Listening with understanding ... EUREKA moment!

Working with a math student yesterday and I had such an interesting experience that caused a few "Ah Ha" moments I wanted to share.

So many struggle with algebra concepts and that whole solving thing, and my student (lets call him Bob) was no different. As Bob struggled to follow some of the suggestions and models I presented, I started sitting back a bit and just asking questions about how he was thinking about the problem. He was slow at first, but once he found I was not judging his statements as stupid or WRONG, he kept at his explanation and he even was able to illustrate how he thought it would work. I was excited about the idea of discovery and I shared that with him. I shared, "I know this is not how people have normally done this and I can't guarantee it works in all cases but lets try that out" We increased the complexity with each try an sure enough, his system seemed to work well for the way he was thinking.  Let me see if I can share what he was doing in a few examples. 

Starting with X + 4 = 12 he would quickly say 8 as many would and his explanation was "What number + 4 = 12?" I am thinking that he is using substitution or guess and check so we continued with X-4 = 12. Same strategy was shared with correct response.

This is where it gets interesting. When presented with 2x + 5 = 25 here is how he started. He removed the 2x completely to say "What number + 5 = 25?" which was consistent with his previous work. Then it was, "So 2x has to somehow end up being 20 and 2x simply is x + x so x has to be 10". OK that's cool so we tried it with 2x-5=25 and the same logic and process worked. I am thinking, fine this works for solutions where x is a whole number lets try something different.

6x +2 = 6, I offer like a kid who has just put a bucket of water over a door and can't wait for their brother to walk through the door! Same logic starts out ...something +2 = 6 so that something is 4....6x must equal 4 then. Now Bob surprised me by pulling in fractions! 6x means 6 times x and 6 could be thought of as a fraction 6/1. Well I need to find x so let me start with the recipricol because I know that comes out nicely to be 1 all the time. So I know if x was 1/6 I would have 1 = 4 which is silly and not true, but I can see that 4 is 4 times my one, that means that if I used 4/6 instead of the 1/6 it should work. He then took 6/1 * 4/6 and demonstrated how it came out to =4 My mind was blown and I was so excited to have had this moment to have Bob articulate his methods with me.  

Next we tried an example like 4x + 3 = 2x + 6. He added this first step into his process. 4x means x+x+x+x and on the other side 2x = x + x. So if I got rid of the two x on the right side I would have to get rid of two of them from the left side leaving me with 2x + 3 = 6 which he proceeded to use his "normal" procedure to solve. 

OK what if that 2x was -2x I thought. To this he responded..."Well, that is like magnets then ... For every negative charge you need a positive charge to cancel it out. He again drew his 4 xs on one side and then put 2 -xs on the right. He added in 2 + xs on the right and circled the pairs that would cancel each other out, then he added a similar 2 +xs on the left. Then counting up the 4xs on the left with the 2 added on he got 6x and proceeded as "normal". 

I was bouncing in my seat and celebrating Bob's thinking as he was sharing that no one had ever understood what he was trying to do. Likewise no one had been able to show him why his process was wrong, they all just kept trying to force him to try to change his process over to theirs because they were just ignorant of why his method does not work. Our time was running down (it took us over 2 hours with all the examples we were trying to test) and I made a deal with Bob that if he kept exploring for different types of algebra equations, then I would post this experience to get the field to help test this more. He was so proud that not only was one person going to hear and understand, there might be a whole flock that could really "get him" and be able to help him see the pitfalls or concerns from how he was processing things. 

So, I offer to the field, where does this student's logic break down? It seems to me he is very much in line with our typical algorithms, but he is approaching each step from a much more conceptual angle it seems to me. This is exciting and fascinating to me and I will continue to explore some more this week before I meet with Bob next week. 

