – how do you know an answer is correct? Is this step an effective step? What are some other thoughts? Etc why did you stop/cross it out? What was going on through your brain? So much better than ✔️ x

Why? <— powerful q

Sharing your own thinking to model how learners can likewise describe what they’re thinking…

8 key strategies

1. Take a step back (look at bigger picture of what is being asked)
2. Draw the problem
3. Find a new approach  (in fact an extension for early work finishers can be to explore other strategies – one of my favourite ‘worksheets’ was a bit of a spider web with a question in the middle (2×3) and arrays spun off to have students show multiple ways to solve the problems… 2? 4? 12…?
4. Reflect on Why?
5. Simplify 19+6 —> 20+5
6. Conjecture – more than a hypothesis…
7. Become a skeptic aka talk in class…
8. Try a smaller case spaces on a 8×8 chessboard? Start with a 2×2 board

Journal! Write about the maths. Reflections connections synthesis funnies etc  it expands understanding BUT takes practice – won’t be great on entry one (just look way back on my own blogging at www.technolandy.wordpress.com … yeesh) so provide prompts!

Love the way some group work strategies are introduced – feels similar to the literature circles I have used in reading – much like a book club…peers interacting can open up understanding and richness that may not be achieved on one’s own… and yes, rubrics work in mathematics. No more ‘survival by half-marks nor reliance that ✔️s – xs= perfect score

 Me: Remember – many of the ‘great mathematicians’ has their week spots – usually the basic components we under appreciate and over emphasize. Seriously – look at Paul Lockharts Arithmetic and tell me that’s not university level doing//thinking/understanding/

Valuing Struggle

Me; even in reading we separate work into levels: independent – instructional – frustration —>knowing that sometimes frustration can be fun (but usually depending on if it was or was not a choice… actually that applies to a lot of life!)

Growth mindsets: mistakes are opportunities to learn; fixed mindsets = mistakes are evidence of weakness. 

A comment in the book is similar to what I often share about mistakes “the mistake is what’s important. If people can do it from the beginning, then they don’t need to come to school.” One approach I have done is ask ‘who doesn’t know how to x’ – usually only one or two raise hands – so I randomly call on students who, by design, don’t know the topic – I build this in a way to show that risk taking means showing vulnerabilities and we aren’t going to learn if we assume people already know things…

I also love, and have started, giving questions to ponder… even questions that stay on the wall for over a week while we work on it… get mad at it… ignore it… and eventually solve it (then I share Kryptos…)

Ooh the part that drives ‘back to basics’ crusaders wild: give the questions before teaching the methodologies… struggle first can be a good thing… but, the lesson from reading: you can’t stay in the frustration zone. 

So…

1. share neuroscience (what we know the brain tends to like and that there ain’t no ‘math brain’
2. Start conversations about ‘struggles’
3. Share metaphors eg avoid math apps/games that only accept ‘one’ answer…
4. Give challenging work but with plenty of access points
5. Celebrate when students struggle and make mistakes
6. Use growth, rather than fixed, praise
7. Change the way you assess (me: look at learning outcomes rather than test scores)
8. Share famous/important mistakes 
9. Crumple paper there’s a reason – great activity in the book – no spoilers 
10. Choose a ‘favourite mistake’ (easy: 2/3 + 1/4 = 3/7)
11. Share videos and articles (me: love the shoutout to Vygotsky and the zone of proximal development)

Mathematics in the World

Big words for education: few things matter, and they matter a lot

Math Matters:

Number Sense (typically arithmetic)

Data Literacy: Data Analysis & Problem Solving

Linear Equations ~how things in life relate to each other

Love love love Jo bringing some of the ‘history of maths’ into the forum! Zero… 24 hours… numbers… tallies… great stories and world wide coverage – not just from Europe (gotta find my copy of Crest of the Peacock)

Math is a creative domain! If we’re not being creative, we are losing an audience and they are becoming math haters. 

Love ish thinking (too many staff will agree with that sentence)

Especially thinking -ish when it comes to estimating (my reflection, I think I gave too easy estimation questions when teaching this… easy entry: how old are you? Precisely?…..)

Better than I can paraphrase: “Suzanne Downes, who teaches mathematics in international schools, shares this reflection:

“I am starting to feel sad about young and older students that when asked to divide 272 by 8 mentally, they will try and do long division, without a real feel for whether the answer makes sense or not, inside their heads. Same for adding or subtracting mixed numbers. When asked to add 19¾ + 27⅓ many students change these into improper fractions with a common denominator. These students will lose most sense of the numbers on the way. Is it that we don’t take enough time for teaching true understanding? Is there not enough time for instilling joy of number sense and logical thinking?”

272/8. Kinda close to 30×8=240 – guess 32 (mental math says 4 8s are 32 so 34 is more precise – thank you Howie Hua for Mental Math Mondays). Precision matters, but does it matter all the time? For everyone?

