Do you think teaching math causes you to over think??? Almost like “going insane “…NaNa Dunn from a post on Visual Math’s FB page

Oh yeah! I used to dream about my lessons… over and over during the night. I’d wake exhausted. Here’s an example of what I do now…

When you realize the students can’t ‘see’ the thing you are teaching them

My kiddos are learning to write quadratic equations. This requires them seeing patterns of growth. The first lesson in this series was to help them see the growth in a pattern of cubes. I used a YouCubed lesson that involved coloring each growth step. Just that. This exposed several issues. The lesson with numbers was the next day, and I was pleased by the intensity of their interest, but here’s why I think that happened: I had prepared them for what they were looking for.

Color each growth step. Image from Stairs to Squares Gr3-5, YouCubed.org

Quotes taken from the original FB post…

Beth Hanna McManus writes “My 8th graders… totally missed the concept that the altitude to the base of the triangle goes through the vertex and hits the base at a 90 degree angle. I explained this in mathematical terms, in layman’s terms, using words like straight to the base, “shortest distance”, had them draw them on the white boards (checking and helping each drawing) multiple times, had them label triangles. When it came time for homework, no one knew what to do. I flipped out and spent the rest of the day in a dither wondering what is wrong with me that I can’t communicate with these kids…”

When I plan a learning experience, I look for the base skill/image/idea a child needs to be able to have to participate in a lesson. It may be a simple warmup – for the triangle problem above, after realizing they didn’t get it, I might have them draw the altitude, pointing out the way the angle looks, and how it starts in the vertex, in several different triangles. Just that. No math, just letting them learn to see what they are looking for.

Starla Adams writes, “Yes, yes, and yes….says the teacher who is out of strategies to teach long division after Friday. I actually thought I was going a little bit insane for an entire hour.”

Teaching long division is challenging. I’ve worked with 9th graders who do not understand what dividing does. If they don’t have an understanding of partitioning and regrouping, long division is just a nonsensical set of steps that they must follow – and memorize. A warmup using manipulatives (coins buttons beads) to set the stage for dividing into groups would help them understand what they are looking for. Then long division can be taught as a routine to get there.

Another common student skill gap with division is poor factoring skills. A warmup (or preview lesson the day before) with a factoring math problem string like this video from Pam Harris, might help strengthen their math fact fluency.

While working on any new skill that requires factoring, try giving students a factor chart. They won’t be using all their working memory on remembering number facts, but on learning the process of the task at hand.

Learning plans should empower students…

I realized that my 9th graders didn’t yet know that they had the power to create their own understanding. They were waiting for me to tell them what to do. Danielle Love and Kay Butler point out ways to shift the heavy lifting (and learning!) to the students!

As teachers, we spend lots of time creating learning plans. Many of us already know what misconceptions kids have, and what errors are going to get made. So lets plan ahead to expose -and remediate and preview- so that these issues don’t cause student failure during the learning of new material. These little discovery sessions and warmups are critical to building understanding and are often worth every minute of time we spend on them!

By all means, overthink and go crazy, in a most productive way! Math teachers, you rock!! I would love to hear how other teachers prepare for these misconceptions and gaps!

(And no, I don’t have those recurring math dreams nearly as often anymore!😂)

Thanks to Shana McKay of Scaffolded Math and Science, and and this really interesting thread on her fantastic FB Visual Math!

I’m Seeing Dots!

Do you number sense?

What I’m trying to ask is how comfortable are you with numbers? As children, we learn to count, 1, 2, 3… Then we begin to realize a one to one correspondence, that one isn’t just a number, it is a value, a penny or one fork, or one person. The bigger the number, the more items. We learn some rules for numbers: That one plus anything makes one more. That if we take away one, it makes the number smaller. Next thing you know we are adding and taking away all kinds of numbers. Then along comes place value and we learn to carry numbers; there are ones, tens, hundreds, thousands, ten thousands, whew! And for many children, each number becomes a solid thing, no longer fluent, no longer one plus one plus one makes three. Three is, well, three. It means a certain amount, and we no longer see it as individual items making up a whole.

I would like you to try an experiment.

Without counting, take a look at image 1.

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How many dots do you see?

Now quickly look at image 2. (Remember, no counting!)

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How many did you see this time?

Now think about how you saw the dots in each image. How did you group them or add them?
Which picture was easier for you? The neatly grouped dots or the randomly spaced ones?

Show the pictures to someone else. They will probably come up with the same number you did. Ask them to describe how they saw the dots, grouped them, made sense of them.

Was their method different? Did you like it better than your method? Will you see the dots differently the next time you look?

Whether you realize it or not, you have just made some new connections about the number eight.

Now solve for x:

4x = 3 + 3 + x

Did your new sense of the number 8 make the problem less complicated? Or did you pull out some algebra skills?

We rarely see dot cards used outside of first, second, or third grade here in the US. I think we may be losing valuable math fluency by not continuing to expose our children to these practices on a regular basis throughout their education. It is especially relevant now, as teachers are asking students to think about the different elements of math problems, asking them to make connections from one set of circumstances to another, expecting them to be able to break apart and re-form numbers like a set of Lego blocks. Yet, from my own observations, we have 7-12 grade students who lack a basic ability to do this.

We give them formulas that have no meaning, theorems for which they do not understand the proofs, and polynomials they struggle to factor or multiply, because they lack the ability to construct and deconstruct numbers. Algebra is difficult because they cannot see that x can be any number. In that exercise above, could there not be more than one right answer? Shouldn’t our kids be curious about that, check for that? (Bravo if you did!!)

Cathy Humphries uses dot card exercises with her 10-12 graders. She has this to say:

“Dot Card Number Talk Commentary:
Cards with configurations of objects, that we often call “dot” card number talks, establish important new principles for mathematics classes. While it may seem that these arrangements of shapes are only for young children, we have found that they are critical for older children – even high school students – because they help to lay the groundwork for changing how students think about mathematics. Dot cards do not suggest procedures that students are “supposed” to follow; instead, they encourage students to think about what they “see” rather than what they are supposed to “do.” This frees up students for learning new ways of interacting in math class.
Some of the things they can learn from dot card number talks:
• Just as people “see” things differently, there are often many ways to approach any mathematical problem.
• Explaining one’s thinking clearly is important. This requires that students to retrace the steps of their answers and learn to use academic language, where possible, to describe what they did to solve the problem.
• It is important for students not only to explain what they did, but why their process makes sense. In the case of dot card number talks, this involves where they “saw” the numbers they used. In the case of arithmetic operations, it involves understanding the mathematics that underlies any procedure that they use.
• The teacher’s job is to ask questions that clarify what the students see rather than how they “should” see.”
Cathy Humphreys.

Boaler, J., & Humphreys, C. (2005). Connecting mathematical ideas: middle school video cases to support teaching and learning. Portsmouth, NH: Heinemann.

Introduce a new fluidity into your classroom. Check out more about dot cards in this piece by Math Coach’s Corner.

If you have good number sense activities resources, I’d be happy to link to them here!