So in this series about assessing math proficiency, I’ve already asked you to consider the number of problems that you are giving students when you are assessing them. Today, I would like you to consider the type of math problems you are giving them.

I’m Christina Tondevold, the Recovering Traditionalist, and I hope you will stick around as we look at assessing math proficiency. Does the type of math problems you use matter? In our quest to build our math mind, so we can build the math minds of our students.

Watch the video or read the transcript below:

Here are links to products/activities mentioned in this vlog. (Some may be affiliate links which just means that if you do purchase using my link, the company you purchased from sends me some money. Find more info HERE about that.)

• Achieve The Core – Rigor in Math
• Teaching the Math Practice Standards video

Other videos in the series:

• Assessing Math Proficiency: How many math problems should kids get?
• Assessing Math Proficiency: Standards Based Grading

Now there’s kind of two different camps on this. There are the types of problems that are story problems, really contextual-based problems. Then there are problems that are just bare problems, when you just give them an equation to solve; just a problem that needs solved with no story problem around it. 

The Use of Contextual Problems

One of the first issues that comes up that I hear a lot, is if we use contextual, story-based problems, does reading get in the way of being proficient with math? That is a huge concern. But I think that contextual problems are a huge part of mathematics. Now it isn’t just me, our math practice standards say the same thing. 

Math Practice #2 talks about having students decontextualize and contextualize mathematics. What that basically means is that for students to really be seen as proficient, they need to be able to do these math practice standards. It’s how they’re engaging with mathematics. To be able to decontextualize means that if you are given a contextual problem, a story problem, you are able to pull the mathematics out of that contextual situation. The reverse of that, which is being able to contextualize a situation, is whether or not kids can take a bare problem and create a story problem out of it. 

We see this far too many times where kids can solve 6 x 7 but then they can’t give you an actual situation that means 6 x 7. 

Do they really understand what 6 x 7 means? If so, they should be able to create a story problem or what the math practice standard calls contextualize that situation. So to me, it is a big part of being mathematically proficient, being able to solve problems when they are in a context. 

The Use of Bare Problems

The other piece of that, a lot of people will say, “Well, what about bare problems? “Kids need bare problems, right?” And when I was in school, that was it. Like that’s all that our problems were about when we were doing math assessments. 

Everything was just bare problems; just solve 6 x 7. And most of the time, it had multiple choice answers down below it, for us to choose from, which most assessments still do. And it sure made for an easy way to grade and to see whether or not kids got them right or not. 

But to me, that is not fully being proficient. As I talked about before, just because they can solve 6 x 7, does not mean they truly understand the mathematics behind it. 

The Three-Legged Stool

So to me, and again, it’s not just me, kids need both. What I prefer is this idea from Achieve the Core. They talk about how kids do need Rigor in mathematics. But rigor does not mean harder. Rigor in mathematics is really this three-legged stool.

They need exposure to problems that are assessing their conceptual understanding, their procedural fluency and application-type problems. If students do not do all 3 of these, and if we are not assessing all 3 of these, it’s not giving us a clear picture of their proficiency. 

So I will link to Achieve the Core’s website so that you can go and learn more, but here’s just a really quick down and dirty example of the three different types of problems and ways you could assess. 

Let’s come back to the 6 x 7. I go to 6 x 7 because that is one of the hardest multiplication problems. It’s the one that even adults tend to get incorrectly or take longer when they are trying to do an assessment. So 6 x 7 for procedural fluency, that is usually just, “What is 6 x 7? “Solve 6 x 7.” 

If I really wanted to assess a child’s conceptual understanding, a problem might be something like, “Sue says that to figure out 6 x 7, “she could do 5 x 7 “and then add 1 more 7. Is she correct? Why or why not?” So conceptual understanding is really problems that dig to the root of the mathematical understanding. 

One of the big ideas around multiplication is helping kids to see groups of and there’s a lot of controversy around this. But when kids are first starting out, they need to see it as groups of.  6 x 7 is 6 groups of 7, right? And they find out, “5 groups of 7, could that help me figure out 6 groups of 7?” That connection is super huge for building a solid foundation of what multiplication is and how we can solve multiplication problems.  So conceptual understanding type problems really need to build and assess that foundational piece. 

Then application problems are the story problems that I talked about in the beginning of this video. If we gave it to them in a context, would they be able to solve that problem?  It may not be a group of them. Maybe in this one, I might want to do something where it’s kind of like arrays. So I might have something like, “Sue has 6 books on every shelf. “She has 7 shelves in her bookcase. “How many books does she have on her bookcase?” And that image, to me, when I’m imagining it, it brings up kind of an array in my mind, but for some kids, it might not. 

So the application-type problem is usually a contextual problem to see whether or not kids can solve this in a contextual, or story problem, format. So we need to assess all 3 of these areas. Kids need exposure to all 3 and we do need to assess using all 3 of these areas to really see whether or not students are proficient. 

If you’re only assessing procedural fluency, you are only getting 1 of those legs. If you’re only giving contextual situations, it’s only 1 of those legs. We need to have kids working in all 3, thus we need to assess all 3 areas. 

All right, I hope that this video has helped you build your math mind, so that you can go build the math minds of your students. Have a great day.