As teachers, we have been exposed to only a very few strategies and procedures to "do math" and very few means of "thinking the math" have been in our training. I am starting to seek out those students that really struggle with math to patiently ask questions and start learning more about their thinking. Sure, I am encountering some areas where there is just a logical flaw and my questions help the learners discover that flaw. The beauty of this is that the learner then make the decision that another way is needed rather than me dictating to him/her, "That is wrong ... you need to do this because ..." The questioning process is helping learners discover their errors and gently introducing them metacognition skills we all need to develop to become good learners. Effectively, Bob is doing our standard algorithm but he is thinking through those steps in a way that most of us teaching it may not immediately understand and we end up invalidating his approach because it does not sound right. Such exciting discoveries! 

If I did not explain Bob's methods or thinking well, I apologize, please offer a sample problem you wish me to answer and I will try again to share how Bob would approach that problem. If you think you get what Bob was doing at each step, can you find problems in algebra where his methods fall short? I anxiously await reactions and insights from others so I can go in well armed to help Bob next week. 

Bob's steps:

  1. If there are variables on each side, try to eliminate the smaller number of variables to get variables only on one side
  2. Remove the variable term completely to use + or - to find what that variable term must be
  3. Using the multiplication of fractions and the idea of recipricols find what x must be
  4. Plug x back into the original equation to verify it works. 


My interpretation of what this student was thinking and doing is different from Ed's.  I think the student needs to become comfortable with using addition/subtraction and multiplication/division; he needs more general methods if he wishes to become a successful and efficient problem solver in contexts which are a little bit more general.

Given X+4 = 12  he asks himself "What # +4 = 12?".  Clearly he is asking exactly the question which the subtraction operation was designed to answer, but he seems reluctant or unable to actually use that operation and get X = 12 - 4 = 8.  He seems not yet to have internalized what subtraction does for the user -- or that this is using the additive inverse to "undo" addition.  He is just not comfortable working with "expressions"-- that is, with recipes for computation.  And similarly for X -- 4 = 12, which should lead immediately to  X = (X --4) + 4 = 12 + 4 = 16.

Thinking ahead, what will be needed when instead of integers the equations have rational numbers, real numbers, other numerical expressions?

Additive inverse:  A + (-1)*A = 0 = A -- A    for all A of the above types of numbers and expressions.

How do we use that?  Suppose that we have some math expression Expr1 and you can evaluate it to get the number 7, in which case we usually write Expr1 = 7.  Then what happens if we add 13 to Expr1?  Of course the value of the new expression  Expr1 +13  is computable as 7+13 = 20, and we write Expr1 + 13 = 20.  Now suppose we subtract 13 from this expression (Expr1 + 13), then we get  (Expr1 +13) --13 = 20 -- 13 = 7. 

More generally, for any expressions Expr1 and Expr2, if we evaluate each of Expr1 and Expr2 and IF we find that  Expr1 + 13 = Expr2, THEN it must also follow that Expr1 = (Expr1 + 13) -- 13 = Expr2 -- 13.   And the same is true for any other number A instead of 13. 

So, if  (6*W --J) -- 3 = 5*Z + 2  then adding 3  gives  6*W --J = 5*Z +2 +3 = 5*(Z + 1).

If 4*X + 3 = 2*X + 6  then adding --3 yields 4*X +3 --3 = 4*X = 2*X +6 --3 = 2*X +3  and then subtracting 2*X gives  4*X -- 2*X = 2*X =  3.

The student's simple counting of 4 x's on one side and 2 x's on the other and then cancelling two x's works in this simple case, but what about if we have 3.2*X + 5 = 1.7*X --13  or what about in the case of  (5/3)*X  = (1/4)*X + 6 ?  Instead of simple counting he will need to think multiplicatively and use associativity to get  3.2*X -- 1.7*X = --13 --5 and  1.5*X = --18 and then dividing by 1.5 to get X = (1/1.5)*(--18) = --18/1.5 = --12.

That is, he will often need to use the multiplicative inverse property:  For any A that is not 0, we have A*(1/A) = 1.  So IF we have A*X = Expr3, THEN also it follows that  (1/A)*A*X = X = (1/A)*Expr3.  And if A = B/C then 1/A = C/B.

