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!

Making Math Formulas Tactile


The credit for this unusual idea goes to Math Giraffe and the Tactile Formula Project . Here’s how it played out in my 9th grade Algebra I classroom.

The challenge? Making Exponential Formulas out of concrete images

I believe the lesson is applicable with any formula and any level of student. Try it yourself… how well do you understand the formulas you teach?

The Lesson

I started by telling the students about ‘this cool idea I saw on the internet’.

The idea is to create formulas out of three dimensional objects, like candy or pennies, or legos, or cubes, or squares that suggest or seem to define the elements in a formula. I didn’t have the luxury of time to collect objects, so I told my students to use drawings of items that they planned to use to make the elements.

To get the creative juices flowing, I shared the images from the Math Giraffe blogpost, and started the conversation about what I saw in the images. The students quickly started noticing and interpreting what they saw in the images, and questioning why students used some of the images in the examples. Some students began interjecting their own ideas for images they would have used. This really primed the pump, so to speak, for the formulas we were going to interpret.

Images from Brigid’s Math Giraffe blog post.

Brigid, of Math Giraffe, used the classic slope-intercept formula,

y = mx + b

well known to my students. The task for my students was to do this with the multiple versions of the exponential formula:

I wrote all of the different forms of exponential formulas on the board, and let each student choose the formula they wanted to use for their visual interpretation. What followed was a deeply enriching -and eye-opening – mathematical and conceptual conversation among all of my students!

Note: We used exponential formulas, but any formula would work with this activity.

The challenge for my students was using pictures to relay the action contained in the formula elements vs just using a picture that looked like an item in the formula. An example of this was incorrectly using bananas for parentheses, because, as the student explained, “they looked like parentheses.”

The items here don’t exactly illustrate the action of the elements.

This student was just a little shaky on choosing an item that revealed the inclusive nature of the parentheses. Another student didn’t use any pictures at all. He wrote out his formula using a color for each letter, basically color-coding the identity of each element in the formula with an elaborate system for understanding each part:

Did they miss the point? Absolutely not! Both were much more able to explain when and how to use the formula after this task.

This task got my students thinking more deeply about the formula itself, than just trying to memorize when to use it!

Interestingly, the exponential growth element was mostly identified as time, with one child drawing a clock face whose hands illustrated the exponent variable.

Y is “get out” because it is output. B is the rate, so percentage signs. The crossed watches were used to show the x as time.
Cars, roads, and streets were the subject of this formula.

This student uses bacteria below the formula to show starting amount (3), the growth multiplier with a question mark, and the x was shown as time. It’s interesting that he used a calculator with an ‘answer’ to illustrate the y variable!
Dinner was the subject here. The ingredients to make the dinner were multiplied. Heat from the burner was the common ratio. X was the amount of time the food was cooked!

One of my favorites: chocolate and peanut butter to make candy, with the exponent increasing the amount of candy produced!

I noticed that after this exercise, students were beginning to talk about the formulas by identifying individual elements and their actions. When explaining why representational items were chosen, students were recalling examples from word and story problems from our practice work.

Research (source articles noted below) shows the value of connecting the abstract ideas of math to concrete items and has been proven to help with retaining the material. In this lesson, connecting physical objects (or in this case, drawing pictures of physical objects) allowed my students to make clear connections to what the formula was representing. More than that, this exercise functioned as a valuable informal assessment for me about what my students really understood about the elements and actions of the formulas we were learning. I look forward to using this task again!

Research that supports connecting concrete and abstract ideas: Allsopp (1999); Baroody (1987); Butler, Miller, Crehan, Babbit, & Pierce (2003); Harris, Miller, & Mercer (1993);  Kennedy and Tips (1998); Mercer, Jordan, & Miller (1996); Mercer and Mercer (2005); Miller, Butler, & Lee (1998); Miller and Mercer, 1995; Miller, Mercer, & Dillon (1992); Peterson, Mercer, & O’Shea. (1988); Van De Walle (2005); Witzel, Mercer, & Miller (2003).

How do you tell parents to expect less from their children?

A recent article on the upcoming Common Core assessments that will be administered in 39 states this spring (2015) predicts that student scores will suffer. The writer cites the experiences of Kentucky and New York, where the new testing took place this past Fall (2014), two very different places and two very different results. The difference? Continue reading “How do you tell parents to expect less from their children?”