Visual Patterns and Missing Figures

Reflections in the Plane

The past few years I’ve used Fawn Nguyen’s Visual Patterns site as the structure for building an 8th graders understanding of linear relationships. Students GET patterns–even the most struggling learners can identify what’s happening in a linear pattern and complete a table of data points. What I’ve done recently is used the patterns to build an understanding of finding the constant rate of change between two points. Here is an example of what the students are presented with… Instead of giving the students something like this:   I give them this:   The students work on their dry erase board to create an input-output table of data points beginning with x is 0.  We’ve spent A LOT of time creating our own tables that it has really started to become second nature.  I didn’t tell them how to figure out the missing figure numbers…they just figured it out on their own…

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Thinking in the Concrete~Manipulatives in Math Class

Amie Albrecht retweeted this the other day over a year ago (I just came back to finish this post):

#MathsConf7
"It is not childish to think in concrete terms" @mikeaskew26
Don't ditch the concrete and pictorial pic.twitter.com/xAp6vPZ7QJ

— Ben Ward (@MrBenWard) June 25, 2016

It reminded of me when I used my son to try out the first part of this NRich problem:

I’ve used this problem with middle school students before…but I wanted to use it in a math for elementary teachers undergraduate class, so I needed to see how an elementary student would think it through (he’s in 5th grade).

He attempted some mental thinking about the problem at first. But, I could tell he was going to get frustrated. I found something that could work as counters for him and asked him to represent the problem. He argued with me about it and finally said “only babies use these.”

I pushed him to use the counters…and he figured the problem out quite simply.

Using my son as guinea pig for the 1st part of this @nrichmaths task-https://t.co/krcdxXLj3l w/ manipulatives #tmwyk pic.twitter.com/SpYgOf0doU

— Bridget Dunbar (@BridgetDunbar) October 3, 2015

his prediction for the task was the 3rd child would get the most. after the 2nd child he asked "is this a trick question?" #noticingpatterns

— Bridget Dunbar (@BridgetDunbar) October 3, 2015

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Using manipulatives makes problems so interesting to think about. But, I just asked him about that day and he agreed that the tool made the problem easier, but he didn’t like using them.

We need to use more manipulatives in math class!

Desmos, WODB, and Open Up

I’m working with teachers as they implement the Illustrative Mathematics Middle School Curriculum released by Open Up Resources.

The first unit in the 8th grade is on Rigid Transformations and Congruence.

This weekend I worked to edit a Desmos Polygraph activity that was originally authored by Mike Waechter. You can check out my version of the activity here. Please provide feedback if you think of ways to refine it.

I also thought it might be nice when using the Polygraph activity to open the class period with a Which One Doesn’t Belong task. This way the teacher may be able to highlight important language to use during the Polygraph activity.

If you create activities/resources that work with the Illustrative Mathematics Curriculum OR just want to see what others have created, access the link in this tweet from @k8nowak.

Hey! Did you make something cool for @IllustrateMath 6–8 Math? https://t.co/tHZqi4ri2g #MTBoS Please share it here! https://t.co/oDDOBwXDKP

— Kate Nowak (@k8nowak) August 30, 2017

Reading Aloud in Math Class

So I’ve been informally experimenting with the effect of reading aloud in math class.

Many years ago, I noticed that when a student couldn’t get started on a task on their own, they’d raise their hand and claim “I don’t know what to do.”  I would ask, “Well, what did the problem say?”  The student would then answer, “I don’t know.”  My next step would then be to read the problem aloud and ask “What do you think you’re supposed to do?”  The student would respond to this question…and most often with the correct response.

I didn’t need to ask the students any questions related to the math at hand.  They just needed to hear the problem aloud.

I started to pay attention to this back and forth that I would have with countless numbers of students.  And then began to explore the question-what if they read aloud to themselves???

An eight grade honors level student came to find me because she couldn’t figure out a problem she had on an assignment.  I said read the problem.  She said “I already did.”  I asked her to read it aloud to me.  I could see the lightbulb go off when she finished and she asked “Am I supposed to _______?” And she was correct!

Two nights ago, my fourth grader that was accepted into the STEM program in our district, was working on an online assignment in the other room.  He came out to my husband and I and asked for help because he was stuck.  He sat down next to my husband and began reading the problem out loud to him.  As soon as he finished, he said, “Oh, never mind! I know what to do.”

I’ve noticed that I will often put my fingers on my ears and read-aloud in a whisper if I’m trying to double check the words that I’ve written.  It’s helpful to hear myself.  How can we explore this more with students?  How can we incorporate this in our classrooms?

I did a quick search attempting to find research on this topic.  I noted this article about reading aloud for English language learners.  But what was interesting was this:

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I’m interested in researching this further and would definitely love to know if anyone has had similar experiences with their students.