***

Jo does a great job taking on traditionalists – and I ‘get’ where they think she is trying to make math easier on kids… but I see it as responding to developmental readiness and new math fields (data science) being more relevant but not having a seat at the 1882 ‘math pathway to grad’ that is algebra + calculus

*****

Data talks: question asking; pattern seeking; communication: meaning making (not sequentially)

Not everything needs to be a line graph…

But the data available and which can be collected… only more expansive each year…month… day. Meaningful could be; music you hear each day (time… location… genre… chosen… band… solo artist etc) then explore who collects data… why… what’s being missed 

Mathematics as a visual Experience

Love the follow up to an earlier group of youcube camp participants to see if math positivity stuck… it did. Especially the mental representations. (Hmm how many people really can visualize an acre… a km… a mile…a cm2

Deliberate practice: you don’t build mental representations by thinking about something; you build them by trying to do something, failing, revising and trying again, over and over – hmmm design thinking framework…

Looking forward to representing-and-re-reading the neuroscience section. Juicy! Especially after reading a book about words and now coming across ‘groupitizing’. Explore: ways to group dots. 

Oooh – challenge: ‘let’s (encourage) kids to use their fingers… it may help (or discourage when told to stop) mathematical development. 

Division of fractions: ours is not to reason why, just to flip and multiply. Ugh. Hated this when my kids brought ‘that’ math homework to me. 

Fraction helper: think of their relationship (-ish thinking)

Kids don’t hate math; they just disliked the ways it had/has been taught to them

Start the day with: a dot talk; a data talk; a shape talk

The Beauty of Mathematical Concepts and Connections

Drill and kill is needed less… number conceptually and flexibly is needed more.  

Research: compression of concepts is needed… focus on learning only rules and methods means compression won’t happen (4+3 is never ‘just 7’ it is 4 and one & one &1) quote: “Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through the same process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics.”

Sigh. I’m still working on my sketchnotes. See Sylvia Duckworth for more…

Beautiful shares as to why math and neurology can be beautiful (but likely not in current textbook formats). More talking and sharing. Love triangle-ish. 

Diversity in Practice and Feedback

Beautiful exemplars (and peeks into Jo’s research in practice…) of the value of going beyond a text with maths. From -ish thinking to precision. Perspective matters. 

I hate to give spoilers, but…

Characteristics of Effective Practice

1. Application of methods: Problems should ask people to use methods in new and different situations.
2. Consideration of contrasting cases
3. Focus on concepts and big ideas, not small methods
4. Development of representational models that include visual or physical referents
5. Nonstandard examples and representations
6. Connections that people can see and learn; connections between mathematical ideas and between mathematics and the world

And my favourite: assess with feedback loops! ❤️

Mathematical Modelling (via WolframAlpha & Mathematica):

1. Define questions
2. Abstract to computable form
3. Compute answers
4. Interpret results

A New Mathematical Future

Model of teaching and assessing:

Learning to Learn

Encouraging Struggle (not staying in zone of frustration)

Concepts & Connections

Multiple Representations

Diverse Practice & Feedback

Sample Lesson Structure (Japan)

Introduction (of a problem)

Research (students work on solutions as a group)

Sharing (ideas shared and developed as class)

Synthesis (summarizing specific mathematical knowledge to be taught)

Narrow Math (tests/contests) vs Full Selves (competitions/projects) <— and of course, how ‘time’ matters/factors into completion success.

Status Quo disruption: relationships matter… diversity is vital… black & white loses to a colourful array of options <— perspectives such as: mathematics is a set of patterns that illuminate the world (of art and music)

There is also a continuation of systemic racism and bias (you can see by looking at who is and who is not in some key courses, competitions, and jobs). While math seems like it would be more unbiased in how results are found… that is not currently the case. Jo has felt that (and shared her stories) first hand with people not liking her findings and ideas and getting very nasty. Yes, people get mad when it is uncovered that math is much more than solving algorithms and memorizing formulas for test completions. 

Mathematicians have long been change agents and revolutionaries… no reason that this shouldn’t continue… 5 principles:

1. Believe in Yourself – strong North American culture in disrespecting educators, bipoc, women, queer, nonbinary, trans people, those with physical disabilities – anyone who seems different than ‘typical’. We can do better. Know that you (educator) are the one with the knowledge and expertise. 
2. Practice Empathy – timing was great with a saying from a staff meeting… when it comes to sharing diverse ideas (and people) – we are not us without you! Sharing ideas is powerful (even if we don’t like some of the ideas – some of us will still 
3. Build a Network – You are not alone. It may feel like that some days… but there are many of us who support this work!
4. Investigate – collect and share data. Data can take many forms (great film example) 
5. Develop a Warrior Mindset – not about fighting or war, but about connecting with the world in new and different ways, oriented toward making a difference (I agree, this can be quite the fight…!)  Find yin/yang and balance between interconnected/opposing ideas how things can work together. 

It’s not always easy, but I’ve seen how and why it is totally worth it. 

I appreciate Jo taking on this work and sharing this information. Controversial? Sure, to the haters who wish that mad minute success led to high scores in math. To those of us with a complicated relationship with math (I love a lot of it but was algorithmed out of alignment) I appreciate the deeper look of what it means to be a mathishmaticIAN