Example.   The circumference C of a circle is pi*D where D is the diameter.  So if we take a tape measure and wrap it around a bike wheel and we get C= 88 inches, then of course we  have 88 = pi*D.  So if we want to find D we must divide C by pi to get D = 88/pi (just about 28 in.).  

Landor, the solutions you offer are of course accurate and include the procedures we wish our learners to have. What I noticed right away with this student is that he was lacking the concept of "Why are we solving for X and what does that mean?"  among some other questions along the line of "Well, why does that work?" As we were working through the problems he was using the correct language (recipricol, additive inverse ...) but the way he was organizing his thoughts were the difference. Towards the end of my post, I tried to summarize his thinking and if you look at our standard algorithms, I think he is not very far off. 

You provide some great problems I will share with him this week, specifically the fractional coefficients in both expressions. He had a strategy of clearing fractions when there was just one fractional coefficient so I am curious to see if he extends that to clearing all fractions at once. My strategy here is to follow a student's concept of what is going on at first to verify that there are no mathematically incorrect or challenging steps. For instance, if the student consistently wanted to deal with the coefficient first, it is very possible to do that, but the learner would quickly learn how important complete mastery of fractions becomes with that approach. When the student feels they have a system, then we can introduce the standard procedures, symbols, terminology and do a side by side comparison. "See when you were doing this fraction stuff over here, that is just like over here when they are multiplying by the recipricol. You were just doing out all of the work in your process which is ok, but you may not see that done out in more formal work ..."

As math instructors, we "get it" and after we explain it, as you have done so well, Landor, we struggle when a learner does not understand. Very often we simply offer similar explanations or simply repeat our explanations over and over which does very little to increase the learner's comfort or ability. I am finding that the more I start to explore their mental rabbit hole of thinking, the easier I can find those bits that are right on but might be out of place, those bits that are right but are just floating around with no connections, and those bits that are just confused or plain wrong or out of context. Then, I can start patching things together in a way that helps the student eventually get to where we all hope they end up. 

A perfect example of this is subtraction. For the first 7 or 8 years of their instruction they learn that the minus sign is an operation. Then those nasty middle school teachers spring this other meaning that it could also be a symbol to change the sign of the term to the right of it (particularly with distribution like 10-2(3x+4) where even their rules of order of operations indicates the 2 might be multiplied first before 'doing the subtraction'). Until learners can make that conceptual shift from minus being an operation to it becoming an indication of sign, Algebra is almost impossible for most of these learners. It is too bad elementary school does not start out with 4-2 conceptually being 4 + (-2). That transition to the Algebraic concept of minus would be so much easier then. 

Landor, and others, as you review the wonderful descriptions Landor offered, think about how you might help a learner that patiently reads all that and just shakes their head, "I still don't get it". You don't need to try to address all the examples he gave, but pick one and offer how you would help support to get that learner up to speed an really understanding what you were describing. I would suggest you try this with someone that does not understand math (there seems to be many to pick from sadly) to see how effective the strategies are and what feedback you get. I can share some ideas and reaction as well once others start the ball rolling :)


Ed, thanks for your wonderful description of your Eureka moment.  You have touched upon something that all of us who teach math should consider. Too often we tell or show students procedures without first developing a conceptual understanding of why that particular algorithm came to be.  I feel your student has a wonderful understanding of equality, that is balancing both sides of an equation so that expressions are "the same as". Too often, students think of equality as answer not the same amounts on both sides.  I think your student's approach is something we should all explore with our students. I might suggest starting with a real life application to give the naked numbers more context but then to let the students grapple with the problem before telling them the steps.  Your student's reasoning was very sound and made sense of the problem which is far more important then following steps from memorization. Remember an algorithm is the explanation of somebody else's idea and sometimes that idea does not work for our students. Perhaps as your student works on more problems with his explanation, he might eventually understand your way of solving, particularly when introduced to the more difficult problems that Landor posed. Even as our students explore more abstract concepts, it is worth the time to let them construct some understanding for themselves. After all, that's how mathematicians came to develop algorithms in the first place, through exploration.

