A recent article published by Jo Boaler on the youcubed website entitled “SEEING AS UNDERSTANDING: The Importance of Visual mathematics for our Brain and Learning” has really got me thinking about the direction that math education is heading and I AM EXCITED!!!

As a young student I recall my teacher constantly telling me to stop counting on my fingers and that I had to memorize the facts. Memorizing became a useful tool for me as I progressed further in my education that I feel as though it wasn’t until I started teaching that I truly began to understand math. I recall during my student teaching experience (in a 6^{th} grade classroom) my master teacher reading the book “Sir Cumference and the First Round Table” and it was during his reading that I first understood what pi represents. Prior to hearing this book pi to me was just a never ending decimal that we used when dealing with circles.

In a nutshell, Jo Boaler’s article focuses on how the brain truly LEARNS math and the importance of visualization and the use of manipulatives in the learning process. It gives three recommendations to teachers and parents: Encourage and celebrate students’ visual approaches, focus on finder discrimination and encourage finger use, and math teaching and learning needs to become more visual.

A few summers ago I had the privilege of taking a course entitled Patterns 1: Foundations for Algebraic Reasoning offered by Ruth Parker and the Mathematics Education Collaborative (MEC). This class completely changed the way that I teach and started moving me down the path of visual mathematics and using manipulatives to help students better understand the concepts of Algebra. This class focused on using tasks to help students explore and make sense of patterns and how those patterns are represented in different functions. Since integrating patterns tasks into my classroom, students have had a much easier time learning the concepts that will lead to upper level mathematics.

The state has jumped on board the visual mathematics bandwagon and thanks to the legislature is offering a paid opportunity for all high school math teachers to learn more about some of the productive routines that can help promote reasoning and student discourse in the classroom. The course (which can be taken online or in-person) is entitled Developing Conceptual Understanding through Productive Routines. The productive routines covered in the training partner nicely with the patterns type tasks promoted by both YouCubed and MEC. (Depending on the time you read this article, you may still be able to register for the course)

This shift from memorization to conceptual understanding with a focus on visualization and using manipulatives is a welcomed change to the rote memorization of formulas and theorems that mathematics has historically been. The Standards of Mathematical Practice that was introduced along with the Common Core State Standards has helped in facilitating this change.

I highly encourage all math teachers to strongly consider implementing more patterns tasks, number talks, and manipulative usage. Ever since I started integrating the concepts and ideas into my classes, I have seen tremendous growth from my students in their understanding of concepts which, I have been told, has continued on into upper level math courses.

### Aaron Brecek

Aside from teaching, I also coach baseball (JV for the high school and AA for American Legion) and enjoy spending time with my wife and son.

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Erin Armstrong says

Aaron, I wonder if we may have met during Ruth Parker’s MEC course a few summers ago in Yakima? I agree that that course was incredibly powerful at challenging me to incorporate more visual math tasks in the classroom.

I also remember being introduced to Jo Boaler’s work during her course and I wondered if you are familiar with her Week of Inspirational Math? She has designed an amazing sequence of lessons for students from K-12 that gives them opportunities to engage with visual math tasks (ideally it could be used the first week of school). You can sign up for access to the free materials at https://www.youcubed.org/ . I used them with my 7th grade students last year and found that it was a very engaging to start the year. There is also a lot of growth mindset research built into the lessons, so students have the chance to think about how they will respond to mistakes and challenges that they encounter in the new school year.

In addition, I’m so glad that you mentioned the Developing Conceptual Understanding through Productive Routines course. I had the opportunity to take the course last year and it does a wonderful job of building on these ideas with conversations about how to develop conceptual understanding of quadratics with visual models. It’s exciting that our teachers have access to professional development that can support them in learning how use this approach!

Aaron Brecek says

The MEC class that I took was here in Spokane, but it did also talk a lot about Jo Boaler and the brain research.

I personally have not used the week of math in my classes, but know that a few of my colleagues have used it and enjoyed it.

I understand that the online course is till available (just no longer for pay) so anyone who hasn’t taken it yet are still able to. I would highly recommend it. My department has this year (so far) focused on My Favorite kNOw and Number Talks and have found both to have a great impact with our students.

Alisa Louie says

A funny story is my nephew learned how to add to 10 at about age 4. He was so proud and quickly became the evening entertainment at a restaurant one evening. My brother asked him probably 6 or 7 facts that he quickly responded correctly to. Then he asked him something like 6+4. At this point, my nephew started using his fingers, then quickly looked at the family, tears welled in his eyes, and hid his face and began to cry. He was too embarrassed to play the “Impress Auntie” game anymore because in his words, “I’m ‘barrassed to use my fingers like a baby.”

