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!

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.

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.

Everything I learned about teaching I learned from teaching students with special needs…

I came about a teaching career in a round about way.  As a math major at a small liberal arts college in Southern Maryland, I earned money by tutoring local middle and high school students.  I realized that I really enjoyed this experience and decided to pursue a teaching degree at the graduate level.

While taking classes towards a graduate degree for secondary math instruction, I fell into a job at a non-public special education facility.  I worked under a conditional certificate with some very wonderful special educators.  I ended up becoming THE math teacher for the entire high school program (the Harbour School is a k – 12 facility).  This meant that I taught all of the students ALL of their high school math.

I believe that I wouldn’t be the educator that I am today without this experience.

At the Harbour School I worked with students with all sorts of ABILITIES.  This school was their SAFE HARBOUR.  The students came to the school because it was found that their home school couldn’t meet their educational needs.

Teaching math at this school was an exercise in flexibility.  I had to really listen to the students to understand their understanding.  I do believe that math was a mystery for most.  I’ve always used this analogy for teaching math…

If I couldn’t get through the front door, I found a way in through the window, the garage, or around the back of the house.

By having to ask the right questions and make the right connections, this experience helped me to understand the math I was teaching at a deeper level.

At the Harbour School I also learned:

• acceptance
• tolerance
• patience
• community
• awareness
• perseverance
• resilience

If you are wondering about how to meet the needs of the students in your classroom with learning challenges–my advice is to listen to them.  ...Then figure out how to get into the house…

My favorite talk about listening from @maxmathforum…

Things that make you go hmmmm…..

Well….this is an interesting place to be.  I’m not sure about how I feel about putting my thoughts out there for the world, but I do love the fact that this community exists.

I came here because I am just as intensely obsessed with education as all of these people that have already begun their journey in the reflective world of blogging.  I like to say that education is not only my “job” but my hobby.  Most of my friends and colleagues (even occasionally my husband) do not understand that.

I enjoy ALL things education…

• Special Education
• Math Education
• Professional Development about Education
• Curriculum Writing
• Universal Design for Learning
• The list can go on…

But…first and foremost, I just plain old LOVE MATH.  I’ve loved math my entire life.  I can’t remember having a bad teacher…only teachers that made me love math more.

I even majored in math for my undergraduate degree and didn’t work towards education until my graduate degree.

Interestingly…my very first teaching job was at a place called The Harbour School (this is a non-public special education facility for students K – 21).

This was the best professional development that I have ever experienced…teaching math to ALL of the students that entered the high school portion of the program–students with Autism, Asperger’s Syndrome, Learning Disabilities, Cerebral Palsy, Speech and Language difficulties, etc.  I taught students that were community based…but also students that worked towards your typical advanced math classes.

So, I am certified in both secondary math and special education.  But, now I work in a public middle school in southern Maryland.  This school pulls from Title I elementary schools and serves underprivileged students.  I work as an Instructional Resource Teacher in my building…and work daily with teachers and students.

So, all of these experiences have created the lens through which I view all things math education.  I love to absorb it ALL!