I’m terrible in areas such as algebra, which rely entirely on abstraction and

provide nothing to visualize.

I couldn’t bring myself to title this newsletter “For Algebra.” As it happens, animal scientist Temple Grandin is right about a lot in her essay “Against Algebra,” especially her critique of the gatekeeping role that advanced algebra plays in our society. High school graduation requirements and university admission policies in math prevent too many people from reaching educational goals that have no meaningful connection to their ability to factor third-order polynomials or conjugate radical expressions.

However, Grandin frequently attributes to “algebra” what we should really attribute to “ineffective algebra curriculum and instruction.”

For example, Grandin writes, “I value my different way of seeing the world, because it’s made me who I am.” Grandin refers here to her tendency to think visually, but she may as well speak for every student who feels like the ways they see the world—the images and experiences and histories they bring to class every day—have no place in algebra.

Grandin is right that in many versions of algebra instruction, students are offered an abstract idea (an equation, for example) and asked to transform it into *another* abstract idea (a *graph*!) with no tangible benefit beyond a grade. Grandin is right that this kind of curriculum and instruction asks students to rush quickly up “the ladder of abstraction,” forsaking ideas they can understand and visualize in favor of ideas that are abstract and alien.

We shouldn’t accept this as an inherent feature of algebra, though. Effective instruction of *any* kind in *any* discipline means teachers are inviting and taking seriously the knowledge students already have–the ideas students can readily and concretely visualize.

Very early into the lessons in our curriculum, for example, we ask students to play, sketch, describe, write a story, predict, notice and wonder—all efforts at inviting a student’s concrete, visual, and sensory knowledge.

We ask students to turn that concrete knowledge into algebraic abstractions, but we help students see how the abstract *fortifies* rather than *replaces* the concrete. Algebra sharpens our intuition. It makes our predictions more accurate and tedious tasks more efficient. It adds new textures to the ideas and images we bring to class.

Grandin is right that this isn’t often the reality, but it can and should be. Not every student will be “for algebra,” but they should know, at least, that algebra is for them.

### What Else?

Come hang out with me next week and experience one of our lessons as a student in an Amplify Step Ahead webinar.

Our team over here just released Challenge Creator to the public, a feature of Desmos activities (previously internal only) that lets students create challenges for one another.

Our newer, beefier “Free Response” component lets students add images and audio responses to their text responses—all an effort to get technology out of the way of student thinking.

I’m extremely enthusiastic about

**Dominic Hill’s**context for cooling curves: what should you make the delivery radius of your pizzeria?“Split 25” came up on my timeline again last week, bringing to mind the late

**Malcolm Swan**and the fantastic math posters from Xi Yu and Joey Kelly at Play With Your Math.

I agree with Dan's notion that a lot of the problems with Algebra can be attributed to “ineffective algebra curriculum and instruction.” But, after 30 years in the math classroom, I'd say that math education can be moving forward in a really positive direction. Platforms such as Desmos Classroom really give students the opportunity to visualize what were once only theoretical and abstract concepts. Also, student ownership in their learning can really be enhanced with the use of interpretational feedback. Math education no longer has to be about "did I get the right answer to this problem"? I'm optimistic that as more teachers come into this mindset, both student achievement AND student perceptions of math will rise.

Funny that just yesterday at our High School Math Dept. meeting, we in the "old guard" were just arguing for more relevance and concrete lessons to help students acquire the skills to move to abstract math (what we call naked math at our building, because when you say naked in class, kids tune in really fast). We were frustrated that we lost the art of making connections due to Covid, and the current plan to fill in the gaps have teachers rushing to teach skills instead of deeper learning.

Luckily for me, our principal said he would rather we slow down to help kids learn instead of keeping pace with the curriculum map for the sake of staying with the calendar. My kids had a fantastic lesson in Algebra 1 experiencing exponential decay in the study of insulin amounts in the bloodstream. Almost every student I had knew someone with diabetes and how they take shots or have a pump and they jumped right in! Another lesson was on radioactive decay and nuclear plants should there be a leak, and it just so happened they knew about Chernobyl and the situation in Ukraine and the war hitting close to a plant. In addition, we studied growth with bacteria and the chessboard/rice story. But finished the unit with paper folding, which let us watch the Myth Busters fold a football field size piece of paper 11 times!

Now, to get them comfortable with presenting math problems is the next step!