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!

CCSS: Key Tasks and Scientific Method

Common Core State Standards (CCSS) are lists of things kids are expected to learn in each subject during each school year. They are not the actual lessons. Your child’s teacher and school system will decide what lessons to teach and how to teach them.

Research shows that one of the best ways to teach children is to give them tasks and have them work through the solutions to the tasks. The child is given directions as needed, or small lessons on specific skills, but the child is allowed to figure out what skills they need, or what they need to know, to accomplish or solve the task. These are called Key Tasks.

To be able to participate in the process of Key Tasks, a child needs to have a way to approach and organize the information. In mathematics, this process is going to feel very different than the existing process of “show the child the problem, work the problem with the child, and then let the child practice problems. Without setting up the structure of the process first, students may feel that the teacher has abandoned them, is not really teaching them anything, or worse. It takes time to transfer responsibility for problem solving when a child has never been asked to shoulder the responsibility (except for remembering how to do some practice problems, or work a formula – not really learning, just memorizing and regurgitating information).

The process of approaching and organizing the information is very similar to the scientific method. Any teacher or parent can help a child learn how to approach these new key tasks by teaching them a “mathematical thinking” process. The steps in the process are simple ones:  Read and think about the problem, draw the problem or restate the problem, discuss the ideas about the problem with others or use resources (group discussion), estimate the answer, “mathematize” the problem (a formula or equation), try/refine/rethink, see if the answer makes sense. These steps may need to be used more than once throughout the process. As you can see, the actual answer is only a tiny part of the process.

For teachers, here is a brief lesson for teaching students how to use mathematical thinking in the class. For parents, you can help your child use this pattern with their math homework:

I love the idea of math as a thinking process. Instead of just giving students the list of actions, , I would approach this the same way I like to approach setting up classroom norms. I would start with a sample problem and just have the students think about it. I would tell them not to try and solve it yet. Then I would have them verbalize their thoughts in small group and then whole group- we would write the thoughts/assumptions on a big sheet of paper titled ‘Thinking’ and tape it to the board. I would facilitate with clarifying and summarizing questions.

The next step would involve visualizing. Students would be asked to draw a picture to illustrate what they saw happening in the problem. Again in small groups, they would create a picture/illustration, titling the poster ‘visualize’. The posters would go up around the room. At this point, I would ask all students to move around the room ( in groups) and visit the posters to see if the illustrations made sense, and if they suggested any mathematical way to look at the problem.

Returning to their seats, each group would create another poster titled ‘mathematics’ showing the calculations / solutions that they came up with. Those would go up on the wall. Each student would then be instructed to visit each mathematics poster and decide if those mathematics/answers made sense in light of the problems. This might be a good place for the students to use sticky notes and place comments or questions onto the posters.

The groups would then go back to their posters, check out the comments (as would I, so that I could come up with more questions) and we would come back to a whole class discussion to examine the various mathematics and reasonableness of answers. I might put up an empty poster titled ‘revisions’ so that students could add ways that they will need to revise their own thinking to solve this and future problems.

The summing up of the lesson will not be right or wrong answers, but a summary of the process itself. The students will be asked to discuss with one other person what steps they went through to solve the problem, and then write a brief paragraph in their math journal about what steps they took to solve the problem.

The final poster, and the one that will remain on the wall for future work, will be the steps the students noticed were common to the process- maybe have each group list a step (different colors, handwriting?). This will give them a roadmap to solving future problems- and as the teacher, I will give them the time to use the steps as we move through the learning process.

Parents can help by giving your children the time to verbalize problems to you or draw what they think the problem represents. You don’t always have to know how to do the math to help your child think through the process. Even if they don’t come up with a “right” answer, or maybe all they have come up with is questions, the thought process is going to give them a way to get in on the next conversation in class.