Make every math homework a test
Make every math homework a test
Use the results of a large meta-analysis of study techniques to help your math-kid
Flashcards for Learning Multiplication Tables
The next obvious step in our kids’ math development was learning the multiplication tables.
We made flashcards for them with them and then tried to do it with them.
They hated doing them.
So we stopped.
Then, their homework caught up to their level, and they struggled with problems that required extensive multiplications (think a 5-digit number times a 4-digit number).
We brought out the flashcards again.
Which they continued to hate.
We worried that they were going to continue struggling.
Given it was an often fractious encounter, we decided that perhaps they would learn them after having to skip-count seventy billion times for each problem they were doing.
So we let it go.
They eventually learned their multiplication tables.
Helping students to better regulate their learning through the use of effective learning techniques
In 2013, John Dunlosky, Katherine A. Rawson, Elizabeth J. Marsh, Mitchell J. Nathan, and Daniel T. Willingham wrote a research article titled: “Improving Students’ Learning With Effective Learning Techniques: Promising Directions From Cognitive and Educational Psychology.”
You can find the paper here:
https://journals.sagepub.com/stoken/rbtfl/Z10jaVH/60XQM/full
The paper covers:
In this monograph, we discuss 10 learning techniques in detail and offer recommendations about their relative utility. We selected techniques that were expected to be relatively easy to use and hence could be adopted by many students. Also, some techniques (e.g., highlighting and rereading) were selected because students report relying heavily on them, which makes it especially important to examine how well they work. The techniques include elaborative interrogation, self-explanation, summarization, highlighting (or underlining), the keyword mnemonic, imagery use for text learning, rereading, practice testing, distributed practice, and interleaved practice.
The methodology of each technique is covered in the paper as follows:
In particular, each review is divided into the following sections:
General description of the technique and why it should work
How general are the effects of this technique?
Learning conditions
Student characteristics
Materials
Criterion tasks
Effects in representative educational contexts
Issues for implementation
Overall assessment
The review for each technique can be read independently of the others, and particular variables of interest can be easily compared across techniques.
The PDF is 58 pages (including the References).
The findings are roughly broken down as follows:
High Effectiveness
Distribute Practice - setting a practice schedule spread out over time.
Practice Testing - taking tests (self-administered work as well) over the material
Moderate Effectiveness
Elaborative Interrogation - creating a statement for why the material is true
Interleaved Practice - setting a schedule (during a single study session) for practicing different parts of the material or different problem types of the materials
Self-explanation - explaining how the new information connects to information that is already known and/or explaining steps in a multi-step problem
Low Effectiveness
Highlighting - coloring parts of the material while reading
Imagery use for text learning - creating mental images of text materials
Re-reading - doing multiple reads of the same material
Summarization - writing summaries (length didn’t matter) of material
The keyword mnemonic - inventing interactive mental imagery to group new material with already known (striking) visual things.
The research takeaway - testing and distributed practice
The takeaway from the paper (which you should read at some point because it’s wonderful and goes into so many details about how they tested these things and what prior research had been done) is that the most effective way to study is to do testing and distribute practice.
Okay, that’s good.
What does that mean?
Note that we use the term practice testing here (a) to distinguish testing that is completed as a low-stakes or no-stakes practice or learning activity outside of class from summative assessments that are administered by an instructor in class, and (b) to encompass any form of practice testing that students would be able to engage in on their own
and
The term distributed- practice effect refers to the finding that distributing learning over time (either within a single study session or across sessions) typically benefits long-term retention more than does massing learning opportunities back-to-back or in relatively close succession.
Why not both?
If both are highly effective, then doing both should be even more effective.
Right?
Right?
Well, yes, with an asterisk.
Cognitive psychologist Thomas Toppino, who chairs the Department of Psychological and Brain Sciences at Villanova University, was interviewed for a segment on KQED-FM (San Francisco NPR station)1, where he shared:
Never test yourself immediately after you study. You’re going to grossly overestimate how well you know the information if you test yourself right away.
What he and other researchers recommend in their 2016 paper2, Relearn Faster and Retain Longer: Along With Practice, Sleep Makes Perfect, is
We found that interleaving sleep between learning sessions not only reduced the amount of practice needed by half but also ensured much better long-term retention. Sleeping after learning is definitely a good strategy, but sleeping between two learning sessions is a better strategy.
So what the students should be doing is distributed practice and testing WHILE sleeping in between the distributed practices.
You just invented homework.
How about that?!
We want the student to do a “low-stakes or no-stakes practice or learning activity outside of class from summative assessments that are administered by an instructor in class, and (b) to encompass any form of practice testing that students would be able to engage in on their own.”
We want the student to do it in a spaced fashion.
We want the student to sleep in between.
You just invented homework.
Seriously, though, that’s neat.
But the way the schools usually do it is that the homework gets turned in, and then days (weeks?!) later, it is returned with marks on the paper of what the student got wrong.
Which my kids may look at? I look at it but I’m not sure they do.
What’s missing is a more immediate feedback loop.
Homework with self-reflection
A simple self-reflection strategy can work wonders while the kid does homework (or outside-of-school work).
Have the student self-reflect (or you ask them) - what took a long time?
That’s what they need to work on.
It was the 7 and 8 times tables for one of our kids.
We didn’t need to create extra practice / work for ourselves.
All we had to do was do the daily homework and ask them what they were struggling with.
If you work with your kid one-on-one, you won’t even have to ask them, as it’ll be (painfully) obvious when they encounter material that hasn’t been mastered.
You wouldn’t believe the mental gymnastics involved in multiplying 7 times 8. It was frequently 6 * 6 = 36, then add a 6, a 7, and another 7—all in their heads.
(Think of it geometrically: 6 rows of 6 unit columns - add another column of 6, so you have 6 rows of 7 unit columns; then you add a row of 7, so you have 7 rows of 7 unit columns, and then you add a final row of 7 so you have 8 rows of 7 unit columns)
It seemed much easier to know that 7 * 8 is 56 rather than doing the above process in your head.
Conclusion - make every homework a test
We get back to the title of this piece.
The key is to make every homework a low-stakes test.
The way homework works in the US is that the teacher goes over the material and then the homework is a way to explore the material further and to practice the problems.
Since homework is done at home and the teacher is at school (unless homeschooling or doing outside-of-school work), there is a space of time between having seen the material and doing the “test.”
With some self-reflection, the kid can figure out what stuck and didn’t.
Then, when they do the next homework (which should build on previous homework), it’ll be evident if they remember the past material and what still needs to be mastered.
Try it out and let me know how it goes :)
That’s all for today :) For more Kids Who Love Math treats, check out our archives.
Stay Mathy!
All the best,
Sebastian Gutierrez
Mazza, S., Gerbier, E., Gustin, M.-P., Kasikci, Z., Koenig, O., Toppino, T. C., & Magnin, M. (2016). Relearn Faster and Retain Longer: Along With Practice, Sleep Makes Perfect. Psychological Science, 27(10), 1321-1330. https://doi.org/10.1177/0956797616659930