Tag: Math

Three for the Week

Saturdays are my time to reflect on what I read, heard, and discussed throughout the week. It is my “exit ticket” for the last seven days. So here are three ideas that made me think this week. (They are not in any particular order.)

1 – The overall health of teachers is so important. We work in a very emotionally, physically, and mentally demanding field. It is critical that we take care of our bodies and minds in order to better take care of our students. So I was doing some digging for the best exercises and workouts. I love running but know that the impact is not great for your body. I found an interesting article which outlines exercises to do at every age group. For example, they recommend boot camp in your 20’s, high-intensity interval training in your 30’s, and running in your 40’s. Guess I don’t have to stop running quite yet!

2 – An article from The University of Virginia really caught my attention. It is a Q&A with NCTM President and UVA professor Dr. Samuel Braley Gray. He outlines what effective math teaching looks like in our schools, touches on some inequities in math education, and even talks about why children should use their fingers in math. (That last point alone got me wondering why we would encourage students to use printed ten frames, but discourage them from using their fingers – which are ten frames.) What really struck me was what Dr. Gray said about the effective ways to teach math. “These ideas are a shift from focusing on memorization. Mathematics is more than getting an answer quickly. Effective mathematics teaching engages students in explaining why their answers make sense and why the strategy they used is appropriate.” Well said, Dr. Gray!

3 – Last week I chose something lighthearted as my third point for the week. I’ll keep that trend going this week. Two ridiculously cute boys show up to a Canadian airport to pick up their grandmother. The boys decide to play a trick on grandma and dress up in full T-Rex costumes. Grandma, as grandmas always seem to do, was one step ahead of the boys. She appeared wearing… a full T-Rex costume of her own. The video is well worth the 2:29 of your time and will definitely put a smile on your face.

Start With the Answer

My students have been working hard this year to improve as problem solvers. We consistently take time to talk about what makes effective problems solvers and practice the skill of problem solving. Last week I talked about the Three Reads strategy we use in our classroom.

Another strategy I like is giving students the answer and having them create the problem. Recently I gave my students three prompts:

  1. Write a division problem where the quotient would be 6r3.
  2. Write a division problem where the answer would be 5 1/3.
  3. Write a division problem where the quotient would be 7r3 but a mathematician would add one to the quotient to report the answer as 8.

Students had the option to brainstorm with a partner before writing. Each student had to create their own problem. My goal was to get students to think about the structure of math problems – narrative and expository text combined.

One of the first things I noticed was students struggled to create the complex narrative structure that exists in most fourth grade word problems. This made me wonder if one of the obstacles for young mathematicians is they struggle with the narrative component of a word problem.

Most of the problems students initially wrote for #2 were similar to this: There are are 16 cookies for 3 kids. How many cookies does each person get? These problems lacked the character names and any extraneous information that often appears in rigorous problems. They also lacked the need for multi-step problem solving.

The conversations with students after they wrote their problems was wonderful. I had some of the students go into their math books and look at similar division word problems. This helped them better understand the structure. Other students practiced writing some problems with me. In both cases, we talked about the “story” at the beginning of problems with characters and a scenario which creates the necessity to solve a math equation. One student actually said, “Ahhh!” The lightbulb went off.

This exercise made me realize the value of students looking at math problems to analyze the structure of how a problem is put together instead of trying to solve it. All of the problems we revisited had already been completed, so the student could focus on how the problem was written.

It’s another tool in the problem solving toolbox which I hope will continue to grow for me and my students.

Problem Solving

I was walking into work one day and a colleague literally came running across the parking lot. She was frustrated and asked, “What in the world is going on with these math word problems?” I looked at her waiting for more detail and trying to not drop my smoothie. “These aren’t math problems,” she said. “They’re reading problems.” She had no idea how correct she was.

Word problems are as much reading problems as they are math problems. One of the challenges with younger mathematicians is getting them to slow down and read problems multiple times to understand the complex structure.

The higher-level thinking problems students are asked to solve go far beyond basic computation (5 x 5 = 25). Students have to read a complex problem, understand the context, know based on that context that they have to multiply 5 x 5, then multiply 5 x 5  to get 25, and finally write an answer of 25 with the correct label. Talk about challenging!

Math word problems are especially challenging for readers because the structure is unlike most of what we teach in reading. The authors of Routines for Reasoning state, “Reading in math – especially reading a math word problem – is different from reading in other subject areas… word problems combine both narrative and expository text… Therefore, word problems must be read several times with a different focus each time…” 

In reading, we generally teach narrative OR expository text. In math, students often encounter both types swirled into one problem.

There is another challenge with the structure of math word problems. Students learn in reading that the main idea is generally at the beginning of a paragraph or section of text. Think of the main idea of a math problem as the question being asked. The main idea – the question – is at the end of the paragraph or section of text.

So how do we help our young mathematicians become effective problem solvers? Routines for Reasoning shares a strategy called the Three Reads. This approach requires mathematicians to read a word problem multiple times and sets a purpose for each read. 

  1. Three Reads
  2. Read 1: Understanding the Context – Focuses on the general idea of what the problem is about. 
  3. Read 2: Interpreting the Question – Determine the question or questions being asked in the problem. 
  4. Read 3: Identifying Important Information – Look for the important information or words in the problem. 

Let’s say students are solving this 4th grade released problem from the Pennsylvania state assessment, known as the PSSA:

David started his coin collection with 14 coins. He added 3 coins to his collection at the end of each month for 5 months. How many coins were in David’s collection at the end of the 5 months?

  • Three Reads
  • First read: David is collecting coins (Don’t worry about any expository text right now. Focus on the narrative. Save the numbers for later.)
  • Second read: How many coins were in David’s collection at the end of the 5 months? 
  • Third read: Collection started with 14 coins; added 3 coins each month; 5 months total

This is a great technique to begin creating effective problem solvers. First, I create an anchor chart, which is pictured, for my students. The anchor chart is displayed in the classroom throughout the year. Next, I model the Three Reads and think aloud my thoughts as a problem solver. This cannot be a once and done process. Students need to see and hear this process multiple times throughout the year with a variety of problems.

I’d love to say that creates problem solvers over night, but it takes time. It takes repetition. It must be persistence and grit. Rome wasn’t built in a day, and neither are problem solvers.