Barbie Bungee Implementation

I think that there are a million and one ways you can do almost anything.  Dan Meyer tweeted this out the other day…

Hey: three versions of Barbie Bungee. What differences do you notice? Which is best and what do you mean by "best"? https://t.co/lGMC1aXWl5

— Dan Meyer (@ddmeyer) June 13, 2016

I was quick to comment.

I think it’s important to note that I work most closely with 8th grade (on grade level) students.

After seeing the subsequent comments…I wanted to add that IMPLEMENTATION can be more than just here’s your worksheet, “get on with it.”

I like to lead up with this…

Can you guess what their answer is???

Then I show them this…

Of course this changes their answer.

Then I move onto this…

And I have students come up to put a point…usually around a y-value of about 10 or so.

Then, I click to reveal additional points.

The point is…I make the point that collecting multiple pieces of data helps to make better predictions.  I also ask the question about what mathematical models can we create to help make predictions in math.

Their answers:

• Graphs
• Tables
• Equations

Just what I want.

Then I show one of the many videos that you can find on YouTube.

The last thing I do is pass out a worksheet.

 

 

Elementary Observations from a Secondary Math Teacher

Here is the interaction that I had with my 5th grade son yesterday:

Me: “What does 6 ÷ 2 mean?”

Son: “Six split into two equal groups.”

Me: “What else could 6 ÷ 2 mean?”

Son: [crickets]

Me: “…how many groups of two are in six… so…what does 6 ÷ ½ mean?”

Son: “…how many groups of one-half are in six…12.”

Me: “What is 6 ÷ ¼?”

Son: “…how many groups of one-fourth are in six…24.  I remember this every time you remind me…but I always forget.”

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My son is very strong with fractions, and he’ll go straight to invert and multiply if I let him.  I often engage him in this same conversation…working on the understanding of dividing a whole number by a unit fraction because I know how difficult division of fractions can be for students.

My hypothesis is that it has to do with the two interpretations of division:

(a) How many groups?                      and                     (b) How many in each group?

***At this point…I will hope that elementary people will tell me I’m wrong if I’m wrong…

I began teaching Math for Teachers at the local college (St. Mary’s College of Maryland~my alma mater) two years ago.  I use Sybilla Beckmann’s Math for Elementary Teachers fourth edition for this course.

The first activity for the division unit is this:

Write a simple word problem and make a math drawing that you could use to help children understand what 10 ÷ 2 means.

Each year I’ve taught this course EVERY single student wrote a how many in each group problem.  This is called the sharing model of division.

I think it’s less natural for students to consider the how many groups (or measurement model) interpretation of division.  But it seems that this is the model that makes the most sense for the division of fractions.

I think this less familiar interpretation of division can also impact a student’s ability to be successful with long division.

A student needs to be able to think, “How many groups of 30 are in 1429??”

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I think it’s important to be purposeful with the language we’re using with students, and how we are exposing them to different interpretations of division specifically.  Secondary math teachers can learn A LOT by digging into the elementary material.

Learning from a 5th Grade Math Team

When my oldest child entered kindergarten I wanted a way to volunteer my time at the school, so I began coaching the 5th grade math team.  I saw this as an opportunity to better understand the math that elementary students bring with them to middle school. The elementary school that my children attend is a feeder school to my middle school.

I used the set of resources provided by the school system to train my mathletes that year. One problem I tasked them with was titled Kicking Tees below:

I didn’t attempt this problem before giving it to the students that day.  I watched as they solved it…easily handling #1, skipping #2, and then answering #3.  In my mind I thought, “Wow! I’d create an equation to solve #2…how would a 5th grader figure this out?”

I can’t be too dissimilar from other secondary certified teachers, where an algebraic approach is the first that comes to mind.  It took me a few moments to think about using a table, or simply guessing and testing given the boundaries offered by the answers to #1 and #3.

This was a revelation for me though…That students might have skills to approach problems in which I’d use a more sophisticated method.

I was curious about what a 7th grade Algebra class in the midst of learning about systems of equations might do with this problem.

They did the EXACT same thing that the 5th graders did~they skipped #2!

This began my thinking about the intersection between the teaching of content, skill, and strategy.  And the connection between elementary math and algebraic thinking.

I used this problem solving experiment to talk with the 7th grade students about math learning and about the connection between arithmetic and algebra.

For teachers, this highlights the importance of horizontal content knowledge…

“a kind of mathematical ‘peripheral vision’ needed in teaching, a view of the larger mathematical landscape that teaching requires”

“According to Ball and Bass (2009), HCK is an awareness of where and how the mathematics being taught fits into the structures and hierarchies of shared collective mathematical knowledge. This awareness serves both to engage students and to provide meaning to the present mathematical experience”

Further, this article reports

“The teachers in our study seemed to be more concerned about the mathematical content at the level they were teaching than the broader (more advanced) mathematical context—which can be referred to as the mathematical horizon”

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It’s become incredibly important for me in the work that I do with teachers, to help them see how the math that they are teaching fits into the learning the students do across a mathematical spectrum.

