Through doing Number Talks with students K-5, I started to realize that one thing I look for students to use in our whole number computation discussions is using known or derived facts to come to a solution. I feel like the problems I have been using are crafted to use the answers from previous problems to reason about the ending problem.
In the younger grades, I would like to see students using the double known fact of 7+7=14 to know 7+8=15. I want them using 23 + 20=43 to get 23 +19 = 42. I don’t want them treating every problem as if they have to “start from scratch” adding all or adding on.
An example in the upper elementary:
18 x 2
18 x 20
18 x 19
This progression leads them to use a known or derived fact (18 x 20) in order to solve 18 x 19. To build efficiency, I don’t want them to the treat the final problem in the progression as a “brand new” problem in order to reason about an answer.
Along these lines of thinking, as I observed students working the other day, I realized that students weren’t using this same use of known/derived facts when working with fractions. For example, a student was adding 3/4 + 7/8. He used 6/8 as an equivalent of 3/4, added that to 7/8 and ended with an answer of 13/8. Don’t get me wrong, I loves his use of equivalency and I am a fan of improper fractions, however I started wondering to myself if it would have been more efficient (or show that he actually thought about the fractions themselves) if he used a fact he may have known such as 3/4 + 3/4=1 1/2 to then add an 1/8 on to get 1 5/8? Or used 3/4 + 1 = 1 3/4 and then took away an 1/8? Is that the flexibility I want them using with fractions like I do with whole numbers?
I thought I would try a Number Talk the following day to see….
1/2 + 1/2
Thumbs went up and they laughed with a lot of “this is too easy”s going around.
1/2 + 1/4
Majority reasoned that 1/2 was the same as 2/4 and added that to 1/4 to get 3/4. Some said they “just knew it because they could picture it in their head” I asked if anyone used what they knew about the first problem to help them with the second problem? Hands went right up and I got an answer that I wish I was recording. It was to the effect of,”I know a 1/4 is half of 1/2 so the answer would be a 1/4 less than 1.”
1/2 + 3/4
Thumbs went up and I got a variety here. Some used 2/4 + 3/4 to get 5/4 while others decomposed the 3/4 to 1/2 + 1/4, added 1/2 + 1/2=1 and added the 1/4 to get 1 1/4.
3/4 + 3/4
Got some grumbles on this one, because it was “too easy” – 6/4…Duh! The class shook their hands in agreement and they were ready to move on to something harder. I noticed that when the denominators are same, they don’t really “think” about the fractions too much. I waited….finally a student said, “It is just a 1/4 more than the previous problem so it is 1 1/2″ and another said each 3/4 is 1/4 more than a 1/2 so if you know 1/2 + 1/2 = 1 then you add 1/2 because 1/4 + 1/4 = 1/2.” I had to record that reasoning for the class bc it was hard for many to visualize.
3/4 + 5/8
Huge variety on this one and I thoroughly enjoyed it! From 6/8 + 5/8 = 11/8 to decomposing to combine 3/4 and 2/8 to get the whole and then 3 more 1/8s = 1 3/8. There were many more students who used problems we had previously done.
What I learned (and questions I still have) from this little experiment:
– Students LOVE having the same denominator when combining fractions.
– Do they really “think” about the fractions when the denominators are the same? Can they reason if that answer makes sense if they are just finding equivalents and adding.
– Students can be flexible with fractions if you push them to be.
– Subtraction will be an interesting one to try out next.
– I would much prefer if I remembered to use the word “sum” instead of “answer”…. I tell myself all of the time, but in the moment I always forget.
– Using known or derived fact and compensation are invaluable for students when working with both whole numbers, fractions and decimals.
– Are there mathematical concepts that present themselves later in Middle School or High School in which known and derived facts would be useful?
Happy Thanksgiving,
Kristin
MathMinds 
To consider 5/8ths meaningfully, one must imagine a whole, consider one of eight equal parts (“one eighth”), and then collect together 5 of these new units, the one-eighths. So “five eighths” is thought of (when understood) as “five one-eights.”
A fraction thus is a unit of units of units. This coordination is rather mentally taxing, so it is of no surprise that your children prefer to reason with just the numerators once they know they simply have a collection of like sized “units”, i.e. “one eighths” in your example.
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Typo, “five one eighths”, not “five one eights”
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I agree Brian and have become more and more of a fan of CCSS wording of “a parts the size of 1/b.” My thought was that once they realize they have like-sized units, do they lose reasoning about the fractions themselves? Is that flexibility something I should really work through more? I feel like I would like them to be more flexible.