I thought "Bob" did a great job of explaining why algebra works. Many of my students have similar intuitive approaches, and I tend to let them practice problems in whatever way works for them until they are confident enough (and fluent enough) to be able to compare their method to the standard algorithm. I present this as another way of notating the steps, and explain that it can be useful with less "friendly" numbers or with more complicated problems. Usually, the students see the connection between what they are doing intuitively and the standard procedure--but only if they have reached a point where they understand and can explain their own problem-solving logic.

I'd also like to note that in the EMPower algebra book, one way they present to solve two-step equations is very similar to Bob's. They recommend that the student cover up the variable and coefficient with a finger and then figure out what number "goes" in that space. Then, they go on to determine the value of the variable using logic very similar to his. (eg: If 3x = 12, then x must be 4 because three fours make 12.) I recognize that students who will be progressing to more advanced algebra will need to know the standard algorithms, but when introducing a subject that many find intimidating, I like to emphasize approaches that make sense to the students until they begin to be interested in the "right" way to do it.

Also, note that if you write down the result of each of Bob's steps, they look very similar to the standard way of solving the problem (eg: 2x + 4 = 10, so 2x = 6, so x = 3). I wonder how many students appear to be doing algebra the way it is taught, but are actually following his line of thought? I would argue that this is better in the long term than the student who copies the procedure correctly without any understanding of why it works. At least Bob can reason his way to an answer if he forgets exactly what his teacher told him to do.

Rachel, I am so glad to have read your post right after I was browsing through the empower stuff today to prepare for Bob tomorrow. I figured I would see what the empower take was and was delighted to see it was so closely aligned Image removed.

Since my post, I have been asked this type of question a bunch and I would be interested to hear how you and anyone else reading this might respond. 

"Ed, you are obviously spending great quality time with a learner, but how would you possibly spend all this time to figure out what a student is thinking if you have 20 learners in the same class? I also know, from your other posts, that you have many individual learning lab type setups, how would you suggest a more traditional teacher approach 10 or 12 different Bobs in their classrooms?"


The only way to find out what students think is to ask them! This doesn't have to be oral, though, and they don't have to tell just the teacher. I try to ask "why/why not?", "how do you know?", or "what are you thinking about?" every time I work individually with students, but I also ask these questions (and make them write/draw) on worksheets, when students work in groups, and when they do work on the board. I also have them explain their thinking to each other. This helps both the explainer (who puts the thoughts into words and generates rules) and the listener (who is exposed to a new way to think about the math).

One timesaver here is that as you see more ways of thinking, you get to recognize the common approaches. You can check in with a student who has trouble verbalizing their thoughts by saying, "Some people solve this problem like this, is that what you did?" I frequently get students who say, "Yes, sort of, but I did [...] like this." I also get to recognize the different barriers that students run into depending on how they are thinking about the problem. I still like to get the students to explain (I don't want to jump to conclusions), but it's easier when I have a good idea already of what is going on.

Along a similar vein, I also make a show of following through a sampling of problems using the students' scratch work, whether they got the answer right or not, so that I can see how they're doing the problem. This helps for diagnosing errors, but it also warns me if a student is using a method that won't continue to work with more complicated problems. With a class, I have the option of  addressing a problem with an individual, a group, or the whole class, depending on how widespread it is.

You don't immediately get the same depth of understanding that you got with Bob when you teach classes, but you do get a lot more breadth, and that can be just as helpful for practical purposes.

    I've got a kiddo in here working through our software tutorial program to try to improve his placement... alas, he's plowing through the tests instead of actually listening to the videos (Modumath has pretty awesome little videos), but he is thinking through it. 