…in addition to helping teachers see the importance of this “horizontal content knowledge” as a way to create instruction that engages, inspires, and makes math a meaningful, connected body of work.

 

Mosvold, R., & Fauskanger, J. (n.d.). Teachers’ Beliefs about Mathematical Horizon Content Knowledge. Retrieved May 27, 2016, from http://www.cimt.plymouth.ac.uk/journal/mosvold2.pdf

 

We Need More Drawing in Math Class

I worked with a 6th grade student in our In School Intervention classroom today.  I do this every day.  A boy had been assigned a set of pages from a workbook and had begun the work on his own. From what he had already completed, I could tell that this student had a strength in math. He was unsure of what he had done, so he was looking for validation from me.  We looked at this next problem together:

He said, “so you gotta subtract right?” I said, “yes you do, but that’s if you’re looking for the answer. This is asking you to write an equation.”

He began to write y = x – 14.

I talked to him about variables, and how many unknowns were in this problem and he could tell that there was only one.

I asked him to draw a picture of the situation.  Here is what he drew.

HMMMM….(maybe they’ve recently worked on area and perimeter???)

I then asked him to reread the problem and prompted him to draw the house and the airport.

He drew the line between the house and the airport, and I had to do some questioning to get him to realize it represented the distance of 29 miles.  We reread the problem and I asked him about the 14 and how he would represent that on the drawing.

He thought to himself for a moment and then began drawing the tick marks.  I could tell that he was counting them.  He labeled the 14th tick mark and the 29th, and then circled the 14.

I drew the bracket and asked him what that would represent.  He said with a questioning tone, “the remaining distance?” Thinking for a minute, he then said, “oh I know…would it be 14 + x = 29?”

We did one more problem together that we represented with a drawing, and then I left him to be independent in the rest of his work.

He came over to show me one of his drawings and the equation he had written:

(In a previous part of the problem they were told there are 150 seats in coach seating.)

We high-fived!

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Teaching students how to create math drawings is a valuable sense-making tool that can be over looked in math class.  I know numerous reading language arts teachers that ask students to draw pictures to portray what’s going on in a story.  We need more of that in math class.

Noticing Subtleties…Reflection

So I wrote this post yesterday, and today we tried it out.

The purpose was to give students an opportunity to notice the subtleties in the language associated with three different mathematical scenarios they may need to represent.

We read each scenario out loud and then asked the students to compare and contrast each problem type with their table partner.

Some students began by discussing the similarities and differences of the contexts…

Others began by making a list of what they noticed on the back…You can see that this student paid attention to more of the mathy parts~understanding what was meant by a one-variable versus a two variable equation.

 

And then you’ll see below where students were able to make sense of each scenario and the math required.  However, the first student used an equation in two variables for the first scenario and created a table to find the solution.

We didn’t get to the whole class conversation part of this lesson…I want to talk about each problem type and how to recognize the differences.  Notice that two of the students above wrote the equation for the two variable scenario, but the third student created a table.  I think we need to talk about why that is.  Also, I think I may want to do three more scenarios that would produce equations in standard form, to see if they would recognize the differences then.

I definitely think that this was a useful exercise and would do it again.

One little shout out…one student pulled out their phone because they wanted to check out the equations for the third scenario on their Desmos app!!!

 

Noticing Subtleties

This just happened…

I walked into an 8th grade classroom finishing up writing and solving systems of equations given word problems.  They began this learning last week.

I overheard the teacher and a student having a discussion about drawing pictures to represent the problem versus trying to write the equations.  This told me that the weekend was too big of a gap from the examples that they had done last Thursday (they had a sub on Friday).

Here is a problem from the set that they were working on:

I briefly walked over to a different student and noticed that she only had one equation to represent this cars and motorcycles problem. She defined the variables correctly, but had written a single equation that mixed the information about the number of wheels and the total number of vehicles.  She hadn’t made sense of what her equation actually meant.

But, I thought that maybe it goes further than that.

I think the students need to see the difference between problems that require a one-variable equation, a two-variable equation, and a system of equations side by side. Here is what I came up with:

It’s nothing fancy or mind-blowing…I just want to see if this gives them a structure to look for when they are deciding how best to represent a given scenario.  I want them to see that there are two unknowns in the third case, and that they need to create an algebraic representation for Eli’s savings and for Lucas’ savings.

I’m wondering if we should ask them to notice and wonder???  Or simply ask…what is the same and what is different…

I’m also wondering if I should remove the headings from the table before we ask them to notice and wonder???

We’re going to try this tomorrow…so I’ll let you know how it goes.