# 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):

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 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!

# 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!

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!!!

# Un-quiz

My first year out of the classroom was 2010.  I was talking to a colleague today about a type of assessment I used to do in my classroom.  I called it an un-quiz.

This is a variation of a typical multiple choice quiz, except that I asked for a WRONG answer. Along with the wrong answer choice, the students had to give a reason why it was a wrong answer.  (Be sure that your wrong answer choices provide valuable information!) This allowed me to see how they reasoned about wrong solutions, language they used, AND the students had multiple means of showing their knowledge.

I haven’t used this strategy in a while…if you do something similar or try it for the first time, I’d be interested in hearing about what you think.