    He got to inequalities and asked if they were basically like equalities.   I have learned to go beyond the procedural rule and stating that if you multiply or divide by a negative number you've got to switch the inequality symbol to the other way.   I show that while 8 < 10,  I ask:   is  -8  less than -10... and show it on a number line if I can.   Then I move to a simple problem with simple numbers (like -3x >12)   and try different numbers and *show* that the statement works.   

I wouldn't wish negative numbers on an elementary kiddo but I *do* think early learning could be structured to map better into later learning, with lots of language connections... 

Susan, I have heard similar feelings "I wouldn't wish negative numbers on an elementary kiddo..." expressed by others and I question why? Can you offer some further rational behind that? 

My initial reaction is that most kids can relate to loss of something. Heck we could give them 5 jelly beans then take 2 of them or even say they owe us 2 back if that language might create a stronger mental foundation of what a negative is (it is when we owe x amount). 

Just curious what everyone thinks. What are the challenges to teaching the concept of negatives very early on in mathematic development? Are there studies out there saying this just does not work?


It's late on a Friday and the resources I want to link to that support this notion aren't coming forth.   Children develop at different rates in their ability to deal with abstractions.  Almost always, when teachers throw abstract ideas like negative integers at young students, it's done as if they were older and it doesn't work.  They build a model of "understanding" with the tools they have that won't hold up later.   When it fails to hold up, they look bad, feel bad, and get off the math success train.  


Hello all (and Sue)

I'm a month behind with Sue's comment about "children develop at different rates."

Yes, they do. The key is - When do "normal" children develop the ability to keep 2 things in mind at the same time? As an example, how old was your child before he/she was allowed to cross a busy street on his/her own? Cars are coming. The child has to think about BOTH the car's speed AND the car's distance. This usually happens around age 8.

This BOTH / AND thinking is what is necessary for children to grasp the PART / WHOLE concept. When they have that concept, they can understand subtraction. So when do children turn 8? All the way through 2nd grade. So the younger ones (late spring and summer birthdays) do not have the brain connections to understand abstract subtraction (without manipulatives) when it is introduced. Therefore, "math methods" courses tell people teaching 2nd grade to teach 7-year-olds to "count up to" the "big number" from the "small number." That means many 2nd graders learn to add in order to subtract.

Now, age 8 is about right for "normal" children. Then you add TOXIC STRESS. [Translation: POVERTY] That is the new research buzzword. Stress has been shown to strip myelin from nerves and/or delay myelin being laid down on children's nerves. Myelin is the material that helps nerves make connections.

Children growing up in poverty or in situations of abuse may be age 9 or later before their brains have the capacity to keep 2 things in mind at the same time.

These are our adult education math students. Some were merely too young to understand the connections that adults make normally. So they felt stupid, hated math, etc. (as Sue mentioned).

The problem is not the children. It is the organization of the primary grade math curriculum. So much as been pushed so early, with no consideration of children's normal biological brain development.

Just came upon another educator sharing some benefits of helping learners develop their mathematical thinking. 

Quick read and quick video view:

What are thoughts, reactions or other examples that help us all feel and taste what this shift is all about. Maybe we need to rename this thread to "Getting Shifty with Math" Image removed.

Hi Ed and all, This brief article and the videos illustrate how teachers "normalize the struggle" for learning math. In the first brief video "Find 3 Ways," the teacher acknowledges that making mistakes is part of the learning process. After presenting the problem, the teacher asks the 3rd graders to find three ways to solve it. The graphic organizer she uses seems quite useful in guiding students to think of different processes to solve the problem. First, students are given time to wrestle with the problem independently. Students then share their solutions and raise any questions in small groups. The teacher circulates while students are working in small groups and selects specific students to present their solution.

I love how this teacher normalizes the struggle. 

Math teachers, take a look at this brief video and/or the other Teaching Channel video on this page on "Conjecturing about Functions" with 8th graders and share your thoughts. In your view, what are the benefits of normalizing the struggle in math?

Thanks for sharing this, Ed.

Cheers, Susan Finn Miller

Moderator, College and Career Standards