Thanks for the reply!
-Kristin
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That CCSS wording reflects what we know about how children learn fractions. Learn, as in come to know and understand for themselves as derived knowledge–as opposed to mimicry.
I think that students showing the ability to reason without the baggage of continuing to coordinate the add’l unit demonstrates the compression of knowing that Jo Boaler spoke of in her DE talk. So that speaks to me as greater flexibility.
But i understand your concern–it would be for students who have a memorized “understanding” of fractions and fraction rules, not a constructed understanding.
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That compression piece is what I needed to hear, makes me more comfortable with the majority of the students’ work. I think my concern stemmed from not feeling comfortable that some are just working with the numerators without the reasoning piece. Maybe selfishly I needed to hear them talk it through for me to have a real grasp on their understanding! I also like them to see fractions as numbers that can be decomposed like the whole numbers they are comfortable in using. Oh the power of mathematical talk!
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Also, you asked about the role for known and derived facts as vehicles for teaching HS Math.
I fear I am misunderstanding the question because virtually all mathematics must be derived (re-invented) from known facts. But this isn’t a property if math so much as it is about how knowledge is constructed.
I like Connie Kamii’s summary of Piaget’s learning theory in her book “Children Reinvent Arithmetic” if you wish to follow up further on this idea.
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I will definitely follow up with that. The focus of my question was more leaning toward using Number Talks as the vehicle to encourage student’s use of derived facts. I know we had talked briefly about using Number Talks at the high school level and that has stuck with me. I know that teachers at all levels pose a problem and talk about multiple solution pathways, but do the upper grade levels structure a number talk to focus on efficient strategies or use of derived facts? Are there areas where that would be useful in the upper grades? Keep in mind that sometimes I pose questions that seem to make sense in my brain but don’t necessarily translate well 🙂
Thanks! Kristin
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I don’t know an answer to how HS teachers use Number Talks.
I also don’t know of many instances in Maths where trying to emphasize one particular strategy w.r.t. thinking with and/or knowing numbers is valuable. Teaching particular strategies may only be useful when the teaching goal is to ensure students know a standard algorithm, that the society may expect them to know. For example, 2 variable 2 equation situations such as the classic #feet & #heads, how many chicken and sheep? A teacher may be looking to develop the “Elimination Method” on the way toward Matrix Algebra, two historical remnants of symbolic Algebra still expected in HS math.
Trying to stay on point, my sense is that the power if Number Talks is that it is a particular strategy that can help teachers learn how to get kids to relearn that there are many ways of thinking mathematically, and that greater and deeper understanding (and greater flexibility) comes when drawing connections between different ways of thinking.
When Number Talks become about teaching kids particular ways to think, I become concerned.
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Agreed. I do think that it is interesting during Number Talks when students realize particular questions are solved more efficiently with different strategies. For example, when breaking on number apart would work best for them versus double/halving. The problem I see with many students in the elementary, is given a 2 digit x 2 digit multiplication problem, they resort back to all four partial products, without reasoning about a more efficient way to solve. So, while not directly teaching kids ways to think, making various strategies available to them is invaluable.
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This is a great post and conversation about number talks. I think part of the power of number talks is in the how you scaffold the exercises posed. I teach 6th grade math and I brought number talks up from intermediate school with me. The protocol is the same, although I don’t ask the students to gather out of their seats. I find teaching the students the gestures for agree, disagree, strategy, more than one strategy, and strategy but no answer yet transfer to better discourse in all aspects of math class not just number talks. Last week I was doing number talks with my students focused on equivalent numbers (fractions, decimals, and percents).
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Thank you Diana for your comment! It is great to hear they are being used in the middle school for student discourse and connecting strategies. What was your number talk like for equivalents? I did one with placing fractions on a number line 0-2 and they used percent and equivalents to place them. It was a great discussion and really pushed students use precision in their descriptions.
-Kristin
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I love the placing fractions on the number line. I use it several times during the year. I think the number line really helps students have a visual representation of rational numbers. I think the double-number line that is new to us with Common Core provides a great bridge for students between the number line and coordinate plane.
The goal of my number talks on equivalents was to help students move beyond the benchmarks they know from elementary and help them conceptually with decimal equivalents. So I started with 1/2, then 1/4 and 3/4. They really talked through out 3/4 was related to the first two problems. Then we worked on eighths. The next day was tenths and fifths. The final day we worked on thirds and sixths. Pairing the number talks with the rich problem-solving task of Connected Math Project 3 really helps students build their rational number fluency.
I am so looking forward to continued conversations 🙂
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