Visualization in Math to Increase Comprehension

We are working on our reading strategies.

And it's been easier to implement in the math classroom than I initially had thought!

Upon our return to school after spring break,

We reviewed the quiz on functions we had taken before we all went our separate ways for a week.

To do this, the students first analyzed where their strengths and weaknesses were.

Using the chart below from BTC, 

Students made notations to find any patterns.

While I was focusing on the horizontal rows of content,

I was surprised when a student piped up that he had mainly checkmarks in the Basic column,

But that he was missing most of the Intermediate and Advanced problems.

Interesting.

For me. 

As his teacher.

While it's important to start at the concrete, basic level,

This was a red flag for me to continue to push them to a higher level of rigor and increase in fluency.

At this point, I started in on how they visualized certain vocabulary words and statements.

We started with the word "Function".

What did they see in their mind when they read the word "function".

After all of my classes, 

I had a much clearer picture of what they were seeing when they encountered the word, function.

We discussed each visualization after they drew them up on the board. 

We again pointed out similarities.

We asked for justifications.

We found mistakes or misconceptions to revise the picture and ultimately, the understanding.

We did this again for the concept of proportional relationships.  

I loved how in each case,

All three representations were included!

I was expecting the graphs,

But not the equations, with them being the most abstract. 

One of the best realizations of this activity,

Was noting how many students do not have a picture in their mind 

when they encounter these vocab words.

My hope is that when students shared their images,

And our discussion followed,

That they students were able to now attach a visual to the word. 

Selecting Tasks - Chapter 3 Rough Draft Math

We had a field trip to an area technical school today.

On the way, I read chapter 3 in the book, Rough Draft Math. 

It was all about selecting tasks that invite rough draft thinking.

• Tasks that ask students to explain their thinking
• Tasks that have more than one good answer
• Tasks that can be solved with more than one good strategy
• Tasks that ask students to predict and then test and revise problems

While on the tour, a student pointed out one of the whiteboards with all the math on it.

As we left the room, he said, "I think I recognize some of that math on there!"

I quickly ran back, took a photo, and forgot about it until we got back on the bus.

Sure enough!  There was math that my 8th graders would recognize!

I cropped the two pieces that I wanted to focus on,

And for the remaining two classes of the day,

We dove right in to seeing what we noticed or wondered about the problem.

We didn't have any context to go by,

But we could make connections to what we had learned,

And then make predictions to what we thought they might have been learning about.

We had just been learning about linear functions,

And the students immediately recognized that the students at WSU Tech had been solving for y.

My next question was, is y = x/5 a linear function?

Their choice of how to solve it was to create a table and pick the inputs.

They quickly found the pattern and we were able to graph it to confirm it was linear.

I then switched it around to y = 5/x. 

Just wondering outloud how that might change things.

This turned out to be a great intro to our lesson tomorrow, linear vs non-linear functions.

We then moved on to the next problem.  

I started this discussion the same way, "What do you notice?"

"Does this remind you of anything?"

"What do you think they are trying to solve?"

The discussion was great!

It started off slow.

"I see a square root sign."

"They changed the fractions to decimals."

"Why are they squaring the decimals?"

And then pretty soon, they connected it to the right triangle,

And the connection to Pythagorean's Theorem was made!

As a class, we then explored why it was written as it was.

This was most definitely not how we had been solving for missing side lengths!

What a great follow up discussion to a great tour!

What great examples of where the things we are learning will be used!

What a great review of grade level content!

Learning Retention

So this year I've gotten my students thinking more.

Plus, we are using rough draft thinking as a stepping stone in our learning,

But when I measure what my students are retaining from all our learning,

We are not where we need to be yet.

I shared this data with my students last week.

We noticed that the fall scores (blue bars) tended towards the upper percentages.

While the spring scores (red bars) tended towards the lower percentages.

When I asked students what this meant,

The looks on their faces said it all.

They weren't remembering what they were learning!

Today, we looked at just their retention during the Functions unit.

This is a large unit covering the five function standards, slope, and proportional relationships.

And we still have more to go...non-linear functions and bivariate data.

The data is showing the same slipping of holding on to this new learning.

So I started researching.

I'm including these websites to reference when I have time.

But will continue looking for more strategies to help my students.

1) https://mathsnoproblem.com/blog/teaching-practice/7-maths-mastery-strategies-to-retain-knowledge/

May need to focus on the revisiting part.  While test trainer revisits all the concepts in a spiral, there is no reteaching during this unless a student looks it up.

Would be interesting to compare McCurdy's students average with the non-McCurdy students to see if more exposure is helping.

2) https://mathsnoproblem.com/blog/teaching-practice/helping-learners-retain-knowledge/?utm_source=blog&utm_medium=blogintlink&utm_campaign=7strategiestohelplearnersretain

Focusing on the forgetting curve and the ideas suggested in this article.

Also, focusing on mastery. https://thirdspacelearning.com/blog/what-is-maths-mastery-teaching/

Choppy and Sloppy Talk - Chapter 2 Building a Culture

Two terms have emerged as go-to's in our math class this week.

When trying to explain our thinking on a problem,

We are noticing that when trying to make sense of something,

The words are not always there.

What comes out is choppy.

Sloppy.

Sloppy and choppy even happens to teachers that get stumped.

Which I think it a very valuable learning tool,

When students can see their teacher struggle to make sense of something.

On Monday, I had asked the students to justify if all three functions were indeed linear.

They could not simply say they were or they weren't.

They had to show evidence of how they knew on the vertical white boards.

On one board, it was brought to my attention, that a student was dabbling in both y- and x- intercepts!

Whoa.

Didn't see that one coming.

So I jumped on board, brought everyone's attention to his board, and started in.

Halfway through, as we worked through justifying the new thinking he brought to the discussion,

I started to notice something wasn't working as it should.

His "new" math idea wasn't jiving with all our work prior to this.

I started to stumble in my thinking,

My words coming out sloppy and choppy

As my brain was in overdrive to figure out where the mistake was.

Why wasn't this working.

The surprising thing was, 

I. Did. Not. Panic.

The first thing I noticed,

Was that the students were practically in my lap.

They had all pressed in to be closer to the board in my struggle.

They were asking clarifying questions.

Our roles had switched.

They weren't understanding the problem either,

But were trying to help ME understand it!!!

The active learning that was taking place was amazing!

Rough. Draft. Thinking.

Allowing ourselves to be vulnerable in the learning process.

A place where we just talk to learn.

I also noticed the benefits of rough draft thinking with one particular student last week.

This student, prior to last week, had been disengaged in the learning process.

Rarely turning to look at the board that was being discussed.

But for some reason, after I talked about this new book I was reading,

And that I was encouraged, because we were already doing what it was suggesting,

He seemed to find courage in talking through his learning.

In fact, every time I came around to his group,

He would engage me in what he was thinking,

How he was thinking.

After about two days of this new behavior,

I watched him walk out of math class with much more confidence than I had seen from him in the past.

My takeaway...

It's important to teach students the content.

But it's also important to teach students how they learn.

By instructing them on these strategies,

They can see for themselves, when implemented, how much these new found tools do help!

Tying in Vocabulary

Students reported that the review of math vocabulary is important to understanding the question.

This was brought to light on Monday when the words "initial value" was used in place of "y-intercept".

Today, while students created graphs, equations, and tables for three space ships in a race,

I had them write their statement using as many of the following vocabulary words that they could.

function

linear

slope

rate of change

y-intercept

initial value

proportional

Here is an example of one student's work.

"Spaceship #3 will win. The slope of the function is greater than the other linear proportional functions. It makes it proportional because the initial, or y-intercept, is 0."