Age 4, I remind you. He didn’t learn that embarrassment from me. He didn’t learn it from his parents. It makes me wonder, what experience did he have to think at age 4, counting on your fingers was inappropriate?

SO glad we’re seeing this movement in math towards a visual and conceptual understanding. No kid should be embarrassed of his math understanding!

Douglas Ferguson says

Thanks Aaron for sharing. I was fortunate enough to take MEC classes 3 summers in a row and found them to be very helpful in how I try to approach the mathematical aspects of my practice. Additionally, I think it’s worth noting that one of the Cross Cutting Concepts from the Next Generation Science Standards is Patterns. We learn through pattern recognition: whether it’s patterns of manipulatives or patterns of fingers… One area where I pushed back for the record, like Johanna, was telling my students to use all of their tools at their disposal including thier fingers and even toes if it helped :).

Aaron Brecek says

I often model counting with my fingers, especially in situations like counting how many months until something (usually winter break, spring break, or summer) or counting how many hours until something (lunch or the end of school).

Keeping manipulatives readily available for students is a good idea and not only encouraging their use, but modeling their use is important as there is still the unfortunate stigma that using them means you are “not as good at math.”

This summer I’m reading a great book entitled Mathematical Mindset that digs deeper into the misrepresentation that quickness means better when it comes to math… My thoughts on the book will probably be the subject of my first after summer blog.

Jill Escalera says

I am so happy to read this!! We have been talking about it non-stop in my primary circles. We adopted Engage New York this year for math, that luckily ENY really emphasizes bringing back the finger counting. I never understood why we pushed using manipulatives but not using our fingers…seems like manipulatives and finger counting provide the same tactile step! Hopefully with the research that is being done and the articles causing conversation about this, finger counting won’t go out of style again.

Aaron Brecek says

I did some piloting of ENY at the high school level and really enjoyed that aspect of the curriculum as well. Their use of patterns to help build understanding of concepts was also better than most curriculums. Ultimately Spokane elementary did adopt ENY and the high schools chose to simply adopt a scope and sequence and take from multiple OERs.

I’m glad my blog helped solidify in your mind the effectiveness of what you are doing.

Johanna Brown says

Ahhh! I have chem students that are embarrassed when they bust out their fingers sometimes. My response is always “use ANY and ALL resources you have.” Thanks for keeping math messy and human.

Aaron Brecek says

Thank you for encouraging students to use any and all manipulatives.

Chris Gustafson says

It’s amazing how revolutionary your phrase ”encourage finger use” sounds to me! Like you, I got much better at math when I started teaching it to sixth graders. Thanks for your encouraging and sensible ideas.

Aaron Brecek says

I wish I could credit for that, but those were the words of Jo Boaler… But I agree it is a revolutionary idea that if followed would have a huge impact on math education.

Patty Reed says

Aaron, I read Jo Boaler’s article recently and was very moved. For years we have been told as teachers not to allow students to count with their fingers, catch them and make them stop. I have always had a hard time with this and let it go with my students, I had no research behind my reasoning only that it felt right and if it helps students in their computation let them use them! Jo Boaler in her expertise has allowed us to go back to the that strategy.

I too love using manipulatives in teaching math. I see my students truly understanding what is happening as we are computing, it’s not a magical operation, it has reasoning.

I also love your information about patterns. That is what i feel math is patters!

Thanks for your post.

Aaron Brecek says

I wish more teachers used manipulative and patterns, it really helps students make sense of otherwise difficult concepts. Like I said at the beginning of my post, I am encouraged by the direction that brain research as it pertains to math is going and the impact it should have on teaching.

Patricia Gustin says

Aaron, Your post reminds me of sitting at the kitchen table with my mom and a box of toothpicks. She taught me so much math at a very young age with such a simple manipulative. Thank you for the links and for your ideas.

I strongly encourage all math teachers to take the state course Developing Conceptual Understanding through Productive Routines. You can even get paid for attending.

Aaron Brecek says

It’s funny how we are encouraged at a young age to build conceptual understanding of the math being done at a young age, then at some point we stop doing that (seemingly just when the concepts become more abstract). I’m glad that the age of “Memorization” is gone and students are now being asked to build their understanding using patterns, manipulatives, or simply talking through their understanding of the math being done. I am SUPER EXCITED about the direction that math is going and the types of students I will start seeing at the high school level.