Why Mathematics?

The mathematics world was in the news this week as Open AI submitted over 700 AI-generated papers on hundreds of previously unsolved mathematics problems. This opens many, many questions about the future of mathematics research and the purpose of mathematics graduate work etc. The real world impact on many mathematicians life work is truly difficult to appreciate for most people. While the remainder of this post centers on the PreK-12 questions this raises for me, I encourage you, if curious, to follow the discourse on this topic and implications to career mathematicians as it is a part of the math world most in PreK-12 aren’t a part of in our day-to-day work.

Undoubtedly mathematics is important to society. It has allowed us to develop new technologies and learn more about the world we live in. With new technologies, inevitably there is discussion of what mathematics matters to who and when. We as a society have seen this with the advent of calculators and computers. So now, with the rise of generative AI we once again should grapple with why are we studying mathematics in PreK-12 and what mathematics should we be studying.

Previously, I wrote a post that encouraged teachers to think about what math matters. I encourage you to go back to that to see my earlier thoughts. Today, instead, I raise wonderings about why mathematics in the first place. For me, we teach mathematics because it allows us to learn to reason logically and problem solve. Mathematics is a vehicle for these practices and allows us to make sense of the world around us by quantifying phenomena and experiences we cannot hold or see (and those we can hold and see as well). I can say with near certainty that I will not ever touch or visit the moon. However, I know a lot about the moon and can compare it to things I can touch or feel to generate a deeper understanding of the moon. We know a lot about how it moves and how it impacts life here on Earth as a result of mathematical and scientific investigations, that have, allowed others to visit and tell us even more. We can also use mathematics to make arguments for change. We can bring data and awareness to issues by quantifying issues that matter to you and your community. These uses often start with curiosity and wonder.

When I visit classrooms or think about how mathematics is taught in school, I wonder how students ever get the notion of mathematics as anything other than calculation and procedures. Now, certainly calculation and procedures are important to understand the mechanics of some aspects of mathematics, however, if that is the majority of what students experience, it is no wonder that this is largely the public view of mathematics. Moreover, it is not surprising that most people fail to understand why we still teach mathematics given our technological advances. When we at the Lastinger Center did a listening tour across Florida and asked adults and students about why we learned mathematics, the majority of answers centered on personal finance and reading clocks or other day-to-day uses. Seldom did we hear about the value of reasoning and sense-making.

As we now have access to tools that can start to do some of the reasoning for us, we must think about what is the role of humans’ engagement in mathematics. It seems understand what is produced is paramount to interpreting and validating results. Moreover, it is asking the questions that we need answered or want answered in our quest to advance, understand and innovate. If this is the why, then we must seriously reconsider what mathematics. Should we continue to spend the vast majority of elementary schoolers’ time on calculation, or should we rebalance it with lots of justification, interesting contexts, and problem posing? We also must consider that the vast quantities of data that we know produce and can access have certainly evidenced a need for greater data literacy, not just in mathematics, but in all areas of study.

I guess it all comes back to what mathematics and how does that align with why we teach mathematics. If we as educators don’t grapple with these questions, students will continue to calculate low level questions for years to come instead of using mathematics as a way to explore wonder and reason.

Instructional Nudge – Yes, No, IDK

This is one of my absolute favorite nudges. Examples and non-examples are hugely important. With the Two-Minute Teacher’s Guide project, we used them as a basis for introducing quadratic functions (check out the video here). To make it a bit more engaging for students, we made a nudge that has students create sets of examples and non-examples of a particular concept. Then, they give those sets to another group who then sorts them into one of three groups:

• YES: It is an example of that concept
• NO: It is not an example of that concept
• IDK: I Don’t Know if it is an example or not

Including the IDK is important. It allows students to share uncertainty which is great information for us as teachers. The IDKs are the potentially interesting cases to explore further. It also lets students understand that it is important to know what you feel confident about and where you think you might need a little more understanding. In math class, we often just talk about right and wrong, but the discussion is infinitely more interesting and productive when we can talk about the things we aren’t yet sure about. The one thing teachers should keep in mind to get to more of the interesting cases, you really need to discuss the interesting examples and non-examples to get the most out of this example. For example, you might seed the set with a figure that almost fits a definition, but is just a little off. If you want to read more about examples and non-examples see this post I wrote a few years ago.

So go ahead and try out Yes, No, IDK and let me know how it goes! You can also see more about this nudge including videos of it in use on the Lastinger Center’s Math Language Routine Nudge page. Find more of our nudges on the Practice-Driven PD website.

UnTeach Project – Favorite Fraction Tasks

In a previous post I discussed fractions. I wanted to share some of my favorite tasks about fractions, these came from or were inspired by John Lannin and his course on teaching fraction to pre-service elementary teachers at Missouri which I also taught after him. These are really nice tasks for driving conversations about fractions and hopefully you find them interesting too. I like them because they force you to think about the whole in each instance and highlight how important it is to define the whole in your models. If you think one of the three examples in the first set could represent 8/7, the important thing you’d have to do is justify what the whole is so that you have indeed shown the equivalent of 8, one-seventh sized pieces. In the second set you have to take what is given and decide what the unit fraction would be which then determines the whole after iterating an appropriate number of times. I think these are fun to think through, but also challenging for many people since we usually don’t dig too deeply into fractions, we just think about them as a symbol and not really what they mean in terms of quantity. So give these a try and see what you think.

Upcoming Events & Happenings

Here are some updates and upcoming events I have planned. Let me know if you plan to be at any of these, it’s always nice to say hi in real life.

ConnectED

On October 13th I get to kick off a math-focused day with Florida Educational Consortia members by giving a keynote talk. I love sharing new ways of thinking about mathematics and learning from people who have different experiences and relationships with mathematics. This should be a great day in Orlando!

NCSM Annual Conference in Denver, Colorado

I’ll be a major speaker (I am so excited!). I’ll be presenting a session titled, “Ideas for Aligning Messages, Actions, Feedback and Evidence and Data to Feedback and Actions” on October 28th.

National Council of Teachers of Mathematics (NCTM) Annual Conference in Denver, Colorado

I’ll be rebooting one of my favorite sessions I’ve done. It’s called “Thinking Outside the Box with Substitution and Elimination” and I’ll be presenting it on October 29th.

Association of Mathematics Teacher Educators (AMTE) Annual Conference in Baltimore, MD

One of my favorite conferences every year! This year I’ll be sharing a session focused on advocacy for those who are eager to get started but may not know how. Should be a good time!