Remember back before everything auto-saved all the time, how many times were you working on something, finally finished up it, and shut down your computer. Immediately after, you would realize that you didn’t save it and all your work was gone. If only we could’ve remembered before we shut everything down, instead of after. Does the fact that we just lost all of our work because of some small mistake mean that we were unable to do the work?
In math assessment, we treat small mistakes as if you simply don’t have the ability. One small error or typo that happens in the first step carries to the second, and to the third… We are judged as if we are unable to do the problem because of a silly mistake we made in step one (or step two, or three). My question is — is that really the case?
In general, I do think it’s important to acknowledge that an answer is incorrect. But it’s also important to think about what specifically the student is getting wrong. Let’s check out some examples and dive a little deeper.
A common error here is to take the difference between the x and y coordinates and get a change in 5 in x and a change in -10 for y.
Wait, but y changed -20!
Yep, but a missed negative and all of a sudden we have a change of -10, not -20.
That causes us to get a slope of -2 instead of -4. Now, when we go to solve for the y-intercept, we get -5 – 3 * -2 = -5 – -6 = 1. y = -2x + 1.
This correctly solves for the y-intercept, but gets the slope incorrect due to a small mistake. In this case, we would normally say that this student doesn’t know how to get the slope or y-intercept… But that just isn’t true!
The student is then told that they did it wrong, sometimes causing them to start doing things a different way (not good in this case!). Zooming out, we might look at the class and think that we’ve got more problems than we actually do. It starts to look like none of our students have learned anything and we’ve got to start over. Not taking into account that our students have cascading errors makes us think that we have bigger problems than we do and our own problems start to cascade!
As we all know, this is no easy problem to solve! At Math ANEX, we’ve built some systems to automatically find the slope in a students equation and then calculate the possible y-intercepts based on that, in order to get a better understanding of students’ abilities. We found that in a group of students where only 18% got the right y-intercept, when using this framing, 37% got the correct y-intercept based on their slope!
Let’s look at another example.
Normally, we’d like to see it solved in the following way. “The change between 4 pots and 2 pots is 32-24 = 8cm, which means that for every 2 pots stacked on top, 8cm is added. 8cm / 2 = 4cm which is how many cm is added per pot. Take that way from a stack of 2 pots (24-4 = 20cm). An individual pot is 20cm tall.”
Over 500 students got 18cm as their answer. How do you think that could’ve happened? Do they not know how solve the problem? Let’s look at some student examples.
I first wanted to figure out the height of one pot, but it is more complicated because they are stacked on top of each other.
2. 28-4= 24
Two pots stacked from the left side equals 24 and the two pots on the right also equal 24.
3. 32-24=12 and there are two more pots stacked up on the left so 12/2=6.
Each pot adds 6 centimeters.Student Example
(% in this explanation means divided by.) I did 32-24=12. Then I saw that the 4 pots has 2 more pots than the 2 pots so i did 12%2=6. So then I did 24-6=18.Student Example
Well I subtracted 32-24=12 then did 12/2 for the other 2 pots that were added then just subtracted 24 and 6 then do 18.Student Example
It looks like these students all have a great idea of how to solve the problem! However, they all make a similar error in the first step. 32-24 = 8, not 12.
But you can see how they’d get 12… 3-2 = 1 and 4-2 = 2. Subtracting by the biggest digit gives you an incorrect answer when subtracting, and then that cascades.
We may be inclined to assume that students who arrived at an answer of 18 aren’t showcasing an understanding of linear relationships. The reality is, they all just made a simple subtraction error. If we had made the assumption that they aren’t showing an understanding of linear relationships, we may think we need to re-teach the topic. This would have used up valuable instruction time and in the end still not solved the issue.
Again, solving this is no easy feat. How can you figure out every missed computation, what answer it resolves to, and then score all of those? Great question. In our case, technology can be a powerful tool. We have programmed our computers to do math both correctly, and in commonly incorrect ways, in order to make sure we can identify what students did!
While we are on the topic of students solving subtraction problems by subtracting the largest digits, check out how we worked with a teacher to better understand how we could effectively help her students with this same issue. We found that almost half of her students were solving subtraction problems using that method. Once she realized that, she was able to really address the issue head on. She utilized the fact that her students were using the same method over and over again to adjust their understanding of place value and use a new method. This led to an improvement of 55% of her students getting it right to almost 80%! Finding and understanding these particular issues that affect our students makes a huge difference in our ability to provide the right supports, to the right students, at the right time.