I am thinking back to when I got my training wheels off on my first bike. My Dad held on to the back of my bike at first for a little ways down the street, then let go but stayed close by. He never said, “make the right foot push the pedal like this, then feel your tummy muscles adjust your balance…” That would have been ridiculous. There was a natural sequence to learning how to ride a two-wheel bike without training wheels. The sequence involved experiences, guidance, opportunities to try and fall and scrape up my knees, and finally the feel the wind on the handlebar tassels...success.
During a recent conversation with Maarten Dolk (Reference 1), we discussed student questioning. I was telling him about my fall action research involving my students engaging in weekly wondering workshops. They had been designed to open up children’s understanding of what mathematics included. I told him that at first, my students had no questions and only observations. Then they began to ask questions but they were often not about the mathematics or were rather shallow. Towards December, the number of questions increased and they became increasingly about mathematics, and demonstrated more complexity and thought. Now, their questions are often investigatable (I may be making up a new word here but it suits my work and children). Maarten told me that this is a process that children go through and as teachers, we need to understand this progression, support it, and trust it. We discussed how often we as teachers sometimes interject and try to over manage students or tell them what to ask. Now I am finding that if I present a context or investigation inviting the students to observe and think about it for themselves, they often ask questions that I would have otherwise told them to go off and investigate. Instead, they now own the questions and are more invested in going off to investigate. THEY are the mathematicians.
When Maarten said the phrase, “trust the process,” it reminded me of my colleague Carly Albee’s work around student questioning because she often says the same thing. Carly and I shared our fall action research work during a recent Virtual Math Coaches PLC. The recording of that session can be found if you check out this wonderful PLC and join. http://mathleadership.org/virtual-math-coaching-plc/
Our session is session 15 held on March 4, 2021.
My students are now demonstrating increased understanding of what mathematics includes and what mathematicians do. Check out their growing web!
They often refer to it and want to add pictures or more ideas to it. I am noticing that many of their comments exhibit the action of questioning or seeking to learn more. For example, “When things don’t go as you expect, keep on trying.” Or, “Is nature made out of math?” Even things like “playing with shapes” suggests the notion of inquiry. In the middle on the top, there is a picture of a wooden board filled completely with small geometric pieces. Ivan did this and had tried to cover the board completely numerous times over 3 days. His classmates suggested the words on the card “never give up, keep trying each day.” I am also struck that no where on the web is anything resembling “we do pluses” or “we write 3 plus 2 equals 5.” These were the kinds of things they said about math back in September.
As I continue to reflect, I am thinking more about sequencing my instructional moves. When I notice my students’ questioning skills growing, how can I build on this and continue to grow their mathematics development along with their questioning? In an article about what mathematics education could prepare students for, Koeno Gravenmeijer writes, “Teachers will have to be able to orchestrate whole class discussions, asking deepening questions, and posing tasks that help students to reflect and build upon their current thinking.” (Reference 2) He points to a problem around individual tasks. When we think about helping students construct ideas in mathematics, develop things like critical thinking and reasoning, or collaborate with peers, we have to think beyond just giving students a series of individual tasks or routines.
I often cringe when I hear the term “spiral” used. This reference often entails going back to some set of “skills” so “kids don’t forget.” I do not often hear people use spiral to mean building upon what students are doing and constructing, and this is exactly what we need to be doing.
Currently, I am finding that my students’ questions are now often connected to what they were noticing about something. I think there is something to stop and think about here. During recent number strings involving doubles (Reference 3) using the 20-bead MathrackTM (Reference 4), my students were noticing things like, “they each have a partner,” or “that one doesn’t have a partner so it’s one more than before.” When they next explored different sizes of chocolate bars (the kind that have the grooves to break squares off), they were asking about sizes that would “look funny.” For example, Ivan said, “9 wouldn’t work because there would be a piece sticking out.” Then Flourine said, “I wonder what other numbers would look funny?” Alejandra asked, “can we draw the ones that don’t work?”
So what are they developing here? What can I do next to support their development? I want to take “the mathematics of the children” as Maarten Dolk often refers to, and support and nurture it. They love wondering workshops, but I must consider the content that would nudge something. Kids will often do “cool” things with lots of great individual tasks, but if they’ve already done those “cool” things, or they haven’t yet constructed some underpinnings then what have I done? I need to consider where my students are in their development and what might nudge them forward.
Please add your thoughts in the discussion below!
For my readers who are parents, ask your child, “what does a mathematician do?” Or, “what is mathematics?” I would be curious what responses you get.
References
1: Maarten Dolk is a mathematics education researcher and developer in The Netherlands. He co-authored the Young Mathematicians at Work series with Cathy Fosnot as well as a number of Contexts for Learning units. Here is a link to a talk Maarten gave at a conference in Indonesia where he talks about the role of context. https://www.youtube.com/watch?v=3IyUutVD7Ac
2: Gravemeijer, K., Stephan, M., Julie, C. et al. What Mathematics Education May Prepare Students for the Society of the Future?. Int J of Sci and Math Educ 15, 105–123 (2017). https://doi.org/10.1007/s10763-017-9814-6
3: Number Strings were originally created by Cathy Fosnot and are designed to support progressive development. The ones I use come from the book: Minilessons for Addition and Subtraction.
4: www.mathrack.com is the best place to purchase mathracks.
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